tex: Draft text for PCA analysis
[gostyle.git] / tex / gostyle.tex
1 \documentclass[journal]{IEEEtran}
3 \usepackage{cite}
4 % cite.sty was written by Donald Arseneau
5 % V1.6 and later of IEEEtran pre-defines the format of the cite.sty package
6 % \cite{} output to follow that of IEEE. Loading the cite package will
7 % result in citation numbers being automatically sorted and properly
8 % "compressed/ranged". e.g., [1], [9], [2], [7], [5], [6] without using
9 % cite.sty will become [1], [2], [5]--[7], [9] using cite.sty. cite.sty's
10 % \cite will automatically add leading space, if needed. Use cite.sty's
11 % noadjust option (cite.sty V3.8 and later) if you want to turn this off.
12 % cite.sty is already installed on most LaTeX systems. Be sure and use
13 % version 4.0 (2003-05-27) and later if using hyperref.sty. cite.sty does
14 % not currently provide for hyperlinked citations.
15 % The latest version can be obtained at:
16 % http://www.ctan.org/tex-archive/macros/latex/contrib/cite/
17 % The documentation is contained in the cite.sty file itself.
22 \ifCLASSINFOpdf
23 % \usepackage[pdftex]{graphicx}
24 % declare the path(s) where your graphic files are
25 % \graphicspath{{../pdf/}{../jpeg/}}
26 % and their extensions so you won't have to specify these with
27 % every instance of \includegraphics
28 % \DeclareGraphicsExtensions{.pdf,.jpeg,.png}
29 \else
30 % or other class option (dvipsone, dvipdf, if not using dvips). graphicx
31 % will default to the driver specified in the system graphics.cfg if no
32 % driver is specified.
33 % \usepackage[dvips]{graphicx}
34 % declare the path(s) where your graphic files are
35 % \graphicspath{{../eps/}}
36 % and their extensions so you won't have to specify these with
37 % every instance of \includegraphics
38 % \DeclareGraphicsExtensions{.eps}
39 \fi
41 \usepackage{algorithm}
42 \usepackage{algorithmic}
43 %\usepackage{algpseudocode}
44 % WICKED: nefunguje ani jedno???
45 % algorithmic.sty can be obtained at:
46 % http://www.ctan.org/tex-archive/macros/latex/contrib/algorithms/
47 % There is also a support site at:
48 % http://algorithms.berlios.de/index.html
49 % Also of interest may be the (relatively newer and more customizable)
50 % algorithmicx.sty package by Szasz Janos:
51 % http://www.ctan.org/tex-archive/macros/latex/contrib/algorithmicx/
55 %\usepackage{array}
56 % http://www.ctan.org/tex-archive/macros/latex/required/tools/
59 \usepackage{amsmath}
60 %\usepackage{mdwtab}
61 % http://www.ctan.org/tex-archive/macros/latex/contrib/mdwtools/
64 % IEEEtran contains the IEEEeqnarray family of commands that can be used to
65 % generate multiline equations as well as matrices, tables, etc., of high
66 % quality.
68 %\usepackage{eqparbox}
69 % Also of notable interest is Scott Pakin's eqparbox package for creating
70 % (automatically sized) equal width boxes - aka "natural width parboxes".
71 % Available at:
72 % http://www.ctan.org/tex-archive/macros/latex/contrib/eqparbox/
77 %\usepackage[tight,footnotesize]{subfigure}
78 % subfigure.sty was written by Steven Douglas Cochran. This package makes it
79 % easy to put subfigures in your figures. e.g., "Figure 1a and 1b". For IEEE
80 % work, it is a good idea to load it with the tight package option to reduce
81 % the amount of white space around the subfigures. subfigure.sty is already
82 % installed on most LaTeX systems. The latest version and documentation can
83 % be obtained at:
84 % http://www.ctan.org/tex-archive/obsolete/macros/latex/contrib/subfigure/
85 % subfigure.sty has been superceeded by subfig.sty.
