morphology and image analysis of some solar …

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MORPHOLOGY AND IMAGE ANALYSIS OF SOME SOLAR

PHOTOSPHERIC PHENOMENA

By

ASMA ZAFFAR

Supervisor:

PROFESSOR (Dr.) MUHAMMAD RASHID KAMAL ANSARI

C0- Supervisor:

PROFESSOR (Dr.) MUHAMMAD JAVED IQBAL

Presented in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

Institute of Space and Planetary Astrophysics (ISPA)

University of Karachi (Pakistan)

2018

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In the Name of Allah, the Most Beneficent, the most Merciful.

May Allah increase in my Knowledge

(Al-Quran Alkarim)

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DEDICATED

TO

My Beloved PARENTS (Zafar Ali and Nasreen Zafar) and

Dr. Shaheen Abbas

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CERTIFICATE

We accept the thesis as conforming to the required standard.

1. Internal Examiner (Supervisor):

2. External Examiner:

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CERTIFICATE

It is to certify that the Ph.D. thesis entitled “Morphology and Image Analysis of

Some Solar Photospheric Phenomena by Asma Zaffar is completed under my

supervision.

Supervisor

Professor Dr. Muhammad Rashid Kamal Ansari

Co- Supervisor

Professor Dr. Muhammad Jawed Iqbal

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CONTENTS

(i) Acknowledgment ix

(ii) Articles Appended to the Thesis x

(iii) Abstract (Englisn) xi

(iv) Abstract (Urdu) xiii

Chapter 0: Introduction 1

0.1 Birth of Sunspots 1

0.2 Sunspots 1

0.3 Group of Sunspots or Active Region 2

0.4 Sunspots Cycles 4

0.5 Solar-Climate Relationship 5

0.6 Enso- Sunspots Relationship 6

0.7 Our Approach 7

0.8 Thesis Outlook 8

Chapter 1: GARCH Model for Sunspot Cycles

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1.1 Data Description and Methodology 12

1.2 Time Series Analysis With Stationary Process 13

1.3 Autocorrelation Function, Partial Autocorrelation Function 14

and Ljung-Box Q-Statistics Of Sunspot Cycles

1.4 Unit Root Test Of Sunspots Cycles Based On Akaike 24

Information Criterion (AIC)

1.5 Tests for Normality 29

1.6 Test of Adequacy 31

1.7 Diagnostic Checking Test 32

1.8 ARMA Model 33

1.9 AR (p)-GARCH (1, 1) Process Of Sunspot Cycle 39

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1.10 ARMA(p, q) – GARCH (1, 1) Methods of Sunspot Cycles 50

1.11 Comparison Of Forecasting Evolution of ARMA, AR-GARCH 60

and ARMA-GARCH of Sunspot Cycles

1.12 Conclusion 68

Chapter 2: Fractal Analysis of Sunspots Cycles and 71

Enso Cycles and Determination of their Relationship


2.2 Mathematical Models and Fractal Dimension Relations of 76

Sunspots Cycles

2.3 Mathematical Relationship between Higuchi Fractal Dimension 88

and Spectral Exponent of Sunspots Cycles

2.4 Fractal Dimension and Hurst Exponent of Enso Cycles by 88

Using Self - Similar and Self - Affine Fractal Dimension

2.5 Comparative Analysis of Self- Similar and Self-Affine Fractal 95

Dimension of Sunspots Cycle and Active Enso Phenomena in

Same Time Interval

2.6 Conclusion 96

Chapter 3: A Comparative Study of Sunspot Cycles and 98

Enso Cycles Analyzing Probability Distributions and Heavy Tail


3.2 Probability Distribution Approach 100

3.3 Best Fitted Probability Distributions of Sunspot Cycle 104

3.4 Best Fitted Probability Distributions of Enso Cycle 110

3.5 Comparative Analysis of Best Fitted Distribution of Enso Active 116

Region and Sunspot Cycles in Same Duration

3.6 Probability Distribution of Fractal Dimension of Sunspot Cycles 118

and Enso Cycles

3.7 Solar Activity Cycles and Enso Activity Cycles in the Perspective 119

of Heavy Tails Parameter

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3.8 Solar Activity Cycles and Enso Activity Cycle in the Perspective 123

Of Long Range Correlation

3.9 Unite Root Stochastic Process of Tail Parameter and Long Range 127

Correlation of Sunspot and Enso Cycles

3.10 Conclusion 129

Chapter 4: A Study of Active Regions of Rotating 131

Sunspots: Image Processing Approach


4.2 Fractal Dimensions, Wave Spectrum and Heavy Tail 136

Analysis of AR12192

4.3 Mathematical Morphology (MM) 137

4.4 Mathematical Morphology (MM) of AR12192 139

4.5 Image Segmentation of Rotating Sunspots Using Genetic 140

Active Contour

4.6 Active Region AR9114 and AR10696 147

4.7 Conclusion 150

Chapter 5: Conclusions and Future Work 152

5.1 Summary and Outlook 152

5.2 Principle Results 156

5.3 Future Outlook 158

References: 160

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ACKNOWLEDGEMENT

ALLAH is the most merciful and beneficent, I am extremely thankful to give me the knowledge

and ability to complete this dissertation. Without the opportunities given by Almighty ALLAH I

could not have complete my dissertation successfully.

I express my gratitude to my supervisor Professor Dr. Muhammad Rashid Kamal Ansari and co-

supervisor Professor Dr. Muhammad Jawed Iqbal, for his animated guidance and encouragement.

I am very thankful for teachers, Professor Dr. M. Ayub Khan Yousufzai (University of Karachi),

Professor Dr. Zakaullah Khan (Late), Shuja Muhammad Quraishi (Sir Syed university of

Engineering and Technology). I specially acknowledged to Abdul Qayoom Bhutto (Pakistan

Meteorological Department), Dr. Muhammad Ali Virsani and Professor Dr. Shahid Shukat.

Specially Thanks to Dr. Fozia Hanif Khan, Dr. Frahat Naz Rehman, Talat Salma.

I would like to acknowledge to all my relatives, friends, class fellows, Colleagues, students,

Specially Abdul Moiz Khan, Syed Moutasim Ali, Javed Hameed Khilji, Adil Hameed, Salman

Shahid, Ovais Siraj, Sana waji, Fazal Akber, Bulbul Jan, Arsalan Khanzada, Ghazanfar Ali Khan,

Muhammad Younus, Muhammad Illyas.

Limitless thanks are due to my parents who motivated, encouraged and supported me throughout

the completion of the work. Many thanks to my sisters Dr. Hina Zafar, Saba Zafar and Farah Naz

who remained keen in every stage of completion of this work. In the end I once again thank

Almighty Allah for the completion of this work.

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ARTICLES APPENDED TO THE THESIS

1. Asma zaffar, Shaheen Abbas and Muhammad Rashid Kamal Ansari (2017). The

Probability Distributions and Fractal Dimension of Sunspots Cycles associated with

ENSO Phenomena. Arabian Journal of Geoscience, Springer (Accepted)

2. Asma zaffar, Shaheen Abbas and Muhammad Rashid Kamal Ansari (2017). Study of

Sunspot cycles using Fractal Dimensions: Wave-Spectrum Scaling. Solar System

Research Springer (Accepted)

3. Asma zaffar, Shaheen Abbas and Muhammad Rashid Kamal Ansari (2017). An Analysis

of Heavy Tail and Long-Range Correlation of Sunspot and ENSO Cycles. Arabian

Journal of Geoscience, Springer (submitted)

4. Asma zaffar, Shaheen Abbas and Muhammad Rashid Kamal Ansari (2017). Model

Estimation and Forecasting of Sunspots Cycles through AR-GARCH Models.

Journal of Forecasting (submitted)

5. Asma zaffar, Shaheen Abbas and Muhammad Rashid Kamal Ansari (2017). A study of

largest Active Region AR12192 of 24th Solar Cycle using Fractal Dimensions and

Mathematical Morphology. Journal of Astronomy and Astrophysics (submitted)

6. Fozia Hanif, Asma zaffar, Shaheen Abbas and Muhammad Rashid Kamal Ansari (2017).

Image Segmentation of Sunspots Using Genetic active contour. Solar System Research

Springer (submitted)

7. Asma zaffar, Shaheen Abbas and Muhammad Rashid Kamal Ansari (2017). Comparison

of Forecasting Evolution of ARMA, AR-GARCH and ARMA-GARCH. Astrophysics

(submitted)

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ABSTRACT

In this dissertation, sunspot cycles (1-23) analyzed together with the total sunspots observational

data from 1755 to 2008. Similarly, each ENSO cycle (cycle 1 to 21) and the total data from 1866

to 2012. In addition, the data of active period ranging from 1981 to 2000 are evaluated and the

relation between sunspot and ENSO are determined and an analysis is performed. The time series

of sunspot cycles has stationarity and holds linear behavior. ARMA (p, q), AR (p) - GARCH (1,

1) and ARMA (p, q) - GARCH (1, 1) models are analyzed and forecasting, evolution and test of

normality are performed for sunspot cycles. Least square estimator are used for ARMA models.

Whereas the quasi maximum likelihood estimator is used for AR (p) -GARCH (1, 1) and ARMA

(p, q) - GARCH (1, 1) models. AIC, SIC, and HQC are calculated for ARMA (p, q), AR (p)-

GARCH (1, 1) and ARMA (p, q) - GARCH (1, 1) models. Comparison of forecasting evolution,

such as Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Mean Absolute

Percentage Error (MAPE) and Theil’s U-Statistics test (U test) for ARMA (p, q), AR (p)-GARCH

(1, 1) and ARMA (p, q)-GARCH (1, 1) models is done. Self-similar and self-affine fractal

dimension (FDS and FDA) of the sunspot and ENSO cycles are analyzed. Active region of ENSO

during 1981 to 2000 corresponded to sunspot cycles 21 to 23. This data was partitioned

corresponding to these cycles. Distributions of both the sunspot cycles data and corresponding

ENSO data were analyzed and compared. Wave-spectrum and universal parameters are calculated

for sunspot cycles. The well-known Higuchi’s relation is also analyzed for sunspot cycles. Results

show that sunspot and ENSO cycles are persistent and long term correlation and dependence exists.

It was also concluded that self-similar (box counting method) is more appropriate than self-affine

(rescaled range analysis method). The wave-spectrum parameters such as the spectral exponent

and autocorrelation coefficient have significant range showing that sunspot cycles are strongly

correlated which shows that the sunspot are heavy tailed and have long range correlation.

Correspondingly ENSO cycles are also heavy tailed and long range correlated. This study also

shows that ENSO cycles are more persistent and strongly correlated than Sunspot cycles data. The

adequate probability distributions of sunspots and ENSO cycles are Generalized Pareto

Distribution (GPD) and Generalized Extreme Value Distribution (GEV). These distributions

behave heavy tailed on the right side. Same period of sunspot and ENSO active episode have best

fitted distributions which are also Generalized Pareto Distribution (GPD) and Generalized Extreme

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Value Distribution (GEV). Moreover, self-similar and self-affine differencing parameter (ds and

dA), heavy tail parameters βS and βA and long range correlation parameters 𝛾𝑆 and 𝛾𝐴 are analyzed

for sunspot and ENSO cycles. The values of Heavy tailed and Long Range correlation parameters

show that the sunspot cycles and ENSO cycles are strongly correlated and each observation is

dependent on the previous observation. The image processing of largest active region AR12192 of

solar cycle 24 is performed using self-similar fractal dimensions (Hausdorff - Besicovich box

dimension and correlation dimension). Mathematical Morphology (MM) of the active region

AR12192 is also studied and analyzed. The persistency of AR12192 is also determined using Hurst

exponent and wave spectrum. The outcomes were analyzed and compared. This analysis

concluded that AR12192 is strongly correlated and spectral exponent and autocorrelation have

significant range. The transformation tools of mathematical morphology such as dilation, erosion,

closing and opening are also analyzed for AR12192. Image segmentation of rotating sunspots

AR9114 and AR10696 of solar cycle23 was performed by using genetic active contours. The

impact of negative and positive helicity and tangential velocity is discussed with the same values

of energy released and temperature.

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CHAPTER 0

INTRODUCTION

The main source of terrestrial energy is the sun. It also governs the variations in the earth’s climate.

Sun has 1.99 x 1030 kg mass, 6.96 x 108 m radius and 3.86 x 1026 W photon luminosity. The average

temperature of the sun's surface is approximately 5810.928 K. The solar activity like solar flare,

coronal mass ejection and sunspots can be associated with the climate variations in the upper

atmosphere and with the surface of the earth (Anderson, 1990; Gray et al., 2010). The Sun spins

and rotates as a globe of gas not rotate like a solid. The Sun spins faster at its equator as compared

to its poles. The Sun rotates once every 27 days at its equator, but only once every 31 days at its

poles. This can be observed by watching the motion of sunspots and other solar features moving

with the Sun. The giant gas planets, Jupiter, Saturn, Uranus, and Neptune, also spin faster at their

equators than at their poles.

0.1: BIRTH OF SUNSPOTS

Different layers of the sun spin at different rates, creating a magnetic field for the solar sphere.

Convection currents create local magnetic fields in hot gas bubbles. Larger local magnetic fields

and bubbles rise to the surface. At the surface, north and south polarity is split into pairs of

disturbances. Large pairs usually create sunspots. Large sunspot groups often create flares and

mass coronal ejections.

0.2: SUNSPOTS

Solar activity is demonstrated by the dark spots on the surface of the Sun called Sunspots. The

number of Sunspots on the solar disc is considered to be a measure of solar activity. The yearly

average of sunspot areas has been recorded since 1700 (Kok K. et al, 2001). Number of Sunspots

have been recorded reliably since 1755. Sunspots are the dark spots that appear on the photosphere

because of the lower temperature there as compared to the remaining surface. The depth of the

photosphere is about 400 km and its surface reveal most of the solar radiation. Its inner and outer

boundary layer have 6,000 degrees K and 4,200 K respectively. The temperature of the sunspots

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is about 4,600 K. Sunspot has an intense magnetic field. The central temperature of a sunspot is

3810.928 K. Galileo first made the drawings of sunspots in 1612 (TJO News, 2006). Sunspots are

associated with active regions. The active regions are areas of locally increased magnetic flux of

the Sun (Kevin R. M. et. al, 2014). Usually the sun’s surface is heated to the orange-white stage,

but the sunspot being cooler appear dark. The deeper regions of sunspots are opaque. The higher

and abundant less dense corona is only one-million as bright as the photosphere in light visible

(Stenning D et al. 2010). Normally sunspots appear in pairs in which one has a north magnetic

polarity and the other has a south magnetic polarity. The line joining the centers of these two spots

is nearly parallel to the sun’s equator (Choudhuri, 2008). Sunspots often appear near 30-35 degree

north and south of the sun’s hemisphere with higher latitudes. The sunspots life consists of either

days or one week or few weeks (Taylor, 1991). The number of sunspots always goes on increasing

and decreasing over time showing a cycles high to low and from low to high again. The number

of sunspots is calculated by the sum of individual sunspots and ten times the number of groups.

Since 2005 there is an active region in the sun that contains dozens of sunspots, including one of

the size of 17 earths. This spot is known as Benevolent Monster (Weiss, 2006).

Figure. 0.1 The internal and external structure of sunspots.

0.3: GROUP OF SUNSPOTS OR ACTIVE REGION

Sunspots appear individually and in groups. Sunspot groups are manually classified according to

a number of schemes (Stenning D et al. 2010). The area of sunspot group is related to magnetic

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fields (E. Forg´acs-Dajka et al 2014). The sunspots growth rate has a linear relation to the

maximum area. The growth rate consists of two parts. First one is leading part and another one is

following part. Leading part is greater than the following part, but the cycle complete of following

parts as one day earlier to the following part. Leading sunspot has southern hemisphere and

following hemisphere has northern hemisphere (Muraközy J et al. 2012 and 2013). Leading

sunspots of all groups are situated closer to the solar equator as compared to the following sunspots

(Baranyi T 2001). Sunspots consist of four parameters viz. Starting point, amplitude, rising time

and asymmetry (between the rise to a maximum and the fall to a minimum is found to be very little

form cycle to cycle and can be fixed at a single). Over the solar sphere, sunspots do not exist

randomly, but develops in groups in the active regions. Active regions are developed by flares and

exist in east-west direction. Sunspots are visual indicator of active regions. Early observation of

magnetic field shows that the active regions of the sun are represented by a bipolar pattern

comprising of both leading and following sunspots. These two sunspots have opposite magnetic

polarities have southern hemisphere and northern hemisphere. Sunspots area plays an initial role

in the sunspot groups and variation in solar irradiance. Systemic and random errors may affect the

result of calculating the area of sunspots (Baranyi T et al 2001). A solar flare is a rapid and intense

variation of brightness and energy. Sunspots and active regions are mostly related with flare (Lin

et al 2005, Zirin and Ligget 1987, Shi and Wang 1994). The image below indicates complex and

numerous sunspot group.

Figure. 0.2 The Groups of sunspots.

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Solar flares, solar storms and coronal mass ejection blast forth from active regions. When magnetic

energy is created in the solar atmosphere and near the sunspots is released, then solar flare occurs

on the surface of the sun. These energies are more than 1000 times stronger than the average

magnetic field of the sun. Though sunspots existing in active regions, but not all active regions

produce sunspots. In the same active region, different spots have opposite magnetic polarization.

Sunspots have a physical connection with X-rays in the active region. In the active region X-rays

exist in the corona just above a sunspot. X-rays and sunspots both have same magnetic field. The

active regions appear bright in the corona.

0.4: SUNSPOTS CYCLES

The solar magnetic activity cycle (solar cycle) is approximately periodic 11-year. The solar cycle

depends on the changes of the activity of the sun, solar material ejection and the level of solar

radiations. The appearance of solar cycle depends on the changes of sunspot numbers, flares and

other manifestation. For each cycle the maximum number of sunspots is known as solar maximum

and minimum number of sunspots as solar minimum. The first solar cycle is considered to be

starting from 1755 though, the solar cycles were first discovered by Samuel Heinrich Schwabe in

1843. The current solar cycle is the 24th cycle. This solar cycle was started from January 08, 2008.

The initiating pair of sunspots was observed at high latitudes. The maximum number of sunspots

in cycle 24 is 90 though it has some large sunspots too. Solar maximum existed in May 2014.There

were no sunspots in the duration of 2008 and 2009. The situation is very unusual in almost a

century. In 24 solar cycle, northern hemisphere has following sunspots of active regions with

negative polarity field while southern hemisphere has leading sunspots with positive polarity. It is

important to note that each cycle has different duration. Average duration of sunspot cycles is

slightly greater than 11 years. In each cycle the number of sunspots vary from maximum to

minimum and again back to maximum. Cycle 1 consists of 11.3 years, cycle 2 has 9 years, cycle

3 (9.3 years), cycle 4 (13.7 years), cycle 5 (12.6 years), cycle 6 (12.4 years), cycle 7 (10.5 years),

cycle 8 (9.8 years), cycle 9 (12.4 years), cycle 10 (11.3 years), cycle 11 (11.8 years), cycle 12 (11.3

years), cycle 13 (11.9 years), cycle 14 (11.5 years), cycle 15 (10 years), cycle 16 (10.1 years),

cycle 17 (10.4 years), cycle 18 (10.2 years), cycle 19 (10.5 years), cycle 20 (11.7 years), cycle 21

(10.3 years), cycle 22 (9.7 years), cycle 23 (11.7 years) and cycle 24 is going on which is start

from January 2008 and till 2018.

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Figure. 0.3 The sunspot cycles indicate reversal of magnetic polarity during the solar cycle.

0.5: SOLAR-CLIMATE RELATIONSHIP

Solar activity plays an important role in the variation of global temperature. The solar-climate

relationship has established that the lower temperatures are associated with below average sunspot

activity. While the higher temperatures are associated with above average sunspot activity. It is

not easy to describe the direct relationship between the sunspot cycle and climate, but some events

of climate coincide with sunspot numbers. For instance, in the duration of 1645 to 1715, the

observation of sunspots was dramatically lower than previous one. This duration is called the

Maunder Minimum. The output of the reduction in solar energy in the minimum duration could

have a slight cooling of the Earth. The thin layer of gases, which surrounds the globe of the earth’s

atmosphere protects and sustains life on earth. The atmosphere is frequently in motion and sun

releases radiative energy. On the earth’s surface, the amount of heat is different between the

equator and the poles. The Earth also rotates itself. The atmosphere consists of moisture content,

temperature, airflow and pressure, which is usually described in terms of climate and weather.

Weather describes the local state of the atmosphere in the duration of time periods ranging from

minutes to days. Weather phenomena, such as heat-waves, rainfall, gales, snowfall and frost, these

are effective in daily life. Weather system brings uncertainties in the atmosphere from time to time.

The hotness of the environment are frequently increased and affected over the Cryosphere (e.g.

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Ice caps, glaciers, etc.) and precipitation in numerous areas across the globe. The climate is

changing over frequently in time durations. The climate change can be expressed as a change in

the average climate from one period averaging to the next. There is no pattern in climate change,

some changes can be large and dramatic, as is clearly shown from reconstructions of temperature

changes between the ice ages and warmer interglacial periods.

Fig. 0.4 Curtsey by (David. Hath away @ NASA.GOV)

0.6: ENSO- SUNSPOTS RELATIONSHIP

The influence of the earth climatic condition of oscillations of solar activity is measurable only in

the long run duration. The Solar cycles (solar activity) and ENSO episode are correlated with each

other. Theory describes the relationship between sunspots and ENSO phenomena is premature,

but now is established by a collection of evidence that the solar cycle moderates wind field in the

stratosphere and troposphere (Anderson 1992, tnislay et.al, 1991) which reveal a potential

mechanism for the link between ENSO phenomena and the solar geometric effect. The change in

sunspot number can describe the association in which ENSO episode in more active when solar

activity is weak (Anderson 1990). The ENSO events can be reduced by long term solar changes

like Mander minimum, medieval grand maximum, similar solar activity is strong then ENSO

period is weak. For instance, the medieval warm period, solar activity is increased (Damon, 1988

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and Anderson 1990). Enso phenomena effects on climate and weather. The strong ENSO episode

brings droughts in the Southeast Asia countries and in the USA these events make a reason to

bring heavy rain and the number of hurricanes and typhoons are increased in the eastern pacific.

The relationship between rainfall and ENSO in different world areas vary from area to area

(Bhalme et. al. 1984). The impact of ENSO of Indian Ocean varies from season to season (Singh

et. al. 1999). In the early nineties climatologist have intent to understand the relation of the Indian

monsoon with the ENSO is ordered to estimate the monsoon rainfall over Asia (wabster et.al 1992)

describe the review the historical background the relation of monsoon and ENSO some qualitative

studies on the coupling of ENSO year and rainfall over Pakistan (Arif et al. 1994; Chaudhary,

1998).

0.7: OUR APPROACH

The purpose of this study is to analyze the behavior and the relationship of sunspots data and to

study the relationship of each cycle with the other cycles. The stationarity of Sunspot cycles is

determined using second differences with the help of correlogram test which studies

autocorrelation function, partial autocorrelation function and Ljung-Q statistics. The stationarity

of sunspot cycles is verified by using unit root test with Augmented Dickey Fuller (ADF) test.

Sunspot data has a stationary nature indicating the existence of a white noise together with the

linear component. Descriptive study of sunspot cycles rejects the existence of normality. The

normality test is based on skewness, kurtosis and Jurque-Bera test. All sunspots cycles are

positively skewed (right tail) and leptokurtic (high peakness). Different appropriate ARMA (p, q)

models are selected by analyzing the least value of the Durbin-Waston test (DW). Least square

estimator is used to estimate the ARMA process. AIC, SIC, HQC and Log likelihood are also

investigated for each model. RMSE, MAE, MAPE and U test are used to predict the sunspots in

each cycle. One of the novelties of this study is to develop a suitable GARCH model for sunspots

data with appropriate ARMA model specifications. The Lagrange Multiplier test is used to detect

the presence of Autoregressive Conditional Heteroscedastic (ARCH) effect on sunspots data.

Moreover GARCH (1, 1) specification with AR (p) process and ARMA (p, q) - GARCH (1, 1)

model are also investigated in this study. Finally, a comparison of forecasting evolution of these

models is discussed. The other novelty of this study is to analyze the complexity of sunspots cycles

and ENSO cycles using fractal approach. Both the self-similar (Hausdorff – Besicovich method)

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and self-affine (rescaled range analysis method) FDS and FDA approaches are used. Hurst

exponents exhibit that each sunspot cycles and ENSO cycles is persistent and long term

predictable. Universal scaling laws are used to explore the complexity of sunspots data. These laws

involve scaling exponents (βS, βA), spectral exponent (αs, αA) and autocorrelation coefficient (CA∇

and CS∇). It is concluded that self-similar approach is more appropriate than the self-affine

approach. The behavior of sunspot cycles and ENSO cycles is determined with the help of most

adequate probability distributions. The probability distributions are very helpful to understand the

cyclic behavior in future for the time series used.

It is found that Generalized Prato distribution and Generalized Mean value distribution are the

most appropriate probability distributions for sunspot cycles and ENSO cycles respectively. It is

to be noted that both these distributions are heavy tailed and long range predictable. Heavy tailed

analysis and long range correlation show that the time series data are strongly correlated and long

term predictable. Performing heavy tail and long range correlation analyses using self-similar and

self-affine techniques also confirmed that the time series data for sunspot cycles and ENSO cycles

are strongly correlated and long term predictable. Heavy tail analysis and long range correlation

indicated stationarity and linear trend. Again, the unit root test is used for verification of the results.

One of the objectives of this study is to analyze the images of solar active regions. For this purpose

the largest active region of 24th cycle viz. 12192AR is selected. This analysis is performed by using

fractal approaches and mathematical morphology. Fractal dimension are calculated by Hausdorff–

Besicovich method or box counting method and correlation dimension method. Universal scaling

laws, Hurst exponent, spectral exponent and autocorrelation coefficient are used for evaluation.

The basic principle of mathematical morphology (Dilation, Erosion, opening and closing) is used

to study the active region. Image segmentation of rotating sunspots is performed to understand the

active contour for edge detection for AR9114 and AR10696 of solar cycle 23rd are used to

understand the active contour with genetic algorithm. .

0.8: THESIS OUTLOOK

This dissertation exhibits the introduction of a brief review of the formation of sunspots, group of

sunspots, the active regions of sunspot and the role of sunspots in climate variability on the earth.

Solar Spots-ENSO relationship is also discussed.

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This dissertation consists of 4 chapters. Chapter zero consists of an introduction and the last chapter

5 is devoted to conclusion and results.

In chapter 1 is concerned with the modeling and forecasting of sunspot cycles. First of all the data

is checked for stationarity. Then, various ARMA (p, q) models are developed with the particular

focus on least value of Durbin-Waston statistics test. Moreover, GARCH (1, 1) model specified

with AR (p) process and GARCH (1, 1) model specified by ARM A (p, q) process are analyzed

for each sunspot cycle. Forecasted values of ARMA (p, q), AR (p) - GARCH (1, 1) and ARM A

(p, q) - GARCH (1, 1) are also compared with RMSE, MAE, MAPE and U test.

Chapter 2 discusses the regular-chaotic and irregular-chaotic behavior in view of the fractal

dimension (FD) (calculated with both self-similar FDS Hausdorff–Besicovich and self-affine FDA

Rescaled Range analysis methods) for Sunspots and ENSO cycles. Higuchi’s Algorithm is also

applied to calculate the fractal dimension FDH for sunspot cycles. Hurst exponents (HES and HEA)

describe the persistency of each cycle. Persistency is calculated to know the correlation and long

term behavior of the time series data. Universal scaling parameter like a scaling exponent is also

evaluated for the sunspots cycles. Wave-spectrum relation like the spectral exponent (αs, αA) and

autocorrelation coefficient (CA∇ and CS∇) for sunspots cycles are also analyzed. Theoretical fractal

scale instruments are developed to find a relationship between Sunspot cycles and ENSO cycles

regarding a long term relationship. The comparison of the two methods shows that self-similar

fractal dimension is more appropriate than the self-affine fractal dimension.

Chapter 3 studies the best fitted and most adequate probability distributions of sunspot and ENSO

cycles. In this connection significant probability distribution is determined for 24 sunspot cycles

in which 24th cycle is still to complete ENSO data against cycles (1-23) is also studied accordingly.

Generalized Pareto and Generalized Extreme Value distribution are found to be most significant

for sunspots and ENSO cycles respectively. These distributions show long term persistence and

heavy tail behavior. The modeling of heavy tail and long term correlation of sunspots and ENSO

cycles are analyzed for the confirmation of heavy tail behavior. Unit root test is applied for

confirmation to linear and stationary behavior.

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Chapter 4 consists of image analyses of the largest active region of 24th cycle viz12192AR. Fractal

technique are used to analyze the active regions. Both the box counting and correlation dimension

methods are used to evaluate fractal dimensions. Hurst exponent are analyzed to show the

persistency of active regions. Scaling parameters (scaling exponent, spectral exponent and

autocorrelation coefficient) are found to evaluate the active region. Mathematical morphology is

used to study the solar active regions. Image segmentation of rotating sunspot of active region

AR9114 and AR10696 of solar cycle 23 using genetic active contour. Helicity, Temperature,

tangential velocity and energy released are used as a parameter to determine active contour.

Chapter 5 is comprised of the conclusion and principal results of the dissertation along with the

future outlook.

Contents of chapters 1, 2, 3, and 4 are submitted for publication. Each chapter exhibits different

significant research study with consequent introduction, methodology, results and discussion.

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CHAPTER 1

GARCH MODEL FOR SUNSPOT CYCLES

Sunspot cycles demonstrate solar activity. Solar activity is a feature of the Sun which is not

properly distributed in time. This fact can be observed from the variability of the number of

sunspots with respect to time. Usually they follow the 11 year’s cycle with a minimum and a

maximum (Schwabe, 1844). Schwabe was the first to explain an irregular behavior and

fluctuations of sunspot cycles as well as individual’s maximum intensity and shapes. The

maximum and minimum number of sunspots varies from cycle to cycle. For example a very few

sunspots were observed during the maunder minimum (1645 to 1715) (Thomas et. al, 2008).

Maunder minimum is considered to impose harmful effects on the earth. Long term variations in

solar activity bring a wide change in the climate of the earth. Whereas, short term variations may

cause fluctuations of certain meteorological parameters. In each 11-year cycle, the polarity of the

sun’s magnetic field reverses (Hale et. al, 1919). Cycles vary both in length and size. One of the

successful models to explain the development of sunspots is the model developed by Babcock

(Babcock 1961). Sunspots have an impact on earth’s climate and rainfall. The solar-climate

relationship has established that the lower temperatures are associated with below average sunspot

activity while the higher temperatures are associated with above average sunspot activity.

In chapter 1, describe stochastic autoregression and moving average (ARMA) process for

modeling time series stationary sunspot cycles mean monthly sunspot numbers (1-23 and 24 cycle

consists of a complete observation range from 1st to 23rd) and ENSO cycles (1-21, 22 cycle

complete data from 1 to 23 and 23 cycle belong to an active episode from 1981 to 2000). The

novelty of this chapter is analyzed the stationary generalized autoregressive conditional

heteroskedasticity GARCH (1, 1) stochastic volatility model with specification autoregressive AR

(p) process are used to estimate and forecast the sunspot cycles. GARCH (1, 1) specification with

ARMA (p, q) process also are used to estimate and forecast the sunspot cycles. Sunspot cycles are

forecasted with the help of a Root mean square error (RMSE), Mean Absolute Error (MAE) and

Mean Absolute Percentage Error (MAPE). The Gaussian quasi maximum likelihood estimation is

used to analysis these processes. The adequacy of each model depends on Akaike information

criterion (AIC), Bayesian Schwarz information criterion (BIC) and Hannan Quinn information

criterion (HIC). Best fitted model of sunspot cycles is chosen with compression of these techniques

24

with the help of Root mean square error (RMSE), Mean Absolute Error (MAE) and Mean Absolute

Percentage Error (MAPE).

1.1: DATA DESCRIPTION AND METHODOLOGY:

Time series sunspot cycle’s data are stationary at second difference. This study describes stochastic

autoregression and moving average (ARMA) process for time series modeling. Stationary sunspot

cycles mean monthly sunspot numbers (1-23 and 24 cycle consists of a complete observation range

from 1st to 23rd). The novelty of this chapter is analyzed the stationary generalized autoregressive

conditional heteroskedasticity GARCH (1, 1) stochastic volatility model with specification

autoregressive AR (p) process are used to estimate and forecast the sunspot cycles. GARCH (1, 1)

specification with ARMA (p, q) process also are used to estimate and forecast the sunspot cycles.

Sunspot cycles are forecasted with the help of a Root mean square error (RMSE), Mean Absolute

Error (MAE) and Mean Absolute Percentage Error (MAPE). The Gaussian quasi maximum

likelihood estimation is used to analysis these processes. The adequacy of each model depends on

Akaike information criterion (AIC), Bayesian Schwarz information criterion (BIC) and Hannan

Quinn information criterion (HIC). Best fitted model of sunspot cycles is chosen by diagnostic

checking with compression of these techniques with the help of Root mean square error (RMSE),

Mean Absolute Error (MAE) and Mean Absolute Percentage Error (MAPE) and Theil’s U-

Statistics.

Sunspot cycles have an appropriate process by selection criteria based on the least value of the

Durbin-Waston stats (DW) are ARMA (2, 2), ARMA (2, 6), ARMA (3, 2), ARMA (3, 3), ARMA

(3, 4), ARMA (4, 3), ARMA (4, 4), ARMA (4, 5), ARMA (4, 6), ARMA (5, 2), ARMA (5, 3),

ARMA (5, 4), ARMA (5, 5), ARMA (5, 6), ARMA (6, 2), ARMA (6, 3), ARMA (6, 4), ARMA

(6, 5), ARMA (6, 6). Moreover, most appropriate model is ARMA (6, 4) process which is

frequently used in sunspot cycles. ARMA (5, 3), ARMA (5, 4), ARMA (3, 3), ARMA (4, 4),

ARMA (6, 5), ARMA (6, 2), these models are maximum use in sunspot cycles. Sunspot cycles

have appropriate GRACH (1, 1) process specification with AR (p) are estimation and forecast.

Sunspot cycles follow AR (2) - GARCH (1, 1) process and AR (3) - GARCH (1, 1). In ARAM (p,

q) - GARCH (1, 1) follows ARAM (1, 1) - GARCH (1, 1), ARAM (2, 2) - GARCH (1, 1), ARAM

(3, 2) - GARCH (1, 1), ARAM (3, 3) - GARCH (1, 1), ARAM (4, 2) - GARCH (1, 1), ARAM (4,

25

4) - GARCH (1, 1), ARAM (5, 1) - GARCH (1, 1), ARAM (5, 3) - GARCH (1, 1) and ARAM (6,

1) - GARCH (1, 1).

1.2: TIME SERIES ANALYSIS WITH STATIONARY PROCESS

Time series are very helpful in many disciplines of the solar physics, climate change and so on.

The stochastic component remains, periodicities and trend have been excluded from a time series.

The large range correlation in the context of the solar and terrestrial physics. Many time series in

the solar physics exhibits persistence, where successive value is positively correlated. Big values

trend to follow big whereas small observations followed the small trend. The correlation is the

statistical dependence of distantly and directly neighbor values.

If the data set is larger than time series analysis is used. Mathematically time series is described as

a series of vector Xt; t = 0, 1, 2, ….. Where t indicate the time elapsed (Raicharoen et. al. 2003,

Hipal et. al. 1994 and John et. al. 1997). Variable Xt is a random variable. Time series analysis

based on various goals. Descriptive is defined as identify the behavior trend of correlated data.

Explanation expresses as modeling and understanding the data. Forecasting explore the predication

the trend either long-term or short-term from previous patterns. Intervention analysis is the changes

in a single event with respect to time. Quality control defines the problem indicate specified size

of deviations.

In a time series, stationery series has over time constant mean, variance, autocorrelation and so on.

Consequently, in time shifting joint probability distribution does not change. Compulsory

condition for developing time series models that are better future forecasting. Moreover, the

mathematical complexity of the best fitted model reduces. Stationary process is strong as well as

a week. Strong stationary process Xt is defined as,

if Xt, Xt + 1, Xt + 2 , … , Xt + n and Xt + S, Xt + 1 + S, Xt + 2 + S , … , Xt + n + S

have some joint distribution of all t and n where s > 0.

Weak stationary explain as, if mean value E (Xt) is time independent and Cov (Xt, XS) depends on

the time difference |𝑡 − 𝑆| (time lag). The figure 1.1: depict the time series of monthly average

sunspot number from cycle 1st to cycle 23rd is plotted.

26

Figure 1.1: The monthly average sunspot number from cycle 1 to 23

1.3: AUTOCORRELATION FUNCTION, PARTIAL AUTOCORRELATION FUNCTION

AND LJUNG-BOX Q-STATISTICS OF SUNSPOT CYCLES

These functions are very useful in order to verify a time series model: Autocorrelation function

(ACF), Partial Autocorrelation function (PACF) and Ljung-Box Q-statistics (Reale 1998).

The autocorrelation function (AC) is used to analysis the linear predictability of the data set Xt.

ℵ(𝑠) = 𝐶𝑜𝑣 (𝑋𝑡,𝑋𝑡−𝑠 )

√𝑉𝑎𝑟(𝑋𝑡)𝑉𝑎𝑟(𝑋𝑡−𝑠) −1 ≤ ℵ(𝑠) ≤ 1 (1.1)

For stationary Var (Xt) = Var (Xt-s). Samples of AC are called a Correlogram.

Partial Autocorrelation function (PAC) defines the correlation between Xt and Xt-s after omit the

effect of the intervening variables Xt, Xt + 1, Xt + 2 , … , Xt – s-1 denoted by 𝜇𝑠𝑠.

𝜇𝑠𝑠 = ℵ(𝑠)− ∑ 𝜇𝑠−1,𝑗 ℵ𝑠−𝑗

𝑠−1𝑗=1

1− ∑ 𝜇𝑠−1,𝑗 ℵ𝑠−𝑗ℵ𝑠−1𝑗=1

(1.2)

Partial Autocorrelation function (PAC) is very helpful to identify the AC and PAC are used to

identify linear time series model.

Ljung-Box Q-statistics are used to analysis the effect of autocorrelation at lags.

Ljung-Box Q-statistics = N (N+2) ∑ℵ(𝑠)2

𝑁−𝑠

𝑠𝑗=1 (1.3)

Where ℵ(s) is the s-th autocorrelation and N is the number of observations. Ljung-Box Q-statistics

are also used as a test of white noise series.

27

Sunspot cycles have stationary in second difference. Differencing is the transformation which is

converted non-stationary the time series data into stationary. Table 1.1 depicts that Autocorrelation

(AC), Partial Autocorrelation (PAC) and Ljung-Box Q-statistics test of sunspot cycles. Each

sunspot cycle has a white noise in second difference. White noise is purely random and simple

stationary process in which sequence {∈𝑡} of uncorrelated random variables with ∈𝑡 = WN (0,𝜎2).

White noise is very useful to construct many processes. Probability is significant in each lag.

Figure 1.2 describe that ACF and PCF of each sunspot cycle (1-23) and 24 cycle based on complete

data set from 1st to 23rd cycles.

35

Figure 1.2: ACF and PACF of Sunspot Cycles (1-23, 24)

36

1.4: UNIT ROOT TEST OF SUNSPOTS CYCLES BASED ON AKAIKE INFORMATION

CRITERION (AIC)

The unit root test can be used for some stochastic process which cause problem is statistic inference

involving in time series. The random walk model

Yt = ρYt-1 + ut −1 ≤ ρ ≤ 1 (1.4)

In this model Yt belongs to the time series and ut represent the white noise error while ρ is the

coefficient. The equation (6) is used to regress the Yt on the lagged value Yt-1estimate for unit root

test. If the predictable value ρ identical to 1 statistically. The series comprises unit root and null

hypothesis rejected (McCarthy, 2015). Transform the equation 6 by subtraction Yt-1 both sides

Yt−Yt -1 = ρYt -1− Yt -1 + ut

Yt−Yt -1 = Yt -1(ρ− 1) + ut

∆ Yt = ɸYt -1 + ut (1.5)

Where ∆ and ɸ represent the operation of the first difference. The null hypothesis (ɸ = 0) for time

series non stationary condition represents the unit root (da silve Lopes, 2006).

The Augmented Dickey Fuller (ADF) test to solve the autocorrelation problem. This test is

implemented by augmenting three following equations, the dependent variable’s (∆Yt) adding

lagged values and Yt represents a random walk with

∆ Yt = 𝛾1+ 𝛾2t + ɸ Yt -1 + ut (1.6)

Where 𝛾2t and 𝛾1represents the intercept and trend respectively.

∆ Yt = ɸYt -1 + ut (1.7)

It shows neither intercept nor trend.

∆ Yt = 𝛾1+ ɸ Yt -1 + ut (1.8)

It shows intercept (𝛾1) only (Kwiatkowski et al., 1992; McCarthy, 2015).

We assume the null hypothesis H0: variables have a unit root (non-stationary data). The condition

where the H0 is rejected if p-value is less than 5%. The critical value of the absolute value of

Augmented Dickey Fuller (ADF) test is greater than at 1% and 5% significance level.

The Augmented Dickey Fuller (ADF) test for stationary explored that the sunspot cycles 1-23

(1755 to 2008) and complete from 1 to 23 cycle’s time series data. In case the data is not stationary

than behave like a unit root. Table 1.2 depicted that sunspot cycles has stationary nature and unit

root test rejected. The reason to reject H0 is p-value is less than 5% or the critical value of the

37

absolute value of Augmented Dickey Fuller (ADF) test is greater than at 1% and 5% significance

level. The stationary of sunspot cycles are calculated at second difference with based on Akaike

Information Criterion (AIC). Sunspot cycle (1-24) reject unit root test with Augmented Dickey

Fuller (ADF) test values -6.181642, -6.091491, -6.893381, -7.579517, -6.823644, -6.875331, -

6.695780, -8.558407, -6.745618, -6.933791, -5.950719, - 6.602668, -7.700490, -8.106940, -

7.087756, -10.74242, -6.524422, -7.167179, -7.122662, -6.852907, -7.960910, -7.082791, -

7.048743, -17.69189 respectively. Table 1.3 described that

41

Figure 1.4: Stationary sunspot cycle (1-24)

1.5: TESTS FOR NORMALITY

Normality test is very useful to determine a normally distribution of well modeled data set. Test

based on analysis the two numerical measures, shapes skewness and excess kurtosis. The data sets

are normally distributed if values are close to zero. The acceptance of Jurque-Bera test also focused

on skewness and kurtosis. Test of normality consists of skewness, kurtosis and Jurque-Bera test.

Skewness measures the degree of asymmetry of the data.

Skewness = ∑ (𝑋𝑖−�̅�)3𝑛

𝑖=1

(𝑛−1)𝑆3 (1.9)

Where �̅� is the mean, S is standard deviation and n is the number of values (Cryer et. al. 2008).

Data is normally distributed if the skewness value is equal to zero. Positive skewed (right tail) if

greater than zero and negative skewed (left tail) with less than zero.

Kurtosis measures the degree of peakness of the data. Kurtosis has been estimated as

Kurtosis = ∑ (𝑋𝑖−�̅�)4𝑛

𝑖=1

(𝑛−1)𝑆4 (1.10)

Where �̅� is the mean, S is standard deviation and n is the number of values of the time series data.

Kurtosis of normal distribution is called mesokurtic if it is equal to 3. Whereas, leptokurtic if the

value is greater than 3. Platokurtic if the value is less than 3.

Jurque-Bera Statistics Test is accepted with normality of the data with skewness is equal to zero

and excess kurtosis is also equal to zero. Jurque-Bera test as follows

Jurque-Bera test = 𝑛(𝑆𝑘𝑒𝑤𝑛𝑒𝑠𝑠)2

6 +

𝑛(𝐾𝑢𝑟𝑡𝑜𝑠𝑖𝑠)2

24 (1.11)

42

Jurque-Bera test statistics is approximately as Chi-squared distribution with two degrees of

freedom. Null hypothesis (HO) is a normal distribution with skewness is zero and excess kurtosis

is zero (which is same as a kurtosis is 3). Alternate hypothesis (HA) gives data is not normally

distributed.

1.5.1: Descriptive Statistics and Normality Test of Sunspot Cycles

Table 3 demonstrates the descriptive statistics of each Sunspot cycle. Mean, median, standard

deviation, skewness, kurtosis and Jurque-Bera test have been calculated of each cycle. 1st cycle

has 128 sunspot numbers with 43.81± 24.35. 2nd cycle follows 144 sunspot observations with

57.04± 39.21. 3rd cycle expresses 107 sunspot observations with 68.95± 52.12. 4th cycle represents

169 sunspot observations with 59.92 ± 54.44. 5th cycle explores 148 sunspot observations with

2305 ± 18.71. 6th cycle has 160 sunspot observations with 17.76 ± 17.98. 7th cycle follows 119

sunspot observations with 40.06 ± 26.01. 8th cycle reveals 120 sunspot observations with 65.39 ±

48.31. 9th cycle represents 139 sunspot observations with 59.37 ± 38.41. 10th cycle explores 144

sunspot observations with 45.86 ± 32.57. 11th cycle expresses 140 sunspot observations with 53.59

± 46.81. 12th cycle has 142 sunspot observations with 32.55 ± 27.15. 13th cycle represents 148

sunspot observations with 37.54 ± 30.63. 14th cycle follows 136 sunspot observations with 32.68

± 26.33. 15th cycle expresses 114 sunspot observations with 46.53 ± 33.97. 16th cycle signifies

125 sunspot observations with 39.79 ± 28.73. 17th cycle follows 125 sunspot observations with

57.64 ± 40.52. 18th cycle has 122 sunspot observations with 74.43 ± 54.71. 19th cycle expresses

129 sunspot observation with 89.04 ± 71.52. 20th cycle represents 140 sunspot observations with

60.26 ± 37.29. 21th cycle explores 119 sunspot observations with 83.57 ± 57.22. 22th cycle has

124 sunspot observations with 76.15 ± 58.50. 23th cycle follows 140 sunspots observations with

57.94 ± 42.67. 24th cycle reveals 3030 sunspot observations with 52.26 ± 44.60. Each sunspot

cycle shows that positive skewed (right tail). Whereas different cycles have kurtosis behavior.

Sunspots cycle 15 explore that approximate mesokurtic (3.021) normally distributed. Moreover,

maximum sunspot cycles follow platokurtic which represents that flat tail (peakness). Sunspot

cycles 6th, 9th and 24th evaluate leptokurtic (heavy tail). In Jurque-Bera test, each sunspot cycles

rejected the null hypothesis (sunspot cycles are normally distributed by any mean and variance).

