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Novel Time Series Analysis and Prediction of Stock

Trading Using Fractal Theory and Time Delayed Neural

Network *

F umi nnr i

Yakuwa

Yasubikn

Dote

Mika

Yoneyama Shinji

U z u r a b a s h i

Panasonicushiro branch D e u a r t m e n t

of

C o m u u t e r D e u a r t m e n t of C o m u u t e r

Hokkaido Elec t r ic Power

8-1, Saiwai-cho,

Kushiro,

Sc ie nc e & S y s t e m s

Co . , Inc . Eng ine e r ing

Muroran Inst i tu te of

050, J A P A N T e c h n o l o g y

Phone: +S1-154-23-1114;

27-1,

Miz umoto -c ho ,

FAX: 181-154-23-2220

Muroran 050,

J A P A N

Phone: +SI-143-46-5432;

FAX: +SI-14346.5499

Abstrac t - The stock markets are well known fo r w ide

variations in prices over short and long terms. These

fluctuations are due

to

a large number of deals produced

by agents and act independently fro m each other.

However, even in the middle of the apparently chaotic

world, there are opportunities fo r m aking good

predictions [ I ] .

In this pap er the Nikkei stock prices over

1500

days

from July to

Oct. 2002 are analyzed and predicted using

a Hurst exponent (H), a fractal dimension (D), and an

autocorre~ation coefficient

(c).

They are

H

=

0.6699

D=2-H=1.3301 and C = 0.26558 over three days. This

obtained knowledge is embedded into the structure of our

developed time delayed neural network

121.

It is

confirmed that the obtained prediction accuracy is m uch

higher than that by a back propagation-type forward

neural network o r the short-term.

Although this predictor works fo r the short term. it

is embedded into our developedfiruy neural network [3]

t construct multi-blended local nonlinear models. It is

applied

to

general long term prediction whose more

accurate prediction is expected than that by the method

proposed in [I].

1

Introduction

The Nikkei

Average

Stock prices over

1500

days are

in the middle of the apparently chaotic world. In this

paper. on the basis of Zadeh’s proposal: i.e., “From

Manipulation of Measurements to Manipulation of

Perceptions-Computations with W ords” [25], that is a data

mining technology, knowledge easily comprehensible by

humans i s extracted by o btaining the features of the time

Sc ie nc e

&

S y s t e m s

Engineering Engineering Co.,Ltd .

Muroran

Inst i tu te of

Technology

27-1,

M i m m o t o - c h o ,

Muroran 050, J A P A N

Phone: +SI-143-46-5432;

FAX +SI-143-46-5499

Mobile &

S y s t e m

series using a Hurst exponent, a fractal analysis method,

and an autocorrelation analysis method .

In

order to extract

the knowledge, decision m aking rules com prehensible by

humans using the features are derived with rough set

theory [ 2 6 ] .Finally the knowledge is embedded into the

stmcture of the Time Delayed Neural Network (TDNN).

Th e accurate prediction i s obtained.

This paper is organized as follows.

In

Section

2

t ime

series analysis using fractal analysis is described. Section

3 illustrates the structure of neural networks for t ime

series. Section 4 des cribes short-term prediction u sing

TDNN . Som e conclusions are drawn in Section 5 .

