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Journal of Engineering Science and Technology Review 4 (3) (2011) 286-290

Special Issue on Econophysics

Research Article

Chaos theory in predicting election results

M. P. Hanias1,* and L. Magafas2

1 Dep. of Electronics, Computers, Telecommunications, and Control,University of Athens, Panepistimiopolis,

Athens 15784, Greece

2Dep. of Electrical Engineering, Kavala Institute of Technology, St. Loukas 65404 Kavala, Greece

___________________________________________________________________________________________

Abstract

To date, accounting and financial sciences have attempted to assess companies using financial ratios and other

techniques that evaluate only the front office, without considering its impact to the environment. Recently, this

assessment has been expanded to include new concepts, known as corporate social responsibility and social balance.

These new concepts are usually viewed and studied using a multidisciplinary perspective with an aim to update the

current and future value of the company. In this evolving scientific field a considerable effort has been made to

objectively record and calculate the environmental impact of a company's activities by integrating these elements into a

new form of economic balance sheet. The present paper seeks to examine the presence of weighted environmental

indexes and explores the process by which they can be used to evaluate a company.

Keywords: Econophysics, election, results

_________________________________________________________________________________________

1. Introduction

In our previous works [1,2] we have applied Physical

models, especially methods from chaos theory, in order to

predict and study the Euro election results of Hellas and the

Hellenic National election results , defining the new

scientific term called “DemoscopoPhysics”, in the sense of

application of physics models to social phenomena

modelling. The term DemoscopoPhysics consists from two

words Demoscopie and Physics. The first word is a Hellenic

ancient word that means political survey. This work was

inspired from the emerging field of economo-physics which

mainly consists of autonomous mathematical physics models

that apply to the financial markets. Now we try to use them

particular aspects of the complex nonlinear dynamics of

political survey in order to predict the election results for

New Democracy (ND), Panhellenic Socialistic Movement

(PASOK), Hellenic Communistic Party (KKE), Coalition of

the Radical Left (SYRIZA) and (Popular Orthodox Rally)

LAOS political parties. We have approached the prediction

with two different ways. Taking into account the results of

opinion polls we have done regression in intension to vote.

These new data are the raw data now. If we had applied

statistical methods to these data we would take static results

with very short horizon forecasting. For this reason we apply

dynamic methods based on chaos theory in order to show the

hidden potential of each political party and make

predictions with a time horizon of 60 days for the Euro

election results of Hellas and 30 days for Hellenic National

election results .

2. Public survey time series

To construct the time series for predicting the the Euro

election results of Hellas we have taken into account the

assessment vote from public surveys in Hellas from 16-1-

2007 to 23-04-2009 the estimation of the election behavior

of the unclarified vote based on previous elections. The

number of raw data is 36 for each political party, and each

data is the average value of 4 polling companies with

relative error 1%. In order to reconstruct of the equivalent

phase space from experimental data, the timeseries that

serves as experimental data should be constituted by

sampled points of equal time-distances. For this purpose we

interpolate with cubic spline so we take N=1000 points with

a sample rate of 0.92day. We have applied the same

procedure to construct the time series for Hellenic National

election results and we have taken into account the

assessment vote from public surveys in Hellas from 16-1-

2007 to 6-09-2009 . The number of raw data is 48 for each

political party, and each data is the average value of 4

polling companies with average relative error 1.5%. for ND

and PASOK political parties and 1% for other political

parties. In order to reconstruct the phase space from

experimental data, these data should be constituted by

sampled points of equal time-distances. In our case this

condition is not fulfilled. For this purpose we interpolate our

raw data with cubic spline, as before, so we have create

N=1057 points with a sample rate of 1 day. The raw data

and the interpolated public survey time series of the ND and

PASSOK political party are shown at Fig.1(a) and Fig 1(b)

respectively, covering the period from16-1-2007 to 23-04-

2009

JOURNAL OF

Engineering Science and

Technology Review

www.jestr.org

______________

* E-mail address: mhanias@gmail.com

ISSN: 1791-2377 2011 Kavala Institute of Technology. All rights reserved.