43

1.6: TEST OF ADEQUACY

Selection for the appropriate and adequate process of ARMA (p, q), AR (p) – GARCH (1, 1) and

ARMA (p, q) – GARCH (1, 1) for accurate and significant test are very important. Test of

adequacy consists of Durbin- Watson (DW) statistics test, Akaike information criterion (AIC),

Bayesian Schwarz information criterion (SIC) and Hannan Quinn information criterion (HQC) and

Log maximum likelihood test. The selection of appropriate sunspot cycles is based on the

minimum value of the Durbin- Watson (DW) statistics test for each method.

Durbin- Watson (DW) statistics is a test for measuring the linear association between adjacent

residual from a regression model. The hypothesis of Durbin- Watson statistics is 𝜏 = 0 is the

specification.

Ut = 𝜏 Ut-1 + ∈𝑡 (1.12)

Durbin-Watson (DW) is equal to 2 shows that no serial correlation. If Durbin- Watson (DW) is

less than 2 indicate that positive correlation and range from 2 to 4 represent that negative

correlation. The series is strongly correlated if the value nearly approaches to zero.

Akaike information criterion used to compare the statistical models set to each other. AIC was

introduced by Hirotogu Akaike in 1973. It is the extension of the maximum likelihood principle.

The selection criterion is focused on the least value of AIC.

AIC = - 2Log (likelihood) + 2S (1.13)

Where S is the model parameter numbers. Likelihood is a measure of fit model. Maximum values

exhibits the best fit.

Schwarz criterion is used to select the best model among finite model. The appropriate model is

based on the least value of SIC. Schwarz criterion (SIC) was developed by Gideon E. Schwarz. It

is closely related to the AIC.

SIC = -2ln (Likelihood) + (S + S ln (N)) (1.14)

Where S is the model parameter numbers. N exhibits the number of observations.

In statistics, the Hannan-Quinn information is the criterion for model selection. HAQ is an

alternative to AIC and SIC.

HQC = -2 Log (Likelihood) + 2 (S + S ln (N)) (1.15)

Where S is the model parameter numbers. N exhibits the number of observations.

44

1.7: DIAGNOSTIC CHECKING TEST

In this study, forecasting performance and evolution of each sunspot cycle is checked by Mean

Absolute Error (MAE), Root Mean Square Error (RMSE), and Mean Absolute Percentage Error

(MAPE) and Theil’s U-Statistics. A little description of these techniques is given in the following.

The Mean Absolute Error (MAE) is expressed as

MAE = 1

𝑛 ∑ |∈𝑡|𝑛

𝑡=1 (1.16)

Mean Absolute Error (MAE) measures the absolute deviation of forecasted values from real ones.

It is also known as Mean Absolute Deviation (MAD). It expresses the magnitude of overall error,

happened due to forecasting. MAE do not cancel out the effect of positive and negative errors.

MAE does not express the directions of errors. It should be as small as possible for a good

forecasting. MAE depends on the data transformations and the scale of measurement. MAE are

not the panelized for the extreme forecast error.

The Mean Absolute Percentage Error (MAPE) is defined as

MAPE = 1

𝑛 ∑ |

∈𝑡

𝑋𝑡|𝑛

𝑡=1 × 100 (1.17)

Mean Absolute Percentage Error (MAPE) provides the percentage of the average absolute error.

It is independent of the scale measurement. Whereas, dependent of the data transformations.

MAPE does not locate the direction of Error. Extreme deviation is not penalized by MAPE.

Opposite signed errors does not parallel to each other in MAPE.

The root mean squared error (RMSE) is defined as

RMSE = √1

𝑛∑ ∈𝑡

2𝑛𝑡=1 (1.18)

RMSE calculate the average squared deviation of forecasted values. Opposed sighed error do not

parallel to each other. RMSE provide the complete idea of the error happened during forecasting.

While forecasting, panelizes extreme error happens. In RMSE, the total forecast error is affected

by the large individual error. For instance, large error is more appropriate than small errors. It does

not reveal the direction of overall errors. RMSE is affected to the data transformation and change

of scale. RMSE is a good measure of overall forecast error.

Theil’s U-Statistics is defined as

U = √

1

𝑛∑ ∈𝑡

2𝑛𝑡=1

√1

𝑛∑ 𝑓𝑡

2𝑛𝑡=1 √

1

𝑛∑ 𝑋𝑡

2𝑛𝑡=1

0 ≤ 𝑈 ≤ 1 (1.19)

45

Where ft represent the forecasted value and Xt shows that the actual value. U is the normalized

measure of the total forecast error. U is equal to 0 exhibits the perfect fit.

1.8: ARMA MODEL

A statistical approach to forecasting involves stochastic models to predict the values of sunspot

cycles by using pervious once. In the linear time series, two methods are frequently used in

literature, viz. Autoregressive AR (p) and Moving Average MA (q) (Jenkins et.al. 1970 and Hipal

et. al. 1994). ARMA models are developed by (Jenkins et. al 1994). An ARMA model is the

combination of an idea of Autoregressive AR (p) and Moving Average MA (q) process. The

concept of ARMA process is strongly relevant in volatility modeling. ARMA model is wieldy

used for forecasting the future values. Autoregressive process (AR) is developed by (yule, 1927).

In stochastic process, Autoregressive process AR (p) can be expressed by a weighted sum of its

previous value and a white noise. The generalized Autoregressive process AR (p) of lag p as follow

Xt = α1 Xt-1 + α 2 Xt-2 + … + α p Xt-p + ∈t (1.20)

Here εt is white noise with mean E (∈t) = 0, variance Var (∈t) = σ2 and Cov (∈t -s, ∈t) = 0, if s ≠ 0.

For every t, suppose that 𝜏t is independent of the Xt-1, Xt-2, ….. 𝜏t is uncorrelated with Xs for each s

< t. AR (p) models regress is past values of the data set. Whereas, MA (q) model relates with error

terms as a descriptive variables (Hipal et. al. 1994). The generalized Moving Average process MA

(q) of lag q as follows

Xt = ∈t + β1 ∈t-1 + β2 ∈t-2+ … + βq ∈t-q (1.21)

The process Xt is defined by the ARMA model

Xt = α1 Xt-1 + α 2 Xt-2 + … + α p Xt-p + ∈t + β1 ∈t-1 + β2 ∈t-2+ … + βq ∈t-q (1.22)

With ∈t is an uncorrelated process with mean zero. The prediction of ARMA (p, q) process shows

the decay to be sinusoidally and exponentially to zero.

1.8.1 ARMA Models of Sunspots Cycles

ARMA models are widely used for the prediction of stationary stochastic second order process.

The ARMA model is a tool for analysis and understanding of the causal structures or getting the

predictions of future values in time series. The suitable models for sunspot cycles are selected

based on Durbin-Watson statistics test. Least square estimation is used to calculate ARMA

process. Coefficient of determination, R2 near to one demonstrates that data values in each cycle

depend on each other. Table1.4 depicts 5 best models for each sunspot cycle. Least square

46

estimation is used to calculate ARMA (p, q) model. The selection of models depends on Durbin-

Watson statistic test. The Durbin- Watson (DW) near to zero is describe the series are positively

correlated. The value 2 shows that the series has no correlation. If DW is ranged between 2 and 4

indicates negative correlation. Adequacy of the models is checked by AIC, SIC, HQC and

Maximum log likelihood tests. The forecasting evolution is checked by RMSE, MAE, MAPE and

U tests. Sunspot cycle (Aug 1755 - Mar 1766) shows that best fitted models are ARMA (4,4),

ARMA (4,5), ARMA (5,4), ARMA (5,5) and ARMA (6,5). According to Durbin- Watson (DW)

test ARMA (6, 5) model possesses the least value of DW (0.798) which implies that the 1st cycle

is strongly correlated. ARMA (6, 5) model is also the best model according to log maximum

likelihood (-554.25). ARMA (4, 5) and ARMA (4, 4) models have least value 8.559, 8.648 and

8.595 of AIC, SIC and HQC respectively and show a best fit. Coefficient of determination, R2

demonstrates that data values in each cycle depend on each other. Forecasted evaluation of 1st

cycle describes that RMSE with least value 23.01596 is most favorable for ARMA (5, 4).

According to MAE with the value 18.43576 shows that best fitted model is ARMA (4, 5). MAPE

with the least value 55.73550 shows that ARMA (4, 4) is the best fitted model. U test with least

value 0.251178 describes that ARMA (6, 5) is best fitted. Sunspot cycle (Mar 1766 - Aug 1775)

exhibits that best fitted model are ARMA (2, 2), ARMA (2, 6), ARMA (3, 2), ARMA (3, 3) and

ARMA (5, 3). ARMA (5, 3) process has least value DW (0.719) which is best fitted and strongly

correlated and predictable. Appropriate model according to log maximum likelihood is also

ARMA (5, 3) model with -547.40. ARMA (2, 2) model has least value 9.402, 9.498 and 9.441 of

AIC, SIC and HQC respectively and show a best fit. Forecasted evolution define that RMSE, MSE

and U test with least values 37.92649, 29.55789 and 0.309054 have best fitted model ARMA (5,

3) respectively. MAPE with least value 145.5506 shows that best fitted model is ARMA (2, 2).

Sunspot cycle (Aug 1775 - Jun 1784) exhibits that best fitted model are ARMA (5, 4), ARMA (5,

5), ARMA (5, 6), ARMA (6, 4) and ARMA (6, 5). ARMA (6, 5) model has least value DW (0.846)

and log maximum likelihood with -508.60 which is shown that best fitted and strongly correlated

and predictable. ARMA (5, 4) process has an appropriate model with least value 9.334, 9.434 and

9.375 of AIC SIC and HQC respectively. Forecasted evolution demonstrates that RMSE with

57.08266 has appropriate model is ARMA (6, 5). According to MAE and U test with least value

43.6452 and 0.411658 shows that best fitted model are ARMA (6, 4) respectively. MAPE with

least value 102.4356 shows that the best fitted model is ARMA (5, 5). Sunspot cycle (Jun 1784 -

47

Jun 1798) exhibits that best fitted model are ARMA (3, 3), ARMA (5, 4), ARMA (5, 6), ARMA

(6, 4) and ARMA (6, 5). According to DW and log maximum likelihood with 0.904 and -784.12

shows that ARMA (6, 5) process is best fitted respectively. ARMA (3, 3) process is best fit with

least value 8.496, 8.570 and 8.526 of AIC SIC and HQC respectively. Forecasted evolution define

that RMSE and MAE with 53.38909 and 41.59299 have appropriate model is ARMA (5, 6)

respectively. According to MAPE with 141.12189 define the appropriate model is ARMA (3, 3)

and U test with 0.477880 shows the best fitted model is ARMA (5, 4). Sunspot cycle (Jun 1798 -

Sep 1810) indicate that best fitted models are ARMA (3, 3), ARMA (4, 4), ARMA (5, 3), ARMA

(6, 2) and ARMA (6, 4). ARMA (6, 4) process has least value DW (0.794) which is best fitted and

strongly correlated and predictable. Suitable model according to log maximum likelihood with -

584.79 is ARMA (5, 3) model. AIC SIC and HQC with smallest values 7.278, 7.359 and 7.311

described that most appropriate model is ARMA (3, 3). Forecasted evolution define that RMSE

and MAE with 19.61195 and 16.77907 have appropriate model is ARMA (5, 3). According to

MAPE with 161.6836 shows that the appropriate model is ARMA (3, 3) and U test with 0.416733

describe that the best fitted model is ARMA (6, 4). Sunspot cycle (Sep1810 - Dec 1823) describe

the best fitted models are ARMA (3, 4), ARMA (6, 2), ARMA (6, 4), ARMA (6, 5) and ARMA

(6, 6). ARMA (6, 5) model is most appropriate with the smallest value of DW with 0.794 and AIC

SIC and HQC with 7.944, 8.021 and 7.975 respectively. According to log maximum likelihood

with -636.60 shows that the best model is ARMA (6, 2) model. Forecasted evolution define that

RMSE and MAE with 17.58113 and 13.25282 have appropriate model is ARMA (6, 4). According

to MAPE with 251.9070 appropriate model is ARMA (6, 2) and U test with 0.418201shows that

the best fitted model is ARMA (3, 4). Sunspot cycle (Dec1823 - Oct 1833) shows that best fitted

models are ARMA (4, 4), ARMA (5, 2), ARMA (5, 3), ARMA (5, 5) and ARMA (6, 4). According

to Durbin-Weston (DW) ARMA (5, 5) model is the best model with least value of DW (0.926).

ARMA (6, 4) model is also the best fitted model according to log maximum likelihood (-523.20).

AIC SIC and HQC described that the appropriate model is ARMA (4, 4) with least value 8.628,

8.721 and 8.666 respectively. Forecasted evaluation describes that RMSE and MAE explore

ARMA (5, 3) are favorable with least value 24.84239 and 20.57404 respectively. MAPE has least

value 125.1342 by ARMA (5, 5). U test describes that ARMA (5, 3) is best fitted with least value

0.294667. Sunspot cycle (Oct1833 - Sep 1843) exhibits that best fitted model are ARMA (3, 3),

ARMA (4, 6), ARMA (5, 6), ARMA (6, 2) and ARMA (6, 3). ARMA (5, 6) process is the best

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fitted model with the smallest value DW (0.767). Appropriate model according to log maximum

likelihood is also ARMA (4, 6) model with -572.46. AIC SIC and HQC described that ARMA (6,

2) process has least value 9.321, 9.413 and 9.358 respectively. Forecasted evolution define that

RMSE, MSE and U test have best fitted model ARMA (4, 6) with 47.89669, 37.64070 and

0.353636 respectively. MAPE shows that best fitted model is ARMA (3, 3) with 101.5025.

Sunspot cycle (Sep1843 - Mar 1855) shows that the best fitted models are ARMA (3, 3), ARMA

(4, 3), ARMA (4, 4), ARMA (5, 3) and ARMA (6, 4). Appropriate model is ARMA (6, 4) with

the least value of Durbin- Watson with 0.779 and log maximum likelihood with -655.49. AIC SIC

and HQC depict that ARMA (3, 3) process is suitable with the smallest values 9.159, 9.243 and

9.193 respectively. Forecasted evolution define that RMSE and U test have best fitted model

ARMA (4, 3) with 36.85102 and 0.296860 respectively. According to the MAE best fitted model

is ARMA (6, 4) with 26.67266. MAPE with 55.33579 shows the best fitted model is ARMA (3,

3). Sunspot cycle (Mar1855 - Feb 1867) depict that best fitted models are ARMA (4, 4), ARMA

(5, 3), ARMA (6, 2), ARMA (6, 4) and ARMA (6, 5). According to DW and log maximum

likelihood with 0.993 and -614.66 shows that ARMA (6, 5) process is suitable. AIC SIC and HQC

with 8.424, 8.486 and 8.449 described that the appropriate model is ARMA (4, 4). Forecasted

evolution demonstrates that RMSE, MAE and U test with 34.05871, 26.32748 and 0.318617 have

appropriate model is ARMA (6, 4). According to MAPE with 149.2235 shows that best model is

ARMA (3, 3). Sunspot cycle (Feb1867 - Sep 1878) indicate that best fitted models are ARMA (3,

3), ARMA (5, 3), ARMA (5, 4), ARMA (6, 4) and ARMA (6, 5). ARMA (6, 5) process is suitable

model with the least value DW (0.680) and log maximum likelihood with -649.72. AIC, SIC and

HQC with smallest value 9.002, 9.086 and 9.036 described that ARMA (3, 3) process is best fitted.

Forecasted evolution define that RMSE and MAE with 51.34360 and 40.82203 have appropriate

model is ARMA (6, 4). According to MAPE with 619.2955 shows that the appropriate model is

ARMA (3, 3) and U test with 0.453341reveal that the best fitted model is ARMA (5, 3). Sunspot

cycle (Sep1878 - Jun 1890) describe the best fitted models are ARMA (4, 4), ARMA (5, 3), ARMA

(5, 4), ARMA (6, 4) and ARMA (6, 6). ARMA (5, 4) process is the suitable model with the least

value of DW with 0.877 and log maximum likelihood with -606.7. AIC, SIC and HQC with

smallest value 8.513, 8.596 and 8.547 show that the appropriate model is ARMA (4, 4). Forecasted

evolution define that RMSE, MAE and U test with 26.94070, 22.33445 and 0.384873 have

appropriate model is ARMA (5, 3). According to MAPE with 317.31740 shows that the

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appropriate model is ARMA (4, 4). Sunspot cycle (Jun1890 - Sep 1902) reveal that the best fitted

models are ARMA (4, 3), ARMA (4, 4), ARMA (5, 4), ARMA (6, 2) and ARMA (6, 4). ARMA

(6, 4) model has the least value of DW is 1.122 so is an appropriate model according to Durbin-

Watson state. Log maximum likelihood with -627.95 shows best model is ARMA (5, 4). AIC SIC

and HQC with least value 8.356, 8.437 and 8.389 described that best model is ARMA (4, 4).

Forecasted evolution define that RMSE and MAE with 34.15608 and 28.14172 have appropriate

model is ARMA (6, 4). MAPE with the least value 424.3318 shows that the suitable model is

ARMA (6, 2) and U test with 0.418577 is best fitted model is ARMA (5, 4). Sunspot cycle (Sep

1902 - Dec 1913) exhibits that best fitted models are ARMA (4, 4), ARMA (4, 6), ARMA (5, 3),

ARMA (5, 5) and ARMA (5, 6). ARMA (6, 4) model is an appropriate model with least value of

DW is 0.980.Log maximum likelihood with maximum value -602.26 shows the best model is

ARMA (5, 4). AIC SIC and HQC with least value 8.711, 8.797 and 8.756 shows that the best

process is ARMA (4, 4). Forecasted evolution shows that RMSE, MAE and U test with 26.33349,

21.89815 and 0.363862 have appropriate model is ARMA (4, 6). According to MAPE with

447.7918 describe that the best model is ARMA (4, 4). Sunspot cycle (Dec 1913 - May 1923)

indicate that best fitted models are ARMA (4, 3), ARMA (5, 3), ARMA (5, 4), ARMA (5, 5) and

ARMA (6, 4). ARMA (6, 4) process has least value DW with 0.687 is best fitted and strongly

correlated. The best model according to log maximum likelihood with -528.49 is ARMA (5, 5)

model. AIC SIC and HQC with smallest value 9.077, 9.123 and 9.115 described that ARMA (4,

3) process is most fitted. Forecasted evolution of RMSE, MAE and U test with 33.02818, 26.04595

and 0.329163 have appropriate model ARMA is (5, 3). According to MAPE with 124.6412 shows

that the appropriate model is ARMA (5, 5). Sunspot cycle (May 1923 - Sep 1933) describe the

best fitted models are ARMA (3, 3), ARMA (4, 4), ARMA (5, 3), ARMA (6, 4) and ARMA (6,

5). ARMA (6, 5) process appropriate model with smallest DW value (0.565) and maximum Log

likelihood value (-547.01). Suitable model ARMA (3, 3) is best fitted according to AIC SIC and

HQC with 8.545, 8.635 and 8.582 respectively. Forecasted evolution define that RMSE, MAE and

U test with 28.41330, 23.46526 and 0.340449 have appropriate model is ARMA (6, 5). According

to MAPE with 225.7652 define that the appropriate model is ARMA (4, 4). Sunspot cycle (Sep

1933 - Jan 1944) shows that best fitted models are ARMA (3, 3), ARMA (4, 4), ARMA (5, 3),

ARMA (6, 2) and ARMA (6, 4). ARMA (6, 4) model is an appropriate model with least DW value

(0.583) and maximum Log likelihood value with -601.22. AIC SIC and HQC with least value

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9.553, 9.645 and 9.591 described that the appropriate model is ARMA (3, 3). Forecasted

evaluation describes that best fitted model in RMSE with 40.50563 is ARMA (6, 2). MAE with

least value 31.48136 demonstrates that the ARMA (5, 3) is favorable model. Appropriate model

in MAPE with least value 79.25677 is ARMA (4, 4). U test with least value 0.355584 describes

that ARMA (6, 4) is best fitted. Sunspot cycle (Jan 1944 - Feb 1954) exhibits that best fitted models

are ARMA (3, 3), ARMA (4, 4), ARMA (5, 3), ARMA (6, 2) and ARMA (6, 4). ARMA (6, 4)

process is an appropriate model with smallest DW value with 0.583 and maximum Log likelihood

value with -601.22. AIC SIC and HQC with least value 9.553, 9.645 and 9.591 described that best

fitted model is ARMA (3, 3). Forecasted evaluation describes that best fitted model in RMSE with

52.26395 is ARMA (6, 2). MAE with 44.02761 and U test with 0.366021 explore that ARMA (5,

3) is favorable model. Appropriate model in MAPE with least value 374.3422 is ARMA (4, 4).

Sunspot cycle (Feb 1954 - Oct 1964) shows that the best fitted models are ARMA (4, 3), ARMA

(5, 3), ARMA (5, 4), ARMA (6, 4) and ARMA (6, 5). Appropriate model is ARMA (6, 5) model

with least value DW with 0.663 and log maximum likelihood with -639.27. AIC SIC and HQC

with least value 9.560, 9.649 and 9.561 depict that best fitted model is ARMA (5, 3). Forecasted

evolution defines that RMSE with 81.95903 explore that the suitable model is ARMA (4, 3). MAE

with 64.21752 and U test with 0.483381 have best fitted model is ARMA (5, 3). According to

MAPE with 125.9374 shows that the best fitted model is ARMA (5, 4). Sunspot cycle (Oct 1964

- May 1976) depict that best fitted models are ARMA (3, 3), ARMA (4, 3), ARMA (5, 3), ARMA

(6, 2) and ARMA (6, 4). ARMA (5, 3) process has least value DW with 0.947 is best fitted and

strongly correlated. The best model according to log maximum likelihood with -626.80 is ARMA

(6, 4) model. AIC SIC and HQC with smallest value 8.645, 8.729 and 8.679 described that the

most appropriate model is ARMA (3, 3). Forecasted evolution reveals that RMSE, MAE and U

test with 38.99557, 32.35205 and 0.328025 have appropriate model is ARMA (5, 3). According

to MAPE with 88.14809 shows that the best model is ARMA (4, 3). Sunspot cycle (May1976 -

Mar 1986) indicate that best fitted models are ARMA (3, 3), ARMA (5, 2), ARMA (5, 3), ARMA

(6, 2) and ARMA (6, 4). An appropriate model is ARMA (6, 4) has least value of DW with 0.828

and log maximum likelihood with -582.02. AIC, SIC and HQC with smallest value 9.551, 9.644

and 9.589 described that the ARMA (3, 3) process is best fitted. Forecasted evolution describes

that RMSE, MAE and U test with 61.20026, 50.64551 and 0.375419 have appropriate model is

ARMA (5, 3). According to MAPE with 99.64765 shows that an appropriate model is ARMA (3,

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3). Sunspot cycle (Mar1986 - Jun1996) describe the appropriate fitted models are ARMA (4, 5),

ARMA (5, 4), ARMA (5, 6), ARMA (6, 4) and ARMA (6, 5). The best fitted ARMA (6, 5) process

has least value DW with 0.834 and log maximum likelihood with -595.39 respectively. AIC, SIC

and HQC with smallest value 9.554, 9.645 and 9.591 shows that an appropriate model is ARMA

(4, 5). Forecasted evolution demonstrates that RMSE with 65.05685 explore the suitable model is

ARMA (5, 6). MAE and U test with 50.90771 and 0.433141 have best fitted model is ARMA (4,

5). According to MAPE with 120.4492 shows that the best fitted model is ARMA (6, 5). Sunspot

cycle (Jun1996 - Jan 2008) describe the best fitted models are ARMA (3, 3), ARMA (3, 4), ARMA

(4, 3), ARMA (5, 4) and ARMA (6, 4). ARMA (5, 4) process has the least value DW with 0.880

and log maximum likelihood with -638.59. AIC, SIC and HQC with smallest value 9.111, 9.195

and 9.145 shows that the appropriate model is ARMA (5, 4). Forecasted evolution define that

RMSE, MAE and U test with 43.79293, 35.46916 and 0.373450 have appropriate model is ARMA

(4, 3). According to MAPE with 178.4829 demonstrate that an appropriate model is ARMA (5, 4).

Sunspot cycle (Aug1755 - Jan 2008) describe the best fitted models are ARMA (5, 4), ARMA (5,

5), ARMA (6, 4), ARMA (6, 5) and ARMA (6, 6). ARMA (5, 4) process is best fitted with the

least value DW with 0.818 and log maximum likelihood and -13941.6. AIC, SIC and HQC smallest

value 9.106, 9.113 and 9.108 shows that the suitable model is ARMA (5, 5). Forecasted evolution

defines that RMSE and U test with 44.54175 and 0.370626 have best fitted model is ARMA (6,

4). According to MAE and MAPE with 35.30753 and 464.3263 shows that best fitted model is

ARMA (5, 5).

1.9: AR (p)-GARCH (1, 1) PROCESS OF SUNSPOT CYCLES

Autoregressive process (AR) is developed by Yule (Yule, 1927). An autoregressive process AR

(p) can be expressed by a weighted sum of its previous values and a white noise. The generalized

Autoregressive process AR (p) of lag p is expressed as follows.

Xt = α1 Xt-1 + α 2 Xt-2 + … + α p Xt-p + 𝜏t (1.23)

Here εt is white noise with mean E (𝜏t) = 0, variance Var (𝜏t) = σ2 and Cov (𝜏t -s, 𝜏t) = 0, if s ≠ 0.

For every t, it is supposed that 𝜏t is independent of the Xt-1, Xt-2, ….. 𝜏t is uncorrelated with Xs for

each s < t.

The GARCH (1, 1) process was developed by Bollerslev (1986). The generalized GARCH (p, q)

stochastic volatility model was developed by (Aquilar et. al. 2000, Kim et. al. 1998). The

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generalized autoregressive conditional heteroskedasticity GARCH model is more appropriate to

analyze the fluctuation of variances. GARCH model is preferred to study the volatility clustering

and the relative volatility forecasts (Andersen et. al. 1998). GARCH (1, 1) process has better

forecasting ability as compared to other traditional models (Akgiray, 1989). GARCH (1, 1) process

is better to construct multi period long term forecasting. GARCH model is widely used for

modeling and forecasting of various other types of data which include economic and financial

modeling too. The GARCH (1, 1) process of Xt is represented as follows

𝜏t = 𝜎𝑡𝜖𝑡 (1.24)

Where ∈𝑡 ~ 𝐼𝐼𝐷(0, 1)

The model defines by itself as

𝜎𝑡2 = 𝛿 + 𝛽 𝜏𝑡−1

2 + 𝛾 𝜎𝑡−12 with 𝛿 + 𝛽 + 𝛾 ≥ 0 (1.25)

GARCH (1, 1) process is a covariance-stationary white noise process if and only if β + γ < 1. The

variance of the covariance-stationary process is represented as follows.

Var (Xt) = 𝛿

1−𝛽−𝛾 (1.26)

GARCH (1, 1) process is stationary if it holds stationary conditions. GARCH (1, 1) process mostly

follows leptokurtic (kurtosis greater than 3) which shows a heavy tailed behavior. GARCH (1, 1)

process specification with AR (p) is defined as follows (t = 0 ± 1, ±2, …).

Xt = α1 Xt-1 + α 2 Xt-2 + … + α p Xt-p + 𝜏t (1.27)

𝜏t = 𝜎𝑡𝜖𝑡 with ∈𝑡 ~ 𝐼𝐼𝐷(0, 1) (1.28)

𝜎𝑡2 = 𝛿 + 𝛽1 𝜏𝑡−1

2 + 𝛽2 𝜏𝑡−22 + ⋯ + 𝛽𝑝 𝜏𝑡−𝑝

2 + 𝛾 𝜎𝑡−12 (1.29)

Where E(𝜏t) = 0, variance Var (𝜏t | 𝜏𝑡−12 , 𝜏𝑡−2

2 … ) = σ2 and Cov (𝜏t -s, 𝜏t) = 0, if s ≠ 0.

Moreover, The Box-Jenkins methodology with GARCH approach is used to develop models, to

estimate the models and to forecast the sunspot cycle’s data.

Table 1.5 shows the results of performing the diagnostic checking test, forecast evaluation and

GARCH equations for AR (p)- GARCH (1,1) for Sunspot Cycles (1-24). Table 1.6 describes the

results of normality test for these models. The Gaussian quasi maximum likelihood estimation is

used to analysis AR (p)-GARCH (1, 1) model. Stationary GRACH (1, 1) model follows leptokurtic

(heavy tail) which represent the strong correlation among the spots and similarly each cycle of

sunspots. In this study AR (p)-GARCH (1, 1) process of each sunspot cycle has kurtosis value

greater than 3 (Leptokurtic) expect (8th, 16th, 17th) cycles follows the platokurtic flat tail. In this

53

study, the goodness of fit of the AR (p)-GARCH (1, 1) process is focused on residuals and more

specification with the standardized residuals. The Lagrange Multiplier, LM-test for

heteroskedasticity and Ljung-Box test is used for serial correlation. In stationary model the

coefficient of determination is used to predict the dependent variable from independent variable.

0 < R2 < 1, large values of R2 show that models demonstrate a closer fit to the time series data.

Durbin- Watson statistics test value of each sunspot cycle is less than 2 which shows that each

sunspot observation are correlated to each other. Akaike information criterion (AIC), Bayesian

Schwarz information criterion (BIC) and Hannan Quinn information criterion (HIC) values of each

cycle is also calculated. Sunspot cycles follow AR (2)-GARCH (1, 1) expect cycles (7th, 15th, 17th)

indicate AR (3)-GARCH (1, 1) model with 0.102 ± 0.995, 0.102 ± 0.997, 0.221 ± 0.973

respectively. Sunspot cycles GRACH (1,1) process specification with AR(p) follow positive

skewness (right tail) expect (2nd, 4th) cycles which is showing negative skewness (left tail) are

depicted in table 3. Sunspot cycle 5th is the best fitted model on the basis of Akaike information

criterion (AIC), Bayesian Schwarz information criterion (BIC) and Hannan Quinn information

criterion (HIC) which has least value 6.837493, 6.938750 and 6.878633 respectively. Each sunspot

value is correlated to other value. Jurque-Bera test failed in each cycle, which is shown that sunspot

cycles are not normally distributed. In the figure 1.6 is display the actual values, fitted values and

residual of Sunspot Cycle (Aug1755 - Jan 2008) as well as in the table is determined the forecasting

evolution of each sunspot cycle analysis in the view of diagnostic check, for each cycle Mean

Absolute Error (MAE) has least value. 6th cycle of sunspot has the smallest value of RMSE, MAE

and MAPE with 23.87355, 16.61689 and 81.56241 respectively. 7th cycle has least value of U test

with 0.342102. Figure 1.5 depicted that GARCH graph with a conditional standard deviation of

Sunspot Cycles (1-23 & 24 (complete sunspot cycle duration Aug1755 - Jan 2008)). Y-axis shows

that values of spots and X-axis represent that number of values. In figure 1.6 actual values, fitted

values and residuals of Sunspot Cycle (1-24) are displayed.

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Figure 1.5: GARCH graphs with a conditional standard deviation of Sunspot Cycles

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Figure 1.6: Actual values, Fitted values and Residual of AR– GARCH model of Sunspot Cycle

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1.10: ARMA (p, q) – GARCH (1, 1) METHODS OF SUNSPOT CYCLES

The concept of ARMA models is strongly relevant in volatility modeling. The generalized

autoregressive conditional heteroscedastic (GARCH) models can be linked as ARMA models.

GARCH Models satisfy an ARMA equation with white noise. In time series, GARCH model

supposition that conditional mean is zero. Generally, conditional mean of ARMA model can be

structured. Identification of GARCH process focused on the square of residuals from the

appropriate ARMA models. Moreover, in the ARAM process the quasi maximum likelihood

estimation is nearly independent of their GARCH process. ARMA estimation and GARCH

estimation are strongly correlated if the ARMA – GARCH process has a skewed distribution

(Csyer et. al 2008). The ARAM process and GARCH process have similar behavior in forecasting.

ARMA – GARCH process provides a good estimation in time series data.

GARCH (1, 1) process specification with ARMA (p, q) is defined as follows (t = 0 ± 1, ±2, …).

Xt = α1 Xt-1 + α 2 Xt-2 + … + α p Xt-p + ∈t + β1 ∈t-1 + β2 ∈t-2+ … + βq ∈t-q (1.30)

𝜏t = 𝜎𝑡𝜖𝑡 with ∈𝑡 ~ 𝐼𝐼𝐷(0, 1) (1.31)

𝜎𝑡2 = 𝛿 + 𝛽1 𝜏𝑡−1

2 + 𝛽2 𝜏𝑡−22 + ⋯ + 𝛽𝑝 𝜏𝑡−𝑝

2 + 𝛾 𝜎𝑡−12 (1.32)

Where E(𝜏t) = 0, variance Var (𝜏t | 𝜏𝑡−12 , 𝜏𝑡−2

2 … ) = σ2 and Cov (𝜏t -s, 𝜏t) = 0, if s ≠ 0.

Moreover, The Box-Jenkins methodology with GARCH approach is used to develop models, to

estimate the models and to forecast the sunspot cycle’s data.

The novelty of this section to analyze the ARMA (p, q) -GARCH (1, 1) process of sunspot cycles.

The selection of ARMA (p, q) -GARCH (1, 1) model based on the least value of Darbin - Waston

statistics test (DW). Least DW value (< 2) shows that the each value of cycles is strongly correlated

and persistence to each other. AIC, SIC, HQC and Log likelihood also estimate to each cycle. In

Table 1.7, and 1.8 are depicted the GARCH (1, 1) model equations to specification ARMA (p, q)

model of sunspot cycles by diagnostic checking test and normality test. The Gaussian quasi

maximum likelihood estimation is used to analysis ARMA (p, q) -GARCH (1, 1) model. Lagrange

multiplier is used to verify the ARCH effect on following time series data. Ljung-Box test is used

for serial correlation of each sunspot cycle. The novelty of this research to analysis the conditional

mean and conditional variance effect on each sunspot cycle. The Correlation determination (R2)

of each cycle is range from 7.5 to 9.75 which are shown that each value are strongly correlated to

previous ones. The appropriate model which is frequently verified in sunspot cycles is GARCH

(1, 1) with specification ARMA (2, 2) cycles (1st, 4th, 12th, 13th, 14th, 15th, 16th, 19th, 20th, 23rd and

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24th). ARMA (3, 3) -GARCH (1, 1) process follows with means and standard deviation

0.153±0.988, 0.190±0.983, 0.097±1.009 and 0.238±0.973 of most appropriate sunspot cycles

(5th, 6th, 7th and 15th) models respectively. ARMA (5, 1) -GARCH (1, 1) model follows cycles (2nd

and 11th) with 0.167±0.991 and 0.249±0.968 respectively. Cycle (Aug 1775 - Jun 1784)

represents ARMA (1, 1) -GARCH (1, 1) model with 0.197±0.983. ARMA (5, 3) -GARCH (1, 1)

process follows cycles (18th and 19th) with 0.258±0.965 and 0.258±0.966 respectively. Sunspot

cycle (Aug 1775 - Jun 1784) represents ARMA (1, 1) -GARCH (1, 1) model with 0.197±0.983.

Sunspot cycle (Oct1833 - Sep 1843) explore ARMA (3, 2) -GARCH (1, 1) process with

0.305±0.954. Sunspot cycle (Sep1843 - Mar 1855) explore ARMA (4, 2) -GARCH (1, 1) model

with 0.181±0.984. Sunspot cycle (Mar1855 - Feb 1867) explore ARMA (4, 4) -GARCH (1, 1)

model with 0.145±0.992. Sunspot cycle (Mar1986 - Jun1996) explore ARMA (6, 1) -GARCH (1,

1) model with 0.206±0.977. Sunspot cycle (Jun 1798 - Sep 1810) appropriate model ARMA (3,

3) -GARCH (1, 1) on the basis of least value of Akaike information criterion (AIC), Bayesian

Schwarz information criterion (BIC) and Hannan Quinn information criterion (HIC) which has

least value 7.075953, 7.197462 and 7.125322 respectively. According to log likelihood sunspot

cycle (Aug1755 - Jan 2008) best model ARMA (2, 2) -GARCH (1, 1) with maximum log

likelihood value -12808.63. Test of normality demonstrates that each sunspot cycles have positive

skewed expect sunspot cycles (4th and 19th) have negative skewed with -0.283 and -0.1425

respectively. Each sunspot cycle has kurtosis value greater than 3 (Leptokurtic) heavy tail expect

(2nd, 7th, 10th, 11th, 18th and 19th,) cycles follows platokurtic flat tail. Jurque-Bera test failed in each

cycle, which is shown that sunspot cycles are not normally distributed. Diagnostic Checking Test

is chosen with compression of these techniques with the help of Root mean square error (RMSE),

Mean Absolute Error (MAE) and Mean Absolute Percentage Error (MAPE) and Theil’s U-

Statistics test. Figure 1.7 displayed that GARCH graphs of sunspot cycles (1-23 & 24 cycle

consists of complete time series data from 1855 to 2008) with conditional variance. In the figure

1.8 displayed that the actual values, fitted values and residual of sunspot cycle (1-24) as well as in

the table is determined the forecasting evolution of each sunspot cycle analysis in the view of

diagnostic check, for each cycle Mean Absolute Error (MAE) has least value. 6th cycle of sunspot

has the smallest value of RMSE, MAE and MAPE are 25.17780, 17.81173 and 79.46162

respectively. Each sunspot cycle has U test values less than 1 which is shown that are cycles are

correlated with others. 7th cycle has least value of U statistics test with 0.486173. It is a value is

64

near to zero describe that strongly correlated. Figure 1.9 depicted that the residual graph of

complete time series data ranges from 1st to 23rd cycles. Figure 1.9 shows that the residual graph

of the sunspot cycle (1-23). Figure 1.10 displayed that the standard residual graph of sunspot cycle

(1-23).

67

Figure 1.7: GARCH graphs with conditional variance of Sunspot Cycles

71

Figure 1.8: Actual values, Fitted values and Residual of ARMA– GARCH model of Sunspot

Cycle

72

Figure 1.9: Residuals ARMA – GARCH models of Sunspot Cycle (1-23)

Figure 1.10: Standard Residuals ARMA – GARCH models of Sunspot Cycle (1-23)

1.11: COMPARISON OF FORECASTING EVOLUTION OF ARMA, AR-GARCH AND

ARMA-GARCH OF SUNSPOT CYCLES

The forecasting evolution is based on RMSE, MAE, MAPE and U-test. The novelty of this section

focused on the comparison of the forecast evolution of ARAM, AR-GARCH and ARMA-GARCH

process of sunspot cycles.

73

1.11.1: Root Mean Square Error (RMSE) of ARMA, AR-GARCH AND ARMA-GARCH

of Sunspot Cycles

In this study, forecasting accuracy of ARMA, AR-GARCH and ARMA-GARCH models is

checked by RMSE. Table 1.9 depicts that the comparative study of the forecasting evolution of

ARMA (p, q), AR (p)-GARCH (1, 1) and ARMA (p, q)-GARCH (1, 1) process of each sunspot

cycle. Sunspot cycles (Aug 1755 - Mar 1766) has best fitted models are ARMA (6, 5), AR (2)-

GARCH (1, 1) and ARMA (6, 5)-GARCH (1, 1) process. Sunspot cycles (Mar 1766 - Aug 1775)

has expressed that the appropriate models are ARMA (5, 3), AR (2)-GARCH (1, 1) and ARMA

(5, 1)-GARCH (1, 1) process. Sunspot cycles (Aug 1775 - Jun 1784) has suitable fitted models are

ARMA (6, 4), AR (2) -GARCH (1, 1) and ARMA (1, 1) -GARCH (1, 1) process. Sunspot cycles

(Jun 1784 - Jun 1798) has best fitted models are ARMA (5, 4), AR (2)-GARCH (1, 1) and ARMA

(2, 2)-GARCH (1, 1) process. Sunspot cycles (Jun 1798 - Sep 1810) shows appropriate models are

ARMA (5, 3), AR (2)-GARCH (1, 1) and ARMA (3, 3)-GARCH (1, 1) process. Sunspot cycles

(Sep1810 - Dec 1823) express the suitable models are ARMA (3, 4), AR (2)-GARCH (1, 1) and

ARMA (3, 3)-GARCH (1, 1) process. Sunspot cycles (Dec1823 - Oct 1833) depict best fitted

models are ARMA (5, 3), AR (3)-GARCH (1, 1) and ARMA (3, 3)-GARCH (1, 1) process.

Sunspot cycles (Oct1833 - Sep 1843) shows best appropriate models are ARMA (4, 6), AR (2)-

GARCH (1, 1) and ARMA (3, 2)-GARCH (1, 1) process. Sunspot cycles (Sep1843 - Mar 1855)

has best fitted models are ARMA (4, 3), AR (2)-GARCH (1, 1) and ARMA (4, 2)-GARCH (1, 1)

process. Sunspot cycles (Mar1855 - Feb 1867) express best fitted models are ARMA (6, 4), AR

(2)-GARCH (1, 1) and ARMA (4, 4)-GARCH (1, 1) process. Sunspot cycles (Feb1867 - Sep 1878)

reveals best fitted models are ARMA (6, 4), AR (2)-GARCH (1, 1) and ARMA (5, 1)-GARCH (1,

1) process. Sunspot cycles (Sep1878 - Jun 1890) has suitable models are ARMA (5, 3), AR (2)-

GARCH (1, 1) and ARMA (2, 2)-GARCH (1, 1) process. Sunspot cycles (Jun1890 - Sep 1902)

shows best fitted models are ARMA (5, 4), AR (2)-GARCH (1, 1) and ARMA (2, 2)-GARCH (1,

1) process. Sunspot cycles (Sep 1902 - Dec 1913) express appropriate models are ARMA (4, 6),

AR (2)-GARCH (1, 1) and ARMA (2, 2)-GARCH (1, 1) process. Sunspot cycles (Dec1913 - May

1923) describe the best fitted models are ARMA (5, 4), AR (3)-GARCH (1, 1) and ARMA (3, 3)-

GARCH (1, 1) process. Sunspot cycles (May 1923 - Sep 1933) represent suitable models are


(Sep 1933 - Jan 1944) has best fitted models are ARMA (6, 2), AR (3)-GARCH (1, 1) and ARMA

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(2, 2)-GARCH (1, 1) process. Sunspot cycles (Jan 1944 - Feb 1954) describe the appropriate


Sunspot cycles (Feb 1954 - Oct 1964) has best fitted models are ARMA (4, 3), AR (2)-GARCH

(1, 1) and ARMA (5, 3)-GARCH (1, 1) process. Sunspot cycles (Oct 1964 - May 1976) reveal best

fitted models are ARMA (4, 3), AR (2)-GARCH (1, 1) and ARMA (2, 2)-GARCH (1, 1) process.

Sunspot cycles (May1976 - Mar 1986) express suitable models are ARMA (5, 3), AR (2)-GARCH

(1, 1) and ARMA (2, 2)-GARCH (1, 1) process. Sunspot cycles (Mar1986 - Jun1996) has best


Sunspot cycles (Jun1996 - Jan 2008) describe appropriate models are ARMA (6, 4), AR (2)-

GARCH (1, 1) and ARMA (2, 2)-GARCH (1, 1) process. Sunspot cycles (Aug1755 - Jan 2008)

represent suitable models are ARMA (6, 4), AR (2)-GARCH (1, 1) and ARMA (2, 2)-GARCH (1,

1) process. Root Mean Square Error (RMSE) shows that ARMA (p, q) model is best appropriate

model as comparative to AR(p)-GARCH (1, 1) and ARMA (p, q)- GARCH (1, 1) models. Sunspot

cycles have least RMSE values in ARMA (p, q) process. Figure 1.11 displayed that RMSE

comparison of ARMA, AR-GARCH and ARMA-GARCH process of sunspot cycles.

Figure 1.11: RMSE of ARMA, AR-GARCH and ARMA-GARCH process of sunspot cycles.

1.11.2: Mean Absolute Error (MAE) of ARMA, AR-GARCH AND ARMA-GARCH of

Sunspot Cycles

The selection of best fitted model is based on MAE. Table 1.10: depict that sunspot cycles (Aug

1755 - Mar 1766) has best fitted models are ARMA (4, 5), AR (2)-GARCH (1, 1) and ARMA (6,

75

5)-GARCH (1, 1) process. Sunspot cycles (Mar 1766 - Aug 1775) has expressed that the

appropriate models are ARMA (5, 3), AR (2)-GARCH (1, 1) and ARMA (5, 1)-GARCH (1, 1)

process. Sunspot cycles (Aug 1775 - Jun 1784) has suitable fitted models are ARMA (6, 4), AR

(2) -GARCH (1, 1) and ARMA (1, 1) -GARCH (1, 1) process. Sunspot cycles (Jun 1784 - Jun

1798) has best fitted models are ARMA (5, 6), AR (2)-GARCH (1, 1) and ARMA (2, 2)-GARCH

(1, 1) process. Sunspot cycles (Jun 1798 - Sep 1810) shows appropriate models are ARMA (5, 3),

AR (2)-GARCH (1, 1) and ARMA (3, 3)-GARCH (1, 1) models. Sunspot cycles (Sep1810 - Dec

1823) express the suitable models are ARMA (6, 4), AR (2)-GARCH (1, 1) and ARMA (3, 3)-

GARCH (1, 1) process. Sunspot cycles (Dec1823 - Oct 1833) depict best fitted models are ARMA

(5, 3), AR (3)-GARCH (1, 1) and ARMA (3, 3)-GARCH (1, 1) process. Sunspot cycles (Oct1833

- Sep 1843) shows best appropriate models are ARMA (4, 6), AR (2)-GARCH (1, 1) and ARMA

(3, 2)-GARCH (1, 1) process. Sunspot cycles (Sep1843 - Mar 1855) has best fitted models are


(Mar1855 - Feb 1867) express best fitted models are ARMA (6, 4), AR (2)-GARCH (1, 1) and

ARMA (4, 4)-GARCH (1, 1) process. Sunspot cycles (Feb1867 - Sep 1878) reveals best fitted


Sunspot cycles (Sep1878 - Jun 1890) has suitable models are ARMA (5, 3), AR (2)-GARCH (1,

1) and ARMA (2, 2)-GARCH (1, 1) process. Sunspot cycles (Jun1890 - Sep 1902) shows best


Sunspot cycles (Sep 1902 - Dec 1913) express appropriate models are ARMA (4, 6), AR (2)-

GARCH (1, 1) and ARMA (2, 2)-GARCH (1, 1) process. Sunspot cycles (Dec1913 - May 1923)

describe the best fitted models are ARMA (5, 3), AR (3)-GARCH (1, 1) and ARMA (3, 3)-

GARCH (1, 1) process. Sunspot cycles (May 1923 - Sep 1933) represent suitable models are


(Sep 1933 - Jan 1944) has best fitted models are ARMA (6, 2), AR (3)-GARCH (1, 1) and ARMA

(2, 2)-GARCH (1, 1) process. Sunspot cycles (Jan 1944 - Feb 1954) describe the appropriate


Sunspot cycles (Feb 1954 - Oct 1964) has best fitted models are ARMA (4, 3), AR (2)-GARCH

(1, 1) and ARMA (5, 3)-GARCH (1, 1) process. Sunspot cycles (Oct 1964 - May 1976) reveal best


Sunspot cycles (May1976 - Mar 1986) express suitable models are ARMA (5, 3), AR (2)-GARCH

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(1, 1) and ARMA (2, 2)-GARCH (1, 1) process. Sunspot cycles (Mar1986 - Jun1996) has best


Sunspot cycles (Jun1996 - Jan 2008) describe appropriate models are ARMA (3, 4), AR (2)-

GARCH (1, 1) and ARMA (2, 2)-GARCH (1, 1) process. Sunspot cycles (Aug1755 - Jan 2008)

represent suitable models are ARMA (6, 4), AR (2)-GARCH (1, 1) and ARMA (2, 2)-GARCH (1,

1) process. The forecasting evolution of Mean Absolute Error (MAE) shows that ARMA (p, q)

model are best fitted and appropriate model as compared to AR(p)-GARCH (1, 1) and ARMA (p,

q)- GARCH (1, 1) models. Sunspot cycles have smallest values of MAE in ARMA (p, q) process.

