[ieee 2010 11th international symposium on computational intelligence and informatics (cinti) -...

Analysis of Financial Indexes with ComputationalTechniques

J. Tenreiro MachadoDept. of Electrical Engineering

Institute of EngineeringPorto, Portugal

Tel.: +(351)228340500Fax: +(351)228321159Email: [email protected]

Goncalo M. DuarteLusofona University

Lisboa, PortugalTel.: +(351)216001515

Email: [email protected]

Fernando B. DuarteDept. of MathematicsSchool of Technology

Viseu, PortugalTel.: +(351)232480500Fax: +(351)232424651

Email: [email protected]

Abstract—This paper applies Multidimensional scaling tech-niques and Fourier Transform for visualizing possible time-varying correlations between twenty five stock market values.The method is useful for observing stable or emerging clustersof stock markets with similar behavior. The graphs may alsoguide the construction of multivariate econometric models.

I. INTRODUCTION

It seems that there are many distinct analogies between thedynamics of complex physical and economical or even socialsystems. The methods and algorithms that have been exploredfor description of physical phenomena become an effectivebackground and inspiration for very productive methods usedin the analysis of economical data [1, 2, 3, 4].

Economical indexes measure the performance of segmentsof the stock market and are normally used to benchmarkthe performance of stock portfolios. This paper proposes adescriptive method which analyzes possible correlations ininternational stock markets. The study of the correlation ofinternational stock markets may have different motivations.Economic motivations to identify the main factors which affectthe behavior of stock markets across different exchanges andcountries. Statistical motivations to visualize correlations inorder to suggest some potentially plausible parameter relationsand restrictions. The understanding of such correlations wouldbe helpful to the design good portfolios [1, 2].

Bearing these ideas in mind the outline of our paper isas follows. In Section 2 we give the fundamentals of themultidimensional scaling (MDS) technique, which is the coreof our method, and we discuss the details that are relevant forour specific application. In Section 3 we apply our methodfor daily data on twenty five stock markets, including majorAmerican, Asian/Pacific, and European stock markets. InSection 4 we conclude the paper with some final remarks andpotential topics for further research.

II. FUNDAMENTAL CONCEPTS

MDS is a set of data analysis techniques for analysisof similarity or dissimilarity data. It is used to represent(dis)similarity data between objects by a variety of distancemodels.

The term similarity is used to indicate the degree of“likeness” between two objects, while dissimilarity indicatesthe degree of “unlikeness”. MDS represents a set of objectsas points in a multidimensional space in such a way thatthe points corresponding to similar objects are located closetogether, while those corresponding to dissimilar objects arelocated far apart. The researcher then attempts to “make sense”of the derived object configuration by identifying meaningfulregions and/or directions in the space.

In this article, we introduce the basic concepts and methodsof MDS. We then discuss a variety of (dis)similarity measuresand the kinds of techniques to be used. The main objective ofMDS is to represent these dissimilarities as distances betweenpoints in a low dimensional space such that the distancescorrespond as closely as possible to the dissimilarities.

Let 𝑛 be the number of different objects and let the dissim-ilarity for objects 𝑖 and 𝑗 be given by 𝛿𝑖𝑗 . The coordinates aregathered in an 𝑛×𝑝 matrix X, where 𝑝 is the dimensionality ofthe solution to be specified in advance by the user. Therefore,row 𝑖 from X gives the coordinates for object 𝑖. Let 𝑑𝑖𝑗 be theEuclidean distance between rows 𝑖 and 𝑗 of X defined as

𝑑𝑖𝑗 =

√√√⎷ 𝑝∑𝑠=1

(𝑥𝑖𝑠 − 𝑥𝑗𝑠)2 (1)

that is, the length of the shortest line connecting points 𝑖 and𝑗. The objective of MDS is to find a matrix X such that𝑑𝑖𝑗 matches 𝛿𝑖𝑗 as closely as possible. This objective can beformulated in a variety of ways but here we use the definitionof raw-Stress 𝜎2, that is,

