long-range correlations of sequential · pdf filekeywords and phras es: power-law...
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Fundamental Journal of Modern Physics
Vol. 8, Issue 2, 2015, Pages 175-186
Published online at http://www.frdint.com/
:esphras and Keywords power-law correlations, detrended fluctuation analysis, Go/Baduk/
Weiqi game.
Received October 21, 2015
© 2015 Fundamental Research and Development International
LONG-RANGE CORRELATIONS OF SEQUENTIAL MOVES
IN A BOARD GAME, GO
HYUN-JOO KIM
Department of Physics Education
Korea National University of Education
Chungbuk 363-791
Korea
e-mail: [email protected]
Abstract
The correlation properties of a board game, go in which the making-
decision in human brain works in each steps by simple rules, were
analyzed by using the detrended fluctuation analysis (DFA). We defined
the sequential distance of moves as time series characterizing the go game
and found that the sequential distance time series shows the power-law
correlation behavior with the average value of DFA scaling exponent
.09.066.0 ±=h Such presence of long-term persistence in sequential
distance time series explains well the characteristics of the go game. We
also considered the black (or white) sequential distance series constructed
by only selecting odd (or even) elements from the sequential distance
series to analyze the correlations between only same sides, black or white,
and found that it shows the same correlation behaviors as the sequential
distance series on average.
1. Introduction
In the past decades, the study of the temporal fluctuations has attracted persisting
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interest in many physical, biological, economic, and social systems. For the analysis
of nonstationary time series the detrended fluctuation analysis (DFA) method was
introduced by C. K. Peng et al. [1-5], and it has successfully been applied to diverse
fields such as biophysics [6], seismology [7], economic time series [8, 9], climatic
phenomena [10-12], music [13], Wikipedia [14], and so on. Thus scrutinizing the
correlations of temporal fluctuations involved in various complex systems can serve
as the basis for understanding the system.
The complexity is inherent in various human activities as well as physical or
natural phenomena so it is meaningful to analyze the human activities using the
physical methods [15-21]. Though aspects of human behaviors appear in a variety of
ways, one of them always being with humankind is a play which has been as many
different kind of ways from traditional plays to modern computer games. Among
them, one of the oldest and most popular games is go, a board game in which two
players occupy areas or capture opposite stones by placing black and white stones on
a board in turn and finally the one who occupies larger area becomes the winner. The
total number of legal positions is about 171
10 despite its relatively simple rules,
which makes a go be a brain game involving rich strategies and complicated fights.
This complexity resulted in the lost of the most powerful go programs to professional
players and computer go programs have remained in about four point lower level
compared to professional players until now [22]. This contrasts with the fact that
computer programs beat world class champions of chess recently.
Because developing go or chess algorithms is to approach the working of
decision-making in the human brain, the studies for this have been done continuously
in related fields and the Monte-Carlo method has also been used in the work [17, 23].
Recently, according to containing the large collections of go game records in online
go sites, the studies have been accomplished to find the statistical properties of go
game [18, 24]. In [18], the go game was studied from the perspective of complex
networks [25, 26] and it was found that the directed network of go is scale free, with
statistical peculiarities different from other real directed network.
In this paper, we study the correlation properties of go games. The process of
two player’s strategic thinking to win the game and the interactions between them is
recorded and reflected in the sequential order of placements of stones of a go game.
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177
A player determines a placement of a stone in his/her turn by judgement for the
positions of the previous stones and future strategy. So there are the correlations of
the positions of placements. We consider the values of sequential distances between
two successive placements of stones as a time series and perform the time series
analysis by using DFA method to find the correlation behaviors of a go game. The
paper is organized as follows. In Section 2, we describe the basic properties of go
and define the appropriate time series for the go. DFA, the method for analyzing the
go time series is explained in Section 3. In Section 4, the data analysis and results are
included and we conclude in Section 5.
2. Sequence of Go
The rules of go is very simple. Two players, in turn, place a stone at a legal point
(nonoccupied intersections) on a go board which is composed of 19 horizontal and
19 vertical lines and finally the one occupies the larger territory becomes the winner.