89 %\usepackage[caption=false]{caption}
90 %\usepackage[font=footnotesize]{subfig}
91 % subfig.sty, also written by Steven Douglas Cochran, is the modern
92 % replacement for subfigure.sty. However, subfig.sty requires and
93 % automatically loads Axel Sommerfeldt's caption.sty which will override
94 % IEEEtran.cls handling of captions and this will result in nonIEEE style
95 % figure/table captions. To prevent this problem, be sure and preload
96 % caption.sty with its "caption=false" package option. This is will preserve
97 % IEEEtran.cls handing of captions. Version 1.3 (2005/06/28) and later
98 % (recommended due to many improvements over 1.2) of subfig.sty supports
99 % the caption=false option directly:
100 %\usepackage[caption=false,font=footnotesize]{subfig}
102 % The latest version and documentation can be obtained at:
103 % http://www.ctan.org/tex-archive/macros/latex/contrib/subfig/
104 % The latest version and documentation of caption.sty can be obtained at:
105 % http://www.ctan.org/tex-archive/macros/latex/contrib/caption/
109 % *** FLOAT PACKAGES ***
111 %\usepackage{fixltx2e}
112 % fixltx2e, the successor to the earlier fix2col.sty, was written by
113 % Frank Mittelbach and David Carlisle. This package corrects a few problems
114 % in the LaTeX2e kernel, the most notable of which is that in current
115 % LaTeX2e releases, the ordering of single and double column floats is not
116 % guaranteed to be preserved. Thus, an unpatched LaTeX2e can allow a
117 % single column figure to be placed prior to an earlier double column
118 % figure. The latest version and documentation can be found at:
119 % http://www.ctan.org/tex-archive/macros/latex/base/
123 %\usepackage{stfloats}
124 % stfloats.sty was written by Sigitas Tolusis. This package gives LaTeX2e
125 % the ability to do double column floats at the bottom of the page as well
126 % as the top. (e.g., "\begin{figure*}[!b]" is not normally possible in
127 % LaTeX2e). It also provides a command:
128 %\fnbelowfloat
129 % to enable the placement of footnotes below bottom floats (the standard
130 % LaTeX2e kernel puts them above bottom floats). This is an invasive package
131 % which rewrites many portions of the LaTeX2e float routines. It may not work
132 % with other packages that modify the LaTeX2e float routines. The latest
133 % version and documentation can be obtained at:
134 % http://www.ctan.org/tex-archive/macros/latex/contrib/sttools/
135 % Documentation is contained in the stfloats.sty comments as well as in the
136 % presfull.pdf file. Do not use the stfloats baselinefloat ability as IEEE
137 % does not allow \baselineskip to stretch. Authors submitting work to the
138 % IEEE should note that IEEE rarely uses double column equations and
139 % that authors should try to avoid such use. Do not be tempted to use the
140 % cuted.sty or midfloat.sty packages (also by Sigitas Tolusis) as IEEE does
141 % not format its papers in such ways.
144 %\ifCLASSOPTIONcaptionsoff
145 % \usepackage[nomarkers]{endfloat}
146 % \let\MYoriglatexcaption\caption
147 % \renewcommand{\caption}[2][\relax]{\MYoriglatexcaption[#2]{#2}}
148 %\fi
149 % endfloat.sty was written by James Darrell McCauley and Jeff Goldberg.
150 % This package may be useful when used in conjunction with IEEEtran.cls'
151 % captionsoff option. Some IEEE journals/societies require that submissions
152 % have lists of figures/tables at the end of the paper and that
153 % figures/tables without any captions are placed on a page by themselves at
154 % the end of the document. If needed, the draftcls IEEEtran class option or
155 % \CLASSINPUTbaselinestretch interface can be used to increase the line
156 % spacing as well. Be sure and use the nomarkers option of endfloat to
157 % prevent endfloat from "marking" where the figures would have been placed
158 % in the text. The two hack lines of code above are a slight modification of
159 % that suggested by in the endfloat docs (section 8.3.1) to ensure that
160 % the full captions always appear in the list of figures/tables - even if
161 % the user used the short optional argument of \caption[]{}.