𝜎2 =

𝑛∑𝑖=2

𝑖−1∑𝑗=1

𝑤𝑖𝑗 (𝛿𝑖𝑗 − 𝑑𝑖𝑗)2 (2)

by Kruskal [5] who was the first one to propose a formalmeasure for doing MDS. This measure is also referred to asthe least-squares MDS model. Note that due to the symmetryof the dissimilarities and the distances, the summation onlyinvolves the pairs 𝑖, 𝑗 where 𝑖 > 𝑗. Here, 𝑤𝑖𝑗 is a userdefined weight that must be nonnegative. The minimization

CINTI 2010 • 11th IEEE International Symposium on Computational Intelligence and Informatics • 18–20 November, 2010 • Budapest, Hungary

978-1-4244-9280-0/10/$26.00 ©2010 IEEE- 23 -

of 𝜎2 is a complex problem. Therefore, MDS programs useiterative numerical algorithms to find a matrix X for which𝜎2 is a minimum. In addition to the raw stress measurethere exist other measures for doing stress. One of them isnormalized raw stress, which is simply raw stress dividedby the sum of squared dissimilarities. The advantage of thismeasure over raw stress is that its value is independent of thescale and the number of dissimilarities. The second measureis Kruskal’s stress-1 which is equal to the square root of rawstress divided by the sum of squared distances. A third measureis Kruskal’s stress-2, which is similar to stress-1 except that thedenominator is based on the variance of the distances insteadof the sum of squares. Another measure that seems reasonablypopular is called S-stress and it measures the sum of squarederror between squared distances and squared dissimilarities.

Because Euclidean distances do not change under rotation,translation, and reflection, these operations may be freelyapplied to MDS solution without affecting the raw-stress.Many MDS programs use this indeterminacy to center thecoordinates so that they sum to zero dimension wise. Thefreedom of rotation is often exploited to put the solution in so-called principal axis orientation. That is, the axis are rotatedin such a way that the variance of X is maximal along the firstdimension, the second dimension is uncorrelated to the firstand has again maximal variance, and so on.

In order to assess the quality of the MDS solution wecan study the differences between the MDS solution and thedata. One convenient way to do this is by inspecting the so-called Shepard diagram. A Shepard diagram shows both thetransformation and the error. Let 𝑝𝑖𝑗 denote the proximitybetween objects 𝑖 and 𝑗. Then, a Shepard diagram plots si-multaneously the pairs (𝑝𝑖𝑗 , 𝑑𝑖𝑗) and (𝑝𝑖𝑗 , 𝛿𝑖𝑗). By connectingthe (𝑝𝑖𝑗 , 𝛿𝑖𝑗) points a line is obtained representing the relation-ship between the proximities and the disparities. The verticaldistances between the (𝑝𝑖𝑗 , 𝛿𝑖𝑗) points and (𝑝𝑖𝑗 , 𝑑𝑖𝑗) are equalto 𝛿𝑖𝑗 − 𝑑𝑖𝑗 , that is, they give the errors of representation foreach pair of objects. Hence, the Shepard diagram can be usedto inspect both the residuals of the MDS solution and thetransformation. Outliers can be detected as well as possiblesystematic deviations.

Measuring and predicting human judgment is an extremelycomplex and problematic task. There have been many tech-niques developed to deal with such type of problems. Thesetechniques fall under a generic category called Multidimen-sional Scaling (MDS). Generally speaking MDS techniquesdevelop spatial representations of psychological stimuli orother complex objects about which people make judgments(e.g. preference, relatedness), that is they represent each objectas a point in a n-dimensional space. What distinguishes MDSfrom other similar techniques (e.g. factor analysis) is that inMDS there are no preconceptions about which factors mightdrive each dimension. Therefore, the only data needed isa measure for the similarity between each possible pair ofobjects under study. The result is the transformation of thedata into similarity measures which can be represented byEuclidean distances in a space of unknown dimensions [6].

The greater the similarity of two objects, the closer theyare in the n-dimensional space. After having the distancesbetween all the objects, the MDS techniques analyze howwell they can be fitted by spaces of different dimensions. Theanalysis is normally made by gradually increasing the numberof dimensions until the quality of fit (measured for exampleby the correlation between the data and the distance) is littleimproved with the addition of a new dimension. In practicea good result is normally reached well before the number ofdimensions theoretically needed to a perfectly fit is reached(i.e. 𝑁 − 1 dimensions for 𝑁 objects) [7, 8, 9, 10].