M. P. Hanias and L. Magafas/Journal of Engineering Science and Technology Review 4 (3) (2011) 286 -290

287

0200 400 600 800 1000

30

32

34

36

38

40

42

44

46

N.D. public survey percentage

Time steps (x 0.92 days)

(a)

0200 400 600 800 1000

26

28

30

32

34

36

38

40

42

P.A.S.O.K. public survey percentage

Time steps (x0.92 days)

(b)

Fig. 1 (a) Interpolated time series for ND public survey for period 16-

1-2007 to 23-04-2009 and interpolated time series for PASOK public

survey for the same period (black line) and raw data (red dots)

3. State Space Reconstruction

For a scalar time series, in our case the gallop poll time

series, the phase space can be reconstructed using the

methods of delays. The basic idea in the method of delays is

that the evolution of any single variable of a system is

determined by the other variables with which it interacts.

Information about the relevant variables is thus implicitly

contained in the history of any single variable. On the basis

of this an „„equivalent” phase space can be reconstructed by

assigning an element of the time series Xi and its successive

delays as coordinates of a new vector time series

i

X

. To

construct a vector

i

X

i=1 to N, in the m dimensional phase

space we use the Equation 1 proposed by [1-3]

i

X

={xi,xi-τ,xi-2τ,…..xi-(m-1)τ} (1)

i

X

represents a point to the m dimensional phase space

in which the attractor is embedded each time, where τ is the

time delay τ=iΔt . Time delay is the time necessary to cancel

the correlation between two time series values. The element

Xi represents a value of the examined scalar time series in

time, corresponding to the i-th component of the time series.

The dimension m of the reconstructed phase space is

considered as the sufficient dimension for recovering the

object without distorting any of its topological properties,

thus it may be different from the true dimension of the space

where this object lies. Use of this method reduces phase

space reconstruction to the problem of proper determining

suitable values of m and τ. The next step is to find time

delay τ and embedding dimension m, without using any

other information apart from the historical values of the

indexes. This is why the methodology is labelled as a

stochastic one. We can calculate the time delay by using the

average mutual [3-5] information.

With the above method we found the τ to be 37, 20, 28,

40, 23 time steps for ND, PASOK, KKE, SYRIZA and

LAOS, respectively. One method to determine the presence

of chaos is to calculate the fractal dimension, which will be

non integer for chaotic systems. Even though there exists a

number of definitions for the dimension of a fractal object

(Box counting dimension, Information Dimension, etc.), the

correlation dimension was found to be the most efficient for

practical applications. Firstly we calculate the correlation

integral [7,8] for the time series for lim r0 and N

by

using the equation 3 [2].

N

ij

iJi

pairs

XXrH

N

rC

1

,1

1

)(

(3)

In this equation, the summation counts the number of

pairs

),( ji XX

for which the distance, (Euclidean norm),

Ji XX

is less than r, in an m dimensional Euclidean

space. Η is the Heaviside step function, with H(u)=1 for

u>0, and H(u)=0 for ,where

Ji XXr

,

Ν denotes the number of points and expressed in equation 4.

2

)1( 2

mN

Npairs

(4)

where r is the radius of the sphere centered on Xi or Xj. If

the time series is characterized by an attractor, then for

positive values of r, the correlation function is related to the

radius with a power law C(r)~αrv , where α is a constant

and ν is the correlation dimension or the slope of the

log2C(r) versus log2r plot. Since the data set will be

continuous, r cannot get to close to zero. To handle this

situation, from log2 C (r) versus log2 r plot we select the

apparently linear portion of the graph. The slope of this

portion will approximate ν. Practically one computes the

correlation integral for increasing embedding dimension m

and calculates the related ν(m) in the scaling region. Using

the appropriate delay times for each political party i.e, 37,

20, 28, 40, 23 time steps for ND, PASOK, KKE, SYRIZA

and LAOS, respectively, we reconstruct the phase space for

each political party. The correlation integral C(r) by

definition. is the limit of correlation sum of Equation (3) for

different embedding dimensions, m=1..10. This is shown

for ND in Fig 2(a) while in Fig.2 (b), the corresponding

average slopes v are given as a function of the embedding

dimension m, indicating that for high values of m, v tends to

M. P. Hanias and L. Magafas/Journal of Engineering Science and Technology Review 4 (3) (2011) 286 -290

288

saturate at the non integer value of v=1.6. The embedding

dimension m is found to be m≥2[v]+1=3 where [v] is the

integer part of v[6].