Figure 1.12 depicts that the compression of MAE values of sunspot cycles by using ARMA, AR-

GARCH and ARMA-GARCH models.

Figure 1.12: MAE of ARMA, AR-GARCH and ARMA-GARCH process of sunspot cycles.

1.11.3: Mean Absolute Percentage Error (MAPE) of ARMA, AR-GARCH AND ARMA-

GARCH of Sunspot Cycles

Mean Absolute Percentage Error (MAPE) has variation in selection of best fitted model among

ARMA, AR-GARCH AND ARMA-GARCH process. Table 1.11: depict that sunspot cycles (Aug

1755 - Mar 1766) has best fitted models are ARMA (4, 4), AR (2)-GARCH (1, 1) and ARMA (6,

5)-GARCH (1, 1) process. Cycle 1 shows that best fitted model is an ARMA model with value

55.73550 among of these processes. Sunspot cycles (Mar 1766 - Aug 1775) has expressed that the


process. Sunspot cycles (Aug 1775 - Jun 1784) has suitable fitted models are ARMA (5, 5), AR

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(2) -GARCH (1, 1) and ARMA (1, 1) -GARCH (1, 1) process. Cycle 2 and 3 represent that the

appropriate model is AR-GARCH with value 115.7679 and 80.41010 respectively. Sunspot cycles

(Jun 1784 - Jun 1798) has best fitted models are ARMA (3, 3), AR (2)-GARCH (1, 1) and ARMA

(2, 2)-GARCH (1, 1) process. Sunspot cycles (Jun 1798 - Sep 1810) shows appropriate models are

ARMA (5, 3), AR (2)-GARCH (1, 1) and ARMA (3, 3)-GARCH (1, 1) models. Sunspot cycles

(Sep1810 - Dec 1823) express the suitable models are ARMA (6, 6), AR (2)-GARCH (1, 1) and

ARMA (3, 3)-GARCH (1, 1) process. Sunspot cycles (Dec1823 - Oct 1833) depict best fitted


Sunspot cycles (Oct1833 - Sep 1843) shows best appropriate models are ARMA (3, 3), AR (2)-

GARCH (1, 1) and ARMA (3, 2)-GARCH (1, 1) process. Cycles 4, 5, 6, 7 and 8 reveal that the

best model is ARMA-GARCH with value 88.91661, 82.29495, 79.46162, 113.0161 and 79.03446

respectively. Sunspot cycles (Sep1843 - Mar 1855) has best fitted models are ARMA (3, 3), AR

(2)-GARCH (1, 1) and ARMA (4, 2)-GARCH (1, 1) process. Sunspot cycles (Mar1855 - Feb

1867) express best fitted models are ARMA (4, 4), AR (2)-GARCH (1, 1) and ARMA (4, 4)-

GARCH (1, 1) process. Cycle 9 and 10 express that best fitted model is ARMA model with value

55.33579 and 149.2235 respectively among of these processes. Sunspot cycles (Feb1867 - Sep

1878) reveals best fitted models are ARMA (3, 3), AR (2)-GARCH (1, 1) and ARMA (5, 1)-

GARCH (1, 1) process. Sunspot cycles (Sep1878 - Jun 1890) has suitable models are ARMA (4,

4), AR (2)-GARCH (1, 1) and ARMA (2, 2)-GARCH (1, 1) process. Sunspot cycles (Jun1890 -

Sep 1902) shows best fitted models are ARMA (4, 4), AR (2)-GARCH (1, 1) and ARMA (2, 2)-

GARCH (1, 1) process. Sunspot cycles (Sep 1902 - Dec 1913) express appropriate models are


(Dec1913 - May 1923) describe the best fitted models are ARMA (5, 5), AR (3)-GARCH (1, 1)

and ARMA (3, 3)-GARCH (1, 1) process. Sunspot cycles (May 1923 - Sep 1933) represent suitable

models are ARMA (4, 4), AR (2)-GARCH (1, 1) and ARMA (2, 2)-GARCH (1, 1) process. Cycles

11, 12, 13, 14, 15 and 16 analyze that the best model is ARMA-GARCH with values 163.9832,

105.1265, 108.7426, 117.5639, 79.70615 and 116.2730 respectively among of these process.

Sunspot cycles (Sep 1933 - Jan 1944) has best fitted models are ARMA (4, 4), AR (3)-GARCH

(1, 1) and ARMA (2, 2)-GARCH (1, 1) process. Cycle 17 shows that best fitted model is ARMA

model with value 79.25677 among of these models. Sunspot cycles (Jan 1944 - Feb 1954) describe

the appropriate models are ARMA (4, 4), AR (2)-GARCH (1, 1) and ARMA (5, 3)-GARCH (1,

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1) process. Sunspot cycles (Feb 1954 - Oct 1964) has best fitted models are ARMA (4, 3), AR (2)-

GARCH (1, 1) and ARMA (5, 4)-GARCH (1, 1) process. Cycle 18 and 19 express that best fitted

model is ARMA-GARCH model with values 122.4058 and 87.83657 respectively. Sunspot cycles

(Oct 1964 - May 1976) reveal best fitted models are ARMA (3, 3), AR (2)-GARCH (1, 1) and

ARMA (2, 2)-GARCH (1, 1) process. Cycle 20 and 3 represent that the appropriate model is AR-

GARCH with value 71.04261. Sunspot cycles (May1976 - Mar 1986) express suitable models are


(Mar1986 - Jun1996) has best fitted models are ARMA (6, 5), AR (2)-GARCH (1, 1) and ARMA

(6, 1)-GARCH (1, 1) process. Sunspot cycles (Jun1996 - Jan 2008) describe appropriate models

are ARMA (5, 4), AR (2)-GARCH (1, 1) and ARMA (2, 2)-GARCH (1, 1) process. Sunspot cycles

(Aug1755 - Jan 2008) represent suitable models are ARMA (5, 5), AR (2)-GARCH (1, 1) and

ARMA (2, 2)-GARCH (1, 1) process. Cycles 21, 22, 23 and 24 reveal that the best model is

ARMA-GARCH with value 80.66910, 75.91525, 93.96324 and 205.0083 respectively. Most of

the cycles follow ARAM-GARCH models are appropriate model except cycles 2, 3 and 20 shows

AR-GARCH models and cycles 1, 9, 10 and 17 express are best fitted model are ARMA models.

Figure 1.13 displayed that the compression of MAPE values of sunspot cycles by using ARMA,

AR-GARCH and ARMA-GARCH models.

Figure 1.13: MAPE of ARMA, AR-GARCH and ARMA-GARCH process of sunspot cycles.

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1.11.4: Theil’s U-Statistics of ARMA, AR-GARCH AND ARMA-GARCH of Sunspot

Cycles

Theil’s U-statistics test is used to analyze the correlation of observations. Strong correlation shows

when U test value near to zero. Table 1.12: depict that sunspot cycles (Aug 1755 - Mar 1766) has

best fitted models are ARMA (6, 5), AR (2)-GARCH (1, 1) and ARMA (6, 5)-GARCH (1, 1)

process. Sunspot cycles (Mar 1766 - Aug 1775) has expressed that the appropriate models are


(Aug 1775 - Jun 1784) has suitable fitted models are ARMA (6, 4), AR (2) -GARCH (1, 1) and

ARMA (1, 1) -GARCH (1, 1) process. Sunspot cycles (Jun 1784 - Jun 1798) has best fitted models


(Jun 1798 - Sep 1810) shows appropriate models are ARMA (6, 4), AR (2)-GARCH (1, 1) and

ARMA (3, 3)-GARCH (1, 1) models. Sunspot cycles (Sep1810 - Dec 1823) express the suitable


Sunspot cycles (Dec1823 - Oct 1833) depict best fitted models are ARMA (5, 3), AR (3)-GARCH

(1, 1) and ARMA (3, 3)-GARCH (1, 1) process. Sunspot cycles (Oct1833 - Sep 1843) shows best


process. Sunspot cycles (Sep1843 - Mar 1855) has best fitted models are ARMA (4, 3), AR (2)-

GARCH (1, 1) and ARMA (4, 2)-GARCH (1, 1) process. Sunspot cycles (Mar1855 - Feb 1867)

express best fitted models are ARMA (6, 4), AR (2)-GARCH (1, 1) and ARMA (4, 4)-GARCH

(1, 1) process. Sunspot cycles (Feb1867 - Sep 1878) reveals best fitted models are ARMA (6, 4),

AR (2)-GARCH (1, 1) and ARMA (5, 1)-GARCH (1, 1) process. Sunspot cycles (Sep1878 - Jun

1890) has suitable models are ARMA (5, 3), AR (2)-GARCH (1, 1) and ARMA (2, 2)-GARCH

(1, 1) process. Sunspot cycles (Jun1890 - Sep 1902) shows best fitted models are ARMA (4, 3),

AR (2)-GARCH (1, 1) and ARMA (2, 2)-GARCH (1, 1) process. Sunspot cycles (Sep 1902 - Dec

1913) express appropriate models are ARMA (4, 6), AR (2)-GARCH (1, 1) and ARMA (2, 2)-

GARCH (1, 1) process. Sunspot cycles (Dec1913 - May 1923) describe the best fitted models are


(May 1923 - Sep 1933) represent suitable models are ARMA (6, 5), AR (2)-GARCH (1, 1) and

ARMA (2, 2)-GARCH (1, 1) process. Sunspot cycles (Sep 1933 - Jan 1944) has best fitted models


(Jan 1944 - Feb 1954) describe the appropriate models are ARMA (5, 3), AR (2)-GARCH (1, 1)

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and ARMA (5, 3)-GARCH (1, 1) process. Sunspot cycles (Feb 1954 - Oct 1964) has best fitted


Sunspot cycles (Oct 1964 - May 1976) reveal best fitted models are ARMA (4, 3), AR (2)-GARCH

(1, 1) and ARMA (2, 2)-GARCH (1, 1) process. Sunspot cycles (May1976 - Mar 1986) express

suitable models are ARMA (5, 3), AR (2)-GARCH (1, 1) and ARMA (2, 2)-GARCH (1, 1)

process. Sunspot cycles (Mar1986 - Jun1996) has best fitted models are ARMA (4, 5), AR (2)-

GARCH (1, 1) and ARMA (6, 1)-GARCH (1, 1) process. Sunspot cycles (Jun1996 - Jan 2008)

describe appropriate models are ARMA (3, 4), AR (2)-GARCH (1, 1) and ARMA (2, 2)-GARCH

(1, 1) process. Sunspot cycles (Aug1755 - Jan 2008) represent suitable models are ARMA (6, 4),

AR (2)-GARCH (1, 1) and ARMA (2, 2)-GARCH (1, 1) process. The forecasting evolution of

Theil’s U-statistics test shows that ARMA (p, q) model are best fitted and appropriate model as

compared to AR(p)-GARCH (1, 1) and ARMA (p, q)- GARCH (1, 1) models. Sunspot cycles have

smallest values of Theil’s U-statistics test in ARMA (p, q) process. Figure 1.14 displayed that the

compression of Theil’s U-statistics test values of sunspot cycles by using ARMA, AR-GARCH

and ARMA-GARCH models.

Figure 1.13: Theil’s U-statistics test of ARMA, AR-GARCH and ARMA-GARCH process of

sunspot cycles.

1.12: CONCLUSION

The novelty of this study is to analyze the sunspot cycles using GARCH models. Sunspot cycles

have stationary nature with second difference. Autocorrelation (AC), Partial Autocorrelation

81

(PAC) and Ljung-Box Q-statistics test are used to check the presence of white noise in the time

series data. Unit root test is rejected for each sunspot cycle, which confirms that each cycle has

stationary nature. To test the normality of sunspot cycles skewness and kurtosis were checked and

Jurque-Bera test was performed. This chapter utilizes the stochastic autoregressive and moving

average (ARMA) models to forecast the evolution of sunspot cycles. Least Square Estimation is

used to investigate the ARMA process. Various best fitted ARMA models are developed to

estimate and forecast the sunspot cycles. The selection of appropriate ARMA model is based on

the smallest value of Durbin- Watson statistics test. Durbin- Watson (DW) statistics test value of

each sunspot cycle is less than 2 which shows that sunspot observations are correlated to each

other. The stationary generalized autoregressive conditional heteroskedasticity GARCH (1, 1)

volatility model with specification autoregressive AR (p) process are used to estimate and forecast

the evolution of sunspot cycles. In addition, the stationary generalized autoregressive conditional

heteroskedasticity GARCH (1, 1) volatility model with specification autoregressive and moving

average ARMA (p, q) models are also utilized to estimate and forecasted evolution of the sunspot

cycles. GARCH (1, 1) stationary volatility model appeared to be the best forecasting model as

compared to the other models. GARCH (1, 1) model is heavy tail (fat tailed). Under model

Identification and Estimation, diagnostic checking and Forecasting the ARMA (p, q), AR (p)-

GARCH and ARAM (p, q)-GARCH models are found to be the most appropriate. The Gaussian

quasi maximum likelihood estimation (QMLE) is used to estimate GARCH (1, 1) process for

specification of AR (p) and ARMA (p, q) models. The selection of models in based on least value

of the Durbin- Watson statistic test. To detect the presence of Autoregressive Conditional

Heteroscedastic (ARCH) effect on sunspot cycle data, Lagrange Multiplier test is used. The

selection of the appropriate model is based on residual diagnostic checking such as ARCH LM,

normality test and correlogram squared residuals. Forecasting evolutions are verified by Root

Mean Squared Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error

(MAPE) and Theil’s U-Statistics test (U test). Moreover, Akaike information criterion (AIC),

Bayesian Schwarz information criterion (BIC) and Hannan Quinn information criterion (HQC),

maximum log likelihood estimation values are also calculated. The residuals of the best fitted

model are chosen by diagnostic checking. The forecasting evolution of each sunspot cycle under

the normality test is based on Skewness, Kurtosis and Jurque-Bera statistic tests. GARCH (1, 1) is

leptokurtic (a process having a kurtosis value greater than 3). The adequate ARMA, AR-GARCH

82

and ARMA-GARCH modeling for fractional Brownian motion of sunspot cycles will be useful to

predict the dynamical variables for 24th cycle and next cycles in the future. Finally, the forecasts

for the evolution of sunspot cycles obtained by ARMA, AR-GARCH and ARMA-GARCH models

are compared. RMSE, MAE and U test are utilized to check the appropriateness of various ARMA

models. Only the MAPE exhibited the appropriateness of ARMA-GARCH model.

83

CHAPTER: 2

FRACTAL ANALYSIS OF SUNSPOTS CYCLES AND ENSO CYCLES

AND DETERMINATION OF THEIR RELATIONSHIP

In this chapter sunspots cycles and ENSO cycles are studied and analyzed using fractal techniques.

Further, it is tried to establish a relationship between Sunspot cycles and corresponding ENSO

cycles. Sunspot cycles and ENSO cycle can be studied as time series using different techniques.

In this study, these time series will be analyzed using fractal techniques. As time series these data

sets can be characterized as linear and non-linear time series. Because these time series are

associated with natural phenomena, like other natural phenomena complexity is associated with

these time series. This complexity can be better explored by using fractals because the fractal

dimension of a curve is the space filling property of a curve. Also, fractal dimensions can be used

to study the persistency of the data. To check the persistency (smoothness and complexity) of the

sunspot cycles and ENSO cycles data, fractal dimension and Hurst exponent are calculated for

each sunspot cycle and ENSO cycle. The novelty of this study is to use the fractal instrument to

determine the statistical correlation between the sunspot cycles and the corresponding ENSO cycle

data. It is a new methodology to determine a correlation between two different data sets using

weak and strong fractal dimensions. In fact the method determines a correlation between the

natures of the two data sets regarding their persistency. The other novelty of this study is to analyze

the universal scaling parameters (Scaling exponent, spectral exponent and autocorrelation

coefficient) using fractal dimensions (both the self-similar and self-affine fractal dimensions) for

sunspot cycles. These results can be helpful to understand the relationship between solar and

terrestrial phenomena.

As for as the fractal dimensions are concerned, two main types of fractal dimensions viz. self-

similar and self-affine fractal dimensions are used in this study. In the self-similar, the geometric

object is composed into a union of rescaled copies of itself, which are uniform in all directions (or

isotropic rescaling). Whereas in the self-affine type, the geometric object is described as a union

of anisotropic (directionally non-uniform) rescaled copies of itself. Both the processes are

84

represented by linear transformation. One is self-similar fractal dimension and another is self-

affine fractal dimension. Self-similar fractal dimension FDS in calculating by box counting

technique or Hausdorff – Besicovich method. Whereas Self-affine fractal dimension FDA in

calculating by rescaled range analysis and another method Higuchi’s fractal dimension FDH is also

analyzed.

The common method to calculate the fractal dimension is Hausdorff-Besicovich method. Different

other methods like rescaled range analysis and Higuchi’s method are also used to calculate fractal

dimension. Box dimension or box counting method is more appropriate than other methods

(Michael, 1988). As mentioned earlier the fractal dimension describes the roughness and

smoothness of the data. To study solar activity fractal dimension is used to evaluate the

predictability of the solar cycle (Gayathri, 2010). Yanguang (Yanguang, 2010) has described three

different analyses involved in the fractal study of parameters. These are universal scaling analysis,

spatial correlation analysis and spectral analysis. Some useful details regarding spectral analysis

can be found in (Y. Chen, 2009).

2.1: DATA DESCRIPTION AND METHODOLOGY

The present study describes the Fractal Dimension (FD) approaches for the association of Sunspot

cycles and El Nino- southern oscillation (ENSO) related to local climate variability. The self-

similar fractal dimension (FDS) which is calculated by the Box counting method and self-affine

fractal dimension (FDA) which is applied by rescaled range for calculating the Hurst exponent long

range persistency of time series. Comparison among the FDS and FDA for each mean monthly

sunspot cycle and ENSO Cycles for the period 1st cycle to 23rd cycle and (1866 to 2012) 1st cycle

to 23rd cycle respectively. The present study, investigate that the association of the Mean monthly

Sunspot cycles from 1755 to 2008 which is consisted of 23 cycles whereas overall data cover in

24 cycles. The communication also examines the climatic variability impact of ENSO cycles from

1980 to 2000, in which duration the featured of the ENSO cycle is very active. All results are

explored the ENSO has five El Nino periods (1982-83, 1986-87, 1991-1993, 1994-95, and 1997-

98) and La Nina has three periods (1984-85, 1988-89, 1995-96) that were obtained from World

Meteorological Organization (WMO) and National Oceanic and Atmospheric Administration

(NOAA). The MATLAB 2016 and FRACTALYSE 2.4 software use tool for fractal dimension

and Hurst exponent analysis.

85

2.1.1: Fractal Dimension

The most appropriate method to calculate fractal dimension is Hausdorff - Besicovich box

counting or box dimension (Michael, 1988). It has belonged to the family of self-similarity. It

reveals empirical estimation and mathematical calculation. It can be calculated by taking a ratio

between log changes in object size and log change in measurement scale. This phenomena,

measurement scale approaches to zero. This expression can be mathematically written as

FDs = ln 𝑅

ln 𝑆 (2.1)

Fractal dimension has an ability to connect real world data which can easily measure by

approximated experiments, like coastlines, dust in the air, neuronal networks in the body, the

distribution of frequencies the colors emitted by the sun and during the storm, wrinkle the surface

of the sea are related with the fractal dimensions (Salakhutdinova, 1998).

This relationship can be summarized by (Sugihara, 1990) which reveals correlation as well as the

nature of the data.

Fractal Dimension = no of small pieces

magnification level (2.2)

The relationship between D and H calculate the analysis of the persistency of any time series data.

D + H = 2 (2.3)

2.1.2: Rescaled Range Analysis

Rescaled range analysis method belongs to the family of self-affine fractal dimension. Harold

Edwin Hurst (1880-1973) was introduced the rescaled range analysis (R/S) which is used to

analyze statistical methods for long term natural phenomena (Hurst, H. E. 1951). The purpose of

rescaled ranged analysis to reveal persistency analysis of time series data. Peters and Lo’s (Peters,

E. E. 1991and Lo, A. 1991) describe the dependence of long term persistence of finite non-period

cycle’s indices of stock market. In this technique, the ratio of two main variables, the range of time

series data (the maximum value is subtracted by minimum value) and the standard deviation is

known as Hurst exponent. In case a trend value exists, Hurst exponent can extrapolate a future

value. (Chen et al. 2002 and He et al. 2010) found that time series and Hurst exponent are

correlated. It generates by one dynamic system that is statistically significant degree does not

change when a segment is randomly cut from the correlated signals.

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The rescaled range R/S is defined as the average of a given sequence of observation (Xt) as

⟨X⟩t = 1

τ∑ Xt

τ−1t=0 (2.4a)

Y = (𝑡, 𝜏) = ∑ [𝑋𝑡 − ⟨𝑋⟩𝑡]𝜏−1𝑡=0 (2.4b)

R (𝜏) = max0≤𝑡≤𝜏−1

𝑌(𝑡, 𝜏) - min0≤𝑡≤𝜏−1

𝑌(𝑡, 𝜏) (2.16)

S (𝜏) = (1

𝜏 ∑ ( 𝑋𝑡 − ⟨𝑋⟩𝑡) 2𝜏−1

𝑡=0 )1

2 (2.4c)

R / S (𝜏) = 𝑅 (𝜏)

𝑆 (𝜏) (2.4d)

We can now apply

R

S∝ τH (2.4e)

Where H is the Hurst exponent.

2.1.3: Persistency

The Hurst exponent is used to calculate persistency analysis (H.E. Hurst, 1951). In the data series,

the geometric (fractal) scaling is calculated by Hurst exponent (Turcotte, D.L., 1997). The Hurst

exponent measures long term memory of time series, which is very useful in forecasting. The

range of the value of Hurst exponent parameter HE lie from 0 to 1 (Hasting et al., 1993; Sun et al.,

2005). Hurst exponent measures the persistency and anti-persistency of regularity or irregularity

(chaos) the time series function respectively. If the value of H approaches to 1 it is indicated that

the time series data is quasi-regular (persistence) and it is approached to zero, so it is shown that

irregular (anti-persistence) (salakhudinova, 1998; Hanslmeier, 1999). The Hurst exponent value is

increased from 0 to 1 then persistency will also increase and similarly behave when a value will

decrease. When the value of HE = 0.5, this known as the Brownian process. If the value lies in the

range from 0 to 0.5, it called random walk (anti-persistence) process. The process stands for

persistent if the value of H lie from 0.5 to 1. In case the value of H approaches to 1 then the trend

of the data strength will also be increased (Hasting et al., 1993; Bundeet al. 1994). This relationship

between fractal dimension and Hurst exponent is described that correlation and the nature of the

data (Thomas. et. al 2008). Calculating Hurst Exponent (HE) = 2 – FD.

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2.1.4: Higuchi’s Algorithm

In the time series data, Higuchi’s Algorithm is used to calculate the fractal dimension FD.

Higuchi’s method (Higuchi 1988) defines the scaling relationship between the mean normalized

length’s curves of the coarse-grained time series.

Whereas in the irregular time series, the power spectrum analysis is used for analysis fractal

dimension. The Higuchi’s Algorithm calculates the fractal dimension (FD), in the finite time series

data.

X(1), X(2), X(3), …, X(n) (2.5a)

Newly constructed 𝑋𝑘𝑚 is written as

𝑋𝑘𝑚 ; X(m), X(m + k ), X(m + 2k), …, X [𝑚 + (

𝑁−𝑚

𝐾) 𝑘] (2.5b)

With m = 1, 2, 3,…, k and [.] represent the Gauss notion, that is the highest integer. Where m

represents as initial time and k express time interval.

Suppose that, if k = 5 and N = 50

𝑋51: X(1), X(6), X(11), …, X(46) (2.5c)

𝑋52: X(2), X(7), X(12), …, X(47)

𝑋53: X(3), X(8), X(13), …, X(48)

𝑋54: X(4), X(9), X(14), …, X(49)

𝑋55: X(5), X(10), X(15), …, X(50)

Where k set of Lm (k) is calculated as follows

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Lm(k) = 1

𝑘(∑ (𝑋(𝑚 + 𝑖𝑘) − 𝑋 (𝑚 + (𝑖 − 1)𝑘))

[𝑁−𝑚

𝑘]

𝑖=1) (2.5d)

Where the term 𝑁−1

[𝑁−𝑚

𝑘]𝑘′

indicate that the normalization factor.

The average value ⟨𝐿(𝑘)⟩ indicates the lengths association to the time series. The relationship

exists as follows

⟨𝐿(𝑘)⟩ ∝ 𝑘−𝐷 (2.5e)

Where D is the fractal dimension by Higuchi’s method. The Higuchi’s method is applied over time

series data which are not stationary.

The fractal dimension is a useful tool to analysis the relationship of some tools like spectral

exponent and autocorrelation coefficient by using fractal dimensions and Hurst exponents in both

self-similar and self-affine methods.

2.2: MATHEMATICAL MODELS AND FRACTAL DIMENSION RELATIONS OF

SUNSPOTS CYCLES

In this section demonstrate the brief information about certain qualities of universal scaling law of

self-similar and self-affine fractal dimensions.

2.2.1: Spatial Correlation Dimensions

A fractal is basically scale- free phenomena, whereas fractal dimension is used to be measured

with a characteristic scale. The fractal dimension of sunspots follows three basic concepts as

describe as, Euclidean plane of sunspots has 2-dimension so the Euclidean dimension of the

embedding of sunspots d = 2. The smallest unit of the sunspot is considered as a point, so the

topological dimension of sunspots is considered to be dt = 0 (Mandelbrot, 1983). So the value of

sunspots fractal dimension range from dt = 0 to d = 2. Fractal analysis of a sunspots time series

presented here is based upon the existence of correlation among sunspot cycles. So, this study

associates fractal analysis to correlation analysis. In this sense the generalized fractal dimension is

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often called correlation dimension (Chen and Jiang, 2010; Grassberger and Procaccia, 1983).

Fractal dimension can characterize non equilibrium system FDS ≥ 0, in which the number of

independent degrees of freedom can be measured. In time series data, the fractal dimension usually

estimate and the relationship between Hurst exponent and fractal dimension are as follow

FDS = 2 – HE (2.6)

In the time series, the fractal dimension FDS is equal to 1.5 shows that unpredictable and no

correlation between amplitude changes between two successive time intervals. The value of the

fractal dimension is approached to 1 indicates that the process becomes more and more predictable.

The fractal dimension has ranged from 1 to 1.5 indicates that the process is persistent. If the fractal

dimension increased from 1.5 to 2 means that the process is anti-persistent. The value of FD = 1.5

indicates that the data is considered to be a normal scalar and is purely random, whereas FDS = 1

reveals that the time series data is smooth curve and nature is purely deterministic (Yousuf H et.al,

2008).

The length of sunspot cycles of self-similar fractal dimension FDS is given as follows.

δ(C) = δ1CFDs - d = δ1C –β (2.7)

Whereas δ1 indicates that the proportionality coefficient and β = d - FDS is the scaling exponent.

The relationship between d and FDS shows that FDS < d (Frankhauser and Sadler, 1998). If the

value of FDS lies between 1 and 2 then scaling exponent range from 0 to 1. If the value is FDS < 1

or FDS > 2 then the value of β > 1 or β < 0 respectively. For solar cycle, the fractal dimension FDS

can be revealed as a one-point correlation dimension which is indicating that zero-order correlation

dimension. Based on the equation (2.6), the density-density correlation function can be constructed

as

ζ (c) = ∫ 𝑓(𝑐)𝛿(𝑥 + 𝑐)𝑑𝑥 ∞

−∞= 2𝛿1

2 ∫ 𝑥𝐹𝐷𝑠−𝑑 (𝑥 + 𝛿) 𝑑𝑥∞

0 (2.8)

Where x define to the distance of the initial point and c to the distance of another point from the

initial point.

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Let x = ηy, where η is the scale factor. A scaling relation can be express as

ζ (ηr) = 2𝛿1 2 ∫ 𝑥𝐹𝐷𝑠−𝑑 (𝑥 + ηr )𝐹𝐷𝑠−𝑑 𝑑𝑥

∞

0 (2.9)

= 2𝛿1 2 ∫ ηy𝐹𝐷𝑠−𝑑 (ηy + ηr )𝐹𝐷𝑠−𝑑 𝑑ηy

∞

0 (2.10)

= η2(𝐹𝐷𝑠−𝑑)+1 2𝛿1 2 ∫ y𝐹𝐷𝑠−𝑑 (y + r )𝐹𝐷𝑠−𝑑 𝑑y

∞

0 (2.11)

ζ (ηr) = η2𝐻 ζ (r) (2.12)

Where 2H = 2 (FDS - d) + 1, and H proved as a generalized Hurst exponent.

2.2.2: The Wave-Spectrum Relation of Sunspots

This section stresses upon the calculation of spectral exponent and spatial autocorrelation

coefficient (spatial scaling). These two play an important role to study the spatial behavior of data.

The data under consideration comprises of 23 sunspot cycles. To perform spatial scaling the

correlation function associated with the data is changed into an energy spectrum using Fourier

transform (Y. Chen, 2009). In addition to other methods Fourier transform can also be used to

study similarity. In this method relations of fractal parameters are determined by calculating

spectral exponents. For this purpose the following scaling law is used.

f (λρ) = λβf (λ) (2.13)

Where λ denotes the scaling factor, β describe the scaling exponent (β = d - FDS) and ρ is called

the length variable of each cycle.

The Fourier transform is applied in (2.3) the following scaling relation is obtained.

F (λγ) = F [f (λρ)] = λ-(1- β) F [f (ρ)] = λ-(1- β) F (γ) (2.14)

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Where F has known as Fourier operator, γ represent the wave number, whereas F (γ) is the image

function of f (λ). Finally, the wave spectrum relation can be described from (2.9) as

S (γ) ∝ γ -2(1- β) (2.15)

The numerical relation between fractal dimension and the spectral exponent can be described by

comparison. The equation (2.7) can be correlated to the wave spectrum scaling by taking β = d -

FDS in (2.15), thus

S (γ) ∝ γ -2(1- d + FDs) = γ -2(FDs -1) = γ –α (2.16)

Here,

α = 2 (FDS - 1) (2.17)

Whereas, α is the spectral exponent and consider to be a constant which is describing the dynamical

behavior of the system in the time series fractal dimension. If α = 0 is described as white noise-

like system, and the system is uncorrelated and power spectrum which we have is independent of

the frequency. If α =1 is known as flicker or 1/f noise system which indicate moderately correlated.

If α = 2 is called Brownain noise-like system, which shows strongly correlated. The fractal

dimension DF lies between the range (1< DF < 2) indicate that the structure of nature is chaotic

and irregular then the spectral exponent, having ranged (1< α < 3) (Van Ness, 1968). The idea case

of chaotic physical structure is DF = 5/3.

The prerequisite of (2.17) is 1< FDS < 2 then the spectral exponent α can be revealed to be the

point-point correlation dimension. It is indicated that the relationship between one-point

correlation dimension (FDS) and the point-point correlation dimension (α). The relation between

the fractal dimension (FD) and spectral exponent (α) was introduced (Higuchi, 1988) and describe

by

FD = (5 - α) / 2 (2.18)

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The parameters to determine the fractal dimension can be calculated by using two methods, self-

similarity and self-affinity consequently two different fractal dimensions are obtained viz. Self-

similar fractal dimension (FDS) and Self-affine fractal dimension (FDA). Here the wave-spectrum

scaling is performed by calculating both the fractal dimensions viz FDS and FDA. Earlier this sort

of analysis was performed by (S. D. Liu et.al. 1992, Benoit B 1999) using FDA. The fractal

dimension is related with the spectral exponent α with Hurst exponent H is known as the

Hausadroff measure, define the following relation (Burlaga L F et.al 1986, Turcotte 1992)

α = 2H + 1 = 5 – 2FDA (2.19)

The Hurst exponent can be calculated by the method of rescaled range analysis (Hurst 1965), which

is used nonlinear random process. HE is describe the power function R(𝜏) / S(𝜏) = (𝜏/2)H (Feder

1988). The Fractal dimension FDS is used to analysis the characters of spatial distribution of

sunspot cycles, whereas FDA is revealed the spatial autocorrelation of sunspot cycles. The local

dimension FDA of self-affine fractal records instead of FDS self-similar fractal trails (Feder 1988,

Takayasu 1990). The relationship between FDS and FDA can be described from (2.12) and (2.14)

FDS = 7

2 - FDA (2.20)

The parameters and equation can be express logically organized into a relation and a system of

fractal parameters (Figure 1).

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Figure 2.1: A sketch map among the relationship of different fractal parameters.

The spatial activity of sunspots is supposed to be expressed as fractal Brownian motion (fBm) thus

the value of FDA of each cycle of sunspots lies between 1 and 2.

The relation between autocorrelation coefficient (C∇) and H of an incremental series on the

based on fBm can be given as

C∇ = 22H-1 – 1 (2.21)

Where C∇ represents the autocorrelation coefficient, which is based on the multiple-lag 1-

dimension spatial autocorrelation. The value of H = ½ then C∇ = 0 which is mentioned Brownian

motion. If H > ½ then C∇ > 0 representing that positive spatial autocorrelation and H < ½ then C∇

< 0 indicating that negative spatial autocorrelation.

The numerical relationship between different fractal parameter like FDS, FDA, HS, HA, αs, CA∇

and CS∇ can be indicated by using following equations (2.19), (2.20) and (2.21). The result which

is described in table 1 show that all values lies in significant scale. The range FDS from 1 to 1.5

which is indicate that number of spots are correlated to each other and have linear behavior. The

Hurst exponent range from 0 to 0.5 indicate that the persistency of the sunspots. The relation (2.20)

is theoretically valid when the fractal dimension FDS values is range from 1.5 to 2.

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In this research, we intend to evaluate the complexity of the fractal geometric parameters of fractal

dimension. The scaling analysis and spectral analysis of sunspot cycles from this research is based

on spatial correlation. This study, the correlation function described the 1-dimension space. All

fractal parameters which are calculated by both techniques, self-similar fractal dimension (FDS)

and self-affine fractal dimension (FDA) are lying in significant scale and persistent. The table 2.1.

Indicate that the values of FDS and FDA in each cycle lies from 1 to 1.5 which are shown that each

cycle is persistent, correlated and predictable whereas the value of FDA > FDs. Similarly, self-

similar Hurst exponent HS and self-affine Hurst exponent HA both values are from 0.5 to 1 which

reveals that each cycle is persistent, the relationship between HS is greater than HA. The scaling

exponent β has ranged from 0 to 1 if fractal dimension lies between from 1 to 2. The values of β

lies its own valid scale which describes the dynamical behavior as 1-point correlation dimension.

The table1: also describe the numerical relation between the spectral exponent (α) and

autocorrelation coefficient (C∇) of self-similar and self-affine fractal dimension. If 1< DF < 2 then

the spectral exponent α having a range from 1 to 3. If α = 0 is indicate as white noise-like system

which describe uncorrelated behavior and power spectrum has frequency independent. If α =1 is

called flicker or 1/f noise system which behave moderately correlated. If α = 2 is called Brownain

noise-like system shows strong correlation. The equation (2.18) is used to calculate self-similar

spectral exponent (αS) and self-affine spectral exponent (αA). All values either αS or αA reveals the

system like Brownain noise. The equation (2.17) reveals that if 1< FDS < 2 then the spectral

exponent α describe point-point correlation dimension. It is mentioned that the relationship

between one-point correlation dimension (FDS) and the point-point correlation dimension (α). The

autocorrelation coefficient (C∇) which is described in the multiple-lag 1-dimension spatial

autocorrelation. The autocorrelation coefficient (C∇) is calculated by the relation of Hurst

exponent from the equation (2.21). The value of C∇ = 0 if H = ½ which is described Brownian

motion. If H > ½ then C∇ > 0 indicate positive spatial autocorrelation and if H < ½ then C∇ < 0

represent the negative spatial autocorrelation. Figure 2.1: indicate a sketch map among the

relationship of different fractal parameters, figure 2.2: represent the behavior of each solar cycle

in one domain. Figure 2.3: describe the relationship between the fractal dimensions of self-similar

and self-affine in each cycle.

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Figure 2.2: Sunspot cycles (1- 24)

Figure 2.3: self-similar and self-affine fractal dimensions of sunspot cycles (1-24)

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Figure 2.4: shows that the fractal dimension (self-similar and self-affine) of each sunspot cycle

(1-23) and the cycle 24 (complete data from Aug 1755 to Jan 2008)

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2.3: MATHEMATICAL RELATIONSHIP BETWEEN HIGUCHI FRACTAL

DIMENSION AND SPECTRAL EXPONENT OF SUNSPOTS CYCLES

Higuchi’s fractal dimension (FDH) is usually used for non-linear dynamics. It is an important tool

to reveal relevant information from time series. Higuchi describes the method which provides the

precise estimate of the fractal dimension, even a small number of data. Higuchi method was

developed as an alternative method to spectral analysis. The relation between Higuchi’s fractal

dimension (FDH) and spectral exponent (α) (the slope is plots of spectral analysis). The Higuchi’s

fractal dimension (FDH) is related to the spectral exponent (α) as

α = 5 – 2FDH (2.22)

If FDH is in the range 1< FDH <2 then 1< α <3 (Van Ness, 1968). Higuchi method describes the

concept of two or more regions in which the graph between log Lm (k) and log k divide by

crossover. For instance the value that divides different scaling region associated with different

values of fractal dimension (FDH) (Higuchi 1988, 1990). Whereas FDH = 1.5, the dynamics of the

system are Brownian, FDH < 2 corresponding to pink noise and FDH = 2 represent white noise

(Galvez-coyt et al, 2011). If the value of α is equal to 0 define white noise-like system, and the

following system is uncorrelated and power spectrum which behave like independent of the

frequency. If α equal to 1 is known as flicker or 1/f noise system which describe moderately

correlated. If α = 2 is called Brownain noise-like system, which shows strongly correlated.

Higuchi’s fractal dimension FDH used to describe the behavior of spectral exponent.

The spectral exponent (α) values lie 1.7 < α < 2.3 which indicates that all cycles are strongly

correlated to each other’s. Sunspot cycles 5, 8, 10, 16, 17 and 23 has the spectral exponent (α)

value 2. The sunspot cycles 1, 2, 7, 11 and 13 shows the spectral exponent (α) value 1.8. The

sunspot cycles 9, 15 and 20 reveal the spectral exponent (α) value 1.9. The sunspot cycles 3, 4, 11,

18 and 21 calculate the spectral exponent (α) value 2.1. The sunspot cycle 6 has the value of α =

1.7, cycle 22 has α = 2.2 and cycle 19 indicate α = 2.3.

2.4: FRACTAL DIMENSION AND HRUST EXPONENT OF ENSO CYCLES BY USING

SELF-SIMILAR AND SELF-AFFINE FRACTAL DIMENSION

El Nino- southern oscillation (ENSO) is an irregular periodical fluctuation in wind and sea surface

temperature (SST) of the covering tropical eastern Pacific Ocean. Tropics and subtropics both are

101

affected on ENSO. There are two phases which are classified by temperature cooling phase known

as La Nino whereas warming phase is called El Nino. Southern oscillation is associated with the

sea surface change and atmospheric components. La Nino is associated with low air pressure

surface in the tropical western Pacific, whereas El Nino is associated with high air pressure surface

(Herring D. 2009). ENSO provide a mechanism which has maintained long-term climate stability

by transferring heat with high latitudes from the tropics. El Nino and La Nino have own regional

effect which is fluctuate, like the extremes of sun (Guhathakurta, M., & Phillips, T. 2013). The

Southern Oscillation is calculated by SOI, which is stand for the Southern Oscillation Index. The

SOI measures the variation of surface air pressure between Tahiti (in the Pacific) and Darwin,

Australia (on the Indian Ocean) (Climate glossary, 2009). El Nino periods have negative SOI,

which indicate a lower pressure over Tahiti and higher pressure at Darwin and La Nino periods

have positive SOI, which show higher pressure in Tahiti and lower in Darwin. The ENSO cycle

consists of averagely four years, but as per historical record ENSO cycle has varied between two

and seven years (NOAA).

This research is calculated the complexity of ENSO phenomena by using Self-similar fractal

dimension and self-affine fractal dimension and comparison both techniques along with long term

memory forecasting for data persistency also determined the Hurst exponent for the twenty four

Sunspots Cycle and 23 ENSO cycle with total cycles and active ENSO cycle periods. Each ENSO

cycle duration is 7 years from 1866 to 2012. ENSO cycle 22 cover 147 years data, although we

have selected 23 cycle as the more active duration from 1981 to 2000 respectively. The twenty

four Sunspot cycles and twenty two ENSO cycles along total both cycles with 23 ENSO cycle

have persistent behavior which is calculated from self-similar fractal dimension (FDS) and self-

affine fractal dimension (FDA), Hurst exponents also computed through FD which is depicted by

table 2.3. Results show that self-affine fractal dimension (FDA) and self-similar fractal dimension

(FDS) for twenty four sunspot cycles and twenty two ENSO cycles all along total data and active

ENSO cycles have fractal dimension value less than 1.5, the values of FDA > FDs, which is

indicating that each cycle is becomes predictable, correlated and persistent . Figure 2.5: represent

the behavior of each ENSO cycle in one domain. Figure 2.6: indicate a sketch map among the

relationship of different fractal parameters of ENSO cycles. Figure 2.7: describe the relationship

between the fractal dimensions of self-similar and self-affine in each cycle.

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Figure 2.5: ENSO cycles (1-23)

Figure 2.6: self-similar and self-affine fractal dimension of ENSO cycle (1-23)

106

Figure 2.6: Shows that the fractal dimension (self-similar and self-affine) of each ENSO cycle (1-

21) and the ENSO cycle 22 (complete data from 1866 to 2012) and ENSO cycle 23 (active region

from 1981 to 2000)

107

2.5: COMPARATIVE ANALYSIS OF SELF- SIMILAR AND SELF-AFFINE FRACTAL

DIMENSION OF SUNSPOTS CYCLE AND ACTIVE ENSO PHENOMENA IN SAME

TIME INTERVAL

The impact of the earth climatic condition of solar activity oscillations is some extent is measurable

only in the long run duration. Solar cycles and ENSO are correlated to each other. Theory about,

the relationship between the sunspots and ENSO phenomena is premature, but now is developed

by collection of evidence that the solar cycle moderates wind field in the stratosphere and

troposphere (Anderson 1992, tnislay et.al, 1991) which provide a potential mechanism for the link

between the solar geometric effect and ENSO phenomena. The change in sunspot number can

reveal the association in which ENSO episode in more active when solar activity is weak

(Anderson 1990). The ENSO events were reduced by long term solar changes (e.g Mander

minimum, medieval grand maximum), similar solar activity is strong then ENSO period is weak.

For instance, the medieval warm period, solar activity is increased (Damen, 1988 and Anderson

1990).

ENSO cycle featured is very active in the duration between 1980’s and 1990’s. In this duration El

Nino has five periods (1982/83, 1986/87, 1991-1993, 1994/95, and 1997/98) and La Nina has three

periods (1984/85, 1988/89, 1995/96). In this duration El Nino has two strongest periods of the

century, which is consisted of 1982/83 and 1997/98 similarly two consecutive periods during

1991-1995 has cold period without intervening. ENSO cycle has varied from one cycle to the next

cycle. The fractal dimension is used in this study to define the complexity of the ENSO cycles.

Here, self-similar and self-affine behavior of same duration of sunspots and ENSO cycles.

Table 2.4 and figure 2.7, indicate our selected same cycle duration both of data also signify the

self-affine fractal dimension (FDA) is greater than self-similar fractal dimension (FDS ) and ENSO

cycles is also less than self-similar fractal dimension (FDs) of Sunspot cycles. The Hurst exponent

for each cycle, calculate with the help of FD to observe that FDA is increasing than, Hurst exponent

of self-affine HEA is decreasing, this relation also provides the smoothness and accurate

persistency of the cycles. Table 2.3 – 2.1 depicts Hurst exponents for smoothness. Figure 2.8 shows

that the self-similar and self-affine fractals of sunspot cycles and ENSO cycles in one domain.

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Figure 2.7: Comparative analysis of self- affine and self-similar fractal dimension of sunspot

cycles and ENSO cycles (May 1976 to Jan 2008)

Figure 2.8: comparison of self-similar and self-affine fractal dimensions of sunspot and ENSO

cycles.

2.6: CONCULSION

This chapter reveals the persistency analysis of solar activities and ENSO cycles by using fractal

dimension involving Hurst exponent. The fractal dimension (FD) represents the roughness (local

property) of the time series data of solar activities (sunspot activity) and ENSO episode, whereas

Hurst exponent (HE) provides the smoothness (universal property) of the data. This study

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investigates the relationship between self-similar fractal dimension (FDS) and self-affine fractal

dimension (FDA). FDS calculated by Box counting method and self-affine fractal dimension (FDA)

is calculated by using rescaled range method. The Hurst exponents are calculated by FD which is

calculated by both the self-similar fractal dimension (FDS) and self-affine fractal dimension (FDA).

The spectral exponent is calculated by using equation (2.13) and the autocorrelation coefficient is

calculated by using equation (2.16). Table 2.1: indicates the numerical relationship between Hurst

exponents, spectral exponent and autocorrelation coefficient. Both fractal dimensions (self-similar

and self-affine) are used. Each cycle is found to be persistent and correlated, the values of Hurst

exponent lie between 0.5 and 1 but HES values are greater than HEA. All values of the spectral

exponent (αs and αA) of each sunspot cycles behave like Brownain noise which indicates the long

term dependency. The autocorrelation coefficient is found to be significant using both the fractal

dimensions FDS and FDA. For sunspot cycles the relationship between self-similar and self-affine

fractal dimensions expressed by equation (2.15) failed as this relation is valid only if the fractal

dimension lies between 1.5 and 2. Here each for each sunspots cycles the fractal dimension is less

than 1.5. The spectral exponent value which is calculated by Higuchi’s Fractal dimension (FDH)

also identifies the strong correlation among Sunspots cycles.