2 Time Series Analysis using Fractal

Analysis

2.1

Fra c t a l

Fractal analysis provides a uniq ue insig ht into a wide

range of natural phenomena. Fractal objects are - those

which exhibit ‘self-similarity’. This m eans that the gen eral

shape of the object is repeated at arbitrarily smaller and

smaller scales. Coas tlines have thi s property: a particular

coastline viewed on a world map has the same character

as a small piece of i t seen on a local map. New details

appear at each smaller scale, so that the coastline always

appear s rough. Although true fractals repeat the detail to a

vanishingly small scale, examples in nature are self-

similar up to so me non-zero limit. Th e fractal dimension

measures how much complexity is being repeated at each

scale. A shape with a higher fractal dimension is more

complicated

or

‘rough‘ than on e with a lo wer dimension,

and

fills

more space. These dimensions are fractional: a

shape with fractal dimension of D=1.2, for example, fills

0-7803-7952-7/03/$17.00 003 IEEE. 134



more space than a one-dimensional curve, but less space

than a two-dimensional area. The fractal dimension

successfully tells much information about the geometry of

an object. Very realistic computer images of mountains,

clouds and plants can be produced by simple recursions

with the appropriate fractal dimension. Time series of

many natural phenomena are fractal. Small sections taken

from these series, one scaled by the appropriate factor,

cannot be distinguished from the whole signal. Being able

to recognize a time series as fractal means being able to

link information at different time scales. We call such sets

'self-affine' instead of self-similar because they scale by

different amounts

in

each axis direction.

There are many methods available for estimating the

fractal dimension of data sets. These lead to different

numerical results, yet little comparison of accuracy has

been made among them in the literature. We combine

two

methods which are known as the most popular for

assigning fractal dimensions to time series, the box-

cou ntin g method and rescaled range analysis.

2.2

Box-counting

The box-counting algorithm is intuitive and easy to

apply.

I t

can be applied to sets in any dimension, and has

been used on images of everything from river systems to

the clusters of galaxies. A fractal curve

is

a curve of

infinite detail, by virtue of its self-similarity. The len gth of

the curve is indefinite, increasing as the resolution of the

measuring instrument increases. The fractal dimension

determines the increase in detail, and therefore length, at

each resolution change. For a fractal, the length

L

as a

function of the resolution of the measurement device

6

is:

L ( 6 ) ocF

where D is an exponent known as the fractal

dimension. (For ordinary curves

~ 6 )

pproaches a

constant value as

6

decreases) Box-counting algorithms

measure L(S) for varying 6 by counting the number of

non-overlapping boxes of size

6

required to cover the

curve. These measurements are fitted to Eq. I )

to

obtain

an estimate of the fractal dimension, known as the box

dimension. A

fractal dimension can be assigned

to a

set of

time series data by plotting it as a function of time, and

calculating the box dimension.

Eq. (1)

will hold over

a

finite range of box-size; the smallest boxes will be of

width r where

r

is the resolution in time, and height 0

where

a is

the resolution of the magnitude of the time

series.

2.3 R I S method

The rescaled range analysis, also called as

R I S

o r

Hurst method, was invented by Hurst for the evaluation of

time dependent hydrological data

[8][9].

His original

work is related

to

the water reservoirs and the design of an

ideal storage on river Nile. After the detailed discussion of

this work by Mandelbrot [lO][ll], the method has

attracted much attention in many fields

of

science. For the

mathematical aspects of the me thod we refer to the papers

of M andelbrot

[19],

Feder

1121,

and Daley

[13].

Since its

earliest days the method was used for a number of

applications, whenever the question was the quantification

of long range statistical interdependence within time

series.

As

examples we can cite the analysis of the

asymmetry of solar activity [7][8], elaxation of stress

[SI,

problems in particle physics [IS], mechanical sliding in

solids

[19]. The

Hurst analysis is also used as a tool to

determine the self-similarity parameter of fractal signals

[20-231,

or

to

detect unwanted correlations in pseudo-

random number generaton

[23].

The Hurst exponent was

calculated for corrosion noise data in the work of Moon

and Skerry [24] where the corrosion resistance properties

of organic paint films was analyzed and a direct

relationship between the Hurst exponent and the corrosion

resistance of different coatings w as established. G reisiger

and Schauer [15] discussed the applicability of different

methods to the electrochemical potential and current noise

analysis. They concluded, that the Hurst exponent allows

the extraction

of

mechanistic information about corrosion

processes, hence suitable for characterizing coatings.

We give a brief introduction to the R I S method, in

lines

of

Feder's [I 91 work. L et the time coordinate,

t ,

he

discredited in terms of the time resolution, A t , as i = r / ,

The discrete time record of a given process is denoted by

x ,

=O,l ,--- ,N

if the total duration of the observation is

T

= N & .