Traditionally, the first player takes black stones and the next player takes white
stones. The territory on the board is the parts of being completely surrounded by the
same color stones, and the stones entirely surrounded by the opponent become dead
and removed from the board and counted as the opponent player’s territory.
Therefore the process occupying territory is achieved by properly attacking opponent
stones and defending opposite attacks rather than just connecting same stones. For
this process, a player firstly need to take key points relatively long way apart such as
(4, 4), (3, 4), (3, 5), etc. positions of points in the early stage, while in fight
successive stones might get closer to each other. That is, the positions of the next
placements are affected by the former placements according to player’s strategy.
In order to find the correlations between the placements in a go game, we define
the sequential distance time series ( )nd for a go game as follows
( ) ( ) ( ) ,1...,,3,2,1,1 −=−+= Nnnrnrnd��
(1)
where ( )nr�
is a position vector of the nth stone placed on a go board where the
origin is at the left-bottom corner and N is the total number of placements of the
game. Thus, d represents the Euclidian distance between two points being
successively placed on the board.
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For this study, we used large collections of go records of SGF format [27] being
freely available from go websites [28]. We used recent (from 2007 to 2012) 623
professional games, each of which has over 300 placements considering recent
tendency in go game and meaningful length of time series.
Figure 1 shows, as examples, the sequential distances time series of six different
professional games of go.
Figure 1. Sequential distance time series of six different professional go games.
3. Detrended Fluctuation Analysis
The DFA method works on finding correlations hidden within noisy and
nonstationary time series. Let us suppose that nx is a series of length N, and that this
series is of compact support which is defined as the set of the indices n with nonzero
values .nx The DFA procedure is composed of the following steps [1, 4, 5].
(1) Determine the ‘profile’
( ) .....,,1,1
NtxxtY
t
n
n =−== ∑=
(2)
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Here x is the mean value of x.
(2) Divide ( )tY into ( )sNNs int= non-overlapping segments of equal length
s. In order to prevent from being omitted a part at the end of the series, the profile is
also divided into segments from the opposite end. So one can obtain sN2 segments.
(3) Find the local trend for each segment by fitting polynomial to the data and
then determine the variance ( )skF ,2 of the differences between the profile and the
fit in each segment .2...,,1, sNkk = Subtracting the fit from the profile makes the
series be detrended. In the fitting procedure, linear, quadratic, cubic, or higher order
polynomials can be used and they are conventionally called as DFA1, DFA2 DFA3,
etc.
(4) Average ( )skF ,2 over all segments and take the square root to obtain the
fluctuation function ( ).sF
( ) ( ).,2
12
1
2∑=
=sN
ks
skFN
sF (3)
(5) The dependence of the fluctuation function ( )sF on s is represented as a
power law such that
( ) ,~ hssF (4)
h is the DFA scaling exponent which quantifies the strength of the long-range power
law correlations of the sequence. DFA can conceptually be understood as
characterizing the motion of random walker whose steps are given by the time series.
( )sF gives the walker’s deviation from the local trend as a function of the segment
length. Because the root mean square of displacement of a walker with no
correlations between his steps scales as ,5.0
s a time series with no correlations yields
an .5.0=h However, the values of h is not equal to 0.5 for a time series with long-
range power-law correlations. For ,5.0>h the correlations of the sequence are
persistent (i.e., large (small) terms is more likely to follow large (small) terms) and
for ,5.0<h the correlations of the sequence are antipersistent (i.e., large (small)
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terms is more likely to follow small (large) terms).
4. Data Analysis, Results
We applied the DFA method to the sequential distance time series of go games.
The process of a go play can be represented as the movement of walker using the
DFA profile. Figure 2 shows that the profiles of six different go games already shown
in Figure 1. In Figure 2(a), ( )tY has rapidly descent slope in the range of about time
100 to 130, which indicates that there are many placements with shorter distance than
the average sequential distance. That is, there was heavy fighting in the middle stage
of the game. The pattern with ascent slope above time 290 in Figure 2(b), might fill
in on a persistent pae-fight [29]. Thus, it could be a good method to map a go play to
the DFA profile showing flows and characteristics of the play.
Figure 2. The DFA profiles of sequential distance time series of six different go
records shown in Figure 1.