The complexity of each cycle of ENSO data, the complete data of all the cycles and the data of the

active ENSO period are analyzed. First, self-similar fractal dimension (FDS) (using Box counting

method) and self-affine fractal dimension (FDA) (by using rescaled range method are calculated.

Then the Hurst exponents are calculated by using both methods FDS and FDA. It is observed that

FDA > FDs in both the cases of sunspot and ENSO cycles. Similarly, self-similar fractal dimension

(FDS) of ENSO cycles is also less than the self-similar fractal dimension (FDs) of Sunspot cycles.

This means that ENSO cycles are more persistent than Sunspot cycles data. Sunspots cycles and

ENSO are correlated with each other. It can be noted easily that if FD increases, then HE decreases.

Each sunspot cycle has greater mean and prolonged tail. The mean-tail assessment confirms the

FD-HE analysis. This study can be useful for further investigation of the impact of Sunspot and

ENSO related local climatic variability.

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CHAPTER 3

A COMPARATIVE STUDY OF SUNSPOT CYCLES AND ENSO CYCLES

ANALYSING PROBABILITY DISTRIBUTIONS AND HEAVY TAIL

This chapter describes the behavior of solar activity or solar cycle (Sunspot cycle) and the El Nino-

southern oscillation (ENSO cycle) data and analysis the influence of it in the earth’s climate. The

study of probability distributions of climatic parameters usually gives imminent into the physical

processes (Hussain et al., 2012). Sunspots cycles and ENSO cycles follow the probabilistic

distribution. In evaluating the nature of the distribution observed by existing data, we check for

the deviation from normality using Kolmogorov–Smirnov test (KST), Anderson-Darling test

(ADT) and Chi-Square test (CST). The analysis of earth’s climate, solar cycle plays an essential

role (Weiss et al., 2007). In the early 1600s, the variation of dark spots which are appearing on

the surface of the sun, known as sunspots were observed. These variations are based on regular

cycle which has peak approximately 11.1 years. The climate change is usually affected by about

11 years or 18.6 years solar cycle (Seleshi et al. 1994). The cycle of solar activity concurs with a

small oscillation as an output in solar energy. The output energy of sun light to some extent higher

during episodes with large numbers of sunspots. The sun has a great influences the atmosphere

and the climate of the earth (Bal et.al 2010, Siingh et.al 2010 and Spinage 2012). The solar activity

increase and the intensity of galactic cosmic ray flux is decreasing, causing the high rainfall (Sun

et al. 2002 and svensmark 1997). The magnetic field of the earth increases with the solar cycle

intensified. The annual rainfall in 10/11 year cycles is described in many regions monsoonal

rainfall received (Seleshi et al. 1994). In the recent research described that tropospheric and

stratospheric weather system are usually influenced by the solar cycle, which is indicating that the

dependency of climates in solar activity (Almedia et al. 2004, Lakshmi et al. 2003, Kumar et al

2010 and Guhathakurta et al. 2007). The correlation between rainfall and solar cycle (22 years) for

long term time scale is positive and 11 year time scale provide very low or negative correlation

(Hiremath 2006). It is difficult to calculate the direct relationship between the sunspot cycle and

climate, but some occasions of climate relate with sunspot numbers. For instance, in the interval

1645 to 1715, the sunspot number was dramatically lower than previous one. This interval known

111

as the Maunder Minimum. The solar energy reduction in the minimum duration could have a slight

cooling of the Earth. The main task of this chapter to analysis the comparative study of solar cycles

and ENSO cycle in the perspective of probabilistic model and heavy tail analyzes and long range

persistence.


The present study investigates the association of the Mean monthly Sunspots cycles and ENSO

cycles for the period from 1755 to 2008 and 1866 to 2012 respectively. The Generalized Pareto

and Generalized extreme value, probability distribution techniques are applied to calculate the

mean and standard deviation and most appropriate distribution for each cycle of sunspots and

ENSO activity on the basis of goodness of fit Kolmogorov–SmirnovD, Anderson-Darling and

Chi-square test. The communication also examines the climatic variability impact of ENSO cycles

from 1980 to 2000, in which duration the featured of the ENSO cycle is very active. All results are

explored the ENSO has five El Nino periods (1982/83, 1986/87, 1991-1993, 1994/95, and

1997/98) and La Nina has three periods (1984/85, 1988/89, 1995/96) that were obtained from

World Meteorological Organization (WMO). This chapter is divided by three main sections,

section one is consists of the best fitted probability distribution with the help of Kolmogorov–

Smirnov test (KST), Anderson-Darling test (ADT) and Chi-Square test (CST). In this section, also

described the probability distribution of fractal dimension of sunspot cycles and ENSO cycles from

the perspective of self-similar and self-affine fractal dimension. Comparative study of the sunspot

cycle and Active region of ENSO cycle also investigate. The second section investigates long

range persistence behavior of the solar cycle (or sunspot cycles) and ENSO cycles and also

compare the same interval of active regions of the ENSO cycle and sunspot cycles. Last section

investigates the heavy tail parameter for long term relation and persistency in the perspective of

Hrust exponent of self-similar and self-affine fractal dimension. The Easy fit version 5.6 and

Matlab 2016 are utilized for the testing of underlying distributions. The techniques which are

described in this chapter will not only be useful in the field of solar phenomena, but also will be

helpful to investigate the solar-terrestrial relations (Hassan and Abbas, 2014).

112

3.2: PROBABILITY DISTRIBUTION APPROACH

In this section, we briefly discuss the probability distribution techniques which are used to check

the best fitted distribution

3.2.1: Goodness of Fit Tests

The goodness of fit test is used to measure random sample with a theoretical probability

distribution function. These tests indicate that which distribution is best for the data. The general

method of goodness of fit based on analysis a test statistic in which some function of the data,

calculating the distance between the hypothesis and the data, and also measuring the probability

of the data.

3.2.2: Kolmogorov-Smirnov Test

If a sample follows from a hypothesized continuous distribution, then Kolmogorov-Smirnov test

is used to decide. The Kolmogorov-Smirnov test is basically based on the empirical cumulative

distribution function (ECDF). The Kolmogorov-Smirnov test is defined as the largest vertical

difference between F (x) and G (x).

D = max|H(x) − G(x)| (3.1)

Where H (x) and G (x) are predetermined cumulative distribution and sample cumulative

distribution for the given sample of size n (Hussain, 2006).

3.2.3: Anderson-Darling Test

Anderson-Darling (ADT) test was intended to detect differences in the tails of the distribution,

especially (Li et al., 2002). The Anderson- Darling test is same as the Kolmogorov-Smirnov test,

but it provides strong information related to the tails of the distribution. This test depends on the

number of intervals. The Anderson-Darling test only applies to the input data sample which is a

drawback of the test (Walck, 2000). The Anderson-Darling statistic (A2) is defined as

A2 = −1

n∑ (2i − 1)[lnαi + ln(1 − α n−i+1)] − n,n

i (3.2)

113

Where 𝛼i represent the theoretical cumulative distribution at the ith largest observation, 𝑛 is the

number of observations (Hussain, 2006).

3.2.4: Chi-Squared

The Chi-Squared test apply to determine where a population sample with a specific distribution.

This test is used for binned data. The nature of binned data depends on the test of statistic. Many

formulas are used to calculate bins (S) which depends on sample size (N).

S = 1 + log2N (3.3)

Grouped data can be used which is divided into equal difference. The statistic of Chi-Squared is

defined as

(3.4)

Where Ei is the expected frequency for bin i and Oi is the observed frequency for bin i, calculated

by Ei = G(x2) - G(x1), where G is the Cumulative Density Function of the probability

distribution being tested, and x1, x2 are the limits for bin i.

3.2.5: Best Fitted Probability Distribution

In many branches of science, experimental analysis may encounter the problems where the

standard probability distributions are useful. These probability distributions are useful to generate

the random numbers (Walck, 2007). In the study of probability distribution the comparative study

sunspots and ENSO cycle are very useful to analyze the change and variation in the earth’s climate.

The advantage probability distribution of random number study for solar and terrestrial parameter

provides the essential knowledge about the physical processes governing in it. The KST

[Kolmogorov-Smirnov D-test] is used to check the deviation of normality to evaluating the nature

of the distribution with the real time data. We have described the suitable and most fitted

distributions for solar cycle and the ENSO cycles and the relation of sunspots and active region of

the ENSO cycle in the same interval. The probability distribution of self-similar and self-affine

fractal dimension also calculated.

114

3.2.5.1: The Generalized Extreme Value (GEV) Distribution

The generalized extreme value (GEV) distribution in probability theory and statistics is a family

of continuous probability distributions which are developed with extreme value theory to combine

the Fréchet, Gumbel and Weibull families also called type I, II and III extreme value distributions.

The location, scale and shape parameters of generalized extreme value are μ R, σ > 0 and k R.

The symmetrically generalized extreme value distributed mathematically it can be expressed as a

probability density function.

f (x) = (1 + 𝑘 (

𝑥− 𝜇

𝜎))−

1

𝑘 𝑖𝑓 𝑘 ≠ 0

𝑒− (𝑥− 𝜇)

𝜎 𝑖𝑓 𝑘 = 0

(3.5)

Mean =

𝜇 + 𝜎(𝑔1−1)

𝑘 𝑖𝑓 𝑘 ≠ 0, 𝑘 < 1

𝜇 + 𝜎𝛾 𝑖𝑓 𝑘 = 0∞ 𝑖𝑓 𝑘 > 1

(3.6)

(3.7)

and γ is Euler’s constant.

Variance =

𝜎2 (𝑔1−𝑔22 )

𝑘2 𝑖𝑓 𝑘 ≠ 0 , 𝑘 <1

2

𝜎2 𝜋2

6 𝑖𝑓 𝑘 = 0

∞ 𝑖𝑓 𝑘 ≥ 1

2

(3.8)

In Eqs. (3.7) and (3.8), E [X] and Var (X) represent mean and variance of samples respectively

(Hussain, 2006).

115

3.2.5.2: Generalized Pareto Distribution

The Generalized Pareto Distribution (GPD) was developed by (Pikands 1975) and further studied

(smit 1984, Castillo 1997 and 2008). The Generalized Pareto Distribution (GPD) plays an essential

role to study extreme events in general.

Parameter of Generalized Pareto distribution is;

𝛼 ∈ (−∞, ∞); Location (real) (3.9)

𝛽 ∈ (0, ∞); Scale (real)

𝛾 ∈ (−∞, ∞); Shape (real)

Support

x ≥ α (𝛾 ≥ 0)

α ≤ x ≤ 𝛼− 𝛽

𝛾 (𝛾 < 0)

f (x) = (1 + 𝑘 (

𝑥− 𝜇

𝜎))−

1

𝑘 𝑖𝑓 𝑘 ≠ 0

𝑒− (𝑥− 𝜇)

𝜎 𝑖𝑓 𝑘 = 0

(3.10)

𝑒− 𝑡 (𝑥) for x ∈ support

Probability density function

f (x) = 1

𝛽 (1 + 𝛾 𝑧)

−(1

𝛾 +1)

where z = 𝑥− 𝛼

𝛽 (3.11)

Cumulative distribution function:

F (x) = 1 – (1 + 𝛾 𝑧)−

1

𝛾 (3.12)

Mean = α + 𝛽

1− 𝛾 (𝛾 < 1) (3.13)

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Variance = 𝛽2

(1− 𝛾)2(1−2𝛾) (𝛾 <

1

2) (3.14)

The probability distribution of sunspots and ENSO cycles are described as below,

3.3: BEST FITTED PROBABILITY DISTRIBUTIONS OF SUNSPOT CYCLE

The sunspot has an intense magnetic field and cool temperature and the center temperature of a

sunspot is 6400°F (TJO News, 2006). The Sunspots are associated with active regions, which are

areas of locally increased magnetic flux of the Sun (Kevin R. et.al 2014). The sunspot is cooler

and look dark on the sun’s surface is heated to the orange-white stage. The sunspots are a large

number of the blotches on the sun that is often larger in diameter than the earth. The yearly average

of sunspot areas has been recorded since 1700 (Kok K. et. al. 2001). A maximum number of

sunspots are known as solar maximum. Normally sunspots appear in pairs in which one has a

positive magnetic polarity and the other has a negative magnetic polarity and join these two spots

centers by a line which is nearly parallel to the sun’s equator. Sunspots often appear near 30-35

degree north and south of the sun’s hemisphere with higher latitudes. The sunspots life consists of

either days or one week or few weeks.

This portion investigates that sunspot data from 1755 to 2016 (25 cycles) which are consist of 1 to

23 cycles, 24 cycle (duration Jan 2008 to continue) prediction and cycle (1-23 complete data). In

this study, two types of probability distribution have been stressed one is Generalized Pareto

(GPD) Distribution and another is Generalized Extreme value (GEV) Distribution. The deviation

is checking from normality using Kolmogorov–Smirnov test (KST), Anderson-Darling test (ADT)

and Chi-Square test (CST). Both distributions have continuous nature and have heavy tail

behavior. The Generalized Pareto Distribution (GPD) is commonly used for those qualities which

are distributed with long right tail. The table 3.1 represent that the mean, standard deviation and

distributions of each sunspot cycle (from 1 to 23, 24 and complete data 1-23) with parameters of

testing probability distributions. From Table 3.1 we observe that different cycles follow different

distributions. This behavior shows that the distributions become increasingly heavy right tailed.

Figures. 3.1 represent the plot of total sunspots data and plot of each cycle from 1755 to 2016 as

well. Figure 3.2 depicts a complete sketch the probability distribution with different peak of

sunspot data.

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Figure 3.1 These figures shows the fitting of probability distribution of 23 sunspot cycles and 24th

incomplete cycle and 1-23 complete data cycle with different peaks and duration along with total

duration 1755-2016.

Figure 3.2: The probability distributions different peaks plots of 24 sunspot cycles (1755-2016)

This study estimates the probability of the mean monthly Sunspot cycle (1st to 24th cycle is in

continuation and total (1stto 24thcycles). The probability distribution (Tables 3.1 and figure 3.1 and

3.2) for Sunspot data series depicts most of the sunspot cycles are followed Generalized Pareto

distribution and Generalized extreme value distribution. In Table: 3.1 indicates the mean and

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standard deviation for the duration of all cycles and total cycles of Sunspots parameters of test

probability distributions. The sunspot cycles reveals the Generalized Pareto Distribution

probability apart from cycle 9 which describes the generalized extreme value distribution. Both

distributions indicates the heavy tail on the right side. In this section, the prediction of 24 cycle

also follows the Generalized Pareto Distribution. Table 3.1 depicted. Variations of the mean and

standard deviation are useful to analysis the changeable behaviors of the sunspot cycles start from

1755 to till 2011, sunspot cycle 24th is still processed. The figure 3.2 indicate the probability

distributions different peaks plots of 24 sunspot cycles (1755-2016). The variation in mean and

standard deviation represents the change in the strength of the sunspot cycle. Figure: 3.2 represents

the plot of twenty four with the total sunspot cycle plot of each cycle from 1755 to 2008 Cycles

1st to8th and 10th to 23rd follow the Generalized Pareto distribution and the remaining cycle 9

follow the Generalized Extreme value probability distribution.

3.4: BEST FITTED PROBABILITY DISTRIBUTIONS OF ENSO CYCLE

Enso phenomena effects on climate and weather. The strong ENSO episode brings droughts in the

Southeast Asia countries. In the USA, these events make a reason to bring heavy rain and

increasing the number of hurricanes and typhoons in the eastern pacific. The relationship between

rainfall and ENSO in different world areas vary from area to area (Bhalme et. al. 1984). The impact

of ENSO of Indian Ocean varies from season to season (Singh et. al. 1999). In the early nineties

climatologist have intent to understand the relation of the Indian monsoon with the ENSO is

ordered to estimate the monsoon rainfall over Asia (Wabster et.al 1992). The review of the

historical background the relation of monsoon and ENSO some qualitative studies on the coupling

of ENSO year and rainfall over Pakistan (Arif et al. 1994; Chaudhary, 1998). The variability of

Rainfall of Indian summer monsoon and East Asian summer monsoon and their relationship with

ENSO has been discussed (Walker, 1923, 1924; Rasmusson et. al, 1983; Shukla et. al. 1983;

Mooley et. al. 1987; Yasunari, 1990; Webster et. al. 1992; Lau et. al. 1996; Ju et. al. 1995; Soman

et. al. 1997; Liu et. al. 2001; Lau et. al. 2000).

This portion investigates that ENSO data from 1866 to 2012 (23 cycles) which are consist of 1 to

21 cycle, 22 cycle (duration from 1866 to 2012 complete data) and 23 cycle (duration from 1980

to 2000) active episode of ENSO cycle. In this study, we stress upon various types of the

probability distribution. Generalized Extreme value Distribution (GEV), Generalized Pareto

123

Distribution (GPD) and Logistic Distribution are used to analyze the behavior of ENSO data with

the help of testing Kolmogorov–Smirnov test (KST), Anderson-Darling test (ADT) and Chi-

Square test (CST). These distributions indicates that the ENSO cycles have continuous heavy tail

persistency on the right side. The table 3.2 shows that the mean, standard deviation and

distributions of each ENSO cycle (from 1 to 21, complete data 1-21 and 22) with parameters used

to test probability distributions. From Table 3.2 we analyze the different cycles follow different

distributions. This behavior reveals that the distributions become increasingly heavy right tailed.

Figure. 3.3 represent the plot of total ENSO cycles and the plot of each cycle from 1866 to 2012

as well

126

Figure 3.3: These figures show the fitting of probability distribution of 21 ENSO cycles, 23th cycle

(1-21) complete data cycle with different peaks and duration along with total duration 1755-2016.

22th cycle (1981-2000) active region of ENSO cycle.

127

Figure 3.4: The probability distributions different peaks plots of 23 ENSO cycles (1866 to 2012)

The probability distributions of ENSO data (from 1866 to 2012) are formed (figure 3.3). In this

study, we analyze the probability of the mean monthly ENSO cycles (1st to 23rdwhile 22 cycle

consists of complete data duration of 1866 to 2012 and 22 cycle duration most active durations

from 1981 to 2000 respectively). The probability distribution (Tables 3.2 and figure 3.3 and 3.4)

ENSO episode series show followed the generalized extreme value distribution, generalized Pareto

distribution and only cycle 6thfollowed the logistic distribution. The ENSO cycles show the

generalized extreme value distribution, the ENSO cycle 13th is revealed Generalized Pareto

distribution which is shown heavy tail similarly cycles 6 seem to Generalized Pareto distribution.

Cycles 22 which is covered for 112 year cycle and cycle 23th is consisting of a 20 year data which

completes the durationfrom1981 to 2000 in which ENSO activity are stronger similar to follow

the generalized extreme value distribution.

In particular, the changes in the distribution of Generalized Extreme Value and Generalized Pareto.

These distributions are becoming an increasingly right side heavy tailed ENSO cycles (figure 3.3

– 3.4). We are able to calculate the distribution parameters mean and standard deviations to

confirm the climatic variability. Table: 3.2 demonstrates the mean and standard deviation for the

duration of all cycles and total cycles ENSO and active region of the ENSO cycle with parameters

of test probability distributions. The mean and standard deviation come out to be for ENSO cycles

128

is ((0.49464 ±1.021), (0.52708 ±1.097) and (0.17989 ±1.107) respectively (table 3.2). The result

(Tables 3.1- 3.2) indicate that mean variation of Sunspots cycles maximum as compared to ENSO

cycles. As figure 3.4 depicts the plot of twenty two and total ENSO cycles plot from 1866 to 2012

cycles, 1st to 5th, 7th to 11th, 13th to 21st follow Generalized Extreme value distribution while cycle

6thand 12thlogistic and generalized Pareto probability distribution show a tail prolonged.

3.5: COMPARATIVE ANALYSIS OF BEST FITTED DISTRIBUTION OF ENSO

ACTIVE REGION AND SUNSPOT CYCLES IN SAME DURATION

It confirms that the period 1980-2000 of ENSO cycles were very active. El Nino was active for

the periods 1982-83, 1986-87, 1991-1993, 1994-95, and 1997-98 these periods include two

strongest periods of the century viz. 1982-83 and 1997-98. When the ENSO active period was

going on, meanwhile the sunspot had cycle 21 (May1976 - Mar 1986), cycle 22 (Mar1986 -

Jun1996) and cycle 23 (Jun1996 - Jan 2008). In this study we intend to analysis the complexity of

the ENSO cycle in the same duration of the sunspot cycle.

In Table 3.3 explore the most active ENSO cycles (May1976 - Mar 1986), (Mar1986 - Jun1996)

and (Jun1996 - Jan 2008) which are mean and standard deviation come out to be three active ENSO

cycles are ((0.49464 ±1.021), (0.52708±1.097) and (0.17989 ±1.107) have positive correlation

and less standard error for these active ENSO cycles, the table results indicates that the Generalized

Extreme value probability distribution become increasingly heavy tailed behavior. In the same

duration sunspots followed the Generalized Pareto distribution (GPD). The nature of the

Generalized Pareto distribution (GPD) reveals that the behavior of time series data is a heavy tail

behavior, strong correlated and long range persistence. Figure 3.5 depicted that same duration of

sunspots and ENSO time series cycles (May1976 - Mar 1986), (Mar1986 - Jun1996) and (Jun1996-

Jan 2008) and probability distributions in one domain.

129

Figure 3.5: same duration of sunspot and ENSO cycles and Probability Distribution of same cycles

of sunspot and ENSO cycles.

130

3.6: PROBABILITY DISTRIBUTION OF FRACTAL DIMENSION OF SUNSPOT

CYCLES AND ENSO CYCLES

This section explores the probability distribution of self-similar and self-affine fractal dimension

of sunspot cycles (1-23 and total 1-23) and ENSO cycles (1-22 and total 1-21) with the help of

checking Kolmogorov–Smirnov test (KST), Anderson-Darling test (ADT) and Chi-Square test

(CST). The table 3.4 indicates the best fit probability distribution of self-similar and self-affine

fractal dimension and the parameter of the mean and standard deviation are also calculated in these

distribution of sunspots and ENSO cycles. The self-similar and self-affine fractal dimension are

also followed Generalized Extreme Value distribution (GEV) and Generalized Pareto distribution

(GPD).

Table 3.4. Shows that the self-similar fractal dimension (FDS) of sunspot cycle followed the

Generalized Extreme Value distribution (GEV) with (1.1928 ± 0.0818). The self-affine fractal

dimension (FDA) of sunspot cycle indicate Generalized Extreme Value distribution (GEV) with

(1.3606± 0.102). Similarly the self-affine fractal dimension (FDA) of the ENSO cycle followed

the Generalized Extreme Value distribution (GEV) with (1.117± 0.0911). Whereas the self-

similar fractal dimension (FDS) of the ENSO cycle followed the Generalized Pareto distribution

(GPD). These distributions are becoming an increasingly right side heavy tailed and persistence.

In the end a relation between probability distribution and fractal dimension persistency is presented

in table 3.4. All the sunspots and ENSO cycles probability distributions indicate persistency. The

research also concludes that twenty four cycles of sunspots will more prolong if it has greater

means and the Gen. Extreme Distribution tail prolongs. Figure 3.6 displayed that the probability

distributions of self-similar (FDS) and self-affine (FDA) fractal dimensions of sunspot cycles.

Figure 3.7 depicted that the probability distributions of self-similar (FDS) and self-affine (FDA)

fractal dimensions of ENSO cycles.

131

Figure 3.6: Probability distributions of self-similar (FDS) and self-affine (FDA) fractal dimensions

of sunspot cycles

Figure 3.7: Probability distributions of self-similar (FDS) and self-affine (FDA) fractal dimensions

of ENSO cycles

3.7: SOLAR ACTIVITY CYCLES AND ENSO ACTIVITY CYCLES IN THE

PERSPECTIVE OF HEAVY TAILS PARAMETER

The heavy tail parameter strongly based on the self-similarity parameter HE which is also called

as Hurst exponent. In the persistent noise, the heavy tail parameter β will be range from 0 to 2.

Whereas, the long-range dependence parameter d is valid in the range from 0 to 1- 1

𝛽. This condition

is applicable if HE < 1. In the stationary time series, the above condition is mandatory. For the

132

finite variance case HE = d + 1

2 , whereas for infinite variance case HE = d +

1

𝛽. The long-range

dependence degree is given by (Feldman and Taqqu, 1998).

HE = (3−𝛽 )

2 (3.15)

The heavy tail parameter (β) range depends on the differencing parameter (d). Case 1, if d > 0,

then the heavy tail parameter (β) will be greater than 1. The limitation of β for any noise is range

from 0 to 2 (Gourieroux and Jasiaky, 1998). Case 2, if d < ½, then it represents the power series

expansion for all the sunspot cycles. Whereas, β < 2, then the tail is asymptotically equivalent to

Pareto law (Sun et al., 2005). For all sunspots cycle β has ranged from 1 to 2. This study develops

the heavy tail parameter (β) calculate by using Hurst exponent in both method self-similar (HES)

which is calculated by Hausdorff – Besicovich method (Box counting Technique) and self-affine

(HEA) obtained by rescaled range analysis technique. dS and dA represented as differencing

parameter of self-similarity and self-affinity respectively. βS is known as self-similar heavy tail

parameter, whereas βA is called as self-affine heavy tail parameter Table 3.5 shows the fractional

differencing parameter and heavy tails parameter of sunspot cycles (1-24) in the perspective of

Hurst exponent.

Table 3.5 indicates that the heavy tail parameter values within range from 1 to 2 which is shows

that the dynamic is more regular. The differencing parameter value of all 24 sunspot cycle (24

cycle represents that complete Aug 1755 –Jan 2008 data) is less than 0.5. The heavy tail parameter

(β) is less than 2 indicate that asymptotically equivalent to Pareto law. The heavy tail parameter

(β) shows that the strength of the dynamics is regular and periodic for all the solar cycles. The

table 3.5 depicted that the solar cycle 5 has smallest value (1.198) of the heavy tails parameter (βS)

whereas, the solar cycle 3 has largest value (1.656) of the heavy tails parameter (βS). Whereas, the

table 3.5 shown that the solar cycle 21 has smallest value (1.374) of the heavy tails parameter (βA)

whereas, the solar cycle 3 and the solar cycle 17 have largest value (2.000) of the heavy tails

parameter (βS). For each sunspot cycle heavy tails are profound. The heavy tail parameter (β) and

differencing parameter (d = HE-0.5) is obtained from Hurst parameter. The heavy tail parameter

(β) value towards 2 depicted that the strength of heavy tail decreases. The novelty of the study

concludes that the heavy tail parameter of self-similar (βS) is more long time persistent and

correlated as compared to self-affine heavy tails parameter (βA).

133

Best fitted probability distribution of maximum ENSO cycles followed Generalized Extreme value

distribution (GVT) and despite of the ENSO cycle shows Generalized Pareto distribution (GPD).

These distributions have heavy tail behavior. For all ENSO cycles β has ranged from 1 to 2. This

study explores the fractional differencing parameter and heavy tails parameter of the ENSO cycles

(1-23) in the perspective of Hurst exponent of self-similar (HES) and self-affine (HEA). Table 3.6

depicts the tail analysis of ENSO cycles.

Table 3.6 develops the heavy tail parameter values (β) within the range from 1 to 2 which are show

that the dynamical theory are more regular and persistent. The differencing parameter value of all

23 ENSO cycle (22 cycle indicates the duration of an active region of the ENSO cycle, whereas,

23 cycle are complete duration 1866-2012) is less than 0.5. The value of the heavy tail parameter

(β) is less than 2 represent the asymptotically equivalent to Pareto law. The heavy tail parameter

(β) reveals that the strength of the dynamics is regular and periodic for all the ENSO cycles. The

table 3.6 explored that the ENSO cycles 22 and 23 have the smallest value (1.000) of the heavy

tails parameter (βS) whereas, the ENSO cycle 1 has largest value (1.118) of the heavy tails

parameter (βS). Whereas, the table 3.6 shown that the ENSO cycle 11 has smallest value (1.002)

of the heavy tails parameter (βA) whereas, the ENSO cycle 3 has largest value (1.720) of the heavy

tails parameter (βA). For each ENSO cycle heavy tails are profound. The novelty of the study

concludes that the heavy tail parameter of self-similar (βS) is more long time persistent and

correlated as compared to self-affine heavy tails parameter (βA). The heavy tail parameter (βS) of

the ENSO cycle values near to 1 which is indicate that ENSO cycle data behave heavy tail increase

as compared to sunspots time series data. Figure 3.8 (a and b) self-similar and self-affine tail

parameter of Sunspot Cycles respectively. Similarly, figure 3.9 (a and b) self-similar and self-

affine tail parameter of ENSO Cycles respectively.

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Figure 3.8 (a): Self-similar Tail Parameter of Sunspot Cycles

Figure 3.8 (b): Self-Affine Tail Parameter of Sunspot Cycles

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Figure 3.9 (a): Self-similar Tail Parameter of ENSO Cycles

Figure 3.9 (b): Self-Affine Tail Parameter of ENSO Cycles

3.8: SOLAR ACTIVITY CYCLES AND ENSO ACTIVITY CYCLE IN THE

PERSPECTIVE OF LONG RANGE CORRELATION:

In the statistical modeling of time series, considered two types of correlation. One is a short range

correlation (persistency) (Priestlay 1981, Box et al. 1999) and another is a long range correlation

(persistency) (Beran 1994, Taqqu and Samorodintsky 1992). Short range correlation (persistency)

is considered by a decay in the autocorrelation function that is bounded by a decreasing of an

exponential for large lags, whereas, long range correlation (persistency) in time series (explicit

subclass often referred to as 1/f noise or functional noise) in which any given value is related by

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all pervious values of the time series. Long range correlation (persistency) is considered by a

decreasing of the Power Law (asymptotic and exact) of all the persistency between values as a

function temporal distance (or lags) between them. The long range correlation can be defined as,

maximum values are correlated to each other at strong, long lags in time (Beran 1994, Taqqu and

Samorodintsky 1992). In the variation of spatially and temporally aggregated variable can be

recorded by the long range persistence (Beran 1994)

Long-range persistence technique follows a Power-Law scaling of the autocorrelation function

[C (ζ) = 1

𝜎𝑥2(𝑁−𝛾)

∑ (𝑥𝑘 − �̅�)(𝑥𝑘+ζ − �̅�)𝑁𝑘=1 ]

Such that

|𝐶(ζ)|~ ζ−(1−𝛾) , ζ→ ∞ , -1 < 𝛾 < 1 (3.16)

Holds for large time lags ζ. Where 𝛾 is known as the strength of long-range persistency. If 𝛾 = 0

depict no long-range persistence between values. If 𝛾 > 0 long-range persistent and 𝛾 < 0 long-

range anti-persistent. In the study, we intents to calculate the strength of long-range persistence

with the help of self-similar and self-affine Hurst exponent. Hurst exponent self-similar (HES)

which is calculated by Hausdorff – Besicovich method (Box counting Technique) and self-affine

(HEA) obtained by rescaled range analysis technique. The self-affine (HEA) is correlated to the

strength of long-range correlation 𝛾 as

𝛾𝐴 = 2HEA – 1 -1 < 𝛾𝐴< 1 (3.17)

(Malamud and Turcotte, 1999). The self-similar (HES) is associated with the strength of long-range

correlation 𝛾 as

𝛾𝑆= 2HES + 1 1 < 𝛾𝑆 < 3 (3.18)

(Burrough 1981 and 1983, Mark and Aronson 1984) . If 𝛾𝑆= 2 depict no long-range persistence

between values. If 𝛾𝑆 > 2 long-range persistent and 𝛾𝑆< 2 long-range anti-persistent.

The novelty of this study to analysis the long range correlation of sunspot cycles (1-24) in the

perspective of self-similar and self-affine Hurst exponent and also compare both of them. The table

3.7 depicts the strength of long term persistence of each solar activity (sunspot cycles).

The table 3.7 explores the analysis of the strength of long-range correlation of the sunspot cycles.

The self-similar strength of long-range correlation (𝛾𝑆) range from 1 to 3, whereas, the self-affine

strength of long-range correlation (𝛾𝐴) range from -1 to 1. The value of 𝛾𝑆 > 2 shows that long-

range persistent whereas, 𝛾𝑆 < 2 indicate long-range anti-persistent. On the contrary, the value of

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𝛾𝐴 > 0 shows that long-range persistent whereas, 𝛾𝐴 < 0 indicate long-range anti-persistent. The

table 3.7 shows that each sunspot cycle has 𝛾𝑆 > 2 which are indicate that every sunspot cycle is

strongly correlated to previous one. Similarly, each sunspot cycle has 𝛾𝐴 > 0 which are indicate

that every sunspot cycle is strongly correlated to previous one, despite of sunspot cycle 3 which is

show that the value of 𝛾𝐴 = 0. In this case, 𝛾𝐴 = 0 shows that no long-range persistence between

values. The conclude of this study that self-similar 𝛾𝑆 has strong strength long-range correlation

as compared to self-affine 𝛾𝐴.

In this study, we also analyze that the strength of long-range correlation of ENSO cycles (1-23).

The table 3.8 explores the correlation of each ENSO cycle in the perspective of self-similar 𝛾𝑆 and

self-affine 𝛾𝐴.

The table 3.8 depicts the analysis of the strength of long-range correlation of the ENSO cycles.

The self-similar strength of long-range correlation (𝛾𝑆) range from 1 to 3, whereas, the self-affine

strength of long-range correlation (𝛾𝐴) range from -1 to 1. The value of 𝛾𝑆 > 2 shows that long-

range persistent whereas, 𝛾𝑆 < 2 indicate long-range anti-persistent. On the contrary, the value of

𝛾𝐴 > 0 shows that long-range persistent whereas, 𝛾𝐴 < 0 indicate long-range anti-persistent. The

table 3.8 shows that each ENSO cycle has 𝛾𝑆 > 2 which are indicate that every ENSO cycle is

strongly correlated to previous one. Similarly, each ENSO cycle has 𝛾𝐴 > 0 which are indicate that

every ENSO cycle is strongly correlated to previous one. The conclusion of this study that self-

similar 𝛾𝑆 and self-affine 𝛾𝐴 both have long-range correlation persistent behavior which is

representing that the value of each cycle is correlated to preceding one. Figure 3.10 (a and b) self-

similar and self-affine Strength of long range-correlation of Sunspot Cycles respectively.

Similarly, figure 3.11 (a and b) self-similar and self-affine Strength of long range-correlation of

ENSO Cycles respectively.

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Figure 3.10 (a): Self-similar Strength of long range-correlation of Sunspot Cycles

Figure 3.10 (b): Self-Affine Strength of long range-correlation of Sunspot Cycles

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Figure 3.11 (a): Self-similar Strength of long range-correlation of ENSO Cycles

Figure 3.11 (b): Self-Affine Strength of long range-correlation of ENSO Cycles

3.9: UNITE ROOT STOCHASTIC PROCESS OF TAIL PARAMETER AND LONG

RANGE CORRELATION OF SUNSPOT AND ENSO CYCLES

The random walk model

Yt = ρYt-1 + ut −1 ≤ ρ ≤ 1 (3.19)

In this model Yt belongs to the time series and ut represent the white noise error while ρ is the

coefficient. The equation (6) is used to regress the Yt on the lagged value Yt-1estimate for unit root

test. If the predictable value ρ identical to 1 statistically. The series comprises unit root and null

hypothesis rejected (McCarthy, 2015). Transform the equation 6 by subtraction Yt-1 both sides

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Yt−Yt -1 = ρYt -1− Yt -1 + ut

Yt−Yt -1 = Yt -1(ρ− 1) + ut

∆ Yt = ɸYt -1 + ut (3.20)

Where ∆ and ɸ represent the operation of the first difference. The null hypothesis (ɸ = 0) for time

series non stationary condition represents the unit root (da silve Lopes, 2006).

The Augmented Dickey Fuller (ADF) test to solve the autocorrelation problem. This test is

implemented by augmenting three following equations, the dependent variable’s (∆Yt) adding

lagged values and Yt represents a random walk with

∆ Yt = 𝛾1+ 𝛾2t + ɸ Yt -1 + ut (3.21)

Where 𝛾2t and 𝛾1represents the intercept and trend respectively.

∆ Yt = ɸYt -1 + ut (3.22)

It shows neither intercept nor trend.

∆ Yt = 𝛾1+ ɸ Yt -1 + ut (3.23)

It shows intercept (𝛾1) only (Kwiatkowski et al., 1992; McCarthy, 2015).

We assume the null hypothesis H0: variables have a unit root (non-stationary data). The condition

where the H0 is rejected if p-value is less than 5% or the critical value of the absolute value of

Augmented Dickey Fuller (ADF) test is greater than at 1% and 5% significance level.

The Augmented Dickey Fuller (ADF) test for stationary explored that the tail parameter and the

strength of long range-correlation of self-similar and self-affine sunspot (1755 to 2008) and ENSO

cycles (1866 to 2012) time series data. In case the data is not stationary than behave like a unit

root. Table 3 (a), 3 (b), 4 (a) and 4 (b) depicted that the tail parameter of self-similar (βS) and self-

affine (βA) of sunspots and ENSO cycles has stationary nature and unit root test rejected. Similarly,

table 5(a), 5(b), 6(a) and 6(b) analyzed that self-similar strength of long-range correlation (𝛾𝑆) and

self-affine (𝛾𝐴) of sunspot and ENSO cycles has stationary nature and unit root test rejected. The

reason to reject H0 is p-value is less than 5% or the critical value of the absolute value of

Augmented Dickey Fuller (ADF) test is greater than at 1% and 5% significance level. These

conditions are satisfied in both cases.

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3.10: CONCLUSION

Different methods have been used to develop the certainty of significant relations among the

Sunspot cycles and some of the terrestrial climate parameters such as temperature, rainfall and

ENSO etc. This study explores the dependence of ENSO cycles on Mean Monthly Sunspots

Cycles. Sunspot cycles range from 1755 to 2008 whereas, ENSO cycles range from 1866 to 2012.

To find the above mentioned dependence probability distribution approach is utilized. In this

regards the appropriateness of distributions is investigated with the help of Kolmogorov–Smirnov

D, Anderson-Darling and Chi-square tests. It is found that most of the sunspot cycle follows

Generalized Pareto Distribution (GPD) whereas, Generalized Extreme Value Distribution (GEV)

were found appropriate for ENSO cycles. This study confirmed that during the period 1980-2000

ENSO cycles were very active. Simultaneously, El Nino was active for the periods 1982-83, 1986-

87, 1991-1993, 1994-95, and 1997-98 these periods include two strongest periods of the century

viz. 1982-83 and 1997-98. Two consecutive periods 1991-1993 and 1994-1995 were cold periods.

Sunspots cycles and ENSO cycles both were found to be persistent. This research is a part of a

larger research project investigating the correlation of Sunspot cycles and ENSO cycles and the

influence of ENSO cycles on variations of the local climatic parameters which in term depend on

solar activity changes. In the time series data, sunspot and ENSO cycles have stressed the

Generalized Pareto Distribution (GPD) and Generalized Extreme Value Distribution (GEV).

These distributions have a heavy tail on the right side.

In the next section of this chapter described the analysis of heavy tail parameter for further analyze

the data behavior. All the solar cycles (1-24) has a stationary nature as the differencing parameter

(0 < d < 0.5) in both perspective self-similar (dS) and self-affine (dA) which represent that the

dynamic is more regular. The heavy tail parameter βS as well as βA exploring that asymptotically

equivalent to Pareto law which is showing that the strength of the dynamics is regular and periodic

for all the solar cycles. For each sunspot cycle heavy tails are profound. The heavy tail parameter

(β) and differencing parameter (d = HE-0.5) are obtained from the Hurst parameter (0.5 < HE < 1)

persistent. The heavy tail parameter (β) value towards 2 depicted that the strength of heavy tail

decreases. Similarly, every persistent data contains the heavy tail since for d > 0 the HT parameter

β > 1. Similarly, All the ENSO cycles (1-23) also has a stationary nature as the differencing

parameter (0 < d < 0.5) in self-similar (dS) and self-affine (dA) respectively. The heavy tail

parameters (βS and βA) of ENSO cycles are depicted asymptotically equivalent to Pareto law. The

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comparative study of solar cycle and the ENSO cycle conclude that the heavy tail parameter (βS)

of the ENSO cycles, values approximately 1 which are explored that ENSO cycle data behave

heavy tail increase as compared to sunspots time series data. This study concludes that ENSO

cycles have a more heavy tail as compare to sunspot cycles.

In the time series data, the tail parameter helps to analysis the persistency and long term

dependency. Statistical modeling based on two types of correlation short range correlation and

long range correlation. All sunspot cycles explored the strength of long-range correlation (𝛾). The

strength of self-similar long-range correlation (1 < 𝛾𝑆 < 3) and the self-affine strength of long-

range correlation (-1<𝛾𝐴 < 1) is persistent in the perspective of 0.5 < HES < 1 and 0.5 < HEA < 1.

The novelty of this study shows that every value of sunspot cycles is strongly correlated to

preceding ones in both manner self-similar as well as self-affine. Similarly, each ENSO cycle

shows that each value is strongly correlated to preceding ones in both manner self-similar (𝛾𝑆)

and self-affine (𝛾𝐴). In all aspects Self-similar techniques are more appropriate as compared to

self-affine.

The unit root test is used for non-stationary data. H0 is rejected when the p-value is less than 5%

or the critical value of the absolute value of Augmented Dickey Fuller (ADF) test is greater than

at 1% and 5% significance level. This study analyzed that the heavy tail parameter (βS and βA) of

Sunspot and ENSO cycles are stationary. Similarly long-range correlation (𝛾𝑆 𝑎𝑛𝑑 𝛾𝐴) of sunspot

and ENSO cycles are also stationary in time series data.

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CHAPTER 4

A STUDY OF ACTIVE REGIONS OF ROTATING SUNSPOTS: IMAGE

PROCESSING APPROACH

This chapter demonstrates the image processing of active regions of solar cycles. Image analysis

is a widely used in image processing technique to analysis the image features such as smoothness,

texture, roughness, area, solidity and persistency. The novelty of this chapter intends to enhance

the study of solar active region by using fractal dimensions, mathematical morphology and Image

segmentation of rotating sunspots using Genetic active contour.

24 solar cycle had been starting from January 08, 2008 with the observation of high latitude of

sunspot pair. Although 24 cycle is considered to be a weak cycle because of below average

intensity. It has maximum 90 sunspots number, but it has some large sunspots. Solar maximum

exists in May 2014.There are no sunspots in the duration of 2008 and 2009. The situation is very

unusual which were not happening for almost a century. In 24 solar cycle, northern hemisphere

has following sunspots of active regions with negative polarity field while southern hemisphere

has leading sunspots with positive polarity. In the 24 cycle, AR12192 of October 18, 2014 hosts

the largest sunspot. According to NASA scientist C. Alex Young and Dean Pasnell demonstrate

that AR12192 has ranked 33rd largest of 32908 active region since 1874. Sunspot AR12192 as

same size as Jupiter. Figure 4.1 depicted that the active region AR 12192.

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Figure 4.1: Active region 12192 AR

This study intends to enhance the study of image analysis and mathematical morphology of the

sunspots. The morphology of sunspot groups not only is a predictor of their own future evolution,

but also of the other associated explosive events like solar flares and coronal mass ejections

occurring higher in the solar atmosphere (Stenning D et al 2010). Further, fractal dimensions, the

space filling properly of the images will be calculated to study the images of sunspots and sunspot

groups. Fractals are developed by Mandelbort (1983). Fractal techniques have been used to analyze

the complexity and quantification of irregular shapes of the geometric structure of the active

region. The fractal dimension is an ideal tool for measuring texture, roughness and persistency.

Images define in two dimension function (x, y). Whereas x and y are spatial coordinates. Fractal

features is a new approach to describe segmentation of the texture gray-scale images (kaspair et.

al 2001).

The theory of mathematical morphology consists of functional and morphological transformation.

Most of the operations of mathematical morphology (MM) are reversible and linear which are

directly related to the shapes. The theory was introduced with binary images by Matheron (1975)

and Serra (1982). Mathematical morphology (MM) is the theory and technique for the analysis and

processing of geometrical structures and for describing shapes using set theory, lattice theory,

topology and random functions.

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Rotating sunspots are usually observed in active region, which is correlated with some solar

eruption events like High flare activity and coronal mass ejection (CME). Observations indicate

that some sunspots rotate about their own vertical axis. Rotating sunspots has both direction

northern hemisphere (shows anticlockwise) and southern hemisphere (shows clockwise). The

magnetic helicity injected due to the rotational motion of sunspots. The magnetic helicity

associated with hemisphere pattern. The positive and negative sign of magnetic helicity represents

that solar magnetic field in the northern and southern respectively hemisphere.


The present study investigates the image processing of solar active region by using fractal

dimensions and mathematical morphology (MM). Fractal dimension is an essential parameter of

fractal geometry that define the significant application in many fields including image processing.

The novelty of this study to compare both fractal dimensions methods Hausdorff - Besicovich box

dimension and correlation dimension belong to self-similar family. The largest active region

AR12192 of solar cycle 24 are used to evaluate fractal dimensions and Mathematical morphology

(MM). Moreover, this study is an attempt to provide a new way of contouring the rotating sunspots

by means of a genetic algorithm which is based on the actual observation. The proposed scheme

is supported by Temperature, Energy, magnetic Helicity and Tangential velocity, and the features

of the sun, which are essential in the formation of sunspots. In this study the active region AR9114

and AR10696 as a case study to analysis the active contour of rotating sunspot by using image

segmentation. For this purpose, we analyze 4 active contours of each active region with variations

of temperature with constant other parameters. MATLAB 2016 is used to calculate transformation

of mathematical morphology. FRACTALYSE 2.4 software is used for the computation fractal

dimension and Hurst exponents and universal scaling parameters.

4.1.1: Fractal Dimensions

Self-similar fractal dimension is defined as the geometric object is composed into a union of

rescaled copies of itself with uniform in all directions or rescaling isotropic. Hausdorff -

Besicovich box dimension or simply box dimension and correlation fractal dimension are belong

to the self-similar fractal dimension family. The plane of Active region has Euclidean dimension

d = 2. The smallest unit of the active region (AR) is considered as a point so the topological

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dimension of active region is considered to be dt = 0. So the fractal dimension of active region

(AR) range from dt = 0 to d = 2.

The Hausdorff - Besicovich box dimension or simply box counting dimension methods are widely

used to calculate the fractal dimensions. It can be calculated by taking a ratio between log changes

in object size and log change in measurement scale (Paul 2005) (FD = 𝑙𝑜𝑔𝑁𝑡

𝑙𝑜𝑔(1

𝑡) ).In the box counting

method, the smallest number of squares of size t is known as reference element which are covered

all black pixels contained in a counting window. Where t is the roughly counting of black pixels

which are existing in the window. Nt is the union of reference element or the size of counting

window. In the infinite time, the algorithm of the box counting method converges to the minimum.