162 % IEEE papers do not typically make use of \caption[]'s optional argument,
163 % so this should not be an issue. A similar trick can be used to disable
164 % captions of packages such as subfig.sty that lack options to turn off
165 % the subcaptions:
166 % For subfig.sty:
167 % \let\MYorigsubfloat\subfloat
168 % \renewcommand{\subfloat}[2][\relax]{\MYorigsubfloat[]{#2}}
169 % For subfigure.sty:
170 % \let\MYorigsubfigure\subfigure
171 % \renewcommand{\subfigure}[2][\relax]{\MYorigsubfigure[]{#2}}
172 % However, the above trick will not work if both optional arguments of
173 % the \subfloat/subfig command are used. Furthermore, there needs to be a
174 % description of each subfigure *somewhere* and endfloat does not add
175 % subfigure captions to its list of figures. Thus, the best approach is to
176 % avoid the use of subfigure captions (many IEEE journals avoid them anyway)
177 % and instead reference/explain all the subfigures within the main caption.
178 % The latest version of endfloat.sty and its documentation can obtained at:
179 % http://www.ctan.org/tex-archive/macros/latex/contrib/endfloat/
181 % The IEEEtran \ifCLASSOPTIONcaptionsoff conditional can also be used
182 % later in the document, say, to conditionally put the References on a
183 % page by themselves.
187 %\usepackage{url}
188 % url.sty was written by Donald Arseneau. It provides better support for
189 % handling and breaking URLs. url.sty is already installed on most LaTeX
190 % systems. The latest version can be obtained at:
191 % http://www.ctan.org/tex-archive/macros/latex/contrib/misc/
192 % Read the url.sty source comments for usage information. Basically,
193 % \url{my_url_here}.
196 % *** Do not adjust lengths that control margins, column widths, etc. ***
197 % *** Do not use packages that alter fonts (such as pslatex). ***
198 % There should be no need to do such things with IEEEtran.cls V1.6 and later.
199 % (Unless specifically asked to do so by the journal or conference you plan
200 % to submit to, of course. )
202 % correct bad hyphenation here
203 \hyphenation{op-tical net-works semi-conduc-tor}
206 \begin{document}
208 % paper title
209 % can use linebreaks \\ within to get better formatting as desired
210 \title{On Move Pattern Trends\\in Large Go Games Corpus}
212 % use \thanks{} to gain access to the first footnote area
213 % a separate \thanks must be used for each paragraph as LaTeX2e's \thanks
214 % was not built to handle multiple paragraphs
215 \author{Petr~Baudis,~Josef~Moudrik% <-this % stops a space
216 \thanks{P. Baudis is student at the Faculty of Math and Physics, Charles University, Prague, CZ, and also does some of his Computer Go research as an employee of SUSE Labs Prague, Novell CZ.}% <-this % stops a space
217 \thanks{J. Moudrik is student at the Faculty of Math and Physics, Charles University, Prague, CZ.}}
219 % note the % following the last \IEEEmembership and also \thanks -
220 % these prevent an unwanted space from occurring between the last author name
221 % and the end of the author line. i.e., if you had this:
223 % \author{....lastname \thanks{...} \thanks{...} }
224 % ^------------^------------^----Do not want these spaces!
226 % a space would be appended to the last name and could cause every name on that
227 % line to be shifted left slightly. This is one of those "LaTeX things". For
228 % instance, "\textbf{A} \textbf{B}" will typeset as "A B" not "AB". To get
229 % "AB" then you have to do: "\textbf{A}\textbf{B}"
230 % \thanks is no different in this regard, so shield the last } of each \thanks
231 % that ends a line with a % and do not let a space in before the next \thanks.
232 % Spaces after \IEEEmembership other than the last one are OK (and needed) as
233 % you are supposed to have spaces between the names. For what it is worth,
234 % this is a minor point as most people would not even notice if the said evil
235 % space somehow managed to creep in.