In the MDS method a small distance between two pointscorresponds to a high correlation between two stock marketsand a large distance corresponds to low or even negativecorrelation [11, 12]. A correlation of one should lead to zerodistance between the points representing perfectly correlatedstock markets. MDS tries to estimate the distances for allpairs of stock markets to match the correlations as close aspossible. MDS may thus be seen as an exploratory techniquewithout any distributional assumptions on the data. The dis-tances between the points in the MDS maps are generally notdifficult to interpret and thus may be used to formulate morespecific models or hypotheses. Also, the distance between twopoints should be interpreted as being the distance conditionalon all the other distances. One possibility to obtain suchan approximate solution is given by minimizing the stressfunction. The obtained representation of points is not uniquein the sense that any rotation or translation of the points retainsthe distances [13].

III. DYNAMICS OF FINANCIAL INDEXES

In this section we study numerically the twenty five selectedstock markets, including seven American, eleven European andseven Asian/Pacific markets.

Our data consist of the 𝑛 daily close values of 𝑆 = 25stock markets,listed in Table I, from January 2, 2000, up toDecember 31, 2009, to be denoted as 𝑥𝑖(𝑡), 1 ≤ 𝑡 ≤ 𝑛, 𝑖 =1, ⋅ ⋅ ⋅ , 𝑆.

The data are obtained from data provided by Yahoo Financeweb site [14] and [15] , and they measure indexes in localcurrencies.

Figure 1 depicts the time evolution, of daily, closing price ofthe twenty five stock markets versus year with the well-knownoisy and ”chaotic-like” characteristics.

The section is organized in three subsections, the first adoptsan analysis based on a Fourier transform (FT) “distance mea-sure”, the second adopts an analysis based on the correlationof the time evolution and the third adopts a metrics based onhistogram distances.

A. MDS analysis based on Fourier transform

We calculate the Fourier transform (FT) for each oneof the indexes, leading to the values Re [ℱ {𝑥𝑘(𝑡)}] +j Im [ℱ {𝑥𝑘(𝑡)}], 𝑘 = {1, ⋅ ⋅ ⋅ , 𝑆}. It is adopted the ”distancemeasure” defined as (𝑖, 𝑗 = 1, ⋅ ⋅ ⋅ , 𝑆):

J. Tenreiro Machado et al. • Analysis of Financial Indexes with Computational Techniques

- 24 -

Table ITWENTY FIVE STOCK MARKETS

i Stock market index Abbrev. Country1 Dutch Euronext Amsterdam aex Netherlands2 Index of the Vienna Bourse atx Austria3 EURONEXT BEL-20 bfx Belgium4 Bombay Stock Exchange Index bse India5 SA£o Paulo (Brazil) Stock bvsp Brazil6 Budapest Stock Exchange bux Hungary7 Dow Jones Industrial dji USA8 Cotation AssistA c⃝e en Continu cac France9 Footsie ftse United Kingdom10 Deutscher Aktien Index dax Germany11 Standard & Poor’s sp500 USA12 Toronto Stock Exchange tsx Canada13 Stock market index in Hong Kong hsi Hong Kong14 Iberia Index ibex Spain15 Jakarta Stock Exchange jkse Indonesia16 Stock Market index of South Korea ks11 South Korea17 Italian Bourse mibtel Italy18 Bolsa Mexicana de Valores mxx Mexico19 Tokyo Stock Exchange nikkei Japan20 NASDAQ ndx USA21 New York Stock Exchange nya USA22 Stock exchange of Portugal psi20 Portugal23 Shanghai Stock Exchange ssec China24 Swiss Market Index ssmi Switzerland25 Straits Times Index sti Singapore

Jan00 Jan02 Jan04 Jan06 Jan08 Jan100

1

2

3

4

5

6

7

8x 10

4

Time

Val

ue

aex

atx

bfx

bse

bux

bvsp

dji

cac

ftse

dax

sp500

tsx

hsi

ibex

jkse

ks11

mibtel

mxx

nikkei

ndx

nya

psi20

ssec

ssmi

sti

Figure 1. Time series for the twenty five indexes from January 2000, up toDecember 2009.