Therefore, the results are only an approximation of the adequate and best coverage. This method

is the generalized form of grid method.

Correlation Dimension was developed by Grassberger and Procaccia (1983) are used to calculate

fractal dimension measurements of a reconstructed attractor. Correlation dimension is closely

related to Hausdorff - Besicovich box dimension. The correlation integral C (I) is defined as

C ( I ) = 2

(𝑛+1)(𝑛) ∑ ∑ ℵ(𝐼 − ‖𝑋(𝑖) − 𝑋(𝑗)‖)𝑛−𝑖

𝑗=𝑖𝑛𝑖=1 (4.1)

Where n = N – (En - 1)η is the embedded number points is a En – space. N represent the length of

the data series and I be the radius of the sphere centered object X (i) or its size of the box. Whereas,

ℵ(𝑥) is the Heaviside step function. The function is defined as

ℵ(𝑥) = 1 𝑥 ≥ 00 𝑥 < 0

(4.2)

If C (I) scale behave like C (I) = ID, where D represent that the correlation dimension of time series

data (x).

Fractal dimension D and Hurst exponent H have relation 𝐹D + HE = 2. Hurst exponents lie

between 0 and 1. If the value of H approaches to 1 it is shown that the time series data is quasi-

regular (persistent). If it approaches to zero represent that irregular (anti-persistent)

(salakhudinova, 1998; Hanslmeier, 1999). Brownian process shows if H = 0.5. The value ranges

between 0 and 0.5, reveal a random walk (anti-persistent) process. The values of H range from 0.5

to 1 shows persistent positively correlated whereas anti persistent negatively correlated. Universal

scaling laws are also analysis in this study. The scaling exponent is described as β = d – FD.

Whereas, β range forms 0 to 1and d > FD. The spectral exponent α = 5- 2FD or α = 2HE + 1. The

relation between the fractal dimension (FD) and spectral exponent (α) was introduced by (Higuchi,

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1988). α = 0 describes a white noise-like system shows that uncorrelated and the power spectrum

is independent of the frequency. α =1 is represented as flicker or 1/f noise system which express

as a moderate correlated. α = 2 is called Brownain noise-like system, which shows a strong

correlation. The relation between spatial autocorrelation coefficient (C∇) and HE based on fractal

brownain motion (fBm) can be given as follows C∇ = 22HE-1 – 1. Where C∇ represents the spatial

autocorrelation coefficient. Spatial autocorrelation is used to measure the correlation of spots with

itself through active region. It should be noted that the value of HE = ½ means that C∇ = 0 which

is considered to be the Brownian motion. If HE > ½ then C∇ > 0 representing that positive spatial

autocorrelation exists. HE < ½ implies that C∇ < 0 which indicates that the spatial autocorrelation

is negative.

4.1.2: Heavy Tail Analysis

The heavy tail parameter strongly based on the Hurst exponent (HE). In the persistent noise, the

heavy tail parameter β will be range from 0 to 2. Whereas, the long-range dependence parameter

d is valid in the range from 0 to 1- 1

𝛽. This condition is applicable if HE < 1. In the stationary time

series, the above condition is mandatory. For the finite variance case HE = d + 1

2 , whereas for

infinite variance case HE = d + 1

𝛽. The long-range dependence degree is given by (Feldman and

Taqqu, 1998).

HE = (3−𝛽 )

2 (4.3)

The heavy tail parameter (β) range depends on the differencing parameter (d). Case 1, if d > 0, then the

heavy tail parameter (β) will be greater than 1. The limitation of β for any noise is range from 0 to 2

(Gourieroux and Jasiaky, 1998). Case 2, if d < ½, then it represents the power series expansion for all the

sunspot cycles. Whereas, β < 2, then the tail is asymptotically equivalent to Pareto law (Sun et al., 2005).

For all AR 12192 β has ranged from 1 to 2. This study develops the heavy tail parameter (β) calculate by

using Hurst exponent in both method Hausdorff – Besicovich method or Box counting Technique (HEB)

and correlation dimension (HEC). dB and dC represented as a differencing parameter of Box counting

technique and correlation dimension respectively. βB is known as heavy tail parameter of box counting,

whereas βC is called as correlation dimension heavy tail parameter.

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4.2: FRACTAL DIMENSIONS, WAVE SPECTRUM AND HEAVY TAIL ANALYSIS OF

AR12192

The novelty of this research is to analyze the complexity and roughness of largest AR 12192 of 24

cycle by fractal dimensions. This calculation is based on scaling analysis and spectral analysis of

AR 12192. Fractal parameters are calculated using box counting (FDB) and correlation dimension

techniques (FDC) lying in a significant range.

Table 4.1: universal scaling parameters and heavy tail analysis of AR12192

Hausdorff - Besicovich box dimension

FDB HEB αB C∇B dB βB

1.336 0.664 2.328 0.2553 0.164 1.672

Correlation Dimension

FDC HEC αC C∇C dC βC

1.15 0.85 2.7 0.6245 0.35 1.30

Table 4.1 demonstrates that the values of FDB and FDC in AR12192 range from 1 to 1.5 which

indicate that AR is persistent, correlated and predictable. The value of FDB was found to be greater

than FDC in AR12192. Similarly, Hurst exponents HEB and HEC both values are from 0.5 to 1

which reveals the persistency. The relationship between HEB is greater than HEC. Since, the fractal

dimension lies between 1 and 2, the scaling exponent β ranges from 0 to 1. This range for β assures

it is significant. Table 1 describes the numerical relation between the spectral exponent (α) and

autocorrelation coefficient (C∇) which are calculated by using both the box counting method and

the correlation dimension. If 1< DF < 2 then the spectral exponent α has a range from 1 to 3. α =

0 is indicates a white noise-like system which describes uncorrelated behavior. α =1 is called

flicker which represents a moderately correlated behavior. α = 2 is called Brownain noise-like

system and shows strong correlation. Spectral exponent (αB is box counting method and αC is

correlation dimension) reveals that the cycles behave like Brownain noise. The autocorrelation

coefficient (C∇) describes the multiple-lag 1-dimension spatial autocorrelation of AR12192. The

autocorrelation coefficient (C∇) is calculated by the relation of Hurst exponent. The value of C∇

= 0 if HE = ½ which describes Brownian motion. If HE > ½ then C∇ > 0 indicates positive spatial

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autocorrelation and if HE < ½ then C∇ < 0 represents the negative spatial autocorrelation. The

autocorrelation coefficients (C𝐵∇ and CC∇) show the positive spatial autocorrelation in AR12192.

This study estimates the long-range dependence parameters dB and dC. The heavy tail parameter

(βB and βC) ranges are influenced by the differencing parameter (dB and dC). Table 4.1 shows the

fractional differencing parameter and heavy tails parameter of AR12192 in the perspective of Hurst

exponent (HE). Table4.1 depicts that the heavy tail parameter values within range from 1 to 2

which is shows that the dynamic is more regular. The differencing parameter value of AR 12192

is less than 0.5. The heavy tail parameter (β) is less than 2 indicate that asymptotically equivalent

to Pareto law (nature of heavy tail any side). The heavy tail parameter (β) shows that the strength

of the dynamics is regular and periodic for AR 12192. The heavy tail parameter (β) value towards

2 depicted that the strength of heavy tail decreases. The novelty of the study concludes that the

heavy tail parameter of βB is more long time persistent and correlated as compared to heavy tails

parameter βC. Figure 4.2 indicates a sketch map of the numerical relationship of box counting and

the correlation dimension of AR12192.

Figure 4.2: indicates a sketch map of the numerical relationship of AR12192

4.3: MATHEMATICAL MORPHOLOGY (MM)

Image processing deals with the manipulation and analysis of the available image information. In

image processing, mathematical morphology is used to investigate the interaction between an

image and a certain chosen structuring element using the basic operations of erosion and dilation.

Mathematical morphology stands somewhat apart from traditional linear image processing, since

the basic operations of morphology are non-linear in nature. It over all deals with image

enhancement algorithms for reconstructing the relative order, geometry, topology, patterns and

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dynamics. The three stages involved in the study of images are image processing, image analysis

and image understanding.

The theory of mathematical morphology consists of functional and morphological transformation.

Most of the operations of mathematical morphology (MM) are reversible and linear which are

directly related to the shapes. The theory was introduced with binary images by Matheron (1975)

and Serra (1982). Mathematical morphology (MM) is the theory and technique for the analysis and

processing of geometrical structures and for describing shapes using set theory, lattice theory,

topology and random functions. It is used to analyze the evolution of images. The main concept

of mathematical morphology (MM) which is copied by the following quotation from (J Serra 1982)

“The perception of the Geometric structure, or texture which is not purely

objective. It does not exist in the phenomenon itself, nor in the observer, but

somewhere in between the two. The mathematical morphology (MM)

quantifies this intuition by developed the idea of structuring elements.

Chosen by the morphology, they interact with the object under study.

Modifying its shape and reducing it to a sort of caricature which is more

expensive than the actual initial phenomenon….”

The transformation of morphological of an image (subset ℤn and Rn) is described as prescribed

manner intersection and union. Constitutes of translation vector is known as structuring element.

The geometric information regarding objects by sequential application of morphological

transformation involving to select structuring elements (the number of possibilities is unlimited).

The morphological transformation has irreversibility: less information provides transform image

as compared to real one. In the discrete case, the transformation works as a cellular automata or

cellular logic. The basic mathematical morphological tools are dilation, erosion, opening and

closing.

4.3.1: Dilation and Erosion, Closing and Opening

The basic mathematical morphological tools are dilation, erosion, opening and closing. In this

study, E be the Euclidean space Rn or discrete space ℤn. Whereas, E is a commutative group. ɸ (E)

is a space which is consists of all subsets of E. A subset S of E is the binary image. Moreover, S is

called the object and ɸ (E) is known as object space. Let S ⊂ E and t 𝜖 E the St is known as

translated by S along h: St = {s + t: s 𝜖 S}. If S, R ⊂ E then S hits R, S R. If S ∩ R ≠ ∅. The

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dilation of a set S by structuring element A is demonstrated by S ⊕ A = {t 𝜖 E: At S}. Whereas,

the erosion of S by A is described as S ⊖A = {t 𝜖 E: At ⊂ S}. Erosion as an impact of shrinking

bright region in the output image. So the eroded image is the subset of the input image. Dilation

has an expanding bright region in the output image. Another two important features, invariant

translation transformation ɸ (E) are the closing and the opening. The closing of a set S by

structuring element A is demonstrated by SA = (S⊕A) ⊖A. Whereas, the opening of S by A is

described as SA = (S ⊖A) ⊕A. The opening and closing has a relation with S such as SA ⊂ S ⊂

SA.

4.4: MATHEMATICAL MORPHOLOGY (MM) OF AR12192

The active region AR12192 shows the distribution complexity of structure with different levels of

intensity and no regular pattern. So, the other novelty to analyze the boundary edges of AR12192.

For this purpose the mathematical morphology basic principal such as erosion, dilation, closing

and opening are applying of AR12192. Figure 4.3 (a, b, c, d) demonstrated that mathematical

morphological operations such as erosion, dilation, closing and opening of AR12192. Figure 4.2

(a) express erosion of AR12192 which shows the subset of actual sunspot. Whereas, Figure 4.3 (b)

expresses dilation of AR12192 which shows the expand edge of actual sunspot. The two important

features such as opening and closing are depicted in figures 4.3 (c and d) respectively show that

the relation which are described that actual sunspot is the subset of closing. Whereas, opening

operation is the sunset of actual sunspot. The purpose of mathematical morphology is the edge

detection of irregular and complex distribution of objects.

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Figure 4. 3(a): Erosion of Active region AR12192 Figure 4.3(b): Dilation of Active region AR12192

Figure 4.3(c): Opening the Active region AR12192 Figure 4.3(d): Closing the active region AR12192

4.5: IMAGE SEGMENTATION OF ROTATING SUNSPOTS USING GENETIC ACTIVE

CONTOUR

Sunspots are associated with high active power regions, which increased magnetic flux on the sun

locally (Kevin et. Al, 2014). It has been observed by many authors that there is a rotational motion

of and around the sunspots and due to this feature it is easier to identify and observe the rotating

behavior sunspots. The magnetic polarity of sunspots being changed with each solar cycle

(approximation 11 years). The phenomenon of rotating sunspots was introduced by J. Evershed in

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1910. The direction of rotating sunspots, rotation in the northern hemisphere indicates anti-clock

wise whereas the rotation of southern hemisphere shows clock wise (Hale, 1927; Bao et. al 2002).

The magnetic helicity are usually associated with hemisphere pattern. The negative and positive

sign of magnetic helicity represents that solar magnetic field in the northern and southern

respectively hemisphere (Pevtsov, 2008). The magnetic helicity has important property is that

conversed in high magnetic Reynolds number plasma, which is also presence even in the deceptive

process (Berger et. al 1984). Magnetic helicity is exists in the solar convection zone, which is

exists is below to the photosphere. The rotating sunspots indicate that different angular and

tangential velocity with radius. The velocity of rotating is calculated by the twist and the

occurrence rate of the twisted flux-tube from lower than the photosphere (Brown et. al 2003). They

are considered to influence many terrestrial environmental and climatic phenomena (TJO News,

2006). Determination of the area of rotating sunspots is important because of their role in the study

of the evolution of rotating sunspots and their effect with solar irradiance (Barangi et. al 2001).

The rotational motion of sunspots is round, its own axis in the plane of the Photosphere. This

rotational motion of the sunspots, which are not usually of a regular circular shape, but frequently

elongated (Knoska., S. 1975).

Segmentation of images is considered as one of the most important and earliest stages of image

processing and serves as a vital role in quantitative and qualitative analysis of sunspots. This study

is intended to provide the image segmentation of rotating sunspots together with a useful technique

of genetic algorithm. The proposed methodology involves the active contouring which has been

one of the widely used methodologies for image segmentation due to the rapid rotational motion

of sunspots. The generation of sunspots depends upon the two components (1) the accumulated

magnetic Helicity (2) Homologous flares that occur in the immediate periphery of the rotating

sunspots. It has been observed by many authors that the evaluation of sunspots reflects the transport

of magnetic energy and the magnetic Helicity from the sub atmosphere to Corona. In case of

unipolar patches the proposed method considers the magnetic Helicity along with the rotating

sunspots. Magnetic Helicity can be viewed as a quantitative measure of global chiral properties of

magnetic field in the atmosphere. For the approximation of sunspots as unipolar patches the

magnetic Helicity injected by the rotational motion is.

𝑑𝐻

𝑑𝑡= −

1

2𝜋∅2(𝜔+ + 𝜔−) (4.4)

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Where 𝜔+and 𝜔−are the angular speed of the positive and negative polarities, respectively, and ∅

is the magnetic flux in equation (4.4).

We are considering only the magnetic Helicity injected by the rotational motion of sunspots and

due to their two homologous flares that occurred in the immediate periphery of the rotating

sunspots and showed an inverse S shape after the ribbon extension. Due to the rotational motion

of sunspots the accumulated magnetic Helicity can be obtained by the equation (4.5),

∆𝐻= −1

2𝜋𝜔 ∙ ∆𝑡 ∙ ∅2 (4.5)

Another associated parameter in the formation of sunspots is energy loss, due to which there is a

sudden change in the temperature of the active region which ranges from 4200 to approximately

4600. For the implementation of genetic algorithm the above mentioned parameters will be highly

appropriate. As described earlier, we have made our calculations on the bases of actual

observations in the active regions. As described by (Attila, 2010) there is heavy loss of energy due

to the magnetic flux and the tangential velocity of rotational sunspots. Since we are dealing with

the rotating sunspots, therefore the loss of energy will be considered as one of the most important

parameters of the rotating sunspots in this study. According to (Attila, 2010) the loss of energy is

ranges from (5.7 1.9) 1033 Ergs and during the active regions of sunspots this loss ranges from

(2.6 0.11) 1033 to (2.6 0.11) 1034ergs.

4.5.1: Description of Algorithm

Homogenous flares are occurring to indicate the periphery of the rotating sunspots. By considering

the rotational motion and calculating the magnetic Helicity are certainly meant that the magnetic

Helicity is injected due to the rotational motion of Sun spots. By considering the contouring point

of each circle we calculate the fitness values of overall contour points to remedy the problem of

local minima. The region is called active region. We can consider only the magnetic Helicity

injected by the rotating sun spots and due to this two homologous flares that occurred in the

indicate periphery of the rotating sun spots and showed an inverse S shape after the ribbon

extension.

The aim of our study is providing a versatile, intelligent tracking algorithm for the identification

and tracking of sunspots together with the motivation of study solar evaluation of sunspots that

make a valuable contribution towards the advance studies of the rotation of sunspots.

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4.5.2: Genetic Algorithm with Active Contours

Genetic Algorithm is a heuristic approach that has captured the attention of many of many

researches, although this algorithm does not provide the surety for optimal solution but with the

help of these algorithms. We may find the near optional solution which may close to optimal

solution in any case. By using the genetic algorithm we obtained the best, most optimal solution

among all the other existing solution. Genetic algorithm starts its operation from the initial

population also called the initial set of chromosomes. These chromosomes are encoded by either

a real number or binary number. Each chromosome is a set of real coded genes or binary coded

gene depends upon the type of the problem. Firstly, we decided the size of the population and the

nature chromosomes, which it is evaluated by using the fitness function further. The fitness

function is the most important factor of the genetic algorithm also known as the bone of GA. The

fitness function is based upon the factor that affect the initially generated chromosomes and the

fitness function will decided which chromosomes is better individual and which is not able to go

for the further produce. After the selection the better individuals and selected chromosomes will

go under the procedure of crossover which provided the most fitted child chromosomes as a better

individual. This process is further enhanced with mutation where the local optima converge to the

global optima. The stopping criteria of genetic algorithm will decide when this procedure will stop

producing further initial population according to the pre decided rule. Figure 4.4 and 4.5 show the

complete picture of genetic algorithm.

Figure 4.4: show that the Active contour of rotating Sunspots

4.5.3: Initial Population

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For the initialization process, here we are applying the active contouring of rotational sun spots as

given in (Kevin et. al, 2014) with a little modification. This algorithm will apply in several steps.

The process will start under these assumptions that the rotation spot is confined within the two

circles, having the minimum radius and maximum radius. The maximum circle with a maximum

radius will cover the complete sunspot of any shape, therefore the region under consideration will

be studied as a concentric circle and the center of this concentric circle will be considered as a

center of the spot of any shape. Each of them will act like a contour as show in the figure 4.3. Each

of these contours contains points from the sun spots edge. Here we can try to separate these points

from each contour and connect them in such away so that we can able to reveal the Sun spots edge.

Initially we generate the population as an initial population to generate these circles with the

inclusion of two genes for every chromosome. The first gene indicates the radius of the circle and

the second gene will specify the radius of each point. After the formation of the contours we

indicate the number of equidistant points on every one. If N denotes the number of points on the

contour, the distance of every point on the contour, the distance of every point from the adjacent

one would be 360/N. The above procedure will help in determining the contour points and now

the next step is to evaluate these contour points for the further selection.

4.5.4: Fitness Function

After the formation of the entire contour and the determination of contour points, each of these

contour points are going to be analysis for the selection of the best one for the further procedure.

As described earlier the evaluation must be done through the fitness function:

Due to the rotational motion of sunspots the Helicity is induced and due to this the internal energy

is released of that portion which directly affect the temperature as well. For calculating the fitness

function we are considering following important features of sunspots:

∅ = Tangential velocity or angular velocity of the selected contour point.

H= Helicity of the initial chromosomes

T= Temperature of the selected contour point

E= energy losses during the time period

Emax = Maximum energy losses during the time period

∅𝑚𝑎𝑥 = Maximum Tangential velocity or angular velocity during the time period

Hmax = Maximum Helicity of the initial chromosomes during the time period

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The Average of temperature will be considered in the calculation since due to the rotational motion

the size the sunspot may decrease or increase that the temperature may not be constant. Therefore

the average of temperature will be considered for the calculation in each contour.

Tavr = 𝑡1+ 𝑡2+𝑡3+𝑡4

4 (4.6)

The idea is to calculate the fitness value of each point one by one from each contour. We calculate

the energy for selected point and selection for next procedure will be done by using the predefined

criteria. After the selection of points from each contour, further procedure of crossover will be

applied.

The above features are mainly involved in any rotated sunspots therefore, our fitness function will

definitely base on these parameters. According to the above features the suggested fitness function

is defined as,

𝐹𝑣𝑎𝑙𝑢𝑒𝑠 = 𝐻∅

𝐸𝑇𝐻𝑚𝑎𝑥∅𝑚𝑎𝑥𝐸𝑚𝑎𝑥𝑇𝑎𝑣𝑔

(4.7)

If the above value is nearer to 0, then that individual is supposed to be more fitted chromosomes

and will be able to go for the further procedure of crossover. After calculating the overall fitness

function from an equation (4.7), we select the contour point that possesses best value according

to our selection criteria as the best solution.

4.5.5: Crossover

At this step the selected contour points from each circle are going to place into the reduction pool

and the members of this pool are paired up and reproduce themselves in order to have best child

chromosomes. We call this procedure, a crossover operation which creates better individual from

the selected parents with the help of the fitness function.

To implement the crossover operation, we will generate the string of each selected contour point

belongs to specific circle which we initially generated in the first step of genetic algorithm. In this

way we consider two selected strings at a time to produce two fitted child chromosomes. Here we

use the two point crossover procedure to generate our children and after that we apply the process

of mutation. Figure 4.5 shows the procedure of crossover for the proposed algorithm.

4.5.6: Mutation

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In order to prevent from the convergence and to have better generation, we apply the process of

mutation. This will help us to obtain the global optimal from the local optima. For the mutation

procedure, we have to set some stopping criteria, usually the process of mutation will automatically

stop when there is no further improvement is possible. In this algorithm the mutations are done by

simply changing the bit of a string and replace it by the bit of the second string. The resultant

string will be further checked by the fitness function until no more improvement is possible.

Parent 1:

P11 P12 P13 P14 P15 P16 P17

Parent 2:

P21 P22 P23 P24 P25 P26 P27

Child 1:

P21 P12 P13 P24 P25 P26 P27

Child 2:

P21 P22 P13 P14 P15 P16 P17

Figure 4.5: Showing the crossover procedure of two parenet choromosomes

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Figure 4.6: is showing the crossover operation

4.6: ACTIVE REGION AR9114 and AR10696

In this study, we consider the rotating sunspots of 23 cycle of four active region AR9114 and

AR10696 as a case study. In 23 cycle, identified 186 rotating sunspots in 153 active regions

observed by SOHO / Mirchclson Dophler image (MDI) and TRACE. The active region AR9114

and AR10696 were injected negative total helicity and located in the northern hemisphere.

AR9114 is located rotation in the northern hemisphere. The leading polarity has a strong magnetic

field. Whereas, the following negative polarity has the appearance of discontinuous. The maximum

tangential velocity of AR9114 is 4O /h (0.13 km /s). The negative helicity dH-/dt is about -5 1040

Mx2 / h. Whereas, the positive helicity dH+ /dt is about -2.4 1041 Mx2 / h. The total negative

helicity injection dH-/dt is about -3.4 1043 Mx2 / h from the photosphere to the upper solar

atmosphere and the total positive helicity injection dH+/dt is approximately -7 1043 Mx2 / h in

this duration when the leading rotating sunspot has maximum tangential velocity. AR10696 is a

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strong, active region (AR) which consist of rotating sunspots. The maximum tangential velocity

of AR10696 is 3O /h (0.098 km /s). The negative helicity dH-/dt is greater than -8 1041 Mx2 / h.

Whereas, the positive helicity dH+ /dt is less than -2 1041 Mx2 / h. The total result is negative

helicity injection dH-/dt is about -6.3 1043 Mx2 / h and total result is dH+/dt is approximately -

5.5 1043 Mx2 / h. The novelty of this study for analyzing the edge detection of rotating sunspot

with the same energy released and temperature of each contour. In this study, the random values

of helicity and tangential velocity are taken to leading and following sunspot which is near to

average values. The temperature and energy released in each active region are considering to be

constant. This study examines the first contour of each active region of rotating sunspot.

In this study, four contours are analyzed with variations of temperature and the rest of parameter

is constant. The temperature of rotating sunspot of very contour to contour which contour near to

the center has minimum temperate as compare to others. Genetic algorithms selects that sufficient

value which is nearer to zero, as per the predefined rules, different points will be selected from

each circle. The crossover will further filtrate the circles, points and mutation will provide the

global optima. The calculations are based on most of the actual observations as given in (Zhu ),

the considered energy loss is supposed to be (2.6 0.11) 1033engrin in our calculations and

temperature effect will be from 4200K to 4600K. The active region AR9114 is located in the

northern hemisphere, with strong magnetic field. AR9114 considering the values of parameters

with negative helicity as follows,

-Hmax = -3.4x1043, ∅𝑚𝑎𝑥= 4, Emax = 0.24x1033 and Tavg = 4350 so −𝐻𝑚𝑎𝑥∅𝑚𝑎𝑥

𝐸𝑚𝑎𝑥𝑇𝑎𝑣𝑔 = -1.30 x 1012

Similarly, AR9114 with positive helicity the maximum values are

+Hmax = -7 x1043, ∅𝑚𝑎𝑥= 4, Emax = 0.24x1033 and Tavg = 4350 so +𝐻𝑚𝑎𝑥∅𝑚𝑎𝑥

𝐸𝑚𝑎𝑥𝑇𝑎𝑣𝑔 = -2.68 x 1012.

The table 4.2 (a, b, c and d) depicted that according to fitness criteria each value of AR9114 with

negative helicity is nearly to zero which shows that each point exactly locate in the contour.

Whereas, the values are far from 0 shows that the observation does not exist in the active contour.

The table 4.3 (a, b, c and d) describes that the behavior of AR9114 with positive helicity. Every

value is fulfilling the fitness criteria has value less than 1. The first observation of each contour

exactly locates in contour with the closest value approaches to zero. This study concludes that the

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image segmentation using genetic active contour of rotating AR9114 with negative helicity is more

appropriate than the results with positive helicity.

Similarly, the active region AR10696 is located in northern hemisphere, with strong magnetic

field. The maximum parameter values of AR10696 with negative helicity describe as,

-Hmax = -6.3x1043, ∅𝑚𝑎𝑥= 3, Emax = 0.24x1033 and Tavg = 4350 so −𝐻𝑚𝑎𝑥∅𝑚𝑎𝑥

𝐸𝑚𝑎𝑥𝑇𝑎𝑣𝑔 = -1.80 x 1012

And AR10696 with positive helicity the maximum values are

+Hmax = -5.5x1043, ∅𝑚𝑎𝑥= 3, Emax = 0.24x1033 and Tavg = 4350 so +𝐻𝑚𝑎𝑥∅𝑚𝑎𝑥

𝐸𝑚𝑎𝑥𝑇𝑎𝑣𝑔 = -1.58 x 1012

The table 4.4 (a, b, c and d) depicted that according to fitness criteria each value of AR10696 with

negative helicity is nearly to zero which shows that each point exactly locate in the active contour.

The table 4. (a, b, c and d) describes that the behavior of AR10696 with positive helicity and

maximum tangential velocity. Every value is fulfilling the fitness criteria has a value less than 1.

First observation of each contour exactly locates in contour with closest value approaches to zero.

Image segmentation using genetic active contour of rotating AR10696 with negative helicity is

found to be more suitable than the results with positive helicity. From the observation of the values

related to sunspots, we may conclude that the values which are greater than one is outside the

contour and contour point with value one will exists on the edge of sunspots. In this way a new

contour will be generated from the above calculation as shown in figure 4.7. By taking the above

mentioned values, promising results came out from these observations.

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Figure 4.7: The steps of Active contours of AR9114

4.7: CONCULSION

This chapter reveals the persistency of AR12192 by using self-similar fractal dimensions involving

Hurst exponent. The fractal dimension (FD) represents the complexity and roughness of the active

region AR12192, whereas Hurst exponent (HE) provides the smoothness of the active region. This

study investigates the relationship between self-similar fractal dimensions FDB and FDC. FDB is

calculated by the Box counting method and FDC is calculated by using the Correlation Dimension

method. The Hurst exponents are calculated by FD which is calculated by both techniques. The

spectral exponent and the autocorrelation coefficient are calculated by both techniques. The active

region AR12192 has a stationarity nature as the differencing parameter follows the inequality 0 <

d < 0.5 in both perspective Box counting (dB) and correlation dimension (dC). This in turn shows

that underlying dynamic is more regular. The heavy tail parameter (β) is less than 2 which confirms

the equivalence of the asymptotic nature of heavy tail (one sided) and the Pareto law which

confirms that the underlying dynamics is strong and regular. For the he active region AR12192

heavy tails are profound. The heavy tail parameter (β) and differencing parameter (d = HE-0.5)

are obtained from the Hurst parameter (0.5 < HE < 1) showing persistency. The heavy tail

parameter (β) tending towards 2 depicts that strength of heavy tail is decreasing. If the data is

persistent, then heavy tails exist. This implies that d > 0 and HT parameter β > 1 heavy tails exist.

Table 1: demonstrates the numerical relationship between Hurst exponents, spectral exponents and

autocorrelation coefficients and differencing parameters and the outcome of the heavy tail analysis

of AR12192. The active region AR12192 is found to be persistent, correlated and heavy tailed.

The spectral exponent (αB and αC) of AR12192 behaves like Brownain noise which indicates the

long term dependency. The autocorrelation coefficient is found to be significant using both the

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fractal dimensions FDB and FDC. Mathematical morphological operations such as erosion, dilation,

closing and opening are also analysis for AR12192.

The proposed study has provided image segmentation of rotating sunspots with the help of genetic

algorithm. We have used the criteria of selection of best contour points from the proposed fitness

criteria and then improvements have been made by the process of crossover and mutation which

are the most important features of Genetic algorithm (GA). Although GA is an approximated

approach, but it may provide very accurate results by making a suitable selection of crossover and

mutation. In the proposed algorithm we have applied crossover in a different way between the

contour points of each cycle the proposed idea is unique and producing promising results. AR9114

and AR10696 of solar cycles 23rd are used as a case study to calculate the image segmentation of

active contour.

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CHAPTER 5

CONCLUSIONS AND FUTURE WORK

This chapter comprises the results obtained from this dissertation. According to the chapter

sequential, principal results are mentioned. Suggestions and future work are exhibited at the end

of this chapter.

5.1: SUMMARY AND OUTLOOK

Sunspot cycles have influenced on earth climates. In the time series, sunspot cycles in second

difference have stationary nature. Autocorrelation (AC), Partial Autocorrelation (PAC) and Ljung-

Box Q-statistics test are used for checking white noise in solar cycles. Moreover, the unit root test

with Augmented Dickey Fuller (ADF) test has applied for verification of stationary. For a selection

of appropriate models, diagnostic checking is used. For checking normality of sunspot cycles test

of normality are used. Test of normality based on skewness, kurtosis and Jurque-Bera test. This

chapter is utilizing the stochastic autoregressive and moving average (ARMA) modeling, AR-

GARCH (1, 1) process and ARMA-GARCH (1, 1) models and forecast evolution of sunspot

cycles. Least Square Estimation is used for ARMA process. Various best fitted ARMA models

estimate and forecast for each sunspot cycle. Least square method is used to calculate ARAM

models and quasi maximum likelihood estimation (QMLE) are used to calculate AR-GARCH and

ARMA-GARCH models. The selection of ARMA, AR-GARCH and ARMA-GARCH models are

focused on smallest value of Durbin-Watson statistics test. Durbin-Watson (DW) statistics test

value of each sunspot cycle is less than 2 which shows that sunspot observations are correlated to

each other. GARCH (1, 1) stationary volatility model has the best forecasting model as compared

with other models. Diagnostic checking is used to identify and estimation of most appropriate

models and confirmation is found by forecasting evolution. The Gaussian quasi maximum

likelihood estimation (QMLE) is used to calculate AR-GARCH and ARMA-GARCH models.

ARCH effect is found by checking Lagrange Multiplier test, correlogram squared residuals and

test of normality. Forecasting evolutions are verified by Root Mean Squared Error (RMSE), Mean

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Absolute Error (MAE), Mean Absolute Percentage Error (MAPE) and Theil’s U-Statistics test (U

test). AIC, SIC and HQC, maximum log likelihood estimation are also calculated. GARCH (1, 1)

is leptokurtic in both perspective AR (p) and ARMA (p, q) for sunspot cycles. The adequate

ARMA, AR-GARCH and ARMA-GARCH modeling for fractional Brownian motion of sunspot

cycles will be useful to predict the dynamical variables for 24th cycle and next coming cycles in

the future. The comparison of forecasting evolution of ARAM, AR-GARCH and ARMA-GARCH

process are also discussed. RMSE, MAE and U test are described that ARMA model is an

appropriate model for sunspot cycles. MAPE exhibits that ARAM-GARCH model is the most

appropriate model of sunspot cycles.

The persistency analysis of solar activities and ENSO cycles by using fractal dimension and Hurst

exponent. The fractal dimension (FD) represents the roughness and complexity of the time series

data of solar activities (sunspot activity) and ENSO cycles. Whereas Hurst exponent (HE) provides

the smoothness of the data. This study investigates the relationship between self-similar fractal

dimension (FDS) and self-affine fractal dimension (FDA). FDS calculated by Box counting method

and self-affine fractal dimension (FDA) is calculated by using rescaled range method. The Hurst

exponents also calculated by FD in both methods, self-similar fractal dimension (FDS) and self-

affine fractal dimension (FDA). The spectral exponent is calculated by using equation (2.13) and

the autocorrelation coefficient to calculate by using equation (2.16). The table 2.1: indicate the

numerical relationship between Hurst exponents, spectral exponent and autocorrelation coefficient

of both fractal dimensions self-similar and self-affine. In the sunspots cycles indicate that each

cycle is persistent and correlated, the values of Hurst exponent lies from 0.5 to 1 but HES values

are greater than HEA. All values of the spectral exponent (αs and αA) of each sunspot cycles behave

like Brownain noise which indicates the long term dependency. The autocorrelation coefficient is

also lies in a significant range of both fractal dimensions. The relation of self-similar and self-

affine fractal dimensions which is described in equation (2.15) in failed in each sunspot cycles

because this relation is valid if fractal dimension is lies from 1.5 to 2 but each sunspots cycles

fractal dimension lies from less than 1.5. The spectral exponent value which is calculated by

Higuchi’s Fractal dimension (FDH) also identify the strong correlation among Sunspots Cycles.

We analyzed the complexity of each cycle of ENSO data along total cycles and active ENSO

period, then compared them by estimating self-similar fractal dimension (FDS) by using Box

counting method and self-affine fractal dimension (FDA) by rescaled range method, and Hurst

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exponents also calculated through FD which is show that self-similar fractal dimension (FDS) for

ENSO cycles and Total data are less than those for self-affine fractal dimension (FDA). Similarly,

self-similar fractal dimension (FDS) of ENSO cycles is also less than self-similar fractal dimension

(FDs) of Sunspot cycles. This means that ENSO cycles are more persistent than Sunspot cycles

data. Sunspots cycles and ENSO are correlated to each other. It is also verifying the fact if FD

increases, then H decreases. The cycle will prolong sunspot activity has greater means and the tail

prolongs. In the end a relation between probability distribution and Fractal Dimension establishes

the persistency approach for both of the data sets. The mean- tail assessment confirm the FD-HE

analysis. This study can be useful for further investigation of the impact of Sunspot and ENSO

related local climatic variability.

Different methods have been used to develop the certainty of significant relations among the

Sunspot cycles and some of the terrestrial climate parameters such as temperature, rainfall and

ENSO etc. This study explores the dependence of ENSO cycles on Mean Monthly Sunspots

Cycles. Sunspot cycles range from 1755 to 2008 whereas, ENSO cycles range from 1866 to 2012.

To find the above mentioned dependence probability distribution approach is utilized. In this

regards the appropriateness of distributions is investigated with the help of Kolmogorov–Smirnov

D, Anderson-Darling and Chi-square tests. It is found that most of the sunspot cycle follows

Generalized Pareto Distribution (GPD) whereas, Generalized Extreme Value Distribution (GEV)

were found appropriate for ENSO cycles. This study confirmed that during the period 1980-2000

ENSO cycles were very active. Simultaneously, El Nino was active for the periods 1982-83, 1986-

87, 1991-1993, 1994-95, and 1997-98 these periods include two strongest periods of the century

viz. 1982-83 and 1997-98. Two consecutive periods 1991-1993 and 1994-1995 were cold periods.

Sunspots cycles and ENSO cycles both were found to be persistent. This research is a part of a

larger research project investigating the correlation of Sunspot cycles and ENSO cycles and the

influence of ENSO cycles on variations of the local climatic parameters which in term depend on

solar activity changes. In the time series data, sunspots and ENSO cycles have stressed the

Generalized Pareto Distribution (GPD) and Generalized Extreme Value Distribution (GEV).

These distributions have a heavy tail on the right side.

In the next section of this chapter described the analysis of heavy tail parameter for further analyze

the data behavior. All the solar cycles (1-24) has a stationary nature as the differencing parameter

(0 < d < 0.5) in both perspective self-similar (dS) and self-affine (dA) which represent that the

167

dynamic is more regular. The heavy tail parameter βS as well as βA exploring that asymptotically

equivalent to Pareto law which is showing that the strength of the dynamics is regular and periodic

for all the solar cycles. For each sunspot cycle heavy tails are profound. The heavy tail parameter

(β) and differencing parameter (d = HE-0.5) are obtained from the Hurst parameter (0.5 < HE < 1)

persistent. The heavy tail parameter (β) value towards 2 depicted that the strength of heavy tail

decreases. Similarly, every persistent data contains the heavy tail since for d > 0 the HT parameter

β > 1. Similarly, All the ENSO cycles (1-23) also has a stationary nature as the differencing

parameter (0 < d < 0.5) in self-similar (dS) and self-affine (dA) respectively. The heavy tail

parameters ( βS and βA) of ENSO cycles are depicted asymptotically equivalent to Pareto law. The

comparative study of solar cycle and the ENSO cycle conclude that the heavy tail parameter (βS)

of the ENSO cycles, values approximately 1 which are explored that ENSO cycle data behave

heavy tail increase as compared to sunspots time series data. This study concludes that ENSO

cycles have more heavy tail as compare to sunspot cycles.

In the time series data, the tail parameter helps to analysis the persistency and long term

dependency. Statistical modeling based on two types of correlation short range correlation and

long range correlation. All sunspot cycles explored the strength of long-range correlation (𝛾). The

strength of self-similar long-range correlation (1 < 𝛾𝑆 < 3) and the self-affine strength of long-

range correlation (-1<𝛾𝐴 < 1) is persistent in the perspective of 0.5 < HES < 1 and 0.5 < HEA < 1.

The novelty of this study shows that every value of sunspot cycles is strongly correlated to

preceding ones in both manner self-similar as well as self-affine. Similarly, each ENSO cycle

shows that each value is strongly correlated to preceding ones in both manner self-similar (𝛾𝑆)

and self-affine (𝛾𝐴). In all aspects Self-similar technique is more appropriate as compared to self-

affine.

The unit root test is used for non-stationary data. H0 is rejected when p-value is less than 5% or

the critical value of absolute value of Augmented Dickey Fuller (ADF) test is greater than at 1%

and 5% significance level. This study analyzed that the heavy tail parameter (βS and βA) of Sunspot

and ENSO cycles are stationary. Similarly long-range correlation (𝛾𝑆 𝑎𝑛𝑑 𝛾𝐴) of Sunspot and

ENSO cycles are also stationary in time series data.

The persistency of AR12192 is determined by using self-similar fractal dimensions (Box counting

FDB and Correlation Dimension FDC) involving Hurst exponent. The fractal dimension (FD)

expresses the complexity and roughness of the active region AR12192, whereas Hurst exponent

168

(HE) provides the smoothness of the active region. The Hurst exponents are calculated by FD

which is calculated by both techniques. The spectral exponent and the autocorrelation coefficient

are also calculated by both techniques. The active region AR12192 has a stationarity nature as the

differencing parameter follows the inequality 0 < d < 0.5 in both perspective Box counting (dB)

and correlation dimension (dC). The heavy tail parameter (β) is less than 2 which confirms the

equivalence of the asymptotic nature of heavy tail (one sided) and the Pareto law which confirms

that the underlying dynamics is strong and regular. For the he active region AR12192 heavy tails

are profound. The heavy tail parameter (β) and differencing parameter (d = HE-0.5) are obtained

from the Hurst parameter (0.5 < HE < 1) showing persistency. The heavy tail parameter (β) tending

towards 2 depicts that strength of heavy tail is decreasing. If the image is persistent, then heavy

tails exist. This implies that d > 0 and heavy tail parameter β > 1 heavy tails exist. The active region

AR12192 is found to be persistent, correlated and heavy tailed. The spectral exponent (αB and αC)

of AR12192 behaves like Brownain noise which indicates the long term dependency. The

autocorrelation coefficient is found to be significant using both the fractal dimensions FDB and

FDC. Mathematical morphological operations such as erosion, dilation, closing and opening are

also analysis for AR12192.

The novelty of this study has delivered image segmentation of rotating sunspots with genetic

algorithm. The criteria of selection of best contour points from the proposed fitness criteria and

then improvements have been prepared by the process of crossover and mutation which are the

most important features of Genetic algorithm (GA). Although GA is an approximated approach,

but it may provide very accurate results by making a suitable selection of crossover and mutation.

In the proposed algorithm we have applied crossover in a different way between the contour points

of each cycle the proposed idea is unique and producing promising results. AR9114 and AR10696

of solar cycles 23rd are used as a case study to calculate the image segmentation of active contour.

5.2: PRINCIPAL RESULTS

The principal results of this dissertation are obtained and discussed which is very useful for the

world of scientific community is demonstrated as follow:

169

1. Describe the stationarity of the time series sunspot cycles (1-24) at second difference with

the help of Autocorrelation (AC), Partial Autocorrelation (PAC) and Ljung-Box Q-

statistics test.

2. Stationarity is confirmed by applying unit root test with Augmented Dickey Fuller (ADF)

test for sunspot cycles.

3. AIC, SIC, HQC and maximum likelihood are estimated for ARMA, AR-GARCH and

ARMA-GARCH models

4. Various ARMA (p, q) models are selected for sunspot cycles. Selection is based on the

minimum value of Durbin-Watson statistics test.

5. Selection for AR-GARCH and ARMA-GARCH models are based on the minimum value

of Durbin-Watson statistics test.

6. AR (p)-GARCH (1, 1) model is estimated for sunspot cycles.

7. ARMA (p, q)-GARCH (1, 1) model is estimated for sunspot cycles.

8. ARCH effect is diagnostic by Lagrange multiplayer, correlogram squared residuals and

test of normality are used to verify ARCH effect in time series data.

9. The least square estimator is used for estimation ARMA models and the Gaussian quasi

maximum likelihood estimator is used to estimate AR-GARCH and ARMA-GARCH

models.

10. Forecasting evolution is used to estimate residual diagnostics checking techniques such as

RMSE, MAE, MAPE and U test for ARAM, AR-GARCH and ARMA-GARCH models.

11. Comparison of residuals diagnostics checking are used for ARAM, AR-GARCH and

ARMA-GARCH models.

12. Fractal dimension and Hurst exponent (self-similar and self-affine) are estimated for

sunspot and ENSO cycles along with the total time series to understand the long term

behavior.

13. Dynamics of sunspot cycles and ENSO cycle’s active episode are compared with the help

of fractal dimension (self-similar and self-affine).

14. Universal scaling parameters like scaling exponent, spectral exponent and autocorrelation

coefficient are estimated by self-similar and self-affine fractal dimension for sunspot

cycles.

170

15. Higuchi’s Fractal dimension (FDH) are used to estimate the spectral exponent for sunspot

cycles.

16. The adequate probability distributions for sunspot and ENSO cycles associated are

obtained and selected with the help of Kolmogorov-Smirnov test.

17. Dynamics of sunspot and ENSO active period compare the adequate probability

distributions with the help of Kolmogorov-Smirnov test.

18. The adequate probability distributions for fractal dimension of sunspot and ENSO cycles

associated are obtained and selected with the help of Kolmogorov-Smirnov test.

19. The heavy tails and fractional differencing parameters (self-similar and self-affine) for

sunspot cycles and ENSO cycles are estimated to understand the peak and strength of each

cycle.

20. Self-similar and self-affine Long-Range correlation (persistency) for sunspot and ENSO

cycles are estimated.

21. Stationarity of heavy tail analysis and Long-Range correlation are confirm by applying unit

root test with Augmented Dickey Fuller (ADF) test for sunspot cycles and ENSO cycles.

22. The complexity and roughness of largest active region AR12192 of 24 solar cycle is

calculated by using fractal dimensions (Box counting FDB and correlation dimension FDC).

23. Persistency and wave spectrum or universal scaling parameters such as scaling exponent,

spectral exponent and autocorrelation coefficient are estimated for AR12192.

24. The principles of Mathematical morphology such as dilation, erosion, opening and closing

are analyzed for AR12192.

25. Image Segmentation of rotating sunspots by using Genetic active contour for AR9114 and

AR10696 of solar cycle 23rd.

5.3: Future Outlook

On the behalf of the principles results obtained in this dissertation some future works and

suggestions are focused. Direction of future studies are mentioned below:

• Forecasting evolution of ENSO cycles are established.

• The effective relationship between ENSO and rainfall and sunspot and rainfall, which are

not discussed in this thesis will be studied.

171

• The spectrum scaling parameter of ENSO cycles and rainfall (self-similar and self-affine)

will be analyzed.

• Time series analysis of ENSO cycles. ARMA (p, q), AR-GARCH and ARMA-GARCH

models are determined for ENSO cycles.