The correlations between sequential distances in go could be quantitatively
investigated by the DFA method. We measured the fluctuation function ( )sF for a
sample play, varying the order of detrending polynomial from 1st to 4th (from DFA1
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to DFA4). In Figure 3, we plot ( )sF as a function of s on double logarithmic scales
and calculate the scaling exponents by a linear fit in the regime 47 Ns << for the
length of the go sequence .331=N It shows that the scaling exponent 1h of DFA1
is different from those of the higher order and the exponent values from DFA2 to
DFA4 are almost same, which indicates that the DFA1 method shows instability in
detrending local trends. Therefore it is appropriate to apply the DFA method with
over order 2 in analysis of correlations between go sequential distances.
Figure 3. The fluctuation function ( )sF for a sample go game, varying the order of
detrending polynomial from 1st to 4th. The ih represents the DFA scaling exponent
obtained by the ith order DFA.
We first measured the fluctuation function ( )sF of the six representative go
sequence considered already in Figure 1 to examine the correlations appeared in
individual plays. Figure 4 shows the fluctuation function ( )sF obtained by DFA2
method. The scaling exponents range from 0.64 to 0.77 larger than 0.5, which
indicates that sequential distance has long-range power-law correlations. This means
that the sequential distances time series persistent, i.e., large distances are more likely
to be followed by large distances and small distances by small distances.
For the entire set of sequential distance time series we measured the fluctuation
functions ( )sF for DFA2 and represented the distribution of the values of the scaling
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exponents as the histograms in Figure 5. It shows that for go plays over 95% the
scaling exponents have values being larger than 0.5 and spread out in the range
9.05.0 << h having the first peak at 0.7 and the second peak at 0.6. It indicates that
most sequential distance time series are clearly correlated following large (small)
distance by large (small) distance and the strengths of the correlations could be
different from each game. We obtained the average DFA scaling exponent
.09.066.0 ±=h
Figure 4. The fluctuation function ( )sF using DFA2 for six different go records
shown in Figure 1. They show the power-law correlation behaviors with the DFA
scaling exponent in ranges from 0.62 to 0.84.
In order to also examine the correlations between sequential distances of same
color stones, we made up times series for same color stones and separately applied
them to the DFA method. The sequential distance time series of black stones bd is
given by
( ) ( ) bb Nnnrnrn
d ...,,5,3,1,22
1=−+=
+ ��
(5)
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and wd for white stones is given by
( ) ( ) ,...,,6,4,2,22 ww Nnnrnrn
d =−+=
��
(6)
( )wb NN is the number of final black (white) stone’s placement which is 1−N or
.2−N Figure 6 shows the distribution of DFA scaling exponents bh and wh for
black and white which are obtained by DFA2 for the entire go records. They look
similar to the distribution of the DFA scaling exponent h. Also the averages of bh
and wh are 13.068.0 ± and ,13.068.0 ± respectively, which are the same as h. It
means that, on average, there are no different correlations between same sides from
entire correlations. However, it does not mean that the correlations between the black
or the white moves are the same as that of the sequential moves in each play.
Figure 5. The distribution of the DFA scaling exponents of sequential distance series
for 623 go games. The average DFA scaling exponent is .09.066.0 ±=h
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Figure 6. The distributions of DFA scaling exponents for (a) the black and (b) the
white sequential distance series. The averages of bh and wh are 11.067.0 ± and
,11.067.0 ± respectively.
5. Conclusions
We tried to examine the statistical characteristics of complex brain games in this
study and analyzed the correlations of time series characterizing the brain game as a
starting point for understanding the brain-strategic games. We chose the go game
which is one of the oldest and popular board game in Asia, as a game for this purpose
and used the DFA method which is appropriate to the analysis of nonstationary time
series, to analyze the correlations of the time series. Because the player's strategies
are revealed from where and in what order stones are placed on the board, we
considered the distance between two successive placements as a basic quantity
characterizing the go game and defined the sequential distance time series of go. We
found that the sequential distance time series are persistent, i.e., display power-law
correlated behavior with the average DFA scaling exponent .66.0=h The
distribution of the exponents showed that the exponent values are spread in ranges
with the value of standard deviation .09.0 This means that although the correlation
of the go sequences is essential for the game, the strengths of correlations in
individual games are not universal. Therefore the next study to understand the
properties of different correlations appearing in individual games is needed, through
which ways to quantify factors deciding the results of games might be found.