-16 -14 -12 -10 -8 -6 -4 -2 0 2

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

log2C(r)

log2r

(a)

0 2 4 6 8 10 12

0

1

2

3

4

5

v

m

(b)

Fig.2 (a) Relation between log2 C(r) and log2 r for different embedding

dimensions m. (b) Correlation dimension v vs. embedding dimension m

for ND

-10 -8 -6 -4 -2 0

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

log2C(r)

log2r

0246810

0

1

2

3

4

5

v

m

Fig.3 (a) Relation between log2C(r) and log2r for different embedding

dimensions m. (b) Correlation dimension v vs. embedding dimension m

for PASOK

Table 1. The correlation dimension for Greek political parties

Political Parties

Correlation dimension ν

ND

1.6

PASOK

1.53

KKE

1.28

SYRIZA

1.29

LAOS

1.23

Applying the same procedure for PASOK political we

found that for high values of m, v tends to saturate at the non

integer value of v=1.53. The embedding dimension m is

found to be m≥2[v]+1=3 [6]. The results are shown in Fig 3.

For KKE v tends to saturate at the non integer value of

v=1.28, while for SYRIZA and LAOS v tends to saturate at

the non integer value of v=1.29 and 1.23 respectively. The

embedding dimension m is found to be=3 for the other

political parties too. [6]. We can see from Table 1 that the

smaller political parties have smaller correlation dimension.

We can interpret it

that the smaller political parties are more robust to keep

there voters but on the other hand they cannot adapt changes

as the larger parties do.

4. Time series prediction

The next step is to predict the evolution of the percentages of

votes, for each political party, by computing weighted

average of evolution of close neighbors of the predicted state

in the reconstructed phase space [1,2]. The reconstructed m-

dimensional signal projected into the state space can exhibit

a range of trajectories, some of which have structures or

patterns that can be used for system prediction and

modeling. To predict the Euro election results of Hellas we

used the values of τ and m from our previous analysis. We

had better results using as embedding dimension the value

of 2*m = 6 [1,2] for all predictions. At table 2 we present

our out of sample estimation. At the same table we present

the results and the devation from Euro election results of

Hellas.

Table 2. Political survey estimation for Euro election results of

Hellas

Political

parties

Political

survey

estimation %,

23/4/

Election

Results %

7/6/2009

Deviation

from election

results

ND

32.33

32.29

0.04

PASOK

37.19

36.65

0.54

KKE

8.52

8.35

0.17

SYRIZA

7.4

4.7

2.7

LAOS

6.8

7.15

0.35

At this point we mark that until 23/4/2009 we had not

data for Ecological Party. This political party, generally

speaking, is in cognation with Coalition of the Radical Left

(SYRIZA) so its presence can affect SYRIZA‟s percentage.

The % percentage for Ecological Party was 3.49 and it is

included to SYRIZA's estimation.

We used the same procedure for the Hellenic National

election results Actual and predicted time series for k=30

time steps ahead are presented at Figs 4.(a),(b) for ND,

PASOK respectively.

In order to capture the polarization of voters we have

decreased the degrees of freedom to 4. We use as embedding

dimension m = 4, keeping the values of delay time the same

M. P. Hanias and L. Magafas/Journal of Engineering Science and Technology Review 4 (3) (2011) 286 -290

289

from our previous analysis and the number of near

neighborhoods nn, as a rule of thumb [2] equal to 3*m=12

,for all political parties. Actual and predicted time series for

k=30 time steps ahead are presented at Figs 5.(a),(b) for ND,

PASOK respectively.

0200 400 600 800 1000 1200

30

32

34

36

38

40

42

44

46

N.D public syrvey percentage

Time steps (x1 day)

(a)

0200 400 600 800 1000 1200

26

28

30

32

34

36

38

40

42

P.A.S.O.K public survey percentage

Time steps (x1 day)

(b)

Fig 4. Actual (crosses) and predicted (solid line) time series for n=30

time steps ahead for ND(a),PASOK (b) political parties. The

embedding dimension is m=6.

0200 400 600 800 1000 1200

30

32

34

36

38

40

42

44

46

N.D. public survey percentage

Time steps (x1 day)

0200 400 600 800 1000 1200

26

28

30

32

34

36

38

40

42

P.A.S.O.K public survey percentage

Time steps (x1 day)

Fig 2. Actual (crosses) and predicted (solid line) time series for n=30

time steps ahead for ND(a),PASOK (b) political parties. The embedding

dimension is m=4.

