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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
M. P. Hanias and L. Magafas/Journal of Engineering Science and Technology Review 4 (3) (2011) 286 -290
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
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prediction of the Hellas Euro election results“ Journal of
Engineering Science and Technology Review Volume 2,
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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).
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Springer, New York.
4. Fraser A.M., Swinney H.L. (1986), Independent coordinates for
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5. Kugiumtzis D., Lillekjendlie B., Christophersen N.: Chaotic time
series, Part I, Modeling Identification and Control 15, 205, (1994).
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