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References
[1] C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley and A. L.
Goldberger, Phys. Rev. E 49 (1994), 1685.
[2] K. Hu, P. C. Ivanov, Z. Chen, P. Carpena and H. E. Stanley, Phys. Rev. E 64 (2001),
011114.
[3] Z. Chen, P. C. Ivanov, K. Hu and H. E. Stanley, Phys. Rev. E 65 (2002), 041107.
[4] J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde and H.
E. Stanley, Physica A 316 (2002), 87.
[5] A. Bunde, S. Havlin, J. W. Kantelhardt, T. Penzel, J-H. Peter and K. Voigt, Phys. Rev.
Lett. 85 (2000), 3736.
[6] J. W. Kantelhardt, Y. Ashkenazy, P. C. Ivanov, A. Bunde, S. Havlin, T. Penzel, J.
Peter and H. E. Stanley, Phys. Rev. E 65 (2002), 051908.
[7] Z. Zheng, K. Yamasaki, J. Tenenbaum, B. Podobnik, Y. Tamura and H. E. Stanley,
Phys. Rev. E 86 (2012), 011107.
[8] R. Gu, Y. Shao and Q. Wang, Physica A 392 (2013), 361.
[9] P. Ch. Ivanov, A. Yuen, B. Podobnik and Y. Lee, Phys. Rev. E 69 (2004), 056107.
[10] A. Bunde, J. F. Eichner, J. W. Kantelhardt and S. Havlin, Phys. Rev. Lett. 94 (2005),
048701.
[11] A. Vecchio and V. Carbone, Phys. Rev. E 82 (2010), 066101.
[12] V. Capparelli, A. Vecchio and V. Carbone, Phys. Rev. E 84 (2011), 046103.
[13] L. Telesca and M. Lovallo, Physica A 391 (2013), 3247.
[14] M. Kämpf, S. Tismer, J. W. Kantelhardt and L. Muchnik, Physica A 391 (2012), 6101.
[15] Y.-M. Choi and H. -J. Kim, Physica A 382 (2007), 665.
[16] H.-J. Kim, Mod. Phys. Lett. B 28 (2012), 1250186.
[17] B. Blasius and R. Tönjes, Phys. Rev. Lett. 103 (2009), 218701.
[18] B. Georgeot and O. Giraud, Europhys. Lett. 97 (2012), 68002.
[19] J. Greenhough, P. C. Birch, S. C. Chapman and G. Rowlands, Physica A 316 (2002),
615; L. C. Malacarne and R. S. Mendes, Physica A 286 (2000), 391.
[20] D. R. Amancio, O. N. Oliveira Jr. and L. da F. Costa, Physica A 390 (2011), 4673.
[21] B. Xu, H.-J. Zhou and Z. Wang, Physica A 392 (2013), 4997; T. Platkowski and J.
Zakrzewski, Physica A 390 (2011), 4219.
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[22] In March 2013 professional player I. Yoshio beat a computer go program Zen and was
defeated to a program Crazy Stone with four handicap stones in the inaugural games at
the 6th Computer Go UEC Cup.
[23] G. Chaslot, M. Winands, J. Uiterwijk, H. J. van den Herik and B. Bouzy, New
Mathmatics and Natural Computation 4 (2008), 343.
[24] P. Baudiš and J. Moudřik, Arxiv, CoRR, October (2012)
http://arxiv.org/abs/1209.5251.
[25] R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74 (2002), 47; M. E. J. Newman,
SIAM Rev. 45 (2003), 167.
[26] H.-J. Kim and J. M. Kim, Phys. Rev. E 72 (2005), 036109.
[27] A. Hollosi, SGF File Format (2006) http://www.red-bean.com/sgf/
[28] http://gomovies.com/pro games/
[29] A pae-fight is also known as ko-fight. The go rule forbids a player to place a stone on
the nearest site of the previous placement by which the same pattern may be repeated
infinitely.