At table 3 we present our out of sample estimation about

political survey estimation for two embedding dimensions

The first is m=6 and the second is m=4. We estimate the

mean error as 1.5% for ND and PASOK and 1% for the

other political parties. At the same table we present the

results and the deviation from Hellenic National election

results

Table 3. Political survey estimation for Hellenic National election

5. Conclusions

In this paper, we use a chaotic analysis to predict Greek

political parties election results. After estimating the

dependence of correlation dimension on embedding

dimension, we point out that the system is a deterministic

chaotic. A separate attractor for each political party,

embedded in 3-dimension space, is derived from the

analysis. However the election system is obviously a

complex multi-variable system with strong inter-relation

between variables. In this sense, we have model each

separate time series and never the whole election system,

whose attractor is obviously much more complex. From

absolute values of correlation dimension we see that the

smaller political parties have smaller correlation dimension.

We can interpret it that the smaller parties are more robust

to keep there voters but on the other hand they cannot adapt

changes as the larger parties do. From reconstruction of the

systems‟ strange attractors, we achieve a 60 time steps out of

sample prediction, for Euro election results of Hellas and a

30 time steps out of sample prediction for the Hellenic

National election results. As the time horizon increases the

prediction becomes weak. This depends on strange

attractor‟s structure and the number of raw data. As this

number increases the influence of cubic spline is reduced

and the results will be more precise. As seen before the in

sample prediction works well so we believe that the out of

sample prediction gives satisfactory results. Of course if we

could include data for Ecological Party our prediction will

be more accurate. The prediction with different degrees of

freedom shown that, when the value of embedding

dimension is 4 instead of 6 there is an increase of the

percentage of larger political parties as N.D. and P.A.S.O.K.

are. This can be interpreted as a polarization of voters which

corresponds to a lesser degrees of freedom. The only

exception is the percentage of SYRIZA political party.

Speculate that this behavior is due to that SYRIZA is

composed of components that tend to come together.

Using tools and principles from Physics, we try modeling

an open humanitarian system as the National and Euro

election system is. In any case this work opens new

Politi

cal parties

Elect

ion

Resu

lts %

Embeddin

g dimension

m=6

political

survey

estimation %

Deviat

ion from

election

results

Embedding

dimension m=4

political

survey

estimation %

for

4/10/2009

Deviat

ion from

election

results

ND

33.4

8

33.2

(Range 31.7 -

34.70)

0.28

34 (Range

32.5-35.5)

0.52

PAS

OK

43.9

2

39

(Range 37.5 -

40.5)

4.92

41.1

(Range 39.6 -

42.6)

2.82

KKE

7.54

8.6

(Range 7.6 -

9.6)

1.06

8.6 ( Range

7.6 -8.6)

1.06

SYRI

ZA

4.60

4

(Range 3-5)

0.6

4.1 (Range

3.1 -5.1)

0.5

LAO

S

5.63

7.8

(Range 6.8 -

7.8)

2.17

7.4 (Range

5.4 -7.4)

1.77

290

directions in order to study social behaviour using Physical

models, since human society obey to Physical laws. This is

the first attempt to predict election results with Physical

models. However our predictions 60 days and 30 days,

before the elections , were the most sucesfull among

predictions and exit polls from Greek gallop companies. To

establish a new term as DemoscopoPhysics we need more

data for testing and more tools from Physics to apply as

entropy and criticality and phase transition are. The future

researches may concentrate on the alternative models (i.e.

parametric and nonparametric ones) for prediction. In

addition, to reflect the time-scaling effects and wavelet

theory which can be combined with the chaos theory.

______________________________

References

1. M.Ηanias and L.Magafas “Application of Physics Model in

prediction of the Hellas Euro election results“ Journal of

Engineering Science and Technology Review Volume 2,

Number 1, Pages 104-111, (2009).

2. M.Ηanias and L.Magafas “Application of Physics Model in

prediction of the Hellas election results“ Journal of Engineering

Science and Technology Review Volume 2, Number 1, Pages 112-

117, (2009).

3. Abarbanel H.D.I. (1996), Analysis of observed chaotic data,

Springer, New York.

4. Fraser A.M., Swinney H.L. (1986), Independent coordinates for

strange attractors from mutual information, Phys. Rev. A, 33, 1134.

5. Kugiumtzis D., Lillekjendlie B., Christophersen N.: Chaotic time

series, Part I, Modeling Identification and Control 15, 205, (1994).

6. Kantz H. and T.Schreiber (1997), Nonlinear Time Series Analysis,

Cambridge University Press, Cambridge