𝑠(𝑖, 𝑗) =

∑Ω

(∣Re𝑖 −𝑅𝑒𝑗 ∣2 + ∣ Im𝑖 − Im𝑗 ∣2)

∑Ω

(∣Re𝑖 ∣2 + ∣ Im𝑖 ∣2) ⋅∑Ω

(∣Re𝑗 ∣2 + ∣ Im𝑗 ∣2] (3)

where Ω is the set of sampling frequencies for the FTcalculation, and Re𝑖, Im𝑖, Re𝑗 and Im𝑗 are the values of realand imaginary parts of the FT. We get the matrix F, whereeach cell represents the distance between a pair of FTs.

In order to reveal possible relationships between the signalsthe MDS technique is used and several distance criteria aretested. The Sammon criterion [16, 17], that tries to optimize acost function that describes how well the pairwise distances in

−50 −40 −30 −20 −10 0 10 20 30 40 50−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

aex

atx

bfx

bse

bux

bvsp

dji

cac

ftsedax sp500

tsx

hsi

ibex

jkse

ks11

mibtel

mxx nikkei

ndx

nya

psi20ssecssmi

sti

Figure 2. Two dimensional MDS graph for the twenty five indexes basedon the FT distance (3).

a data set are preserved, revealed good results and is adoptedin this work.

So, the F matrix was subjected to a MDS analysis [7] withthe following parameters:

∙ Metric multidimensional scaling∙ Uniform weighting∙ Absolute scaling model∙ Stress method: sammon∙ Dimension of the representation space: 2-5∙ Repetitions: 20∙ Iterations: 200∙ Convergence: 0.0001

Figures 2 and 3 show the 2D and 3D MDS locus ofindex positioning in the perspective of the expressions (3),respectively. Figures 4 depicts the stress as function of thedimension of the representation space, revealing that, as usual,a high dimensional space would probably describe slightlybetter the ”map” of the 25 indexes. However, the three di-mensional representations were adopted because the graphicalrepresentation is easier to analyze while yielding a reasonableaccuracy. Moreover, the resulting Sheppard plot, representedin figures 5 shows that a good distribution of points aroundthe 45 degree line is obtained.

Analyzing figures 2 and 3 is visible that the MDS indexesforms an arc shapes in all the stock market indexes underanalysis. How can we interpret these regularities? This isan interesting question for further research and a matter ofdecision of the stock market handlers.

B. MDS analysis based on time correlation

In this subsection, we apply the MDS method to the timecorrelation of all the stock markets.

For the twenty five markets, we consider the time corre-lations between the daily close values. We first compute thecorrelations among the twenty five stock markets obtained a


- 25 -

−50

0

50

−3

−2

−1

0

1

2−1.5

−1

−0.5

0

0.5

1

1.5

2

aex

mibtel

sp500

ndx

bfx

cacdax

tsx

ftse

ssec

psi20

ibex

ssmi

djinya

sti

nikkei

buxmxx

hsi

jkse

atx

ks11bse

bvsp

Figure 3. Three dimensional MDS graph for the twenty five indexes basedon the FT distance (3).

1 1.5 2 2.5 3 3.5 4 4.5 50.293

0.2935

0.294

0.2945

0.295

0.2955

0.296

0.2965

Number of dimensions

Str

ess

Figure 4. Stress plot of the MDS representation, for the twenty five indexes,𝑣𝑠 number of dimensions,using the FT distance (3).

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Original Dissimilarities

Dis

tanc

es/D

ispa

ritie

s

Distances1:1 Line

Figure 5. Shepard plot of the three dimensional MDS representation, for thetwenty five indexes,using the FT distance (3).

−0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03−0.02

−0.01

0

0.01

0.02

0.03

0.04

aex

atx

bfx

bsebux

bvsp

djicac

ftse

dax

sp500

tsx hsi

ibex

jkse

ks11

mibtel

mxx

nikkei

ndx

nya psi20ssecssmi

sti

Figure 6. Two dimensional MDS graph for the twenty five indexes usingtime correlation, according (4).