172

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Table 1.1: ACF, PACF, and Ljung-Box Q-statistics test of sunspot cycles at second differences

Sunspot Cycle (Aug 1755 - Mar 1766)

Lag ACF PACF Q-Stat Prob

1 -0.412 -0.412 23.900 0.000

2 -0.313 -0.581 37.813 0.000

3 0.223 -0.399 44.942 0.000

4 0.047 -0.396 45.265 0.000

5 -0.010 -0.247 45.280 0.000

6 -0.019 -0.009 45.331 0.000

7 -0.180 -0.161 50.103 0.000

8 0.129 -0.212 52.567 0.000

9 0.084 -0.126 53.613 0.000

10 -0.199 -0.379 59.601 0.000

11 0.243 0.004 68.547 0.000

12 -0.199 -0.218 74.626 0.000

13 0.010 -0.101 74.641 0.000

14 0.139 -0.151 77.662 0.000

15 -0.030 -0.005 77.808 0.000

16 -0.130 0.068 79.474 0.000

17 0.042 0.053 79.761 0.000

18 -0.051 -0.094 80.184 0.000

19 0.207 0.167 87.112 0.000

20 -0.202 -0.114 93.768 0.000

21 -0.074 -0.095 94.678 0.000

22 0.275 -0.034 107.27 0.000

23 -0.126 0.056 109.93 0.000

24 -0.107 -0.001 111.87 0.000

25 0.093 -0.097 113.35 0.000

26 0.056 -0.017 113.89 0.000

27 -0.066 -0.010 114.64 0.000

28 0.028 0.049 114.78 0.000

29 -0.155 -0.129 119.05 0.000

30 0.249 -0.039 130.11 0.000

31 -0.057 0.110 130.69 0.000

32 -0.158 0.124 135.22 0.000

33 0.078 0.042 136.35 0.000

34 0.086 0.106 137.72 0.000

35 -0.082 0.053 138.99 0.000

36 0.007 -0.024 139.00 0.000



1 -0.495 -0.495 28.166 0.000

2 -0.112 -0.473 29.629 0.000

3 0.117 -0.237 33.280 0.000

4 -0.078 -0.215 33.997 0.000

5 -0.054 -0.263 34.350 0.000

6 0.116 -0.154 35.961 0.000

7 0.003 -0.019 35.962 0.000

8 -0.143 -0.140 38.462 0.000

9 0.070 -0.188 39.069 0.000

10 0.024 -0.215 39.141 0.000

184

11 0.133 0.157 41.380 0.000

12 -0.252 -0.057 49.459 0.000

13 0.090 -0.112 50.499 0.000

14 0.078 -0.060 51.299 0.000

15 -0.048 0.083 51.606 0.000

16 0.010 0.061 51.619 0.000

17 0.070 0.112 52.282 0.000

18 -0.162 -0.065 55.845 0.000

19 0.076 0.003 56.636 0.000

20 0.133 0.117 59.091 0.000

21 -0.168 -0.002 63.040 0.000

22 0.024 0.100 63.118 0.000

23 0.027 -0.072 63.222 0.000

24 0.047 0.054 63.542 0.000

25 -0.068 -0.028 64.216 0.000

26 0.065 -0.040 64.837 0.000

27 -0.006 0.094 64.844 0.000

28 -0.154 -0.091 68.431 0.000

29 0.163 -0.008 72.508 0.000

30 0.020 -0.030 72.573 0.000

31 -0.061 0.070 73.153 0.000

32 -0.075 0.006 74.057 0.000

33 0.129 0.056 76.743 0.000

34 -0..064 0.005 77.413 0.000

35 -0.042 -0.074 77.710 0.000

36 0.124 0.036 80.301 0.000

Sunspot Cycle (Aug 1775 - Jun 1784)


1 -0.610 -0.610 40.144 0.000

2 0.167 -0.326 43.198 0.000

3 -0.210 -0.487 48.057 0.000

4 0.172 -0.485 51.353 0.000

5 0.036 -0.359 51.497 0.000

6 0.020 -0.122 51.545 0.000

7 -0.170 -0.151 54.843 0.000

8 0.072 -0.174 55.447 0.000

9 0.055 -0.062 55.804 0.000

10 0.003 -0.021 55.805 0.000

11 -0.056 -0.059 56.177 0.000

12 -0.041 -0.059 56.376 0.000

13 0.130 -0.0159 58.456 0.000

14 -0.110 -0.102 59.951 0.000

15 0.114 0.027 61.568 0.000

16 -0.134 0.146 63.824 0.000

17 0.042 0.085 64.051 0.000

18 0.051 0.030 64.080 0.000

19 0.093 0.122 65.218 0.000

20 -0.182 -0.028 69.574 0.000

21 0.137 -0.018 72.083 0.000

22 -0.106 -0.036 73.614 0.000

23 0.107 -0.074 75.177 0.000

24 -0.052 0.126 75.555 0.000

185

25 0.042 0.009 75.806 0.000

26 -0.096 0.622 77.125 0.000

27 0.043 -0.139 77.391 0.000

28 0.112 0.089 79.207 0.000

29 -0.173 -0.018 83.625 0.000

30 0.136 0.048 86.413 0.000

31 -0.131 0.027 89.008 0.000

32 0.099 -0.041 90.516 0.000

33 -0.009 -0.023 90.528 0.000

34 0.017 0.027 90.579 0.000

35 -0.116 0.002 92.732 0.000

36 0.093 -0.102 94.126 0.000

Sunspot Cycle (Jun 1784 - Jun 1798)


1 -0.591 -0.519 59.370 0.000

2 -0.037 0.593 59.604 0.000

3 0.249 0.325 70.232 0.000

4 -0.131 -0.196 73.196 0.000

5 -0.084 -0.303 74.434 0.000

6 0.169 -0.250 79.433 0.000

7 -0.606 -0.157 80.064 0.000

8 -0.094 -0.227 81.628 0.000

9 0.179 -0.056 87.339 0.000

10 -0.197 -0.264 94.304 0.000

11 0.154 -0.134 98.616 0.000

12 -0.045 -0.105 98.982 0.000

13 -0.069 -0.165 99.982 0.000

14 0.133 0.004 103.09 0.000

15 -0.082 0.093 104.33 0.000

16 -0.112 -0.142 106.66 0.000

17 0.229 -0.085 116.49 0.000

18 -0.101 -0.029 118.43 0.000

19 -0.095 -0.005 120.15 0.000

20 0.178 0.064 126.25 0.000

21 -0.123 0.046 129.19 0.000

22 -0.022 0.022 129.19 0.000

23 0.133 0.097 132.74 0.000

24 -0.098 0.140 134.64 0.000

25 -0.065 -0.036 135.47 0.000

26 0.168 0.102 141.11 0.000

27 -0.092 -0.023 142.82 0.000

28 -0.037 -0.036 143.10 0.000

29 0.118 0.058 145.93 0.000

30 -0.128 0.005 149.32 0.000

31 0.069 0.006 150.29 0.000

32 -0.002 -0.024 150.29 0.000

33 -0.029 0.032 150.47 0.000

34 -0.025 0.047 150.61 0.000

35 0.006 0.034 150.61 0.000

36 -0.063 -0.059 151.48 0.000

Sunspot Cycle (Jun 1798 - Sep 1810)

186


1 -0.596 -0.596 53.017 0.000

2 0.133 -0.346 55.662 0.000

3 -0.096 -0.346 57.064 0.000

4 0.052 -0.322 57.469 0.000

5 0.017 0.259 57.514 0.000

6 0.019 0.167 57.567 0.000

7 -0.016 -0.092 57.607 0.000

8 -0.075 -0.204 58.496 0.000

9 0.097 -0.154 59.579 0.000

10 -0.020 -0.085 60.040 0.000

11 -0.057 -0.021 60.565 0.000

12 0.057 -0.214 61.081 0.000

13 0.008 -0.141 61.092 0.000

14 -0.024 0.131 61.188 0.000

15 0.013 -0.101 61.215 0.000

16 -0.014 -0.073 61.245 0.000

17 0.015 0.004 61.281 0.000

18 -0.028 -0.010 61.409 0.000

19 0.032 -0.000 61.581 0.000

20 -0.056 -0.073 62.123 0.000

21 0.059 -0.081 62.729 0.000

22 -0.000 -0.049 62.729 0.000

23 0.010 0.060 62.746 0.000

24 -0.115 -0.109 65.079 0.000

25 0.120 -0.131 67.639 0.000

26 0.025 0.039 67.754 0.000

27 -0.069 0.061 68.612 0.000

28 0.000 0.036 68.612 0.000

29 -0.039 -0.058 68.886 0.000

30 0.063 -0.081 69.623 0.000

31 0.029 -0.023 69.780 0.000

32 -0.063 -0.018 70.539 0.000

33 0.026 -0.023 70.672 0.000

34 -0.041 -0.051 70.999 0.000

35 0.070 -0.057 71.941 0.000

36 -0.020 0.013 72.024 0.000

Sunspot Cycle (Sep1810 - Dec 1823)


1 -0.159 -0.159 55.885 0.000

2 0.070 -0.425 56.668 0.000

3 -0.043 -0.440 56.972 0.000

4 0.120 0.298 59.324 0.000

5 -0.137 -0.425 62.448 0.000

6 0.262 -0.001 73.830 0.000

7 -0.336 -0.152 92.771 0.000

8 0.215 -0.043 100.77 0.000

9 -0.070 -0.002 101.42 0.000

10 0.033 -0.016 101.58 0.000

11 -0.055 0.044 107.43 0.000

12 0.081 -0.034 103.23 0.000

13 -0.072 0.083 104.12 0.000

187

14 -0.024 -0.178 104.23 0.000

15 0.134 0.044 107.43 0.000

16 -0.109 0.022 109.55 0.000

17 0.036 0.125 109.78 0.000

18 -0.069 -0.027 110.64 0.000

19 0.109 0.025 112.81 0.000

20 -0.060 0.064 113.47 0.000

21 0.045 0.002 113.85 0.000

22 -0.133 -0.082 117.21 0.000

23 0.178 -0.046 123.03 0.000

24 -0.088 0.076 124.50 0.000

25 0.005 -0.021 124.51 0.000

26 -0.050 -0.043 124.99 0.000

27 0.099 -0.054 126.88 0.000

28 -0.081 -0.092 128.17 0.000

29 0.083 0.051 129.53 0.000

30 -0.034 0.142 129.76 0.000

31 -0.082 0.039 131.09 0.000

32 0.092 0.046 132.79 0.000

33 -0.052 0.000 132.92 0.000

34 -0.013 -0.071 132.96 0.000

35 0.072 0.048 134.03 0.000

36 -0.080 0.048 135.36 0.000

Sunspot Cycle (Dec1823 - Oct 1833)


1 -0.558 -0.558 37.357 0.000

2 0.018 -0.426 37.396 0.000

3 0.026 -0.350 37.479 0.000

4 -0.035 -0.390 37.633 0.000

5 0.170 -0.113 41.211 0.000

6 -0.211 -0.217 46.819 0.000

7 0.090 -0.215 47.847 0.000

8 0.050 -0.099 48.171 0.000

9 -0.101 -0.169 49.481 0.000

10 0.041 -0.259 49.699 0.000

11 -0.016 -0.334 49.734 0.000

12 0.159 -0.065 53.071 0.000

13 -0.224 -0.181 59.759 0.000

14 0.100 -0.106 61.103 0.000

15 0.028 0.018 61.211 0.000

16 -0.113 0.159 62.957 0.000

17 0.132 -0.167 65.693 0.000

18 -0.046 0.016 65.693 0.000

19 -0.081 0.209 66.613 0.000

20 0.180 0.028 71.262 0.000

21 -0.201 0.053 77.132 0.000

22 0.103 -0.069 78.694 0.000

23 -0.002 -0.114 78.694 0.000

24 0.002 -0.035 78.695 0.000

25 -0.002 -0.098 78.695 0.000

26 -0.015 -0.003 78.730 0.000

27 -0.023 0.014 78.811 0.000

188

28 0.083 0.157 79.879 0.000

29 -0.110 -0.005 81.806 0.000

30 0.074 0.051 82.692 0.000

31 -0.033 0.004 82.866 0.000

32 0.049 -0.042 82.256 0.000

33 -0.084 0.022 84.426 0.000

34 0.060 -0.005 85.035 0.000

35 0.028 -0.046 85.169 0.000

36 -0.083 -0.107 86.353 0.000

Sunspot Cycle (Oct1833 - Sep 1843)


1 -0.645 -0.645 50.398 0.000

2 0.285 -0.226 60.284 0.000

3 -0.220 -0.255 66.261 0.000

4 0.079 -0.291 67.037 0.000

5 -0.112 0.433 68.604 0.000

6 0.188 0.259 74.491 0.000

7 -0.028 -0.061 73.169 0.000

8 -0.101 -0.259 74.491 0.000

9 0.077 -0.308 75.268 0.000

10 0.016 -0.044 75.301 0.000

11 -0.095 -0.065 76.485 0.000

12 0.083 -0.120 77.398 0.000

13 0.017 0.072 77.438 0.000

14 -0.199 -0.249 82.831 0.000

15 0.284 -0.086 93.900 0.000

16 -0.165 0.029 97.961 0.000

17 0.104 0.020 99.204 0.000

18 -0.117 0.022 101.51 0.000

19 0.038 -0.166 101.35 0.000

20 -0.029 0.089 101.48 0.000

21 0.130 0.227 103.93 0.000

22 -0.103 0.072 105.49 0.000

23 0.034 0.086 105.60 0.000

24 0.107 -0.011 107.37 0.000

25 0.087 -0.179 108.52 0.000

26 0.052 0.149 108.95 0.000

27 -0.058 -0.035 109.47 0.000

28 0.105 0.099 111.20 0.000

29 -0.232 0.069 119.47 0.000

30 0.158 -0.142 123.76 0.000

31 -0.109 -0.108 125.69 0.000

32 0.177 0.109 130.83 0.000

33 -0.065 -0.032 131.53 0.000

34 -0.056 -0.047 132.05 0.000

35 0.065 0.033 032.78 0.000

36 -0.131 -0.115 135.76 0.000

Sunspot Cycle (Sep1843 - Mar 1855)


1 -0.595 -0.595 49.611 0.000

2 0.039 0.448 49.827 0.000

189

3 0.117 0.282 51.779 0.000

4 -0.101 -0.265 53.241 0.000

5 0.041 -0.231 53.482 0.000

6 0.040 -0.117 53.718 0.000

7 -0.062 -0.088 54.276 0.000

8 -0.032 -0.215 54.426 0.000

9 0.137 0.069 57.209 0.000

10 -0.168 -0.190 61.433 0.000

11 0.104 -0.187 63.079 0.000

12 0.001 -0.165 63.080 0.000

13 -0.048 -0.153 63.429 0.000

14 0.072 -0.044 64.221 0.000

15 -0.122 -0.210 66.536 0.000

16 0.112 -0.194 68.154 0.000

17 0.030 -0.002 58.658 0.000

18 -0.117 0.000 70.832 0.000

19 0.029 -0.056 70.965 0.000

20 0.068 0.066 72.161 0.000

21 -0.138 -0.017 75.266 0.000

22 0.087 -0.065 76.527 0.000

23 0.107 0.255 78.433 0.000

24 -0.337 -0.079 97.563 0.000

25 0.335 0.050 116.61 0.000

26 -0.097 0.026 118.24 0.000

27 -0.085 0.023 119.48 0.000

28 0.098 0.034 121.15 0.000

29 -0.023 0.023 121.24 0.000

30 -0.010 0.095 121.26 0.000

31 -0.060 -0.139 121.91 0.000

32 0.140 -0.043 125.47 0.000

33 -0.134 -0.018 128.76 0.000

34 0.110 0.015 130.99 0.000

35 -0.132 -0.122 134.22 0.000

36 0.108 -0.040 136.41 0.000

Sunspot Cycle (Mar1855 - Feb 1867)


1 -0.625 -0.625 56.633 0.000

2 0.127 0.432 59.004 0.000

3 -0.02 -0.359 59.077 0.000

4 -0.012 -0.379 59.097 0.000

5 0.089 -0.239 60.285 0.000

6 -0.070 -0.167 61.022 0.000

7 0.029 -0.167 61.110 0.000

8 -0.063 -0.172 61.722 0.000

9 0.047 -0.026 62.068 0.000

10 0.119 0.023 64.270 0.000

11 -0.0223 -0.075 72.043 0.000

12 0.116 -0.164 74.175 0.000

13 0.055 0.036 74.650 0.000

14 -0.128 -0.048 77.268 0.000

15 0.179 0.182 82.411 0.000

16 -0.266 -0.059 93.854 0.000

190

17 0.263 0.078 105.19 0.000

18 -0.156 0.067 109.20 0.000

19 0.049 -0.035 109.60 0.000

20 0.013 -0.043 109.63 0.000

21 0.000 0.175 109.63 0.000

22 -0.089 -0.100 110.97 0.000

23 0.127 -0.086 113.72 0.000

24 -0.043 0.029 114.03 0.000

25 -0.059 -0.133 114.65 0.000

26 0.102 0.059 116.49 0.000

27 -0.051 0.094 116.95 0.000

28 -0.025 0.123 117.06 0.000

29 -0.022 0.021 117.15 0.000

30 0.148 0.114 121.15 0.000

31 -0.190 0.093 127.82 0.000

32 0.085 -0.062 129.15 0.000

33 0.058 0.006 129.78 0.000

34 -0.120 -0.109 132.50 0.000

35 0.083 0.130 133.83 0.000

36 0.030 -0.076 134.01 0.000

Sunspot Cycle (Feb1867 - Sep 1878)


1 -0.464 -0.464 30.353 0.000

2 -0.195 -0.522 35.743 0.000

3 0.226 0.270 43.066 0.000

4 -0.084 -0.0296 44.086 0.000

5 -0.051 -0.328 44.463 0.000

6 0.133 -0.218 47.063 0.000

7 -0.021 -0.098 47.127 0.000

8 -0.142 -0.210 50.105 0.000

9 -0.142 -0.210 56.574 0.000

10 -0.213 -0.207 63.429 0.000

11 0.051 0.234 63.823 0.000

12 0.253 0.078 73.634 0.000

13 -0.338 -0.163 91.255 0.000

14 0.143 0.016 94.430 0.000

15 0.057 -0.008 99.442 0.000

16 -0.144 -0.093 98.204 0.000

17 0.088 -0.008 99.442 0.000

18 0.073 -0.018 100.30 0.000

19 -0.147 -0.055 103.83 0.000

20 0.062 0.017 104.64 0.000

21 0.065 -0.039 105.16 0.000

22 -0.215 -0.206 112.88 0.000

23 0.257 -0.093 123.94 0.000

24 0.003 0.011 123.94 0.000

25 -0.239 0.016 133.69 0.000

26 0.133 -0.080 136.72 0.000

27 0.082 0.062 137.88 0.000

28 -0.174 -0.083 143.20 0.000

29 0.095 -0.117 144.81 0.000

30 0.115 0.051 147.19 0.000

191

31 -0.200 0.025 154.41 0.000

32 0.066 -0.031 155.20 0.000

33 0.089 0.152 156.67 0.000

34 -0.168 0.008 161.90 0.000

35 0.126 -0.074 164.87 0.000

36 0.036 0.050 165.11 0.000

Sunspot Cycle (Sep1878 - Jun 1890)


1 -0.592 -0.592 50.136 0.000

2 0.047 -0.467 50.126 0.000

3 0.107 -0.250 52.126 0.000

4 -0.154 -0.348 55.610 0.000

5 0.168 -0.183 59.763 0.000

6 -0.145 -0.282 62.866 0.000

7 0.162 -0.041 66.801 0.000

8 -0.221 -0.309 74.147 0.000

9 0.224 -0.096 81.747 0.000

10 -0.121 -0.170 83.975 0.000

11 0.043 0.029 -84.257 0.000

12 -0.037 -0.130 84.473 0.000

13 -0.035 -0.140 -84.664 0.000

14 0.112 -0.207 86.640 0.000

15 -0.056 -0.017 87.142 0.000

16 0.012 -0.078 87.164 0.000

17 -0.036 0.031 87.370 0.000

18 0.014 -0.095 87.403 0.000

19 -0.023 -0.126 87.490 0.000

20 0.116 -0.017 89.727 0.000

21 -0.115 0.060 91.926 0.000

22 0.027 0.101 92.050 0.000

23 -0.048 -0.078 92.441 0.000

24 0.106 0.018 94.363 0.000

25 -0.051 -0.019 94.933 0.000

26 -0.044 -0.042 95.264 0.000

27 0.109 0.008 97.370 0.000

28 -0.126 0.010 100.20 0.000

29 0.121 -0.001 102.81 0.000

30 -0.086 -0.016 104.16 0.000

31 0.034 -0.034 104.36 0.000

32 -0.008 0.053 104.37 0.000

33 -0.024 0.023 104.38 0.000

34 0.084 0.038 105.82 0.000

35 -0.075 0.054 106.88 0.000

36 0.020 0.077 106.96 0.000

Sunspot Cycle (Jun1890 - Sep 1902)


1 -0.680 -0.680 68.957 0.000

2 0.231 -0.432 76.952 0.000

3 -0.128 -0.450 79.428 0.000

4 0.179 -0.217 84.319 0.000

5 -0.212 -0.318 91.207 0.000

192

6 0.214 -0.153 98.278 0.000

7 -0.171 -0.137 102.83 0.000

8 0.098 -0.138 104.34 0.000

9 -0.056 -0.121 104.84 0.000

10 0.051 0.127 104.84 0.000

11 -0.073 -0.192 106.10 0.000

12 0.084 -0.212 107.24 0.000

13 -0.027 0.100 107.36 0.000

14 -0.029 -0.091 107.50 0.000

15 0.034 -0.011 107.68 0.000

16 -0.001 0.092 107.69 0.000

17 -0.078 -0.062 108.71 0.000

18 0.128 -0.014 111.47 0.000

19 -0.061 0.091 112.10 0.000

20 -0.045 0.040 112.46 0.000

21 0.066 0.077 113.22 0.000

22 -0.057 -0.065 113.79 0.000

23 0.060 -0.054 114.42 0.000

24 0.006 0.102 114.43 0.000

25 -0.109 -0.081 116.55 0.000

26 0.140 0.019 120.10 0.000

27 -0.111 -0.083 122.32 0.000

28 0.180 0.046 124.44 0.000

29 -0.117 0.076 126.95 0.000

30 0.079 0.013 128.12 0.000

31 -0.032 0.062 128.32 0.000

32 0.016 0.015 128.36 0.000

33 -0.015 0.077 128.41 0.000

34 0.015 0.059 128.45 0.000

35 -0.029 0.041 128.61 0.000

36 0.038 0.006 128.89 0.000

Sunspot Cycle (Sep 1902 - Dec 1913)


1 -0.622 -0.622 52.968 0.000

2 0.240 -0.239 60.926 0.000

3 -0.255 0.377 69.945 0.000

4 0.175 -0.306 74.219 0.000

5 -0.002 -0.098 74.220 0.000

6 -0.145 -0.559 77.220 0.000

7 0.145 -0.319 80.249 0.000

8 -0.069 -0.305 80.940 0.000

9 0.115 -0.279 82.887 0.000

10 -0.098 -0.219 84.281 0.000

11 0.050 -0.097 84.645 0.000

12 -0.035 0.030 84.827 0.000

13 -0.083 -0.108 85.870 0.000

14 0.096 -0.133 87.266 0.000

15 -0.002 -0.042 87.266 0.000

16 0.041 0.028 87.526 0.000

17 -0.107 -0.052 89.309 0.000

18 0.036 0.241 89.514 0.000

19 0.154 0.038 93.261 0.000

193

20 -0.238 -0.131 102.31 0.000

21 0.208 0.141 109.27 0.000

22 -0.204 0.204 118.66 0.000

23 0.197 0.089 125.05 0.000

24 -0.151 -0.108 128.85 0.000

25 0.182 -0.051 134.39 0.000

26 -0.056 0.048 134.92 0.000

27 -0.057 0.044 135.47 0.000

28 0.021 0.008 135.54 0.000

29 -0.023 -0.006 135.63 0.000

30 0.068 -0.038 136.44 0.000

31 -0.115 -0.018 138.79 0.000

32 0.081 0.095 139.69 0.000

33 -0.016 0.041 140.01 0.000

34 0.083 0.063 141.27 0.000

35 -0.097 -0.004 143.00 0.000

36 -0.002 -0.058 143.00 0.000

Sunspot Cycle (Dec1913 - May 1923)


1 -0.459 -0.459 24.279 0.000

2 -0.207 -0.530 29.258 0.000

3 0.108 -0.496 30.624 0.000

4 0.264 -0.098 38.840 0.000

5 -0.249 -0.104 46.214 0.000

6 -0.103 0.246 47.500 0.000

7 0.223 -0.161 53.551 0.000

8 -0.053 -0.276 53.900 0.000

9 -0.045 -0.178 54.149 0.000

10 -0.002 -0.123 54.149 0.000

11 0.107 0.003 55.590 0.000

12 -0.129 -0.052 57.723 0.000

13 -0.001 -0.146 57.723 0.000

14 0.087 -0.142 58.699 0.000

15 -0.007 -0.082 58.705 0.000

16 -0.085 -0.067 59.660 0.000

17 0.032 -0.088 59.801 0.000

18 0.090 -0.064 60.899 0.000

19 -0.050 0.023 61.243 0.000

20 -0.092 -0.022 62.416 0.000

21 0.058 -0.080 62.894 0.000

22 0.072 -0.072 63.633 0.000

23 -0.075 -0.077 64.440 0.000

24 -0.020 -0.050 64.499 0.000

25 0.018 -0.175 64.544 0.000

26 0.082 -0.154 65.548 0.000

27 -0.004 0.091 65.550 0.000

28 -0.175 -0.063 70.207 0.000

29 0.107 -0.135 71.980 0.000

30 0.076 -0.153 72.886 0.000

31 0.018 0.152 72.938 0.000

32 -0.277 -0.045 85.659 0.000

33 0.286 0.115 98.385 0.000

194

34 -0.036 0.086 98.601 0.000

35 -0.131 -0.036 101.47 0.000

36 0.038 0.024 101.71 0.000

Sunspot Cycle (May 1923 - Sep 1933)


1 -0.576 -0.576 41.859 0.000

2 0.137 -0.293 44.233 0.000

3 -0.138 -0.336 46.673 0.000

4 0.034 -0.372 46.822 0.000

5 0.055 -0.302 47.215 0.000

6 0.077 -0.085 47.985 0.000

7 -0.112 -0.091 49.650 0.000

8 0.028 -0.066 49.755 0.000

9 -0.051 -0.096 50.104 0.000

10 0.013 -0.202 50.129 0.000

11 0.105 -0.096 51.654 0.000

12 -0.042 -0.003 51.900 0.000

13 -0.067 -0.069 52.533 0.000

14 -0.036 -0.221 52.713 0.000

15 0.070 -0.237 53.411 0.000

16 0.030 -0.241 53.538 0.000

17 0.092 -0.017 54.762 0.000

18 -0.200 -0.082 60.622 0.000

19 0.073 -0.081 61.406 0.000

20 0.008 0.027 61.415 0.000

21 0.041 0.118 61.673 0.000

22 -0.069 -0.038 62.400 0.000

23 0.119 0.136 64.586 0.000

24 -0.233 -0.059 73.107 0.000

25 0.247 0.110 82.577 0.000

26 -0.193 -0.012 88.478 0.000

27 0.144 -0.055 91.806 0.000

28 -0.018 0.054 91.858 0.000

29 -0.102 -0.013 93.556 0.000

30 0.028 -0.065 93.868 0.000

31 0.069 -0.039 94.475 0.000

32 0.006 0.040 94.462 0.000

33 -0.053 -0.056 94.955 0.000

34 0.016 -0.058 95.001 0.000

35 -0.016 0.032 95.046 0.000

36 0.038 0.019 95.302 0.000

Sunspot Cycle (Sep 1933 - Jan 1944)


1 -0.592 -0.592 44.118 0.000

2 0.094 -0.394 45.281 0.000

3 0.005 -0.274 45.284 0.000

4 -0.035 -0.266 45.443 0.000

5 -0.013 -0.324 45.467 0.000

6 0.115 -0.157 47.190 0.000

7 -0.000 0.103 47.190 0.000

8 -0.271 -0.307 57.014 0.000

195

9 0.284 -0.260 67.921 0.000

10 -0.053 -0.063 68.306 0.000

11 -0.062 -0.074 68.835 0.000

12 0.042 -0.155 69.078 0.000

13 -0.012 -0.144 69.098 0.000

14 -0.025 -0.032 69.188 0.000

15 0.017 -0.099 69.229 0.000

16 0.116 -0.006 71.153 0.000

17 -0.208 -0.026 77.402 0.000

18 0.154 0.126 80.867 0.000

19 -0.121 -0.091 83.047 0.000

20 0.136 0.037 85.811 0.000

21 -0.073 0.149 86.624 0.000

22 -0.076 -0.161 87.495 0.000

23 0.213 0.142 94.453 0.000

24 -0.307 -0.123 109.12 0.000

25 0.345 0.116 127.75 0.000

26 -0.249 -0.011 137.60 0.000

27 0.101 -0.055 139.23 0.000

28 -0.043 0.070 139.53 0.000

29 0.068 0.065 140.28 0.000

30 -0.081 -0.067 141.36 0.000

31 -0.035 -0.213 141.56 0.000

32 0.200 0.008 148.34 0.000

33 -0.193 0.117 154.71 0.000

34 0.087 -0.007 156.01 0.000

35 -0.013 0.020 156.04 0.000

36 -0.044 0.062 156.38 0.000

Sunspot Cycle (Jan 1944 - Feb 1954)


1 -0.547 -0.547 36.806 0.000

2 0.067 -0.332 37.354 0.000

3 -0.129 -0.420 39.432 0.000

4 0.206 -0.181 44.789 0.000

5 -0.141 -0.211 47.333 0.000

6 0.103 -0.066 48.685 0.000

7 -0.153 -0.212 51.702 0.000

8 0.115 -0.218 53.420 0.000

9 -0.041 -0.277 53.645 0.000

10 0.068 -0.181 54.259 0.000

11 -0.021 -0.030 54.321 0.000

12 -0.072 -0.127 55.031 0.000

13 0.044 -0.112 55.293 0.000

14 0.020 -0.119 55.349 0.000

15 -0.010 -0.119 55.363 0.000

16 -0.017 -0.094 55.402 0.000

17 0.016 -0.060 55.440 0.000

18 -0.054 -0.156 55.682 0.000

19 0.149 0.031 59.061 0.000

20 -0.129 0.034 61.500 0.000

21 -0.011 -0.061 61.518 0.000

22 0.031 -0.039 61.660 0.000

196

23 0.061 0.019 62.231 0.000

24 -0.061 0.057 62.527 0.000

25 -0.069 -0.140 63.527 0.000

26 0.114 -0.058 65.563 0.000

27 -0.005 -0.029 65.567 0.000

28 0.016 0.106 65.609 0.000

29 -0.103 0.076 67.302 0.000

30 0.014 0.114 69.115 0.000

31 0.104 0.114 69.115 0.000

32 -0.112 -0.068 71.194 0.000

33 0.086 0.047 72.452 0.000

34 -0.071 0.029 73.309 0.000

35 -0.008 -0.110 73.319 0.000

36 0.091 -0.002 74.762 0.000

Sunspot Cycle (Feb 1954 - Oct 1964)


1 -0.981 -0.981 30.139 0.000

2 -0.258 -0.637 38.845 0.000

3 0.398 0.245 59.778 0.000

4 -0.203 -0.371 65.275 0.000

5 0.046 -0.118 65.557 0.000

6 0.005 -0.232 65.560 0.000

7 0.003 -0.037 65.561 0.000

8 -0.030 -0.187 65.686 0.000

9 -0.008 -0.163 65.695 0.000

10 0.122 0.043 67.767 0.000

11 -0.212 -0.264 74.140 0.000

12 0.133 -0.210 76.668 0.000

13 0.142 0.008 79.583 0.000

14 -0.302 -0.062 92.839 0.000

15 0.138 -0.089 95.264 0.000

16 0.105 -0.057 97.264 0.000

17 -0.099 0.140 98.722 0.000

18 -0.063 0.015 99.310 0.000

19 0.025 -0.142 99.406 0.000

20 0.154 0.000 103.06 0.000

21 -0.199 -0.119 109.16 0.000

22 0.092 -0.023 110.84 0.000

23 0.071 0.028 111.28 0.000

24 -0.195 0.006 117.35 0.000

25 0.169 -0.066 121.94 0.000

26 0.023 0.088 122.03 0.000

27 -0.191 -0.010 128.02 0.000

28 0.145 -0.012 131.51 0.000

29 0.030 -0.029 131.66 0.000

30 -0.082 0.130 132.81 0.000

31 -0.056 -0.010 133.35 0.000

32 0.113 0.094 135.55 0.000

33 -0.021 0.267 135.82 0.000

34 -0.022 -0.022 135.71 0.000

35 0.061 0.007 136.37 0.000

36 -0.109 -0.009 138.53 0.000

197

Sunspot Cycle (Oct 1964 - May 1976)


1 -0.601 -0.601 59.950 0.000

2 0.066 -0.463 51.562 0.000

3 0.079 -0.282 52.453 0.000

4 -0.125 -0.364 54.700 0.000

5 0.180 0.146 59.399 0.000

6 -0.192 -0.270 64.793 0.000

7 0.183 -0.069 69.719 0.000

8 -0.142 -0.140 72.719 0.000

9 0.019 -0.201 72.774 0.000

10 0.108 -0.116 74.545 0.000

11 -0.114 -0.071 76.514 0.000

12 0.044 -0.115 76.814 0.000

13 -0.018 -0.115 76.887 0.000

14 0.052 -0.026 77.282 0.000

15 -0.082 -0.106 78.330 0.000

16 0.073 -0.032 79.166 0.000

17 -0.017 -0.007 79.213 0.000

18 -0.073 -0.106 80.069 0.000

19 0.131 -0.005 82.867 0.000

20 -0.131 -0.100 85.696 0.000

21 0.112 -0.031 87.755 0.000

22 -0.078 -0.047 88.777 0.000

23 0.046 0.024 89.138 0.000

24 -0.029 -0.049 89.277 0.000

25 -0.077 -0.234 90.299 0.000

26 0.247 -0.011 100.79 0.000

27 -0.267 -0.087 113.04 0.000

28 0.116 -0.130 115.38 0.000

29 0.067 0.060 116.18 0.000

30 -0.169 -0.026 121.30 0.000

31 0.170 0.050 126.51 0.000

32 -0.118 0.095 129.05 0.000

33 0.078 0.128 130.18 0.000

34 -0.111 -0.014 132.48 0.000

35 0.091 -0.020 134.02 0.000

36 0.056 0.034 134.62 0.000

Sunspot Cycle (May1976 - Mar 1986)


1 -0.546 -0.546 35.741 0.000

2 0.048 -0.355 36.024 0.000

3 -0.108 0.433 37.441 0.000

4 0.148 0.305 40.141 0.000

5 -0.002 0.171 40.142 0.000

6 -0.085 0.230 41.042 0.000

7 0.053 -0.168 41.392 0.000

8 -0.012 -0.145 41.410 0.000

9 -0.045 -0.287 41.666 0.000

10 0.099 -0.192 42.954 0.000

11 -0.092 -0.283 44.066 0.000

198

12 0.156 -0.029 47.309 0.000

13 -0.159 0.084 50.711 0.000

14 -0.045 -0.109 50.989 0.000

15 0.115 -0.091 52.789 0.000

16 -0.031 -0.205 52.920 0.000

17 0.135 0.075 55.457 0.000

18 -0.253 0.034 64.487 0.000

19 0.127 0.020 66.789 0.000

20 0.006 0.034 66.794 0.000

21 0.008 -0.019 66.804 0.000

22 -0.012 -0.044 66.825 0.000

23 -0.026 -0.011 66.925 0.000

24 0.014 -0.050 66.954 0.000

25 0.013 0.005 66.978 0.000

26 -0.050 -0.086 67.360 0.000

27 0.162 0.073 71.427 0.000

28 -0.213 -0.046 78.519 0.000

29 0.124 -0.059 80.936 0.000

30 -0.105 -0.108 82.703 0.000

31 0.148 -0.026 86.254 0.000

32 -0.037 0.079 86.474 0.000

33 -0.113 -0.046 88.596 0.000

34 0.112 -0.003 90.710 0.000

35 -0.038 0.083 90.953 0.000

36 -0.007 -0.001 90.962 0.000

Sunspot Cycle (Mar1986 - Jun1996)


1 -0.639 -0.639 51.124 0.000

2 0.139 -0.456 53.567 0.000

3 0.001 0.343 53.567 0.000

4 0.018 0.222 53.609 0.000

5 -0.115 -0.390 55.314 0.000

6 0.193 0.250 60.166 0.000

7 -0.132 -0.211 62.472 0.000

8 0.070 -0.082 63.128 0.000

9 -0.047 -0.011 63.420 0.000

10 -0.058 -0.215 63.863 0.000

11 0.099 -0.254 65.199 0.000

12 0.088 0.120 66.270 0.000

13 -0.285 -0.096 77.526 0.000

14 0.252 -0.142 86.455 0.000

15 -0.054 0.036 86.872 0.000

16 -0.110 0.007 88.610 0.000

17 0.117 0.028 90.597 0.000

18 0.000 0.149 90.597 0.000

19 -0.145 -0.173 93.699 0.000

20 0.208 -0.052 100.13 0.000

21 -0.139 0.010 103.03 0.000

22 0.071 0.126 103.80 0.000

23 -0.085 -0.108 104.91 0.000

24 0.087 -0.072 106.07 0.000

25 -0.090 -0.035 107.32 0.000

199

26 0.160 0.032 111.38 0.000

27 -0.215 -0.145 118.72 0.000

28 0.213 0.035 125.99 0.000

29 -0.187 -0.112 131.66 0.000

30 0.116 -0.009 133.86 0.000

31 -0.056 -0.010 134.39 0.000

32 0.120 0.040 136.79 0.000

33 -0.190 0.046 142.90 0.000

34 0.128 0.043 145.73 0.000

35 -0.057 0.002 146.29 0.000

36 0.044 -0.022 146.62 0.000

Sunspot Cycle (Jun1996 - Jan 2008)


1 -0.142 -0.142 23.900 0.000

2 -0.313 -0.581 37.813 0.000

3 0.223 0.399 44.942 0.000

4 0.047 -0.396 45.265 0.000

5 -0.010 -0.247 45.280 0.000

6 0.019 0.009 45.331 0.000

7 -0.180 -0.161 50.103 0.000

8 0.129 -0.121 52.567 0.000

9 0.084 -0.126 53.613 0.000

10 -0.199 -0.379 59.601 0.000

11 0.243 0.004 68.547 0.000

12 -0.199 -0.218 74.626 0.000

13 0.010 -0.101 74.641 0.000

14 0.139 -0.151 77.662 0.000

15 -0.030 -0.005 77.808 0.000

16 -0.103 0.068 79.474 0.000

17 0.042 0.053 79.761 0.000

18 -0.051 -0.090 80.184 0.000

19 0.207 0.167 87.112 0.000

20 -0.202 -0.114 93.768 0.000

21 -0.074 -0.095 94.678 0.000

22 0.275 -0.034 107.27 0.000

23 -0.126 0.056 109.93 0.000

24 -0.107 -0.001 111.87 0.000

25 0.093 -0.097 113.35 0.000

26 0.056 -0.017 113.89 0.000

27 -0.066 -0.010 114.64 0.000

28 0.028 0.049 114.70 0.000

29 -0.155 -0.129 119.05 0.000

30 0.249 -0.039 130.11 0.000

31 -0.057 0.110 130.69 0.000

32 -0.158 0.124 135.22 0.000

33 0.078 0.042 136.65 0.000

34 0.086 0.106 137.72 0.000

35 -0.082 0.050 138.99 0.000

36 0.007 -0.024 139.00 0.000

Sunspot Cycle (Aug1755 - Jan 2008)


200

1 -0.570 -0.570 986.15 0.000

2 0.033 0.433 989.03 0.000

3 0.029 -0.349 992.03 0.000

4 0.010 -0.284 992.26 0.000

5 -0.012 -0.260 992.80 0.000

6 0.030 -0.188 995.54 0.000

7 -0.017 -0.136 996.41 0.000

8 -0.042 -0.196 100.17 0.000

9 0.062 -0.162 1013.6 0.000

10 -0.028 -0.161 1016.0 0.000

11 0.007 -0.144 1016.1 0.000

12 0.013 -0.088 1016.6 0.000

13 -0.037 -0.106 1020.7 0.000

14 0.014 -0.130 1021.2 0.000

15 0.034 -0.082 1024.8 0.000

16 -0.046 -0.111 1031.3 0.000

17 0.049 -0.030 1038.6 0.000

18 -0.054 -0.052 1047.3 0.000

19 0.025 -0.054 1049.2 0.000

20 0.023 0.006 1050.8 0.000

21 -0.035 0.003 1054.4 0.000

22 -0.010 -0.047 1054.7 0.000

23 0.064 0.040 1067.2 0.000

24 -0.084 -0.031 1088.9 0.000

25 0.045 -0.055 1095.1 0.000

26 0.022 -0.026 1096.6 0.000

27 -0.028 -0.007 1099.1 0.000

28 0.003 0.024 1099.1 0.000

29 -0.013 -0.014 1099.7 0.000

30 0.031 0.019 1102.5 0.000

31 -0.037 -0.022 1106.8 0.000

32 0.037 -0.025 1111.0 0.000

33 -0.020 -0.013 1112.2 0.000

34 0.008 0.004 1112.4 0.000

35 -0.006 0.009 1112.5 0.000

36 0.001 0.015 1112.5 0.000

Table 1.2: unit root test of sunspot Cycles

Null Hypothesis H0: Sunspots Cycle (Aug 1755 - Mar 1766) has a unit root

Exogenous: Constant

Lag Length: 12 (Automatics-based on AIC, MaxLag = 12)

t-statistics prob*

Augmented Dickey Fuller (ADF) test -6.181642 0.0000

Test Critical vales 1% level

5% level

10% level

-3.489117

-2.887190

-2.50525

201

Null Hypothesis H0: Sunspots Cycle (Mar 1766 - Aug 1775) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.497727

-2.890926

-2.582514

Null Hypothesis H0: Sunspots Cycle (Aug 1775 - Jun 1784) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.499167

-2.891550

-2.582846

Null Hypothesis H0: Sunspots Cycle (Jun 1784 - Jun 1798) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.473096

-2.880211

-2.576805

Null Hypothesis H0: Sunspots Cycle (Jun 1798 - Sep 1810) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.480425

-2.883408

-2.578510

Null Hypothesis H0: Sunspots Cycle (Sep1810 - Dec 1823) has a unit root

Exogenous: Constant


202

t-statistics prob*



5% level

10% level

-3.476143

-2.881541

-2.577514

Null Hypothesis H0: Sunspots Cycle (Dec1823 - Oct 1833) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.494378

-2.889474

-2.581741

Null Hypothesis H0: Sunspots Cycle (Oct1833 - Sep 1843) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.491345

-2.888157

-2.581041

Null Hypothesis H0: Sunspots Cycle (Sep1843 - Mar 1855) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.483751

-2.884856

-2.579282

Null Hypothesis H0: Sunspots Cycle (Mar1855 - Feb 1867) has a unit root

Exogenous: Constant


t-statistics prob*


203


5% level

10% level

-3.482035

-2.884109

-2.578884

Null Hypothesis H0: Sunspots Cycle (Feb1867 - Sep 1878) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.483312

-2.884665

-2.579180

Null Hypothesis H0: Sunspots Cycle (Sep1878 - Jun 1890) has a unit root

Exogenous: Constant


t-statistics prob*

Augmented Dickey Fuller (ADF) test - 6.602668 0.0000


5% level

10% level

-3.482879

-2.884477

-2.579080

Null Hypothesis H0: Sunspots Cycle (Jun1890 - Sep 1902) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.480485

-2.883408

-2.578510

Null Hypothesis H0: Sunspots Cycle (Sep 1902 - Dec 1913) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.484198

-2.885051

-2.579368

204

Null Hypothesis H0: Sunspots Cycle (Dec1913 - May 1923) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.495021

-2.889753

-2.581890

Null Hypothesis H0: Sunspots Cycle (May 1923 - Sep 1933) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.486551

-2.886074

-2.579931

Null Hypothesis H0: Sunspots Cycle (Sep 1933 - Jan 1944) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.490772

-2.887909

-2.580908

Null Hypothesis H0: Sunspots Cycle (Jan 1944 - Feb 1954) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.490772

-2.889079

-2.580908

Null Hypothesis H0: Sunspots Cycle (Feb 1954 - Oct 1964) has a unit root

Exogenous: Constant

205


t-statistics prob*



5% level

10% level

-3.488063

-2.886732

-2.580281

Null Hypothesis H0: Sunspots Cycle (Oct 1964 - May 1976) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.482035

-2.884109

-2.578884

Null Hypothesis H0: Sunspots Cycle (May1976 - Mar 1986) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.493129

-2.888932

-2.581453

Null Hypothesis H0: Sunspots Cycle (Mar1986 - Jun1996) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.490210

-2.887665

-2.580778

Null Hypothesis H0: Sunspots Cycle (Jun1996 - Jan 2008) has a unit root

Exogenous: Constant


t-statistics prob*


206


5% level

10% level

-3.483751

-2.884856

-2.579282

Null Hypothesis H0: Sunspots Cycle (Aug1755 - Jan 2008) has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.432338

-2.862304

-2.567221

Table 1.3: Descriptive Statistics of Sunspot Cycles

C Duration N Mean Median Std.D Skewness Kurtosis Jur-Bera

1 Aug 1755 - Mar 1766 128 43.81 43.40 24.35 0.376 2.428 4.782

2 Mar 1766 - Aug 1775 144 57.04 51.25 39.21 0.582 2.648 7.035

3 Aug 1775 - Jun 1784 107 68.95 54.30 52.12 0.665 2.686 8.319

4 Jun 1784 - Jun 1798 169 59.92 53.00 54.44 0.512 2.122 12.82

5 Jun 1798 - Sep 1810 148 23.05 23.75 18.71 0.273 1.721 11.96

6 Sep1810 - Dec 1823 160 17.76 14.40 17.98 1.269 4.932 67.81

7 Dec1823 - Oct 1833 119 40.06 42.30 26.01 0.304 2.331 4.055

8 Oct1833 - Sep 1843 120 65.39 59.40 48.31 0.568 2.518 8.018

9 Sep1843 - Mar 1855 139 59.37 54.80 38.41 0.930 3.480 21.37

10 Mar1855 - Feb 1867 144 45.86 40.85 32.57 0.320 1.991 8.558

11 Feb1867 - Sep 1878 140 53.59 39.60 46.81 0.718 2.395 14.18

12 Sep1878 - Jun 1890 142 32.55 27.15 26.99 0.512 2.054 11.52

13 Jun1890 - Sep 1902 148 37.54 30.55 30.63 0.553 2.234 11.16

14 Sep 1902 - Dec 1913 136 32.68 31.95 26.33 0.596 2.793 8.293

15 Dec1913 - May 1923 114 46.53 42.15 33.97 0.703 3.021 9.380

16 May 1923 - Sep 1933 125 39.79 34.40 28.73 0.372 1.922 8.936

17 Sep 1933 - Jan 1944 125 57.64 54.60 40.52 0.400 2.159 7.034

18 Jan 1944 - Feb 1954 122 74.43 61.45 54.71 0.378 1.985 8.145

19 Feb 1954 - Oct 1964 129 89.04 63.60 71.52 0.431 1.816 11.54

20 Oct 1964 - May 1976 140 60.26 57.35 37.29 0.159 1.738 9.887

21 May1976 - Mar 1986 119 83.57 82.70 57.22 0.154 1.616 9.985

22 Mar1986 - Jun1996 124 76.15 60.35 58.50 0.410 1.754 11.49

23 Jun1996 - Jan 2008 140 57.94 48.90 42.67 0.425 2.018 9.834

24 Aug1755 - Jan 2008 3030 52.26 42.10 44.60 1.066 3.741 643.62

Table 1.4: Best Fitted ARMA Model based on DURBIN WATSON STAT


No# Model R2 ADJ R2 SE Reg Log L AIC SIC HQC DWS

207

1 ARMA (6,5) 0.437 0.424 18.509 -554.25 8.723 8.812 8.759 0.798

2 ARMA (5,5) 0.474 0.461 17.848 -550.72 8.667 8.757 8.704 0.959

3 ARMA (5,4) 0.523 0.511 17.049 -543.86 8.560 8.649 8.597 0.976

4 ARMA (4,5) 0.523 0.511 17.050 -543.79 8.559 8.648 8.595 1.051

5 ARMA(4,4) 0.527 0.516 16.972 -543.77 8.559 8.648 8.595 1.096

Dynamics Forecasted Evolution of Sunspot Cycle (Aug 1755 - Mar 1766)