According to the hasic idea of the R I S method

the time record is evaluated for a certain time interval,

called time lag, the length of which is

r

= j A t and begins

at to= luAt

.

Obviously, j <

N

and I < N hold. The

average of xI over the time lag is calculated as

3)

Next the accumulative deviation from the mean,

J ~ ~s evaluated as

(4)

1 = 4 . >

where

k

takes the values 15 k

5

135



In order to visualize the meaning of Eq. (4) let us

refer to the hydrological context in which the method was

devised by Hurst. Here

x is

the annual water input into a

reservoir in the ith year of a series of

N

years, and . v ~ ~

events with Gaussian distribution and zero mean. The

Hurst exponent for such a time record

is 1

2 .

For 112 <

H <

1 the time series is called

persistent,

i s

the

net

gain

or

loss

of stored water in the year

1

+ k .e.

i.e. an increasing trend in the past implies,

on

the average,

a continued increasing trend in the future,

or

a decreasing

trend in the past implies a decrease in the future. If,

O < H <1/2 prevails, the time series observed is anti

persistent, i.e. an increasing trend in the past implies

decreasing trend in the future and vice versa. Persistency

is found also in cases where the time series exhibit clear

trends

with relatively little noise

[ I

1][14][22].

some time within the time lag in question. That is, the

annual increment is the object of analysis. The ideal

reservoir never empties and never overflows,

so

the

required storage capacity is equal to the difference

between the maximum and minimum value of Y ~ , , , ~

over j . This difference is called the range, R , h 3 ,

R,

I , )

=

Yk,k. , 1

Yk ,.lo

1

5)

The variance of x, for the same period, r S given as

and the quotient R,, is called rescaled range. The

above expressions are referred to a given position of the

time lag in the time axis However, the time lag can be

shifted and the procedure giv en by I ) , (2), 3) and (4) can

be repeated for each position. Thus a series of rescaled

ranges is obtained the ave rage of which c an be evaluated.

As

a non-unique but rational choice the lag is shifted by

steps Eqs. ( 3 ) - ( 6 ) , hus a series of non-overlapping but

contacting intervals is constructed. I n other words a series

of R,,, /S,. " is evaluated with j fixed and I vaned as

I = ( l , + m , ) where m= 1,2 ;.. ,[N /j] with the square

bracket denoting integer part. Then the rescaled range for

the time lag

,

s calculated as the average:

2.4 In te rpre ta t ion of f rac ta l d imension

We have already mentioned that the fractal

dimension of an object

is

a measure of complexity and

degree of spac e filling. When the object is

a

series in time,

the dimension also tells us something about the relation

between increments. It is a useful and meaningful insight

into series o f natural processes.

2.5 Frac t ion a l Brownian mot ion

A particle undergoing Brownian motion moves by

jumping step-lengths which are given by independent

Gaussian random variables. For one-dimensional motion

the position of the particle in time, A'( /) , is given by the

addition of all past increments. The function X ( / ) is a

self-affine fractal, whose graph has dimension 1 5

Fractional Brownian motion Eh) generalizes

X ( f ) by allowing the increments to be correlated.

Ordinary Brownian motion can be defined by:

X r) ~ ( t , )

g1t

- tOl2

9 )

1 I = [ N i j ]

where

H =

I12

5

is

a

normalized independent

Gaussian process and X ( t J the initial position [4][5].

Replacing the exponent

H

= 1 / 2 in Eq.5 with any other

number in the range O < H < l defines an fBm function

X , ( r ) . The exponent H here corr espon ds to the statistic H

that R I S analysis calculates.

(I

/ S = -

C ( R j l S j , , )

IN 1

M

Hurst observed that there

is

a great number of

natural phenomena, for which the ratio

R I S

obeys the

rule

RISocr '

8)

The correlation function of future increments with

past increments for the motion X , ( t ) can be shown to be

where H is called Hurst exponent. The Hurst [5]:

exponent was seen to be between 0 and 1 The value

H

=

112 has a special significance, because this reflects

that the observations are statistically independent of each

other. This is the random noise case. For example the

increment series, i.e. the series of displacem ents, in Clearly, C ( t ) = O for H = I 1 2 ; increments in

Brownian motion

is

a seque nce of uncorrelated random ordinary Brownian motion are independent. For

H

>

1

1 2 ,

C r) s

positive for all . This m eans that after a positive

C t)=

22x 1

-

(10)

136



increment, future increments are more likely to be positive.