𝑆 × 𝑆 matrix and then apply MDS. In this representation,points represent the stock markets.

In order to reveal possible relationships between the marketstocks index the MDS technique is used. In this perspectiveseveral MDS criteria are tested. The Sammon criterion re-vealed good results and is adopted in this work [18]. Forthis purpose we calculate 25 × 25 matrix M based on acorrelation coefficient 𝑐(𝑖, 𝑗), that provides a measurement ofthe similarity between two indexes and is defined in equation(4). In matrix M each cell represents the time correlationbetween a pair of indexes, 𝑖, 𝑗 = 1, ⋅ ⋅ ⋅ , 𝑆.

𝑐(𝑖, 𝑗) =

⎛⎜⎜⎜⎜⎝

1𝑛

𝑛∑𝑡=1

𝑥𝑖(𝑡) ⋅ 𝑥𝑗(𝑡)√1𝑛

𝑛∑𝑡=1

(𝑥𝑖(𝑡))2 ⋅ 1𝑛

𝑛∑𝑡=1

(𝑥𝑗(𝑡))2

⎞⎟⎟⎟⎟⎠

2

(4)

Figures 6 and 7, show the 2D and 3D locus of each indexpositioning in the perspective of expression (4), respectively.Figure 8 depicts the stress as function of the dimension ofthe representation space, revealing that a three dimensionalspace describe a with reasonable accuracy the ”map” of thetwenty five signal indexes. Moreover, the resulting Sheppardplot, represented in figure 9, shows that a good distribution ofpoints around the 45 degree line is obtained.

There are several empirical conclusions one can draw fromthe graphs in figures 6 and 7, and we will mention just afew here. We can clearly observe that there seem to emergeclusters, which show similar behavior [19]. Hence, there doesnot seem to be a single world market, but perhaps thereare several important regional markets. This last observationwould match with standard financial theory which tells usthat higher (lower) volatility corresponds with higher (lower)returns. Indeed, if this would be the case, one would expectto see similar patterns over time across returns and volatility.


- 26 -

−0.04−0.03

−0.02−0.01

00.01

0.020.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

hsijkseibex

ssmi

sti

mibtel

ks11

sp500ssec

mxx

tsx

psi20

daxnya

nikkei

ftse

ndx

bux

cac

bse

bvsp

dji

bfx

aex

atx

Figure 7. Three dimensional MDS graph for the twenty five indexes usingtime correlation, according (4)

1 1.5 2 2.5 3 3.5 4 4.5 50

0.02

0.04

0.06

0.08

0.1

0.12


Str

ess

Figure 8. Stress plot of MDS representation of the twenty five indexes vsnumber of dimension using time correlation, according (4)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1


Dis

tanc

es/D

ispa

ritie

s

Distances1:1 Line

Figure 9. Shepard plot for MDS with a three dimensional representation ofthe twenty five indexes using time correlation, according (4)

2000 4000 6000 8000 10000 12000 140000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

xi

Rel

ativ

e F

requ

ency

bfxpsi20sti

Figure 10. Histogram for the BFX, STI and PSI20 indexes over all time.

1 1.5 2 2.5 3

x 104

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

xi

Rel

ativ

e F

requ

ency

buxdjitsx

Figure 11. Histogram for the BUX, DJI and TSX indexes over all time.

C. MDS analysis based on histograms

For each of the twenty five indexes we draw the corre-sponding histogram of relative frequency and we calculatestatistical descriptive parameters like the arithmetic mean(𝜇𝑖), the standard deviation (𝜎𝑖) and the Pearson’s Kurtosiscoefficient 𝛾𝑖.

Figures 10 and 13 depict the histograms of the NDX, DAX,ATX, BFX, STI, PSI20, DJI, BUX, TSX, NIKKEI, MIBTELand MXX indexes. The values of the statistical descriptiveparameters are listed in Table II.