No# Model RMSE MAE MAPE U

1 ARMA (6,5) 22.96202 18.55645 64.93243 0.251178

2 ARMA (5,5) 25.78878 20.41369 58.87236 0.306819

3 ARMA (5,4) 23.01596 18.60317 66.29410 0.252782

4 ARMA (4,5) 22.59682 18.43576 61.99027 0.255410

5 ARMA(4,4) 25.49785 20.23572 55.73550 0.303910

Sunspot Cycle (Mar 1766 - Aug 1775)

No# Model R2 ADJ R2 SE Reg Log L AIC SIC HIC DWS

1 ARMA (5,3) 0.439 0.424 29.763 -547.40 9.674 9.770 9.713 0.719

2 ARMA (3,3) 0.524 0.511 27.407 -539.57 9.519 9.615 9.557 0.855

3 ARMA (3,2) 0.497 0.483 28.190 -541.00 9.561 9.657 9.600 0.949

4 ARMA(2,2) 0.575 0.563 25.912 -531.90 9.402 9.498 9.441 1.104

5 ARMA (2,6) 0.540 0.528 26.942 -536.02 9.474 9.560 9.513 1.197

Dynamics Forecasted Evolution of Sunspot Cycle (Mar 1766 - Aug 1775)


1 ARMA (5,3) 37.92649 29.55789 161.1660 0.309054

2 ARMA (3,3) 43.21607 33.35958 149.9696 0.395928

3 ARMA (3,2) 38.15855 30.36035 185.7616 0.309936

4 ARMA(2,2) 43.71936 33.78659 145.5506 0.405274

5 ARMA (2,6) 38.07863 30.05593 179.8060 0.312458



1 ARMA (6,5) 0.722 0.714 27.882 -508.60 9.581 9.681 9.622 0.846

2 ARMA(5,6) 0.751 0.743 26.402 -502.95 9.746 9.576 9.516 0.993

3 ARMA (5,4) 0.783 0.777 24.607 -495.37 9.334 9.434 9.375 1.007

4 ARMA (5,5) 0.736 0.728 27.176 -506.63 9.544 9.644 9.585 1.155

5 ARMA (6,4) 0.736 0.728 27.174 -505.94 9.532 9.631 9.572 1.162

Dynamics Forecasted Evolution of Sunspot Cycle (Aug 1775 - Jun 1784)


1 ARMA(6,5) 57.08266 43.80487 121.4064 0.415818

2 ARMA(5,6) 60.72247 46.10220 130.2028 0.465018

3 ARMA (5,4) 58.41590 44.52910 107.1816 0.436066

4 ARMA (5,5) 61.03474 46.27304 102.4356 0.469253

5 ARMA(6,4) 56.72805 43.6452 122.6841 0.411658



1 ARMA (6,5) 0.810 0.807 19.962 -784.12 8.901 8.975 8.931 0.904

2 ARMA(6,4) 0.820 0.816 19.476 -743.78 8.849 8.824 8.879 0.977

3 ARMA (5,6) 0.813 0.809 19.849 -746.37 8.880 8.954 8.910 1.078

4 ARMA (3,3) 0.782 0.869 16.415 -713.90 8.496 8.570 8.526 1.150

5 ARMA (5,4) 0.807 0.803 20.153 -749.04 8.912 8.986 8.942 1.172

208

Dynamics Forecasted Evolution of Sunspot Cycle (Jun 1784 - Jun 1798)


1 ARMA (6,5) 55.57598 43.24828 171.1271 0.498712

2 ARMA(6,4) 54.53012 42.47551 175.9627 0.481473

3 ARMA (5,6) 53.38909 41.59299 180.1554 0.465585

4 ARMA (3,3) 59.78222 45.79778 141.12189 0.587859

5 ARMA (5,4) 54.15024 42.15394 176.6212 0.477880



1 ARMA (6,4) 0.670 0.693 10.358 -556.64 7.576 7.657 7.609 0.794

2 ARMA(4,4) 0.749 0.743 9.475 -543.74 7.402 7.483 7.435 0.875

3 ARMA (5,3) 0.729 0.723 9.826 -584.79 7.470 7.551 7.503 0.994

4 ARMA (3,3) 0.777 0.772 8.933 -534.55 7.278 7.359 7.311 0.998

5 ARMA (6,2) 0.731 0.723 9.797 -548.48 7.466 7.547 7.499 0.999

Dynamics Forecasted Evolution of Sunspot Cycle (Jun 1798 - Sep 1810)


1 ARMA (6,4) 19.64111 16.86298 224.7709 0.416733

2 ARMA(4,4) 22.96767 18.54974 172.6923 0.559893

3 ARMA (5,3) 19.61195 16.77907 222.5072 0.418899

4 ARMA (3,3) 23.56660 18.78161 161.6836 0.592059

5 ARMA (6,2) 19.71194 16.93144 225.7594 0.418814



1 ARMA (6,5) 0.521 0.511 12.570 -631.50 7.944 8.021 7.975 1.187

2 ARMA(6,6) 0.503 0.494 12.793 -635.00 7.987 8.064 8.018 1.229

3 ARMA (6,2) 0.490 0.480 12.957 -636.60 8.007 8.084 8.039 1.283

4 ARMA (3,4) 0.507 0.498 12.734 -632.97 7.962 8.039 7.993 1.300

5 ARMA (6,4) 0.543 0.534 12.269 -627.70 7.896 7.973 7.927 1.325

Dynamics Forecasted Evolution of Sunspot Cycle (Sep1810 - Dec 1823)


1 ARMA (6,5) 17.85740 13.28661 261.7866 0.429515

2 ARMA(6,6) 18.43876 13.58944 215.0192 0.482890

3 ARMA (6,2) 17.67344 13.27179 251.9070 0.437088

4 ARMA (3,4) 17.58213 13.69847 296.0727 0.418201

5 ARMA (6,4) 17.58113 13.25282 261.6172 0.429342



1 ARMA (5,5) 0.531 0.518 18.051 -513.30 8.694 8.788 8.732 0.926

2 ARMA(6,4) 0.435 0.421 19.798 -523.20 8.861 8.954 8.898 0.971

3 ARMA (5,3) 0.523 0.511 18.189 -513.19 8.692 8.786 8.730 0.993

4 ARMA (4,4) 0.559 0.548 17.492 -509.35 8.628 8.721 8.666 1.054

5 ARMA (5,2) 0.529 0.516 18.092 -512.59 8.682 8.775 8.720 1.112

Dynamics Forecasted Evolution of Sunspot Cycle (Dec1823 - Oct 1833)


1 ARMA (5,5) 28.54622 23.13027 125.1342 0.376961

2 ARMA(6,4) 25.01170 20.63166 160.3185 0.297296

3 ARMA (5,3) 24.84239 20.57404 159.1863 0.294667

209

4 ARMA (4,4) 29.44770 23.99430 142.4501 0.396488

5 ARMA (5,2) 24.99444 20.66547 156.7645 0.297884



1 ARMA (5,6) 0.663 0.654 28.398 -571.85 9.598 9.690 9.635 0.767

2 ARMA(4,6) 0.657 0.648 28.648 -572.46 9.608 9.701 9.645 0.907

3 ARMA (6,3) 0.688 0.680 27.324 -567.40 9.523 9.616 9.561 0.918

4 ARMA (3,3) 0.695 0.687 27.031 -565.92 9.499 9.592 9.536 0.963

5 ARMA (6,2) 0.746 0.741 24.583 -555.23 9.321 9.413 9.358 0.978

Dynamics Forecasted Evolution of Sunspot Cycle (Oct1833 - Sep 1843)


1 ARMA (5,6) 49.58514 38.8095 121.2569 0.371537

2 ARMA(4,6) 47.89669 37.64070 125.6561 0.353636

3 ARMA (6,3) 52.16269 40.57367 114.0542 0.400505

4 ARMA (3,3) 59.74495 45.34109 101.5025 0.518348

5 ARMA (6,2) 51.83747 40.41983 116.4244 0.396347



1 ARMA (6,4) 0.511 0.500 27.162 -655.49 9.489 9.574 9.523 0.779

2 ARMA(4,3) 0.583 0.574 25.080 -644.24 9.327 9.412 9.362 0.913

3 ARMA (4,4) 0.587 0.578 24.946 -644.38 9.329 9.414 9.364 0.929

4 ARMA (5,3) 0.589 0.579 24.912 -643.51 9.317 9.401 9.351 0.981

5 ARMA (3,3) 0.651 0.643 22.946 -632.53 9.159 9.243 9.193 0.987

Dynamics Forecasted Evolution of Sunspot Cycle (Sep1843 - Mar 1855)


1 ARMA (6,4) 37.01347 26.67266 62.85951 0.297524

2 ARMA(4,3) 36.85102 26.74996 66.02246 0.296860

3 ARMA (4,4) 43.45420 30.97244 55.55537 0.393127

4 ARMA (5,3) 36.91985 26.71517 64.07458 0.297030

5 ARMA (3,3) 43.32231 30.79417 55.33579 0.393812



1 ARMA (6,5) 0.729 0.723 17.138 -614.66 8.593 8.675 8.626 0.993

2 ARMA(6,4) 0.728 0.722 17.161 -614.71 8.593 8.676 8.627 1.178

3 ARMA (4,4) 0.771 0.768 15.689 -306.53 8.424 8.486 8.449 1.222

4 ARMA (6,2) 0.733 0.728 16.997 -613.37 8.574 8.657 8.608 1.222

5 ARMA (5,3) 0.760 0.757 16.067 -607.35 8.477 8.539 8.502 1.285

Dynamics Forecasted Evolution of Sunspot Cycle (Mar1855 - Feb 1867)


1 ARMA (6,5) 34.69173 26.74974 165.8685 0.393126

2 ARMA(6,4) 34.05871 26.32748 173.7933 0.318617

3 ARMA (4,4) 53.58256 43.77341 149.2235 0.856167

4 ARMA (6,2) 34.27413 26.50601 172.0539 0.385227

5 ARMA (5,3) 55.30789 45.35533 122.1337 0.903059



210

1 ARMA (6,5) 0.724 0.717 24.884 -649.72 9.339 9.423 9.373 0.680

2 ARMA(6,4) 0.750 0.745 23.624 -642.34 9.233 9.316 9.268 0.710

3 ARMA (5,4) 0.737 0.732 24.238 -645.69 9.281 9.365 9.315 0.873

4 ARMA (3,3) 0.801 0.797 21.115 -626.13 9.002 9.086 9.036 0.886

5 ARMA (5,3) 0.769 0.764 22.742 -636.62 9.152 9.236 9.188 0.905

Dynamics Forecasted Evolution of Sunspot Cycle (Feb1867 - Sep 1878)


1 ARMA (6,5) 52.74751 41.71703 774.1111 0.490512

2 ARMA(6,4) 51.34360 40.82203 807.2583 0.467277

3 ARMA (5,4) 51.64772 40.83656 788.2092 0.475558

4 ARMA (3,3) 56.57545 42.82194 619.2955 0.576678

5 ARMA (5,3) 50.26837 39.90352 818.8082 0.453341



1 ARMA (5,4) 0.596 0.587 17.341 -606.7 8.603 8.686 8.637 0.877

2 ARMA(5,3) 0.627 0.619 16.656 -600.50 8.514 8.597 8.548 0.888

3 ARMA (6,6) 0.570 0.560 17.898 -611.79 8.673 8.756 8.707 0.908

4 ARMA (6,4) 0.570 0.560 17.897 -610.87 8.660 8.743 8.694 0.977

5 ARMA (4,4) 0.631 0.623 16.574 -600.41 8.513 8.596 8.547 1.056

Dynamics Forecasted Evolution of Sunspot Cycle (Sep1878 - Jun 1890)


1 ARMA (5,4) 28.20895 23.20957 384.5670 0.413804

2 ARMA(5,3) 26.94070 22.33445 409.9513 0.384873

3 ARMA (6,6) 31.12448 25.16658 329.3777 0.494468

4 ARMA (6,4) 27.37615 22.67076 394.1263 0.394094

5 ARMA (4,4) 31.58116 25.16165 317.31740 0.516661



1 ARMA (6,4) 0.720 0.714 16.375 -624.57 8.494 8.575 8.527 1.122

2 ARMA (6,2) 0.723 0.717 16.282 -623.79 8.484 8.565 8.517 1.244

3 ARMA (4,4) 0.756 0.751 15.278 -614.38 8.356 8.437 8.389 1.254

4 ARMA (4,3) 0.748 0.743 15.519 -616.10 8.380 8.461 8.413 1.269

5 ARMA (5,4) 0.707 0.698 16.817 -627.95 8.540 8.621 8.573 1.272

Dynamics Forecasted Evolution of Sunspot Cycle (Jun1890 - Sep 1902)


1 ARMA (6,4) 34.15608 28.14172 429.9099 0.440684

2 ARMA(6,2) 34.40949 28.24120 424.3318 0.446793

3 ARMA (4,4) 36.86275 29.26016 351.6125 0.520870

4 ARMA (4,3) 33.39690 27.52467 429.3716 0.429832

5 ARMA (5,4) 33.19800 27.39173 441.9179 0.418577



1 ARMA (5,6) 0.422 0.409 20.242 -600.95 8.896 8.982 8.931 0.980

2 ARMA(4,6) 0.436 0.423 19.993 -599.17 8.870 8.956 8.905 1.007

3 ARMA (5,5) 0.486 0.474 19.092 -594.02 8.794 8.880 8.829 1.060

4 ARMA (4,4) 0.526 0.515 18.333 -588.36 8.711 8.797 8.756 1.061

5 ARMA (5,3) 0.419 0.406 20.289 -602.26 8.916 9.001 8.950 1.064

211

Dynamics Forecasted Evolution of Sunspot Cycle (Sep 1902 - Dec 1913)


1 ARMA (5,6) 26.60654 22.16956 582.2514 0.369668

2 ARMA(4,6) 26.33349 21.89815 586.2954 0.363862

3 ARMA (5,5) 30.20454 25.05934 469.8335 0.471314

4 ARMA (4,4) 30.33743 25.05251 447.7918 0.480269

5 ARMA (5,3) 27.75297 23.29113 547.1076 0.400474

Sunspot Cycle (Dec 1913 - May 1923)


1 ARMA (6,4) 0.483 0.469 24.750 -526.71 9.311 9.407 9.350 0.687

2 ARMA (5,4) 0.468 0.454 25.110 -528.20 9.337 9.433 9.376 0.743

3 ARMA (4,3) 0.592 0.580 22.006 -513.36 9.077 9.123 9.115 0.744

4 ARMA (5,5) 0.473 0.459 24.984 -528.49 9.342 9.438 9.380 0.810

5 ARMA (5,3) 0.546 0.533 23.207 -519.35 9.182 9.278 9.221 0.953

Dynamics Forecasted Evolution of Sunspot Cycle (Dec 1913 - May 1923)


1 ARMA (6,4) 33.70195 26.41578 142.1956 0.341615

2 ARMA(5,4) 33.40646 26.17204 143.7501 0.338521

3 ARMA (4,3) 33.59338 26.19764 139.4187 0.345463

4 ARMA (5,5) 38.64505 30.04723 124.6412 0.434757

5 ARMA (5,3) 33.02818 26.04595 149.5811 0.329163



1 ARMA (6,5) 0.559 0.548 19.307 -547.01 8.816 8.907 8.853 0.565

2 ARMA(6,4) 0.584 0.574 18.760 -543.59 8.761 8.852 8.798 0.722

3 ARMA (4,4) 0.628 0.618 17.749 -537.13 8.658 8.746 8.695 0.792

4 ARMA (3,3) 0.666 0.658 16.814 -530.06 8.545 8.635 8.582 0.914

5 ARMA (5,3) 0.628 0.619 17.742 -536.60 8.650 8.740 8.686 0.920

Dynamics Forecasted Evolution of Sunspot Cycle (May 1923 - Sep 1933)


1 ARMA (6,5) 28.41330 23.46526 286.6359 0.340449

2 ARMA(6,4) 28.92684 23.72642 284.4390 0.351530

3 ARMA (4,4) 34.14833 27.18319 225.7652 0.470814

4 ARMA (3,3) 34.83879 27.46117 226.5581 0.491752

5 ARMA (5,3) 28.94486 23.72149 276.0542 0.354408



1 ARMA (6,4) 0.601 0.592 25.898 -584.19 9.411 9.502 9.448 0.709

2 ARMA(4,4) 0.654 0.646 24.117 -575.64 9.274 9.365 9.311 0.908

3 ARMA (3,3) 0.710 0.703 22.091 -564.41 9.094 9.185 9.131 0.937

4 ARMA (5,3) 0.644 0.635 24.468 -576.77 9.292 9.383 9.329 0.940

5 ARMA (6,2) 0.684 0.676 23.073 -569.45 9.182 9.272 9.218 0.951

Dynamics Forecasted Evolution of Sunspot Cycle (Sep 1933 - Jan 1944)


1 ARMA (6,4) 41.14730 32.48182 90.65688 0.355584

2 ARMA(4,4) 49.84429 38.45141 79.25677 0.491490

3 ARMA (3,3) 51.63485 39.71464 103.6560 0.526209

212

4 ARMA (5,3) 39.64346 31.48136 94.59616 0.336577

5 ARMA (6,2) 40.50563 32.10883 92.99856 0.346311



1 ARMA (6,4) 0.638 0.629 33.330 -601.22 9.922 10.014 9.959 0.583

2 ARMA(4,4) 0.719 0.713 29.323 -586.07 9.673 9.765 9.711 0.661

3 ARMA (6,2) 0.677 0.669 31.478 -594.25 9.806 9.898 9.843 0.867

4 ARMA (3,3) 0.750 0.743 27.733 -578.74 9.553 9.645 9.591 0.869

5 ARMA (5,3) 0.675 0.667 31.563 -594.15 9.806 9.898 9.843 0.907

Dynamics Forecasted Evolution of Sunspot Cycle (Jan 1944 - Feb 1954)


1 ARMA (6,4) 58.17556 46.03504 487.9669 0.395454

2 ARMA(4,4) 69.35137 53.93403 374.3422 0.539219

3 ARMA (6,2) 52.26395 45.41299 496.7139 0.385152

4 ARMA (3,3) 69.75429 53.79824 380.6224 0.549886

5 ARMA (5,3) 55.25653 44.02761 508.1254 0.366021



1 ARMA (6,5) 0.781 0.775 33.895 -639.27 9.973 10.062 10.009 0.663

2 ARMA(5,4) 0.815 0.808 31.334 -628.68 9.809 9.898 9.845 0.836

3 ARMA (5,3) 0.854 0.850 27.611 -612.63 9.560 9.649 9.561 0.855

4 ARMA (6,4) 0.794 0.792 32.655 -643.31 9.896 9.985 9.932 0.887

5 ARMA (4,3) 0.848 0.844 28.216 -614.84 8.594 9.683 9.630 0.978

Dynamics Forecasted Evolution of Sunspot Cycle (Feb 1954 - Oct 1964)


1 ARMA (6,5) 86.23169 66.88760 126.8783 0.522011

2 ARMA(5,4) 82.96249 64.61203 125.9374 0.491867

3 ARMA (5,3) 82.12713 64.21752 127.4141 0.483381

4 ARMA (6,4) 83.16715 65.28142 132.5322 0.489885

5 ARMA (4,3) 81.95903 64.00049 155.9173 0.485348



1 ARMA (5,3) 0.751 0.745 18.814 -609.96 8.771 8.855 8.805 0.947

2 ARMA(6,4) 0.684 0.677 21.205 -626.80 9.011 9.096 9.046 0.980

3 ARMA (4,3) 0.739 0.733 19.278 -613.09 8.816 8.900 8.850 1.048

4 ARMA (3,3) 0.781 0.776 17.662 -601.12 8.645 8.729 8.679 1.085

5 ARMA (6,2) 0.717 0.711 20.050 -619.01 8.900 8.984 8.934 1.114

Dynamics Forecasted Evolution of Sunspot Cycle (Oct 1964 - May 1976)


1 ARMA (5,3) 38.99557 32.35205 88.63447 0.328025

2 ARMA(6,4) 39.73977 32.97053 88.18282 0.335997

3 ARMA (4,3) 39.00213 32.19808 88.14809 0.329886

4 ARMA (3,3) 47.63696 38.07346 78.84340 0.454724

5 ARMA (6,2) 39.89365 33.13202 88.41728 0.337784



213

1 ARMA (6,4) 0.695 0.687 31.995 -582.02 9.849 9.943 9.887 0.828

2 ARMA(6,2) 0.705 0.697 31.486 -580.04 9.816 9.909 9.854 1.026

3 ARMA (3,3) 0.773 0.767 27.641 -564.29 9.551 9.644 9.589 1.050

4 ARMA (5,3) 0.731 0.724 30.069 -574.29 9.719 9.813 9.757 1.014

5 ARMA (5,2) 0.746 0.739 29.228 -570.97 9.663 9.757 9.701 1.126

Dynamics Forecasted Evolution of Sunspot Cycle (May1976 - Mar 1986)


1 ARMA (6,4) 63.33039 52.49356 120.4046 0.393470

2 ARMA(6,2) 62.69861 52.05456 121.8634 0.386909

3 ARMA (3,3) 75.57275 60.66088 99.64765 0.540982

4 ARMA (5,3) 61.20026 50.64551 121.8423 0.375419

5 ARMA (5,2) 61.73283 51.04290 121.2593 0.380368



1 ARMA (6,5) 0.758 0.752 29.114 -595.39 9.667 9.759 9.704 0.834

2 ARMA(6,4) 0.775 0.770 28.083 -590.87 9.594 9.686 9.631 0.946

3 ARMA (5,6) 0.766 0.760 28.653 -592.93 9.628 9.719 9.665 1.011

4 ARMA (5,4) 0.764 0.758 28.777 -593.41 9.636 9.727 9.673 1.076

5 ARMA (4,5) 0.781 0.776 27.690 -588.33 9.554 9.645 9.591 1.117

Dynamics Forecasted Evolution of Sunspot Cycle (Mar1986 - Jun1996)


1 ARMA (6,5) 67.60252 53.31430 120.4492 0.466438

2 ARMA(6,4) 66.42465 52.50765 122.5996 0.452597

3 ARMA (5,6) 65.05685 51.96270 132.4873 0.439658

4 ARMA (5,4) 65.13699 52.01062 132.6014 0.440693

5 ARMA (4,5) 64.18852 50.90771 126.8260 0.433141



1 ARMA (6,4) 0.714 0.708 23.056 -638.59 9.180 9.264 9.214 0.880

2 ARMA(5,4) 0.733 0.727 22.274 -633.79 9.111 9.195 9.145 0.964

3 ARMA (4,3) 0.788 0.783 19.866 -617.50 8.879 8.963 8.913 1.008

4 ARMA (3,4) 0.791 0.787 19.712 -615.96 8.857 8.940 8.890 1.022

5 ARMA (3,3) 0.792 0.787 19.666 -616.22 8.860 8.944 8.894 1.073

Dynamics Forecasted Evolution of Sunspot Cycle (Jun1996 - Jan 2008)


1 ARMA (6,4) 45.06462 36.50313 187.5552 0.384967

2 ARMA(5,4) 46.62530 37.34643 178.4829 0.410093

3 ARMA (4,3) 45.43849 36.59640 194.0759 0.394859

4 ARMA (3,4) 43.79293 35.46916 204.3410 0.373450

5 ARMA (3,3) 52.87249 41.31675 160.2933 0.518691



1 ARMA (6,6) 0.709 0.708 24.088 -13941.6 9.205 9.213 9.208 0.818

2 ARMA(6,5) 0.712 0.711 23.969 -13926.0 9.195 9.203 9.196 0.823

3 ARMA (6,4) 0.724 0.724 23.438 -13858.0 9.150 9.158 9.153 0.881

4 ARMA (5,5) 0.736 0.740 22.921 -13790.9 9.106 9.113 9.108 0.893

5 ARMA (5,4) 0.732 0.732 23.105 -13814.7 9.121 9.129 9.124 0.933

214

Dynamics Forecasted Evolution of Sunspot Cycle (Aug1755 - Jan 2008)


1 ARMA (6,6) 44.59329 35.32464 465.4666 0.372824

2 ARMA(6,5) 44.55660 35.32846 468.5673 0.371180

3 ARMA (6,4) 44.54175 35.33142 469.6662 0.370626

4 ARMA (5,5) 44.59715 35.30753 464.3263 0.373367

5 ARMA (5,4) 44.54582 35.31252 468.8461 0.370987

Table 1.5: AR(p) - GRACH(1, 1) diagnostic checking and forecasting

evolution of sunspots cycles

Dynamics Forecasted Evolution

RMSE MAE MAPE U

27.61325 21.41193 56.70953 0.348243


Method : ML ARCH- Normal Distribution

GARCH = 11.338 + 0.227*Resid(-1)^2 + 0.775*GARCH(-1)

Variable Coefficient Std. Error z-statistics Prob

AR(2)

Resid(-1)^2

GARCH(-1)

0.824378

0.226696

0.774968

0.047120

0.140947

0.110833

17.49543

1.608383

6.992243

0.0000

0.1078

0.0000

R-Square 0.51076 Durbin-Watson Stat 1.360450

Adjusted R-Square 0.506394 Akaike Info Criterion 9.108496

S.E. of Regression 27.54760 Schwarz – Criterion 9.228505

Log likelihood -514.1843 Hannan-Quinn Criter 9.157201


RMSE MAE MAPE U

48.95338 37.48065 115.7679 0.501998









AR(2)

Resid(-1)^2

GARCH(-1)

0.833626

0.181855

0.776212

0.045517

0.077604

0.080125

18.31450

2.340945

9.687482

0.0000

0.0192

0.0000





215

AR(2)

Resid(-1)^2

GARCH(-1)

0.927650

0.273756

0.740552

0.043225

0.141865

0.112860

21.46112

1.929694

6.561700

0.0000

0.0536

0.0000






RMSE MAE MAPE U

81.25105 62.90952 80.41010 0.859162





AR(2)

Resid(-1)^2

GARCH(-1)

0.999993

0.350577

0.639280

1.39E-05

0.095717

0.066112

7213617

3.662653

9.669657

0.0000

0.0002

0.0000






RMSE MAE MAPE U

69.70664 53.90236 84.66272 0.837476



GARCH = -0.0848 + 0.5254*Resid(-1)^2 + 0.6533*GARCH(-1)


AR(2)

Resid(-1)^2

GARCH(-1)

0.923855

0.525379

0.653287

0.019588

0.136829

0.058282

47.16485

13.83969

11.20897

0.0000

0.0001

0.0000






RMSE MAE MAPE U

29.84910 23.36523 87.63108 0.9998639



GARCH = 19.466+ 0.1187*Resid(-1)^2 + 0.8839*GARCH(-1)

216


AR(2)

Resid(-1)^2

GARCH(-1)

0.779258

0.118733

0.883948

0.065885

0.023108

0.021690

11.82746

5.138217

40.75431

0.0000

0.0000

0.0000






RMSE MAE MAPE U

23.87355 16.61689 81.56241 0.867214





AR(3)

Resid(-1)^2

GARCH(-1)

0.717154

0.197108

0.512512

0.065084

0.088165

0.197635

11.01844

2.23579

2.593227

0.0000

0.0254

0.0095






RMSE MAE MAPE U

26.97423 22.29147 148.6877 0.342102





AR(2)

Resid(-1)^2

GARCH(-1)

0.999994

0.357004

0.621699

2.15E-05

0.168878

0.162796

46403.42

2.113973

3.818886

0.0000

0.0345

0.0001






RMSE MAE MAPE U

76.51305 59.98412 82.81522 0.863572



217



AR(2)

Resid(-1)^2

GARCH(-1)

0.892649

0.169634

0.826009

0.0406181

0.049536

0.057499

19.32935

3.424440

14.25403

0.0000

0.0006

0.0000






RMSE MAE MAPE U

54.20801 41.01302 59.41300 0.590749





AR(2)

Resid(-1)^2

GARCH(-1)

0.939019

0.162706

0.823016

0.031614

0.077090

0.068796

29.70280

2.110597

12.09390

0.0000

0.0348

0.0000






RMSE MAE MAPE U

46.84207 36.86088 163.9772 0.677314





AR(2)

Resid(-1)^2

GARCH(-1)

0.939751

0.267000

0.767531

0.029445

0.101809

0.069816

31.91558

2.622550

10.99364

0.0000

0.0085

0.0000






RMSE MAE MAPE U

66.48149 48.35055 184.8052 0.845125


218




AR(2)

Resid(-1)^2

GARCH(-1)

0.890694

0.306069

0.750207

0.044479

0.087816

0.062365

20.02497

3.485335

12.02933

0.0000

0.0005

0.0000






RMSE MAE MAPE U

39.33113 29.41939 110.6451 0.841212





AR(2)

Resid(-1)^2

GARCH(-1)

0.903458

0.240492

0.767205

0.039764

0.117877

0.090512

22.72052

2.040189

8.476297

0.0000

0.0413

0.0000






RMSE MAE MAPE U

41.43307 30.79801 168.5865 0.703145





AR(2)

Resid(-1)^2

GARCH(-1)

0.803072

0.250610

0.789312

0.040800

0.065314

0.035812

19.68333

3.836991

22.04036

0.0000

0.0001

0.0000






RMSE MAE MAPE U

37.13987 28.22192 163.2702 0.755452

219





AR(3)

Resid(-1)^2

GARCH(-1)

0.856972

0.218223

0.758773

0.05336

0.120566

0.103634

16.05851

1.809981

7.321672

0.0000

0.0703

0.0000






RMSE MAE MAPE U

50.69620 39.37540 80.53190 0.747014





AR(2)

Resid(-1)^2

GARCH(-1)

0.908219

0.291231

0.693905

0.036128

0.137997

0.129705

25.13905

2.110422

5.349893

0.0000

0.0348

0.0000






RMSE MAE MAPE U

41.61186 32.01947 136.7743 0.701884





AR(3)

Resid(-1)^2

GARCH(-1)

0.920729

0.184035

0.828438

0.040200

0.101441

0.074076

22.90396

1.81214

11.18357

0.0000

0.0696

0.0000






220

RMSE MAE MAPE U

66.90030 53.93148 94.93236 0.874060





AR(2)

Resid(-1)^2

GARCH(-1)

0.955562

0.139156

0.857469

0.035750

0.066739

0.055771

26.72916

2.085097

15.37487

0.0000

0.0371

0.0000






RMSE MAE MAPE U

88.43862 70.49887 127.5823 0.627014





AR(2)

Resid(-1)^2

GARCH(-1)

0.933207

0.234765

0.785609

0.025720

0.089860

0.068461

36.28382

2.612577

11.47520

0.0000

0.0090

0.0000






RMSE MAE MAPE U

110.3128 84.95945 101.2019 0.910967





AR(2)

Resid(-1)^2

GARCH(-1)

0.953305

0.044846

0.895546

0.034193

0.049779

0.087291

27.88030

0.900918

10.25934

0.0000

0.3676

0.0000





221


RMSE MAE MAPE U

59.36716 47.77713 71.04261 0.921958





AR(2)

Resid(-1)^2

GARCH(-1)

0.99992

0.183786

0.183309

1.41E-05

0.086960

0.060714

70702.72

2.113467

12.19096

0.0000

0.0346

0.0000






RMSE MAE MAPE U

92.03096 73.19906 81.00142 0.908955





AR(2)

Resid(-1)^2

GARCH(-1)

0.937443

0.184360

0.833166

0.046924

0.083106

0.061641

19.97779

2.218375

13.51636

0.0000

0.0265

0.0000






RMSE MAE MAPE U

84.07413 63.14946 90.53969 0.901397





AR(2)

Resid(-1)^2

GARCH(-1)

0.915932

0.201586

0.789028

0.048648

0.082006

0.074668

18.81267

2.458183

10.56714

0.0000

0.0140

0.0000



222




RMSE MAE MAPE U

60.54590 46.23687 105.7920 0.925559





AR(2)

Resid(-1)^2

GARCH(-1)

0.907941

0.238439

0.781679

0.007499

0.015721

0.011738

121.0810

15.16704

66.59540

0.0000

0.0000

0.0000






RMSE MAE MAPE U

52.01005 37.04478 238.9409 0.612406

Table 1.6: Normality test for AR (p)-GARCH (1, 1) model of sunspot cycles

Duration AR-GARCH Mean Median Std.D Skewness Kurtosis Jur-Bera

Aug 1755 - Mar 1766 AR(2)-G(1,1) 0.142 0.073 0.992 0.3053 3.1963 2.1936

Mar 1766 - Aug 1775 AR(2)-G(1,1) 0.152 0.196 0.993 -0.481 3.0194 0.0450

Aug 1775 - Jun 1784 AR(2)-G(1,1) 0.177 0.012 0.985 1.1670 7.1835 102.31

Jun 1784 - Jun 1798 AR(2)-G(1,1) -0.04 -0.115 1.025 -0.1061 4.0362 7.8780

Jun 1798 - Sep 1810 AR(2)-G(1,1) 0.179 0.051 0.960 0.4299 5.0149 30.269

Sep1810 - Dec 1823 AR(2)-G(1,1) 0.266 0.032 0.964 0.9403 4.1162 31.8829

Dec1823 - Oct 1833 AR(3)-G(1,1) 0.102 0.061 0.997 0.4802 3.5677 6.1715

Oct1833 - Sep 1843 AR(2)-G(1,1) 0.044 0.014 1.017 0.3433 2.7908 2.5764

Sep1843 - Mar 1855 AR(2)-G(1,1) 0.178 0.073 0.985 0.882 5.0451 42.226

Mar1855 - Feb 1867 AR(2)-G(1,1) 0.120 0.020 0.993 0.4680 3.292 5.5845

Feb1867 - Sep 1878 AR(2)-G(1,1) 0.146 0.095 0.990 0.4649 3.9492 10.299

Sep1878 - Jun 1890 AR(2)-G(1,1) 0.226 0.083 0.974 0.5547 3.2357 7.6100

Jun1890 - Sep 1902 AR(2)-G(1,1) 0.149 0.149 0.990 0.4791 4.1853 14.327

Sep 1902 - Dec 1913 AR(2)-G(1,1) 0.243 0.169 1.010 0.2638 3.1268 1.6680

Dec1913 - May 1923 AR(3)-G(1,1) 0.102 0.061 0.997 0.4800 3.5680 6.1715

May 1923 - Sep 1933 AR(2)-G(1,1) 0.169 0.078 0.986 0.034 2.8421 0.1552

Sep 1933 - Jan 1944 AR(3)-G(1,1) 0.227 0.119 0.973 0.1460 2.6545 1.0660

Jan 1944 - Feb 1954 AR(2)-G(1,1) 0.145 0.106 0.988 0.8278 4.5516 26.171

Feb 1954 - Oct 1964 AR(2)-G(1,1) 0.198 0.104 0.981 0.1059 3.1181 0.3159

Oct 1964 - May 1976 AR(2)-G(1,1) 0.131 -0.023 0.990 0.5212 3.4326 7.3407

May1976 - Mar 1986 AR(2)-G(1,1) 0.143 0.053 0.989 0.2668 3.5184 2.7442

Mar1986 - Jun1996 AR(2)-G(1,1) 0.129 0.060 1.001 0.6264 4.1593 15.0531

Jun1996 - Jan 2008 AR(2)-G(1,1) 0.151 -0.013 0.988 0.6279 3.8123 13.046

Aug1755 - Jan 2008 AR(2)-G(1,1) 0.135 0.050 0.991 0.5309 4.123 301.6

223

TABLE 1.7: ARMA (p,q) - GRACH(1, 1) of sunspots cycles


RMSE MAE MAPE U

32.67503 25.67541 56.50119 0.453746





AR(5)

MA(1)

GARCH(-1)

0.533846

0.536496

0.819233

0.095292

0.101409

0.081786

5.602184

5.290412

10.01677

0.0000

0.0000

0.0000






RMSE MAE MAPE U

43.75567 33.31524 124.7012 0.406759





AR(1)

MA(1)

GARCH(-1)

0.971002

-0.653490

0.761148

0.012483

0.071574

0.080710

77.78319

-9.130225

9.430652

0.0000

0.0000

0.0000









AR(2)

MA(2)

GARCH(-1)

0.942377

-0.344875

0.796377

0.031142

0.120105

0.092550

30.26041

-2.871453

8.604828

0.0000

0.0041

0.0000





224


RMSE MAE MAPE U

82.97762 64.99252 88.91661 0.907096





AR(2)

MA(2)

GARCH(-1)

0.963098

-0.460583

0.646251

0.011744

0.087155

0.108109

82.00897

-5.284628

5.975009

0.0000

0.0000

0.0000






RMSE MAE MAPE U

69.78911 53.91451 82.29495 0.841223





AR(3)

MA(3)

GARCH(-1)

0.935858

-0.283606

0.68167

0.031534

0.098061

0.101790

29.67735

-2.892135

6.696870

0.0000

0.0038

0.0000






RMSE MAE MAPE U

28.93611 22.58619 85.55745 0.922685





AR(3)

MA(3)

GARCH(-1)

0.948818

-0.504034

0.892899

0.037621

0.088419

0.018509

25.22040

-5.700533

48.24127

0.0000

0.0000

0.0000



225




RMSE MAE MAPE U

25.17780 17.81173 79.46162 0.972678





AR(3)

MA(3)

GARCH(-1)

1.000000

-0.492451

0.744837

0.0145516

0.104489

0.206923

68.89037

-4.712929

3.599420

0.0000

0.0000

0.0003






RMSE MAE MAPE U

35.06270 27.96897 113.0161 0.486173





AR(3)

MA(2)

GARCH(-1)

0.717264

0.457177

0.665718

0.074861

0.102429

0.060208

9.604436

4.463361

7.722201

0.0000

0.0000

0.0000






RMSE MAE MAPE U

70.49102 53.86759 79.03446 0.727303





AR(4)

MA(2)

GARCH(-1)

0.820607

0.713805

0.828213

0.067158

0.084455

0.059027

12.21903

8.451875

14.03098

0.0000

0.0000

0.0000

226






RMSE MAE MAPE U

55.68955 42.46238 61.37514 0.609577





AR(4)

MA(4)

GARCH(-1)

0.935145

-0.261549

0.823638

0.027516

0.096453

0.085286

33.90542

-2.711680

4.657320

0.0000

0.0067

0.0000






RMSE MAE MAPE U

50.33423 40.31416 150.4605 0.766811





AR(5)

MA(1)

GARCH(-1)

0.810985

0.372377

0.800595

0.040918

0.097022

0.075716

19.81984

2.105510

10.57360

0.0000

0.0001

0.0000






RMSE MAE MAPE U

68.21752 50.46720 163.9832 0.874876





227

AR(2)

MA(2)

GARCH(-1)

0.977805

-0.480838

0.726032

0.020228

0.086600

0.087548

48.33850

-5.52371

8.292922

0.0000

0.0000

0.0000






RMSE MAE MAPE U

40.16201 30.27203 105.1265 0.876150





AR(2)

MA(2)

GARCH(-1)

0.960991

-0.535407

0.741708

0.013212

0.090477

0.072713

72.73458

-5.917594

10.20053

0.0000

0.0000

0.0000






RMSE MAE MAPE U

44.99864 33.86295 108.7426 0.836094





AR(2)

MA(2)

GARCH(-1)

0.986754

-0.552862

0.784486

0.024632

0.080533

0.051384

40.05999

-6.244711

15.26708

0.0000

0.00000

.0000






RMSE MAE MAPE U

36.86236 27.61363 117.5639 0.744690




228


AR(3)

MA(3)

GARCH(-1)

0.913874

-0.467862

0.552189

0.023667

0.115770

0.119920

38.61419

-4.041320

4.604632

0.0000

0.0000

0.0001






RMSE MAE MAPE U

53.34212 41.91097 79.70615 0.826760





AR(2)

MA(2)

GARCH(-1)

0.958244

-0.459230

0.677347

0.020393

0.111471

0.123939

46.98964

-4.119713

5.465159

0.0000

0.0000

0.0000






RMSE MAE MAPE U

44.927819 35.31193 116.2730 0.814867





AR(2)

MA(2)

GARCH(-1)

0.974799

-0.419687

0.840734

0.023533

0.100495

0.106100

41.42327

-4.176213

7.923969

0.0000

0.0000

0.0000






RMSE MAE MAPE U

67.16046 54.07226 96.16188 0.888186


229




AR(5)

MA(3)

GARCH(-1)

0.854521

0.346542

0.812699

0.052139

0.098825

0.072919

16.38938

3.506625

11.14520

0.0000

0.0005

0.0000






RMSE MAE MAPE U

89.57113 72.19864 122.4058 0.895654





AR(5)

MA(3)

GARCH(-1)

0.854124

0.497641

0.695872

0.035637

0.108373

0.102716

23.96715

4.591918

6.774746

0.0000

0.0000

0.0000






RMSE MAE MAPE U

113.2087 88.62173 87.83657 0.939852



GARCH = 618.4769 + 0.1576*Resid(-1)^2 – 1.0532*GARCH(-1)


AR(2)

MA(2)

GARCH(-1)

0.981676

-0.364651

0.902341

0.020894

0.100502

0.073421

46.98375

-3.628281

12.28997

0.0000

0.0003

0.0000






RMSE MAE MAPE U

63.56454 51.95468 74.98162 0.333498

230





AR(2)

MA(2)

GARCH(-1)

0.969742

-0.420841

0.818978

0.20513

0.116761

0.071133

47.27442

-3.604301

11.51331

0.0000

0.0003

0.0000






RMSE MAE MAPE U

91.74100 72.87236 80.66910 0.800143





AR(6)

MA(1)

GARCH(-1)

0.834506

0.380998

0.838039

0.047834

0.108274

0.045692

17.44584

3.518828

18.34105

0.0000

0.0000

0.0000






RMSE MAE MAPE U

87.08569 66.45792 75.91525 0.772314





AR(2)

MA(2)

GARCH(-1)

0.969016

-0.498636

0.819252

0.021416

0.110312

0.059092

45.24690

-4.520227

13.86391

0.0000

0.0000

0.0000






231

RMSE MAE MAPE U

65.31769 50.61469 93.96324 0.802811





AR(2)

MA(2)

GARCH(-1)

0.895023

0.083322

0.780889

0.008475

0.018512

0.011874

105.6073

4.500889

65.76430

0.0000

0.0000

0.0000






RMSE MAE MAPE U

54.37098 38.55961 205.0083 0.778433

Table 1.8: Test of Normality ARMA (p, q)-GARCH (1,1) process of Sunspot Cycles

cycles ARMA-GARCH Mean Median Std.D Skewness Kurtosis Jur-Bera

1 ARMA(2,2)-G(1,1) 0.143 -0.202 0.991 0.3875 3.4860 4.4639

2 ARMA(5,1)-G(1,1) 0.167 0.107 0.991 0.256 2.6522 1.913

3 ARMA(1,1)-G(1,1) 0.197 0.045 0.983 0.9322 6.1950 61.007

4 ARMA(2,2)-G(1,1) 0.140 0.121 0.991 -0.283 4.226 12.839

5 ARMA(3,3)-G(1,1) 0.153 0.061 0.988 0.1280 3.501 1.9499

6 ARMA(3,3)-G(1,1) 0.190 -0.027 0.983 1.1371 4.882 58.097

7 ARMA(3,3)-G(1,1) 0.097 0.018 1.009 0.4005 2.8365 3.3141

8 ARMA(3,2)-G(1,1) 0.305 0.178 0.954 0.6251 3.3824 8.5454

9 ARMA(4,2)-G(1,1) 0.181 0.093 0.984 0.8470 4.5254 30.096

10 ARMA(4,4)-G(1,1) 0.145 0.045 0.992 0.089 2.8840 0.2716

11 ARMA(5,1)-G(1,1) 0.249 0.322 0.968 0.046 2.9809 0.2716

12 ARMA(2,2)-G(1,1) 0.130 -0.003 0.994 0.6397 3.0275 9.6901

13 ARMA(2,2)-G(1,1) 0.140 0.0615 0.992 0.4077 3.8820 808934

14 ARMA(2,2)-G(1,1) 0.030 -0.112 0.996 0.9258 4.1931 27.4940

15 ARMA(3,3)-G(1,1) 0.238 0.098 0.973 0.4546 3.1964 4.1101

16 ARMA(2,2)-G(1,1) 0.165 0.148 0.987 0.1321 2.6787 0.9011

17 ARMA(2,2)-G(1,1) 0.167 0.095 0.985 0.2786 3.2299 1.8924

18 ARMA(5,3)-G(1,1) 0.258 0.100 0.965 0.2909 2.6303 2.4153

19 ARMA(5,3)-G(1,1) 0.258 0.297 0.966 -0.1425 2.4166 2.2661

20 ARMA(2,2)-G(1,1) 0.125 0.017 0.996 0.4562 3.1727 5.0298

21 ARMA(2,2)-G(1,1) 0.142 0.129 0.989 0.0006 3.7090 2.4922

22 ARMA(6,1)-G(1,1) 0.206 0.108 0.977 0.1528 3.6409 2.6051

23 ARMA(2,2)-G(1,1) 0.118 -0.050 0.993 0.6868 4.5192 24.4696

1-23 ARMA(2,2)-G(1,1) 0.142 0.318 0.990 0.5121 3.8258 218.54

Table 1.9: Root Mean Square Error (RMSE) of ARMA, AR-GARCH and ARMA-GARCH of sunspot

cycles

232

Duration ARMA (p, q) AR(p)–GARCH (1,1) ARMA(p,q) )–GARCH (1,1)