This is known as persistence. When H

<

I / 2 , increments

are negatively correlated, which means an increase in the

past makes

a

decrease more likely in the future. This is

called anti-persistence.

Now it is true for self-affine functions such as

X, , ( t )

that the fractal dimension, D is related to H by

[41:

We can then identify persistence

or

anti-persistence

in data sets whose graphs are fractal. Persistent time series

show long-term memory effects. An increasing trend in

the past is likely to continue in the future because future

increments are positively correlated to past ones.

Similarly, a negative trend will persist. This means that

extreme values in the series tend to be more extreme than

for uncorrelated series. In the context of climatic data,

droughts or extended rain periods are more likely for

persistent data.

In order to determine the Hurst exponent,

log(R/S)

is

plotted against logr and the slope renders

H .However, not

all

the points of this plot have the same

statistical weights: when P is very small, a large number

of

R

/ S data can be calculated but their scatter is large;

when

r

is very large, only few R I S data are at hand,

so

the statistics is poor. For this reason the first and the last

few points of the double logarithmic plot are usually

discarded.

To begin with, we verified whether change of a

stock price time series follows the random walk

hypothesis using the rescaled range analysis. We analyzed

stock prices time series of Nikkei Stock Average. The

analysis period used the data for 1500 days from July,

1996 to October, 2002.

In

the analysis, the logarithm

profitability was applied to the original analysis object.

. 0.66Pn-010 l

i_ __I

0 s

0 3

0 .

0 IO 10 , .o ,e

m

1 so I_

N

-

Figure 1 Rescaled range analysis of Nikkei Stock

Average time series

Figure

1

shows the result of analyzing the Nikkei

Stock Average Prices for 1500 days. From the gradient of

this obtained straight line, the Hurst exponent H) in the

Figure

2.

Relationship

of

scaling interval (N) versus Hurst

exponent (H)

As shown

in

this Figure 2, H

=0.88

orresponding

to N =

3

is the maximum. Therefore, it is found that Data

for 3 days show the strongest correlation using the fractal

analysis of the Nikkei Stock Average Prices. This

knowledge is discovered .using a feature.de map with

rough set theory [26] shown in Table I . Firstly a Hurst

exponent is obtained. Then the fractal dimension and the

autocorrelation coefficient are calculated from the Hurst

exponent.

Table I . Knowledge extraction with feature-rule map

with rough set theory

137

http://feature.de/

http://feature.de/



Table

2.

Pawlak’s Lower and Upper Approximation

N u m b e r

of

objects:

Cla ss Brownian Motion

Lower approximat ion

Upper approximat ion

Aooroximat ion accuracv

1

3

Class

N = 3 I

Upper approximat ion

Approximat ion accuracy

Class

1

1

Similaritv (Fractal)

N u m b e r of objects

1

Upper approximat ion 1

Aooroximat ion accuracv 1

E r r o r 0)

Epochs

3 Structure of Neural Network for

Time

Series

3 Inpu ts Nodes 5 Inpu t s N odes

59.4803 304.9743

201 447

In

order to embed the discovered know ledge into the

structure of neural networks, it is found that our

developed delayed neural network is suitable 121.

3.1

Tim e Delayed Neur a l Network (TDNN)

In order to handle dynamical systems time delay

elements representing the obtained knowledge are put into

the inputs of neural networks 121. The structure

configuration of FIR filter is shown in Figure 3. It is a

finite impulse response (FIR) digital filter which is

connected to each input of a back propagation type

forward neural network (BPNN).

A

time delay element is

also put between the inputs of the filter.