For all the twenty five indexes we calculate the ”his-togram’s distance” [20, 21], 𝑑1, 𝑑2, 𝑑3 and 𝑑4 using theequations, where 𝑖, 𝑗 = 1, ⋅ ⋅ ⋅ , 𝑆, 𝐴 = [max (𝜇𝑖 − 𝜇𝑗)]

2,𝐵 = [max (𝜎𝑖 − 𝜎𝑗)]

2, 𝐶 = [max (𝛾𝑖 − 𝛾𝑗)]2 :


- 27 -

1000 2000 3000 4000 5000 6000 7000 80000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

xi

Rel

ativ

e F

requ

ency

atxdaxndx

Figure 12. Histogram for the NDX, DAX and ATX indexes over all time.

0.5 1 1.5 2 2.5 3

x 104

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

xi

Rel

ativ

e F

requ

ency

nikkeimibtelmxx

Figure 13. Histogram for the NIKKEI, MIBTEL and MXX indexes over alltime.

𝑑1(𝑖, 𝑗) =

√(𝜇𝑖 − 𝜇𝑗)

2

𝜇2𝑖 + 𝜇2

𝑗

+(𝜎𝑖 − 𝜎𝑗)

2

𝜎2𝑖 + 𝜎2

𝑗

(5a)

𝑑2(𝑖, 𝑗) =


2

𝐴+

(𝜎𝑖 − 𝜎𝑗)2

𝐵(5b)

𝑑3(𝑖, 𝑗) =


2

𝜇2𝑖 + 𝜇2

𝑗

+(𝜎𝑖 − 𝜎𝑗)

2

𝜎2𝑖 + 𝜎2

𝑗

+(𝛾𝑖 − 𝛾𝑗)

2

𝛾2𝑖 + 𝛾2

𝑗

(5c)

𝑑4(𝑖, 𝑗) =


2

𝐴+

(𝜎𝑖 − 𝜎𝑗)2

𝐵+

(𝛾𝑖 − 𝛾𝑗)2

𝐶(5d)

Figures 14-21, show the 2D and 3D locus of each indexpositioning in the perspective of the expressions (5a), (5b),(5c) and (5d), respectively demonstrating differences betweenthe corresponding MDS plots.

Figures 22-25 depict the stress 𝑣𝑠 the dimension of therepresentation space, for 𝑑1, 𝑑2, 𝑑3 and 𝑑4, revealing that a

−50 −40 −30 −20 −10 0 10 20 30 40−30

−20

−10

0

10

20

30

40

aex

atx

bfx

bse

bux

bvsp

dji

cacftse

dax

sp500

tsx

hsi

ibex

jkse

ks11

mibtelmxx

nikkei

ndx

nya psi20

ssec

ssmi

sti

Figure 14. Two dimensional MDS graph for the twenty five indexes usinghistogram’s distance 𝑑1.

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

aexatx

bfx

bse

bux

bvsp

dji

cac

ftse

dax

sp500

tsx

hsi

ibex

jkseks11

mibtel

mxx

nikkei

ndx

nya psi20

ssec

ssmi

sti


−80 −60 −40 −20 0 20 40−40

−30

−20

−10

0

10

20

30

aex

atx

bfxbse

bux

bvsp

dji

cacftse

dax

sp500

tsx

hsi

ibex

jkse

ks11

mibtel

mxx

nikkei

ndx

nya

psi20

ssec

ssmi

sti



- 28 -

−0.5 0 0.5 1 1.5 2−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

aexatx

bfx

bse

bux

bvsp

dji

cacftsedax

sp500

tsx

hsi

ibex

jkseks11

mibtel

mxx

nikkeindx

nya

psi20

ssec

ssmi

sti


−50−40

−30−20

−100

1020

3040

−30−20

−100

1020

3040

−40

−30

−20

−10

0

10

20

30

tsxibex

psi20

bse

nikkei

bux

djihsi

nyassmi

mxx

mibtel

daxbvsp

ftsecac

sp500aex

atx

ssec

ks11

bfxsti

ndx

jkse

Figure 18. Three dimensional MDS graph for the twenty five indexes usinghistogram’s distance 𝑑1.