Aug 1755 - Mar 1766 (6,5) 22.96202 AR(2) 27.61330 (6,5) 32.67503

Mar 1766 - Aug 1775 (5,3) 37.92649 AR(2) 48.9534 (5,1) 43.75567

Aug 1775 - Jun 1784 (6,4) 56.72805 AR(2) 81.25105 (1,1) 82.97762

Jun 1784 - Jun 1798 (5,4) 54.15024 AR(2) 69.70664 (2,2) 69.78911

Jun 1798 - Sep 1810 (5,3) 19.61195 AR(2) 29.84910 (3,3) 28.93611

Sep1810 - Dec 1823 (3,4) 17.58213 AR(2) 23.87355 (3,3) 25.17780

Dec1823 - Oct 1833 (5,3) 24.84239 AR(3) 26.97423 (3,3) 35.06270

Oct1833 - Sep 1843 (4,6) 47.89669 AR(2) 76.51305 (3,2) 70.49102

Sep1843 - Mar 1855 (4,3) 36.85102 AR(2) 54.20801 (4,2) 55.68955

Mar1855 - Feb 1867 (6,4) 34.05871 AR(2) 46.84207 (4,4) 50.33423

Feb1867 - Sep 1878 (6,4) 51.34360 AR(2) 66.48149 (5,1) 68.21752

Sep1878 - Jun 1890 (5,3) 26.94070 AR(2) 39.33113 (2,2) 40.16201

Jun1890 - Sep 1902 (5,4) 33.19800 AR(2) 41.43307 (2,2) 44.99864

Sep 1902 - Dec 1913 (4,6) 26.33349 AR(2) 37.13987 (2,2) 36.86236

Dec1913 - May 1923 (5,4) 33.40646 AR(3) 50.69620 (3,3) 53.34212

May 1923 - Sep 1933 (6,5) 28.41330 AR(2) 41.61186 (2,2) 44.92782

Sep 1933 - Jan 1944 (6,2) 40.50563 AR(3) 66.90030 (2,2) 67.16046

Jan 1944 - Feb 1954 (6,4) 58.17556 AR(2) 88.43862 (5,3) 89.57113

Feb 1954 - Oct 1964 (4,3) 81.95903 AR(2) 110.3128 (5,3) 113.2087

Oct 1964 - May 1976 (4,3) 39.00213 AR(2) 59.36716 (2,2) 63.56454

May1976 - Mar 1986 (5,3) 61.20026 AR(2) 92.03096 (2,2) 91.74100

Mar1986 - Jun1996 (5,6) 65.05685 AR(2) 84.07413 (6,1) 87.08569

Jun1996 - Jan 2008 (6,4) 45.06462 AR(2) 60.54590 (2,2) 65.31769

Aug1755 - Jan 2008 (6,4) 44.54175 AR(2) 52.01005 (2,2) 54.37098

Table 1.10: Mean Absolute Error (MAE) of ARMA, AR-GARCH and ARMA-GARCH of sunspot

cycles


Aug 1755 - Mar 1766 (4,5) 18.43576 AR(2) 21.4119 (6,5) 25.67541

Mar 1766 - Aug 1775 (5,3) 29.55789 AR(2) 37.4807 (5,1) 33.31524

Aug 1775 - Jun 1784 (6,4) 43.64520 AR(2) 62.90952 (1,1) 64.99252

Jun 1784 - Jun 1798 (5,6) 41.59299 AR(2) 53.90236 (2,2) 53.91451

Jun 1798 - Sep 1810 (5,3) 16.77907 AR(2) 23.36523 (3,3) 22.58619

Sep1810 - Dec 1823 (6,4) 13.25282 AR(2) 16.61689 (3,3) 22.58619

Dec1823 - Oct 1833 (5,3) 20.57404 AR(3) 22.29147 (3,3) 27.96897

Oct1833 - Sep 1843 (4,6) 37.64070 AR(2) 59.98412 (3,2) 53.86759

Sep1843 - Mar 1855 (6,4) 26.67266 AR(2) 41.01302 (4,2) 42.46238

Mar1855 - Feb 1867 (6,4) 26.32748 AR(2) 36.86088 (4,4) 40.31416

Feb1867 - Sep 1878 (6,4) 40.82203 AR(2) 48.35055 (5,1) 50.46720

Sep1878 - Jun 1890 (5,3) 22.33445 AR(2) 29.41939 (2,2) 30.27203

Jun1890 - Sep 1902 (5,4) 27.39173 AR(2) 30.79801 (2,2) 33.86295

Sep 1902 - Dec 1913 (4,6) 21.89815 AR(2) 28.22192 (2,2) 27.61363

Dec1913 - May 1923 (5,3) 26.04595 AR(3) 39.37540 (3,3) 41.91097

May 1923 - Sep 1933 (6,5) 23.46526 AR(2) 32.01947 (2,2) 35.31193

Sep 1933 - Jan 1944 (6,2) 31.48136 AR(3) 32.01947 (2,2) 54.07226

Jan 1944 - Feb 1954 (5,3) 44.02761 AR(2) 70.49887 (5,3) 72.19864

Feb 1954 - Oct 1964 (4,3) 64.00049 AR(2) 84.95945 (5,3) 88.62173

Oct 1964 - May 1976 (4,3) 32.19808 AR(2) 47.77713 (2,2) 51.95468

May1976 - Mar 1986 (5,3) 50.64551 AR(2) 73.19906 (2,2) 72.87236

233

Mar1986 - Jun1996 (4,5) 50.90771 AR(2) 63.14946 (6,1) 66.45792

Jun1996 - Jan 2008 (3,4) 35.46916 AR(2) 46.23687 (2,2) 50.61469

Aug1755 - Jan 2008 (6,4) 35.30753 AR(2) 37.04478 (2,2) 38.55961

Table1.11: Mean Absolute Percentage Error (MAPE) of ARMA, AR-GARCH and ARMA-GARCH of

sunspot cycles


Aug 1755 - Mar 1766 (4,4) 55.73550 AR(2) 56.7095 (6,5) 56.50119

Mar 1766 - Aug 1775 (2,2) 145.5506 AR(2) 115.7679 (5,1) 124.7012

Aug 1775 - Jun 1784 (5,5) 102.4356 AR(2) 80.41010 (1,1) 88.91661

Jun 1784 - Jun 1798 (3,3) 141.1219 AR(2) 84.66272 (2,2) 82.29495

Jun 1798 - Sep 1810 (5,3) 222.5072 AR(2) 87.63108 (3,3) 85.55745

Sep1810 - Dec 1823 (6,6) 215.0192 AR(2) 81.56241 (3,3) 79.46162

Dec1823 - Oct 1833 (5,5) 125.1342 AR(3) 148.6877 (3,3) 113.0161

Oct1833 - Sep 1843 (3,3) 101.5025 AR(2) 82.81522 (3,2) 79.03446

Sep1843 - Mar 1855 (3,3) 55.33579 AR(2) 59.41300 (4,2) 61.37514

Mar1855 - Feb 1867 (4,4) 149.2235 AR(2) 163.9772 (4,4) 150.4605

Feb1867 - Sep 1878 (3,3) 619.2955 AR(2) 184.8052 (5,1) 163.9832

Sep1878 - Jun 1890 (4,4) 317.3174 AR(2) 110.6451 (2,2) 105.1265

Jun1890 - Sep 1902 (4,4) 351.6125 AR(2) 168.5865 (2,2) 108.7426

Sep 1902 - Dec 1913 (5,5) 469.8335 AR(2) 163.2702 (2,2) 117.5639

Dec1913 - May 1923 (5,5) 124.6412 AR(3) 80.53190 (3,3) 79.70615

May 1923 - Sep 1933 (4,4) 225.7652 AR(2) 136.7743 (2,2) 116.2730

Sep 1933 - Jan 1944 (4,4) 79.25677 AR(3) 94.93236 (2,2) 96.16188

Jan 1944 - Feb 1954 (4,4) 374.3422 AR(2) 127.5823 (5,3) 122.4058

Feb 1954 - Oct 1964 (5,4) 125.9374 AR(2) 101.2019 (5,3) 87.83657

Oct 1964 - May 1976 (3,3) 78.84340 AR(2) 71.04261 (2,2) 74.98162

May1976 - Mar 1986 (3,3) 99.64765 AR(2) 81.00142 (2,2) 80.66910

Mar1986 - Jun1996 (6,5) 120.4492 AR(2) 90.53969 (6,1) 75.91525

Jun1996 - Jan 2008 (5,4) 178.4829 AR(2) 105.7920 (2,2) 93.96324

Aug1755 - Jan 2008 (5,5) 464.3263 AR(2) 238.9409 (2,2) 205.0083

Table 1.12: Theil’s U-Statistics (U test) of ARMA, AR-GARCH and ARMA-GARCH of sunspot cycles


Aug 1755 - Mar 1766 (6,5) 0.251178 AR(2) 0.34824 (6,5) 0.453746

Mar 1766 - Aug 1775 (5,3) 0.309054 AR(2) 0.501998 (5,1) 0.406759

Aug 1775 - Jun 1784 (6,4) 0.411658 AR(2) 0.859162 (1,1) 0.907096

Jun 1784 - Jun 1798 (5,6) 0.465585 AR(2) 0.837476 (2,2) 0.841223

Jun 1798 - Sep 1810 (6,4) 0.416733 AR(2) 0.999864 (3,3) 0.922685

Sep1810 - Dec 1823 (3,4) 0.418201 AR(2) 0.867214 (3,3) 0.972678

Dec1823 - Oct 1833 (5,3) 0.294667 AR(3) 0.342102 (3,3) 0.486173

Oct1833 - Sep 1843 (4,6) 0.353636 AR(2) 0.863572 (3,2) 0.727303

Sep1843 - Mar 1855 (4,3) 0.296860 AR(2) 0.590749 (4,2) 0.609577

Mar1855 - Feb 1867 (6,4) 0.318617 AR(2) 0.677314 (4,4) 0.766811

Feb1867 - Sep 1878 (6,4) 0.467277 AR(2) 0.845125 (5,1) 0.874876

Sep1878 - Jun 1890 (5,3) 0.384873 AR(2) 0.841212 (2,2) 0.876150

Jun1890 - Sep 1902 (4,3) 0.429832 AR(2) 0.703145 (2,2) 0.836094

Sep 1902 - Dec 1913 (4,6) 0.363862 AR(2) 0.755452 (2,2) 0.744690

Dec1913 - May 1923 (5,3) 0.329163 AR(3) 0.747014 (3,3) 0.826760

234

May 1923 - Sep 1933 (6,5) 0.340449 AR(2) 0.701884 (2,2) 0.814867

Sep 1933 - Jan 1944 (5,3) 0.336577 AR(3) 0.874060 (2,2) 0.888186

Jan 1944 - Feb 1954 (5,3) 0.366021 AR(2) 0.627014 (5,3) 0.895654

Feb 1954 - Oct 1964 (5,3) 0.483381 AR(2) 0.910967 (5,3) 0.939852

Oct 1964 - May 1976 (4,3) 0.329886 AR(2) 0.921958 (2,2) 0.333498

May1976 - Mar 1986 (5,3) 0.375419 AR(2) 0.908955 (2,2) 0.800143

Mar1986 - Jun1996 (4,5) 0.433141 AR(2) 0.901397 (6,1) 0.772314

Jun1996 - Jan 2008 (3,4) 0.373450 AR(2) 0.925559 (2,2) 0.802811

Aug1755 - Jan 2008 (6,4) 0.370626 AR(2) 0.612406 (2,2) 0.778433

Table 2.1: The numerical relationship between different fractal dimensions, Hurst exponent,

autocorrelation coefficient and spectral exponents can be express as,

Cy

cles

Duration FDS FDA HS HA αS αA CS∇ CA∇ Persistency

1 Aug 1755 - Mar 1766 1.181 1.367 0.819 0.633 2.638 2.266 0.556 0.202 Persistent

2 Mar 1766 - Aug 1775 1.27 1.407 0.73 0.593 2.246 2.186 0.376 0.138 Persistent

3 Aug 1775 - Jun 1784 1.327 1.587 0.672 0.413 2.344 1.826 0.269 -0.114 Persistent

4 Jun 1784 - Jun 1798 1.318 1.446 0.682 0.554 2.364 2.108 0.287 0.078 Persistent

5 Jun 1798 - Sep 1810 1.099 1.380 0.901 0.620 2.802 2.240 0.744 0.181 Persistent

6 Sep1810 - Dec 1823 1.136 1.363 0.864 0.637 2.728 2.274 0.656 0.209 Persistent

7 Dec1823 - Oct 1833 1.177 1.163 0.823 0.837 2.646 2.674 0.565 0.595 Persistent

8 Oct1833 - Sep 1843 1.296 1.223 0.704 0.772 2.408 2.554 0.327 0.458 Persistent

9 Sep1843 - Mar 1855 1.275 1.470 0.725 0.530 2.45 2.060 0.366 0.042 Persistent

10 Mar1855 - Feb 1867 1.267 1.313 0.733 0.687 2.466 2.374 0.381 0.296 Persistent

11 Feb1867 - Sep 1878 1.298 1.347 0.702 0.653 2.404 2.306 0.323 0.236 Persistent

12 Sep1878 - Jun 1890 1.165 1.463 0.835 0.537 2.67 2.074 0.591 0.053 Persistent

13 Jun1890 - Sep 1902 1.252 1.376 0.748 0.624 2.496 2.248 0.410 0.188 Persistent

14 Sep 1902 - Dec 1913 1.141 1.400 0.859 0.600 2.718 2.200 0.645 0.149 Persistent

15 Dec1913 - May 1923 1.235 1.479 0.765 0.521 2.53 2.042 0.443 0.030 Persistent

16 May 1923 - Sep 1933 1.149 1.391 0.851 0.604 2.702 2.218 0.627 0.155 Persistent

17 Sep 1933 - Jan 1944 1.206 1.526 0.794 0.474 2.588 1.948 0.503 -0.035 Persistent

18 Jan 1944 - Feb 1954 1.194 1.373 0.806 0.627 2.612 2.254 0.528 0.193 Persistent

19 Feb 1954 - Oct 1964 1.266 1.410 0.734 0.590 2.468 2.180 0.383 0.133 Persistent

20 Oct 1964 - May 1976 1.193 1.355 0.807 0.645 2.614 2.290 0.530 0.223 Persistent

21 May1976 - Mar 1986 1.198 1.187 0.802 0.813 2.604 2.626 0.520 0.543 Persistent

22 Mar1986 - Jun1996 1.225 1.436 0.775 0.563 2.55 2.128 0.464 0.091 Persistent

23 Jun1996 - Jan 2008 1.213 1.451 0.787 0.549 2.574 2.098 0.489 0.070 Persistent

24 Aug1755 - Jan 2008 1.002 1.354 0.998 0.649 2.996 2.110 0.994 0.229 Persistent

Table 2.2: The numerical relationship between different Higuchi’s fractal dimensions FDH and spectral

exponents αH of sunspot cycles can be express as,

Cycles Duration FDH αH

1 Aug 1755 - Mar 1766 1.621 1.8

2 Mar 1766 - Aug 1775 1.620 1.8

3 Aug 1775 - Jun 1784 1.432 2.1

4 Jun 1784 - Jun 1798 1.454 2.1

5 Jun 1798 - Sep 1810 1.486 2.0

6 Sep1810 - Dec 1823 1.648 1.7

7 Dec1823 - Oct 1833 1.623 1.8

235

8 Oct1833 - Sep 1843 1.507 2.0

9 Sep1843 - Mar 1855 1.550 1.9

10 Mar1855 - Feb 1867 1.465 2.0

11 Feb1867 - Sep 1878 1.455 2.1

12 Sep1878 - Jun 1890 1.580 1.8

13 Jun1890 - Sep 1902 1.588 1.8

14 Sep 1902 - Dec 1913 1.679 1.6

15 Dec1913 - May 1923 1.564 1.9

16 May 1923 - Sep 1933 1.521 2.0

17 Sep 1933 - Jan 1944 1.503 2.0

18 Jan 1944 - Feb 1954 1.448 2.1

19 Feb 1954 - Oct 1964 1.354 2.3

20 Oct 1964 - May 1976 1.542 1.9

21 May1976 - Mar 1986 1.442 2.1

22 Mar1986 - Jun1996 1.402 2.2

23 Jun1996 - Jan 2008 1.481 2.0

Table 2.3: The numerical relationship between self-similar and self-affine fractal dimensions, Hurst

exponent, Probability distribution persistency of ENSO cycles

Cycles Duration FDS HES axd+ c FDA HEA Persistency

1 1866-1872 1.059 0.941 19.95x1.059 + 0.17 1.208 0.792 Persistent

2 1873-1879 1.010 0.990 84.75x1.01 + 0.03 1.090 0.910 Persistent

3 1880-1886 1.002 0.998 83.43x1.002 + 0.005 1.361 0.640 Persistent

4 1887-1893 1.006 0.994 84.08x1.006+ 0.02 1.156 0.844 Persistent

5 1894-1900 1.004 0.996 83.79x1.004+ 0.01 1.129 0.871 Persistent

6 1901-1907 1.005 0.995 83.45x1.005+ 0.01 1.171 0.829 Persistent

7 1908-1914 1.003 0.997 83.55x1.003+ 0.008 1.014 0.986 Persistent

8 1915-1921 1.007 0.993 84.37x1.007+ 0.02 1.041 0.959 Persistent

9 1922-1928 1.003 0.997 83.5x1.003+ 0.009 1.264 0.736 Persistent

10 1929-1935 1.001 0.999 83.24x1.001+ 0.004 1.314 0.686 Persistent

11 1936-1942 1.005 0.995 83.91x1.005+ 0.01 1.001 0.999 Persistent

12 1943-1949 1.002 0.998 83.29x1.002+ 0.006 1.291 0.709 Persistent

13 1950-1956 1.003 0.997 83.51x1.003+ 0.008 1.016 0.984 Persistent

14 1957-1963 1.001 0.999 83.28x1.001+ 0.004 1.160 0.840 Persistent

15 1964-1970 1.005 0.995 83.76x1.005+ 0.015 1.202 0.798 Persistent

16 1971-1977 1.006 0.994 84.09x1.006+ 0.02 1.088 0.913 Persistent

17 1978-1984 1.005 0.995 83.8x1.005+ 0.016 1.122 0.879 Persistent

18 1985-1991 1.005 0.995 83.96x1.005+ 0.017 1.113 0.887 Persistent

19 1992-1998 1.007 0.993 84.32x1.007+ 0.025 1.118 0.882 Persistent

20 1999-2005 1.001 0.999 83.19x1.001+ 0.002 1.072 0.930 Persistent

21 2006-2012 1.006 0.994 84.12x1.006+ 0.022 1.178 0.822 Persistent

1-21 1866-2012 1 1 1763.8x1+ 0.0005 1.224 0.776 Persistent

22 1981-2000 1 1 239.1x1+ 0.001 1.268 0.732 Persistent

Table 2.4: The numerical relationship between self-similar and self-affine fractal dimensions, Hurst

exponents and persistency in the same interval of sunspots and ENSO

Parameters Sunspots ENSO Persistency

Cycle Duration FDS FDA HS HA FDS FDA HS HA Persistent

1 May1976-Mar 1986 1.198 1.187 0.802 0.813 1.001 1.205 0.999 0.795 Persistent

236

2 Mar1986- Jun1996 1.225 1.436 0.775 0.563 1.002 1.080 0.998 0.920 Persistent

3 Jun1996- Jan 2008 1.213 1.451 0.787 0.549 1.002 1.105 0.998 0.895 Persistent

Table 3.1: Sunspot Cycles probability distribution

Cycles Duration MEAN Std.

Dev

Distribution ADT KST CST Parameters

1 Aug 1755 -

Mar 1766

42.001 27.925 Generalized Pareto 12.039 0.463 -- α =

β = 63.963

γ = -0.700

2 Mar 1766 -

Aug 1775

56.959 40.33 Generalized Pareto 4.4108 0.065 -- α =

β = 85.884

γ = -0.5299

3 Aug 1775 -

Jun 1784

68.504 52.54 Generalized Pareto 4.559 0.0741 -- α = −

β = 101.44

γ = -0.42722

4 Jun 1784 -

Jun 1798

60.801 43.02 Generalized Pareto 0.717 0.049 3.642 α = −

β = 95.497

γ = -0.4873

5 Jun 1798 -

Sep 1810


β = 49.461

γ = -0.6527

6 Sep1810 -

Dec 1823


β = 24.961

γ = -0.17961

7 Dec1823 -

Oct 1833

39.303 27.17 Generalized Pareto 12.387 0.081

--- α = −1.1533

β = 72.997

γ = -0.77102

8 Oct1833 -

Sep 1843


--- α = −2.6559

β = 101.25

γ = -0. 48812

9 Sep1843 -

Mar 1855

59.369 40.15 Gen. Extreme Value 0.385 0.052 5.845 k = 0.03061

σ = 29.598

μ = 0. 03061

10 Mar1855 -

Feb 1867


9.065 α = −4.037

β = 83.244

γ = -0. 66846

11 Feb1867 -

Sep 1878


9.837 α = −6.1558

β = 77.048

γ = -0. 28971

12 Sep1878 -

Jun 1890


16.321 α = −4.6614

β = 53.533

γ = -0. 43861

13 Jun1890 -

Sep 1902


25.193 α = −4.8239

β = 61.465

γ = -0. 45076

14 Sep 1902 -

Dec 1913


-- α = −4.907

β = 57.984

γ = -0. 5426

15 Dec1913 -

May 1923


-- α = −0.89396

β = 70.49

γ = -0. 48655

237

16 May 1923 –

Sep 1933


10.4 α = −2.9088

β = 68.284

γ = -0. 59905

17 Sep 1933 -

Jan 1944


-- α = −2.9908

β = 97.886

γ = -0. 61441

18 Jan 1944 -

Feb 1954


6.982 α = −7.0494

β= 130.47

γ = -0. 60125

19 Feb 1954 -

Oct 1964


13.806 α = −12.116

β= 151.2

γ = -0. 49471

20 Oct 1964 -

May 1976


-- α = −0.22406

β = 109.56

γ = -0. 81137

21 May1976 -

Mar 1986


8.389 α = −7.8826

β = 162.73

γ = -0. 77936

22 Mar1986 -

Jun1996


7.543 α = −7.1456

β= 125.54

γ = -0. 50715

23 Jun1996 -

Jan 2008


-- α = −4.2933

β = 96.833

γ = -0. 55605

24 Jan 2008-

Process

30.289 46.53 Gen. Extreme Value 5.508 0.154 -- k = -0.62362

σ = 49.473

μ = 22.155

1-23 Aug1755 -

Jan 2008


15.904 α = 1.8212

β = 67.815

γ = -0. 24514

Table 3.2: The Probability distributions of ENSO cycles (1866-2012)

Cycles Duration MEAN Std.

Dev


1 1866-1872 0.159 1.141 Gen. Extreme Value 0.296 0.067 4.265 k = -0.0971

σ = 1.0101

μ= -0. 3343

2 1873-1879 0.748 2.295 Generalized Pareto 22.606 0.087 -- α = −1.513

β = 7.4476

γ = -2.8151

3 1880-1886 -0.127 1.063 Gen. Extreme Value 0.1704 0.051 2.123 k= -0.54425

σ = 1.1686

μ = -0.3677

4 1887-1893 0.166 1.172 Gen. Extreme Value 0.407 0.064

2.725 k = -0. 4319

σ = 1.232

μ = -0. 163

5 1894-1900 -0.325 1.230 Gen. Extreme Value 0.235 0.087 3.488 k =-0. 2931

σ = 1.2269

μ =-0.754

6 1901-1907 -0.092 1.280 Logistic 0.208 0.060 3.023 σ = 621

μ =-0.0918

238

7 1908-1914 -0.200 1.076 Gen. Extreme Value 0.286 0.050 5.924 k =-0.13346

σ = 0.9908

μ =-0.6531

8 1915-1921 0.373 1.381 Gen. Extreme Value 0.282 0.061 3.390 k = 0.17923

σ = 1.3295

μ =-0.186

9 1922-1928 0.035 0.905 Gen. Extreme Value 0.526 0.078 5.526 k = -0.4696

σ = 0.99794

μ = -0.2058

10 1929-1935 0.070 0.881 Gen. Extreme Value 4.394 0.072

-- k =-0. 3838

σ = 0.87404

μ =-0. 1935

11 1936-1942 -0.274 1.154 Gen. Extreme Value 0.123 0.055 0.683 k = -0. 2916

σ = 1.1584

μ = -0. 679

12 1943-1949 -0.267 1.024 Generalized Pareto 8.366 0.070 -- α = −1.3903

β = 2.7067

γ = -1.0708


β = 3.8763

γ = -1.0454

14 1957-1963 -0.100 0.775 Gen. Extreme Value 4.044 0.044 -- k =-0. 4323

σ = 0.81599

μ = -0.3173

15 1964-1970 -0.127 0.892 Gen. Extreme Value 0.283 0.054

2.79 k =-0. 2158

σ = 0.87581

μ = -0.4734


= 4.8623

γ = -1.1491

17 1978-1984 -0.494 1.067 Gen. Extreme Value 8.564 0.101

-- k = -0. 7049

σ = 1.152

μ = -0.6404

18 1985-1991 -0.261 1.082 Gen. Extreme Value 0.232 0.056 4.020 k = -0.268

σ = 1.1043

μ = -0.6579

19 1992-1998 -0.714 1.182 Gen. Extreme Value 0.230 0.050 1.070 k = -0. 3241

σ = 1.253

μ = -0.1198


σ = 0.96223

μ = -0.4649

21 2006-2012 0.390 1.109 Gen. Extreme Value 0.272 0.058 2.845 k = -0. 2063

σ = 1.1028

μ= -0.0493


σ = 1.3029

μ = -0.7295

1-21 1866-2012 -0.057 1.147 Gen. Extreme Value 8.982 0.021 -- k = -0. 2773

σ = 1.1037

μ = -0.4582

Table 3.3: The probability distribution of active ENSO cycles (1981-2000)

239

Cycles Duration Mean Std.

Dev


1 May1976 -

Mar 1986

0.500 1.021 Gen. Extreme Value 11.881 0.06571 -- k = -0. 5799

σ = 1.1017

μ = -0. 7021

2 Mar1986 -

Jun1996

0.527 1.097 Gen. Extreme Value 0.25186 0.04856

3.3612 k = -0. 2392

σ = 1. 0941

μ = -0. 9415

3 Jun1996 -

Jan 2008

0.180 1.107 Gen. Extreme Value 0.21305 0.03841

0.21305 k= -0.40288

σ = 1. 0818

μ = -0. 1928

Table 3.4: Probability distribution of Fractal dimension of Sunspot cycles (1-23 and total 1-23) and Enso

cycles (1-22 and total 1-21)

Cycles FD Mean St. D Distribution ADT KST CST Parameters

Sunspots

FDS 1.212 0.0751 Gen. Extreme Value 0.17822 0.0791

0.17981 k = -0. 491

σ = 0.0818

μ = 1.1928

Sunspot FDA 1.387 0.0974 Gen. Extreme Value 4.2742 0.15377 -- k =-0. 4567

σ = 0.102

μ = 1.3606

ENSO FDS 0.580 0.3135 Generalized Pareto 8.2478 0.07311

-- α = 1.0007

β =0.00288

γ = 0.4908

ENSO FDA 1.158 0.093 Gen. Extreme Value 0.09756 0.0685

0.16655 k= -0. 1378

σ = 0.0911

μ = 1.117

Table 3.5: the fractional differencing parameter and heavy tails parameter of sunspot cycles

Cycles Duration HES 0 < dS <1/2 1< βS < 2 HEA 0 < dA <1/2 1< βA < 2

1 Aug 1755 - Mar 1766 0.819 0.319 1.362 0.633 0.133 1.734

2 Mar 1766 - Aug 1775 0.73 0.230 1.540 0.593 0.093 1.814

3 Aug 1775 - Jun 1784 0.672 0.172 1.656 0.500 0.00 2.000

4 Jun 1784 - Jun 1798 0.682 0.182 1.636 0.554 0.054 1.892

5 Jun 1798 - Sep 1810 0.901 0.401 1.198 0.620 0.120 1.760

6 Sep1810 - Dec 1823 0.864 0.364 1.272 0.637 0.137 1.726

7 Dec1823 - Oct 1833 0.823 0.323 1.354 0.837 0.337 1.326

8 Oct1833 - Sep 1843 0.704 0.204 1.592 0.772 0.272 1.456

9 Sep1843 - Mar 1855 0.725 0.225 1.550 0.530 0.030 1.940

10 Mar1855 - Feb 1867 0.733 0.233 1.534 0.687 0.187 1.626

11 Feb1867 - Sep 1878 0.702 0.202 1.596 0.653 0.153 1.694

12 Sep1878 - Jun 1890 0.835 0.335 1.330 0.537 0.037 1.926

13 Jun1890 - Sep 1902 0.748 0.248 1.504 0.624 0.124 1.752

14 Sep 1902 - Dec 1913 0.859 0.359 1.282 0.600 0.100 1.800

15 Dec1913 - May 1923 0.765 0.265 1.470 0.521 0.021 1.958

16 May 1923 - Sep 1933 0.851 0.351 1.298 0.604 0.104 1.792

17 Sep 1933 - Jan 1944 0.794 0.294 1.412 0.500 0.00 2.000

18 Jan 1944 - Feb 1954 0.806 0.306 1.388 0.627 0.127 1.746

240

19 Feb 1954 - Oct 1964 0.734 0.234 1.532 0.590 0.090 1.820

20 Oct 1964 - May 1976 0.807 0.307 1.386 0.645 0.145 1.710

21 May1976 - Mar 1986 0.802 0.302 1.396 0.813 0.313 1.374

22 Mar1986 - Jun1996 0.775 0.275 1.450 0.563 0.060 1.874

23 Jun1996 - Jan 2008 0.787 0.287 1.426 0.549 0.049 1.902

24 Aug1755 - Jan 2008 0.998 0.498 1.004 0.649 0.149 1.702

Table 3.6: the fractional differencing parameter and heavy tails parameter of ENSO cycles

Cycles Duration HES 0 < dS <1/2 1< βS < 2 HEA 0 < dA <1/2 1< βA < 2

1 1866-1872 0.941 0.441 1.118 0.792 0.292 1.416

2 1873-1879 0.990 0.490 1.020 0.910 0.410 1.180

3 1880-1886 0.998 0.498 1.004 0.640 0.140 1.720

4 1887-1893 0.994 0.494 1.012 0.844 0.344 1.312

5 1894-1900 0.996 0.496 1.008 0.871 0.371 1.258

6 1901-1907 0.995 0.495 1.010 0.829 0.329 1.342

7 1908-1914 0.997 0.497 1.006 0.986 0.486 1.028

8 1915-1921 0.993 0.493 1.014 0.959 0.459 1.082

9 1922-1928 0.997 0.497 1.006 0.736 0.236 1.528

10 1929-1935 0.999 0.499 1.002 0.686 0.186 1.628

11 1936-1942 0.995 0.495 1.010 0.999 0.499 1.002

12 1943-1949 0.998 0.498 1.004 0.709 0.209 1.582

13 1950-1956 0.997 0.497 1.006 0.984 0.484 1.032

14 1957-1963 0.999 0.499 1.002 0.840 0.340 1.320

15 1964-1970 0.995 0.495 1.010 0.798 0.298 1.404

16 1971-1977 0.994 0.494 1.012 0.913 0.413 1.174

17 1978-1984 0.995 0.495 1.010 0.879 0.379 1.242

18 1985-1991 0.995 0.495 1.010 0.887 0.387 1.226

19 1992-1998 0.993 0.493 1.014 0.882 0.382 1.236

20 1999-2005 0.999 0.499 1.002 0.930 0.430 1.140

21 2006-2012 0.994 0.494 1.012 0.822 0.322 1.356

22 1981-2000 1 0.50 1 0.732 0.232 1.536

23 1866-2012 1 0.50 1 0.776 0.276 1.448

Table 3.7: The strength of long range-correlation of sunspot cycles

Cycles Duration HES 1 < 𝛾𝑆 < 3 HEA -1 < 𝛾𝐴< 1

1 Aug 1755 - Mar 1766 0.819 2.638 0.633 0.266

2 Mar 1766 - Aug 1775 0.73 2.460 0.593 0.186

3 Aug 1775 - Jun 1784 0.672 2.344 0.500 0.00

4 Jun 1784 - Jun 1798 0.682 2.364 0.554 0.108

5 Jun 1798 - Sep 1810 0.901 2.802 0.620 0.240

6 Sep1810 - Dec 1823 0.864 2.728 0.637 0.274

7 Dec1823 - Oct 1833 0.823 2.646 0.837 0.674

8 Oct1833 - Sep 1843 0.704 2.408 0.772 0.544

9 Sep1843 - Mar 1855 0.725 2.450 0.530 0.060

10 Mar1855 - Feb 1867 0.733 2.466 0.687 0.374

11 Feb1867 - Sep 1878 0.702 2.404 0.653 0.306

12 Sep1878 - Jun 1890 0.835 2.670 0.537 0.074

241

13 Jun1890 - Sep 1902 0.748 2.496 0.624 0.248

14 Sep 1902 - Dec 1913 0.859 2.718 0.600 0.200

15 Dec1913 - May 1923 0.765 2.530 0.521 0.042

16 May 1923 - Sep 1933 0.851 2.702 0.604 0.208

17 Sep 1933 - Jan 1944 0.794 2.588 0.500 0.00

18 Jan 1944 - Feb 1954 0.806 2.612 0.627 0.254

19 Feb 1954 - Oct 1964 0.734 2.468 0.590 0.180

20 Oct 1964 - May 1976 0.807 2.614 0.645 0.290

21 May1976 - Mar 1986 0.802 2.604 0.813 0.626

22 Mar1986 - Jun1996 0.775 2.550 0.563 0.126

23 Jun1996 - Jan 2008 0.787 2.574 0.549 0.098

24 Aug1755 - Jan 2008 0.998 2.996 0.649 0.298

Table 3.8: The strength of long range-correlation of ENSO cycles (1-23)

Cycles Duration HES 1 < 𝛾𝑆 < 3 HEA -1 < 𝛾𝐴< 1

1 1866-1872 0.941 2.882 0.792 0.584

2 1873-1879 0.990 2.980 0.910 0.820

3 1880-1886 0.998 2.996 0.640 0.280

4 1887-1893 0.994 2.988 0.844 0.688

5 1894-1900 0.996 2.992 0.871 0.742

6 1901-1907 0.995 2.990 0.829 0.658

7 1908-1914 0.997 2.994 0.986 0.972

8 1915-1921 0.993 2.986 0.959 0.918

9 1922-1928 0.997 2.994 0.736 0.472

10 1929-1935 0.999 2.998 0.686 0.372

11 1936-1942 0.995 2.990 0.999 0.998

12 1943-1949 0.998 2.996 0.709 0.418

13 1950-1956 0.997 2.994 0.984 0.968

14 1957-1963 0.999 2.998 0.840 0.680

15 1964-1970 0.995 2.990 0.798 0.596

16 1971-1977 0.994 2.988 0.913 0.826

17 1978-1984 0.995 2.990 0.879 0.758

18 1985-1991 0.995 2.990 0.887 0.774

19 1992-1998 0.993 2.986 0.882 0.764

20 1999-2005 0.999 2.998 0.930 0.860

21 2006-2012 0.994 2.988 0.822 0.644

22 1981-2000 1 3 0.732 0.464

23 1866-2012 1 3 0.776 0.552

Table 3.9 (a): Augmented Dickey Fuller (ADF) test (stationary)

Self-Similar Tail Parameter (βS) Sunspots Cycles

Null Hypothesis H0: Self-Similar Tail Parameter (βS) of Sunspots Cycles has

a unit root

Exogenous: Constant

Lag Length: 0 (Automatics-based on SIC, MaxLag = 5)

t-statistics prob*


242


5% level

10% level

-3.759246

-2.98064

-3.638752

Table 3.9(b): Augmented Dickey Fuller (ADF) test (stationary) Self-Affine Tail

Parameter (βA) Sunspots Cycles

Null Hypothesis H0: Self-Affine Tail Parameter (βA) of Sunspots Cycles has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.759246

-2.998064

-2.638752

Table 3.10 (a): Augmented Dickey Fuller (ADF) test (stationary) Self-Similar Tail

Parameter (βS) ENSO Cycles

Null Hypothesis H0: Self-Similar Tail Parameter (βS) of ENSO Cycles has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.769597

-3.004861

-2.642242

Table 3.10 (b): Augmented Dickey Fuller (ADF) test (stationary) Self-Affine Tail

Parameter (βA) ENSO Cycles

Null Hypothesis H0: Self-Affine Tail Parameter (βA) of ENSO Cycles has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.769597

-3.004861

-2.642242

Table 3.11 (a): Augmented Dickey Fuller (ADF) test (stationary) Self-Similar strength

of long range-correlation (𝛾 S) Sunspots Cycles

243

Null Hypothesis H0: Self-Similar strength of long range-correlation (𝛾 S) of Sunspots Cycles has a unit

root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.752946

-2.998064

-2.638752

Table 3.11 (b): Augmented Dickey Fuller (ADF) test (stationary) Self-Affine strength of long

range-correlation (𝛾 A) Sunspots Cycles

Null Hypothesis H0: Self-Affine strength of long range-correlation (𝛾 A) of Sunspots Cycles has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.759246

-2.998064

-2.638752

Table 3.12 (a): Augmented Dickey Fuller (ADF) test (stationary) Self-Similar strength of long

range-correlation (𝛾 S) ENSO Cycles

Null Hypothesis H0: Self-Similar strength of long range-correlation (𝛾 S) of ENSO Cycles has a unit root

Exogenous: Constant


t-statistics prob*



5% level

10% level

-3.769597

-3.004861

-2.642242

Table 3.12 (b): Augmented Dickey Fuller (ADF) test (stationary) Self-Affine strength of long

range-correlation (𝛾 A) ENSO Cycles

Null Hypothesis H0: Self-Affine strength of long range-correlation (𝛾 A) of ENSO Cycles has a unit root

Exogenous: Constant


t-statistics prob*


244


5% level

10% level

-3.769597

-3.004861

-2.642242

Table 4.2(a): showing the first contour of AR9114

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

−𝐻𝜑

𝐸𝑇

||

−𝐻∅𝐸𝑇

𝐻𝑚𝑎𝑥∅𝑚𝑎𝑥

𝐸𝑚𝑎𝑥𝑇𝑎𝑣𝑔

||

1 -5×1040 3.5 0.16×1033 4200 -2.60×108 0.00020

2 -4.5×1040 3 0.15×1033 4200 -2.14×108 0.000164

3 -4×1040 2.5 0.14×1033 4200 -3.06×108 0.00017

4 -3.5×1040 2 0.13×1033 4200 -1.28×108 0.000071

5 -3×1040 1.5 0.12×1033 4200 -8.929×1010 0.0496

Table 4.2(b): showing the second contour of AR9114

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

−𝐻𝜑

𝐸𝑇

||

−𝐻∅𝐸𝑇



||

1 -5×1040 3.5 0.16×1033 4300 -2.54×108 0.00014

2 -4.5×1040 3 0.15×1033 4300 -2.09×108 0.000161

3 -4×1040 2.5 0.14×1033 4300 -1.66×108 0.000922

4 -3.5×1040 2 0.13×1033 4300 -1.25×108 0.000694

5 -3×1040 1.5 0.12×1033 4300 -8.721×1010 0.04845

Table 4.2(c): showing the third contour of AR9114

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

−𝐻𝜑

𝐸𝑇

||

−𝐻∅𝐸𝑇



||

1 -5×1040 3.5 0.16×1033 4400 -2.49×108 0.0001382

2 -4.5×1040 3 0.15×1033 4400 -2.05×108 0.0001139

3 -4×1040 2.5 0.14×1033 4400 -1.62×108 0.00090

4 -3.5×1040 2 0.13×1033 4400 -1.22×108 0.0000678

5 -3×1040 1.5 0.12×1033 4400 -8.523×1010 0.04735

Table 4.2(d): showing the forth contour of AR9114

245

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

−𝐻𝜑

𝐸𝑇

||

−𝐻∅𝐸𝑇



||

1 -5×1040 3.5 0.16×1033 4500 -2.43×108 0.000135

2 -4.5×1040 3 0.15×1033 4500 -2×109 0.000111

3 -4×1040 2.5 0.14×1033 4500 -1.59×109 0.000833

4 -3.5×1040 2 0.13×1033 4500 -1.197×109 0.0000665

5 -3×1040 1.5 0.12×1033 4500 -8.333×1010 0.04629

Table 4.3(a): showing the first contour of AR9114

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

+𝐻𝜑

𝐸𝑇

||

+𝐻∅𝐸𝑇

+𝐻𝑚𝑎𝑥∅𝑚𝑎𝑥


||

1 -2.4×1041 3.5 0.16×1033 4200 -1.25×1010 0.004661

2 -2×1041 3 0.15×1033 4200 -9.523×1011 0.3553

3 -1.5×1041 2.5 0.14×1033 4200 -6.377×1011 0.23795

4 -1×1041 2 0.13×1033 4200 -3.663×1011 0.13668

5 -0.5×1041 1.5 0.12×1033 4200 -1.488×1011 0.0555

Table 4.3(b): showing the second contour of AR9114

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

+𝐻𝜑

𝐸𝑇

||

+𝐻∅𝐸𝑇



||

1 -2.4×1041 3.5 0.16×1033 4300 -1.22×1010 0.004552

2 -2×1041 3 0.15×1033 4300 -9.302×1011 0.34709

3 -1.5×1041 2.5 0.14×1033 4300 -6.229×1011 0.23242

4 -1×1041 2 0.13×1033 4300 -3.578×1011 0.13351

5 -0.5×1041 1.5 0.12×1033 4300 -1.453×1011 0.054216

Table 4.3(c): showing the third contour of AR9114

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

+𝐻𝜑

𝐸𝑇

||

+𝐻∅𝐸𝑇



||

1 -2.4×1041 3.5 0.16×1033 4400 -1.193×1010 0.004451

2 -2×1041 3 0.15×1033 4400 -9.091×1011 0.33922

3 -1.5×1041 2.5 0.14×1033 4400 -6.088×1011 0.22716

4 -1×1041 2 0.13×1033 4400 -3.497×1011 0.13049

5 -0.5×1041 1.5 0.12×1033 4400 -1.420×1011 0.052985

246

Table 4.3(d): showing the forth contour of AR9114

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

+𝐻𝜑

𝐸𝑇

||

+𝐻∅𝐸𝑇



||

1 -2.4×1041 3.5 0.16×1033 4500 -1.167×1010 0.004354

2 -2×1041 3 0.15×1033 4500 -8.889×1011 0.331679

3 -1.5×1041 2.5 0.14×1033 4500 -5.952×1011 0.220896

4 -1×1041 2 0.13×1033 4500 -3.419×1011 0.127575

5 -0.5×1041 1.5 0.12×1033 4500 -1.389×1011 0.051828

Table 4.4 (a): showing the first contour of AR10696

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

−𝐻𝜑

𝐸𝑇

||

−𝐻∅𝐸𝑇



||

1 -2×1041 2.5 0.16×1033 4200 -3.16×1010 0.0176

2 -1.5×1041 2 0.15×1033 4200 -2.86×1010 0.0159

3 -1×1041 1.5 0.14×1033 4200 -2.42×1010 0.0134

4 -10×1041 1 0.13×1033 4200 -1.83×1010 0.0102

5 -10.5×1041 0.5 0.12×1033 4200 -1.04×1010 0.0578

Table 4.4 (b): showing the second contour of AR10696

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

−𝐻𝜑

𝐸𝑇

||

−𝐻∅𝐸𝑇



||

1 -8.5×1041 2.5 0.16×1033 4300 -3.03×1010 0.0168

2 -9×1041 2 0.15×1033 4300 -2.79×1010 0.0155

3 -9.5×1041 1.5 0.14×1033 4300 -2.367×1010 0.0132

4 -10×1041 1 0.13×1033 4300 -1.789×1010 0.0099

5 -10.5×1041 0.5 0.12×1033 4300 -1.017×1010 0.0057

Table 4.4 (c): showing the third contour of AR10696

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

−𝐻𝜑

𝐸𝑇

||

−𝐻∅𝐸𝑇



||

1 -8.5×1041 2.5 0.16×1033 4400 -3.018×1010 0.01677

247

2 -9×1041 2 0.15×1033 4400 -2.727×1010 0.01515

3 -9.5×1041 1.5 0.14×1033 4400 -2.313×1010 0.01285

4 -10×1041 1 0.13×1033 4400 -1.748×1010 0.0971

5 -10.5×1041 0.5 0.12×1033 4400 -9.943×1011 0.552

Table 4.4 (d): showing the forth contour of AR10696

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

−𝐻𝜑

𝐸𝑇

||

−𝐻∅𝐸𝑇



||

1 -8.5×1041 2.5 0.16×1033 4500 -2.951×1010 0.01394

2 -9×1041 2 0.15×1033 4500 -2.667×1010 0.01482

3 -9.5×1041 1.5 0.14×1033 4500 -2.262×1010 0.01257

4 -10×1041 1 0.13×1033 4500 -1.709×1010 0.00949

5 -10.5×1041 0.5 0.12×1033 4500 -9.722×1011 0.5401

Table 4.5 (a): showing the first contour of AR10696

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

+𝐻𝜑

𝐸𝑇

||

+𝐻∅𝐸𝑇



||

1 -2×1041 2.5 0.16×1033 4200 -7.44×1011 0.4089

2 -2.5×1041 2 0.15×1033 4200 -7.937×1011 0.5023

3 -3×1041 1.5 0.14×1033 4200 -7.653×1011 0.4844

4 -3.5×1041 1 0.13×1033 4200 -6.410×1011 0.4057

5 -4×1041 0.5 0.12×1033 4200 -3.968×1011 0.5211

Table 4.5 (b): showing the second contour of AR10696

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

+𝐻𝜑

𝐸𝑇

||

+𝐻∅𝐸𝑇



||

1 -2×1041 2.5 0.16×1033 4300 -7.27×1011 0.4601

2 -2.5×1041 2 0.15×1033 4300 -7.75×1011 0.4905

3 -3×1041 1.5 0.14×1033 4300 -7.48×1011 0.4734

4 -3.5×1041 1 0.13×1033 4300 -6.26×1011 0.3962

5 -4×1041 0.5 0.12×1033 4300 -3.88×1011 0.2456

Table 4.5 (c): showing the third contour of AR10696

248

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

+𝐻𝜑

𝐸𝑇

||

+𝐻∅𝐸𝑇



||

1 -2×1041 2.5 0.16×1033 4400 -7.10×1011 0.4494

2 -2.5×1041 2 0.15×1033 4400 -7.58×1011 0.4797

3 -3×1041 1.5 0.14×1033 4400 -7.31×1011 0.4627

4 -3.5×1041 1 0.13×1033 4400 -6.12×1011 0.3873

5 -4×1041 0.5 0.12×1033 4400 -3.79×1011 0.2399

Table 4.5 (d): showing the forth contour of AR10696

S.no Helicity (H)

unit Mx2/h

Tangential

velocity (φ)

unit ∅o/h

Energy (E)

unit egrs

Temperature

(T) unit K

+𝐻𝜑

𝐸𝑇

||

+𝐻∅𝐸𝑇



||

1 -2×1041 2.5 0.16×1033 4500 -6.94×1011 0.4392

2 -2.5×1041 2 0.15×1033 4500 -7.41×1011 0.4690

3 -3×1041 1.5 0.14×1033 4500 -7.14×1011 0.4519

4 -3.5×1041 1 0.13×1033 4500 -5.98×1011 0.3785

5 -4×1041 0.5 0.12×1033 4500 -3.70×1011 0.2342

morphology and image analysis of some solar …

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