Zl elay element

Figure 3. FIR filter

where,

f

is a sigmoid function and the weights are

corrected by the Back Propagation algorithm (BP).

4

Short-term Prediction

of

using

BPNN

We obtained the features (knowledge: N = 3 ) of the

time series by the fractal analysis. Two kinds of

3

layers

BP neural networks which have a delay element between

each input node are considered. No filter at each node has

connected

in

the first one. The second one has a 3-order

FIR at each input node.

4.1

Simula t ion by 3 layer BPNN without f i l te rs

No filter is connected at each input node. The

structure of the neural network is shown in Table

3.

Table 3. BP networks structure

O u t u t N o d e

E ochs

500 300

4.1.1 BPNN simulation with

3

input nodes

The structure of the neural network is illustrated in

Figure 4. The simulation result with 3 input nodes is

shown in Figure

5

We predicted for seven days from the

1501st. The error and the number of epochs are given in

Table 4.

Table 4. BPNN without filters with

3

inputs

I

E r r o r

0)

I

59.4803

Epoc hs 201

4.1.2

BPNN simulation with 5 inpu t Nodes

In the same way, the structure of the neural network

is illustrated in Figure 6. The simulation result with 5

input nodes is shown in Figure 7. The error and the

number of epochs are listed in Table 5

Table 5 . BPNN without filters with

5

input nodes

E r r o r

0)

I

304.9743

Epochs 447

138



input

Three-layer

Back Propagation

Neural Network

JUtpUt

K)

Figure 4. The structure of th e BPNN w ithout filter with

3

input nodes

I

I

Figure 5. BPNN simulation without filters with 3 input

nodes

input Three-layer

Neural Network

Figure 6. The structure of the B PNN w ithout filte r with

5

input nodes

-

-*./-

--d -

I

-

>-

i

input

Three-layer

FIR filters

Zl

.Pm

Z 1 2-1

Back

Pmpagation output

Neural Nework

2-1. . 1

2-1.. 2 ~ '

Figure 8 . The structure of the BPNN with filter with

3

input nodes.

. ..__.. ..

I

I

Figure 9. BPNN Simulation with filter with 3 input nodes

input Three-layer

Figure 10. The structure of the BPN N with filter with 5

input nodes

Figure 7. BPNN simulation without filters with 5 input

nodes

Figure 11. BPNN Simulation with tilter with 5 input nodes

139



4.2 Simula t ion by the 3 layer BPNN with fil ters.

Table 7 shows the structure of the 3 layer BPNN

with filters.

5 Input Node s 304.9743

fi l te rs

Table 7. 3 layer BPNN with filters network structure

447

Inpu t s N odes 5 Inpu ts Nodes

3-order

FIR

f i l te r

connected connected

Hidden Node 3 3

5

Input Nodes

f i l te rs

4.2.1 Simulation w ith 3 inp ut Nodes

The structure of the neural network is illustrated in

Figure 8. The simulation result is shown in Figure 9.

Table 8 lists the prediction error and the number of

e mc hs .

1548.2962

Table 8. Simulation with 3 input nodes

O u t p u t N o d e I 1

I

Table 9. Simulation with 5 input nodes

Epochs I 500

I

r r o r

0)

88.2962

300

I Epoc hs 154 I

Table

10

tells that with both 3 input 5 input nodes

prediction accuracy is fairly high.

E r r o r 0)

Table IO . Conparison ofboth

47.3381

r r o r 0 )

Epochs

Table

1

I , Conparison of both networks

r

E r r o r o ) I Epoc hs

3

Inpu t s N ode 5 Inpu t s N ode

47.3381 88.2962

3 7

154

I T T i n p u t odes

I

59.4803

I

201 I

5

Conclusion

A data mining technique is applied to time series

analysis and prediction. From a large amount of data

understandable knowledge is extracted using a Hurst

exponent, a fractal analysis method and

an

autocorrelation analysis method. Then it is embedded into

the suitable network, BPFN . The accurate prediction is

obtained in the Nikkei Average Stock price time series by

the B PNN w ith filters.

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