−0.4−0.2

00.2

0.40.6

0.81

1.2

−0.3

−0.2

−0.1

0

0.1

0.2−5

0

5x 10

−3

mibtel

bvsphsinikkei

dji

bux

ibex

tsx

mxx

psi20

nya

ssmi

ftsedax

bse

cac

bfx

sti

atx

ndx

ssec

sp500

ks11

jkse

aex


Table IISTATISTICAL DESCRIPTIVE PARAME AND TERS

i 𝜇𝑖 𝜎𝑖 𝛾𝑖1 438.05 121.8 2.29132 2377.9 1274.1 1.80393 2981.3 791.52 2.29934 8080.3 4839.8 2.27245 14495 6925.1 1.72246 29624 17092 2.24767 10497 1474 2.82078 4512.9 1073.5 1.95119 5259 885.31 1.714

10 5323.3 1465.4 1.936811 1192.5 200.4 2.156512 9739.2 2388.6 2.063113 15403 4617.4 3.509514 9990.2 2615 2.278615 1154.2 730.5 2.128516 1057.4 410.42 2.162517 24519 5499.4 2.022318 14822 8876.3 1.87919 12748 3153.3 2.104420 1756.4 715.21 6.7621 7039 1429.5 2.329822 8792.5 2350.9 2.09123 2048.2 1036.2 6.28524 6706.6 1359 1.984925 2160.2 618.29 2.9058

−80−60

−40−20

020

40

−40−30

−20−10

010

2030

−20

−10

0

10

20

30

bvsp

mibtel

mxx

hsi

ndx

ssec

nikkei

bux

ibex

psi20

bsesti

tsx

ssminya

jkse

dji

ks11

dax

ftse

cac

aex

atx

sp500

bfx


three dimensional space describes with reasonable accuracythe ”map” of the twenty five indexes. The resulting Sheppardplots, represented in figures 26-29, show that a good distribu-tion of points around the 45 degree line is obtained for theindices [22].

Curiously in the chart corresponding to the MDS based oncorrelation (figure 6) we can see a parabolic shape with theS&P500 index at the vertex, and the NDX and ATX at thetop. However, in the chart corresponding to the MDS basedon the histogram distances (figures 14-17 and in the chartcorresponding to the MDS based on the FT distance (figure2) such parabolic shape form cannot be found.


- 29 -

−0.50

0.51

1.52

−1.5

−1

−0.5

0

0.5−0.2

−0.1

0

0.1

0.2

0.3

ndxssec

bvsp

hsi

dji

mibtel

sti

mxx

bse

ibexnikkei

bux

nyatsxpsi20

bfxssmi

aexdaxks11jksesp500cac

ftse

atx


1 1.5 2 2.5 3 3.5 4 4.5 50.47

0.472

0.474

0.476

0.478

0.48

0.482

0.484

0.486


Str

ess

Figure 22. Stress plot of MDS representation of the twenty five indexes vsnumber of dimension, using histogram’s distance 𝑑1.

1 1.5 2 2.5 3 3.5 4 4.5 50

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01


Str

ess


1 1.5 2 2.5 3 3.5 4 4.5 50.44

0.445

0.45

0.455

0.46

0.465

0.47

0.475


Str

ess


1 1.5 2 2.5 3 3.5 4 4.5 50

0.01

0.02

0.03

0.04

0.05

0.06


Str

ess


0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5


Dis

tanc

es/D

ispa

ritie

s

Distances1:1 Line

Figure 26. Shepard plot for MDS with a three dimensional representationof the twenty five indexes, using histogram’s distance 𝑑1.


- 30 -

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5


Dis

tanc

es/D

ispa

ritie

s

Distances1:1 Line


0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5


Dis

tanc

es/D

ispa

ritie

s

Distances1:1 Line


0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5


Dis

tanc

es/D

ispa

ritie

s

Distances1:1 Line


IV. CONCLUSION

In this paper, we proposed simple graphical tools to visual-ize time-varying correlations between stock market behavior.We illustrated our MDS-based method daily close values offifteen stock markets. There are several issues relevant forfurther research. A first issue concerns applying our method toalternative data sets, with perhaps different sampling frequen-cies or returns and absolute returns, to see how informativethe method can be in other cases. A second issue concernstaking the graphical evidence seriously and incorporating itin an econometric time series model to see if it can improveempirical specification strategies.

REFERENCES

[1] P. V., G. P. R. B., A. L.A.N., and S. H.E, “Econophysics:financial time series from a statistical physics point ofview,” Physica A, vol. 279, pp. 443–456, 2000.

[2] R. Nigmatullin, “Universal distribution function for thestrongly-correlated fluctuations: General way for descrip-tion of different random sequences,” Communications inNonlinear Science and Numerical Simulation, vol. 15,no. 3, pp. 637–647, 2010.

[3] R. V. Mendes and M. J. Oliveira, “A data-reconstructed fractional volatility model,” Eco-nomics: The Open-Access, Open-Assessment E-Journal, vol. 2, no. 2008-22, 2008. [On-line]. Available: http://www.economics-ejournal.org/economics/discussionpapers/2008-22

[4] R. V. Mendes, “A fractional calculus interpretation of thefractional volatility model,” Nonlinear Dynamics, vol. 55,no. 4, pp. 395–399, 2009.

[5] J. Kruskal, “Multidimensional scaling by optimizinggoodness of fit to a nonmetric hypothesis,”Psychometrika, vol. 29, no. 1, pp. 1–27, 1964. [Online].Available: http://dx.doi.org/10.1007/BF02289565

[6] I. Borg and P. Groenen, Modern Multidimensional Scal-ing:Theory and Applications. New York: Springer, 2005.

[7] T. Cox and M. Cox. New York: Chapman & HallCrc,2001.

[8] J. Kruskal and M. Wish, Multidimensional Scaling.Newbury Park, CA: Sage Publications, Inc., 1978.

[9] J. Woelfel and G. A. Barnett, “Multidimensional scalingin riemann space,” Quality and Quantity, vol. 16, no. 6,pp. 469–491, 1982.

[10] J. O. Ramsay, “Some small sample results for maximumlikelihood estimation in multidimensional scaling,” Psy-chometrika, vol. 45, no. 1, pp. 139–144, 1980.

[11] S. Nirenberg and P. E. Latham, “Decoding neuronal spiketrains: how important are correlations?” Proc. NationalAcademy of Sciences, vol. 100, no. 12, pp. 7348–7353,2003.

[12] F. B. Duarte, J. T. Machado, and G. M. Duarte, “Dy-namics of the dow jones and the nasdaq stock indexes,”Nonlinear Dynamics, vol. 61, no. 4, pp. 691–705, 2010.

[13] A. Buja, D. Swayne, M. Littman, N. Dean, H.Hofmann,and L.Chen, “Data visualization with multidimensional


- 31 -

scaling,” Journal of Computational and Graphical Statis-tics, vol. 17, no. 85, pp. 444–472, 2008.

[14] “http://finance.yahoo.com.”[15] “http://www.bse.hu.”[16] J. W. Sammon, “A nonlinear mapping for data structure

analysis,” IEEE Trans. Comput., vol. 18, no. 5, pp. 401–409, 1969.

[17] G. A. F. Seber, Multivariate Observations. New York:J. Wiley & Sons, 1986.

[18] B. Ahrens, “Distance in spatial interpolation of dailyrain gauge data,” Hydrology and Earth System Sciences,no. 10, pp. 197–208, 2006.

[19] J. T. Machado, G. M. Duarte, and F. B. Duarte,“Identifying economic periods and crisis with themultidimensional scaling,” Nonlinear Dynamics, DOI10.1007/s11071-010-9823-2,ISSN: 0924-090X.

[20] F. Serratosa and A. Sanroma, G.and Sanfeliu, “Anew algorithm to compute the distance between multi-dimensional histograms,” Lecture Notes in ComputerScience, vol. 4756, no. 1, pp. 115–123, 2008.

[21] B. Sierra, E. Lazkano, E. Jauregi, and I. Irigoien, “His-togram distance-based bayesian network structure learn-ing: A supervised classification specific approach,” Decis.Support Syst., vol. 48, no. 1, pp. 180–190, 2009.

[22] J. T. Machado, F. B. Duarte, and G. M. Duarte, “Mul-tidimensional scaling analysis of stock market values,”in Proceedings of 3rd Conference on Nonlinear Scienceand Complexity, D. Baleanu, Ed. Cankaya University,July 2010.


- 32 -

[ieee 2010 11th international symposium on computational intelligence and informatics (cinti) -...

Documents