238 % The paper headers
239 \markboth{Transactions on Computational Intelligence and AI in Games}%
240 {On Pattern Feature Trends in Large Go Game Corpus}
241 % The only time the second header will appear is for the odd numbered pages
242 % after the title page when using the twoside option.
244 % *** Note that you probably will NOT want to include the author's ***
245 % *** name in the headers of peer review papers. ***
246 % You can use \ifCLASSOPTIONpeerreview for conditional compilation here if
247 % you desire.
252 % If you want to put a publisher's ID mark on the page you can do it like
253 % this:
254 %\IEEEpubid{0000--0000/00\$00.00~\copyright~2007 IEEE}
255 % Remember, if you use this you must call \IEEEpubidadjcol in the second
256 % column for its text to clear the IEEEpubid mark.
260 % use for special paper notices
261 %\IEEEspecialpapernotice{(Invited Paper)}
266 % make the title area
267 \maketitle
270 \begin{abstract}
271 %\boldmath
273 We process a~large corpus of game records of the board game of Go and
274 propose a~way to extract per-player summary information on played moves.
275 We then apply several basic data-mining methods on the summary
276 information to identify the most differentiating features within the
277 summary information, and discuss their correspondence with traditional
278 Go knowledge. We show mappings of the features to player attributes
279 like playing strength or informally perceived ``playing style'' (such as
280 territoriality or aggressivity), and propose applications including
281 seeding real-work ranks of internet players, aiding in Go study, or
282 contribution to discussion within Go theory on the scope of ``playing
283 style''.
285 \end{abstract}
286 % IEEEtran.cls defaults to using nonbold math in the Abstract.
287 % This preserves the distinction between vectors and scalars. However,
288 % if the journal you are submitting to favors bold math in the abstract,
289 % then you can use LaTeX's standard command \boldmath at the very start
290 % of the abstract to achieve this. Many IEEE journals frown on math
291 % in the abstract anyway.
293 % Note that keywords are not normally used for peerreview papers.
294 \begin{IEEEkeywords}
295 board games, go, data mining, pattern recongition, player strength, playing style
296 \end{IEEEkeywords}
303 % For peer review papers, you can put extra information on the cover
304 % page as needed:
305 % \ifCLASSOPTIONpeerreview
306 % \begin{center} \bfseries EDICS Category: 3-BBND \end{center}
307 % \fi
309 % For peerreview papers, this IEEEtran command inserts a page break and
310 % creates the second title. It will be ignored for other modes.
311 \IEEEpeerreviewmaketitle
315 \section{Introduction}
316 % The very first letter is a 2 line initial drop letter followed
317 % by the rest of the first word in caps.
319 % form to use if the first word consists of a single letter:
320 % \IEEEPARstart{A}{demo} file is ....
322 % form to use if you need the single drop letter followed by
323 % normal text (unknown if ever used by IEEE):
324 % \IEEEPARstart{A}{}demo file is ....
326 % Some journals put the first two words in caps:
327 % \IEEEPARstart{T}{his demo} file is ....
329 % Here we have the typical use of a "T" for an initial drop letter
330 % and "HIS" in caps to complete the first word.
331 \IEEEPARstart{T}{he} field of Computer Go usually focuses on the problem
332 of creating a~program to play the game, finding the best move from a~given
333 board position. We will make use of one method developed in the course
334 of such research and apply it to the analysis of existing game records
335 with the aim of helping humans to play the game better instead.
337 Go is a~two-player full-information board game played
338 on a~square grid (usually $19\times19$ lines) with black and white
339 stones; the goal of the game is to surround the most territory and
340 capture enemy stones. We assume basic familiarity with the game.
342 Many Go players are eager to play using computers (usually over
343 the internet) and review games played by others on computers as well.
344 This means that large amounts of game records are collected and digitally
345 stored, enabling easy processing of such collections. However, so far
346 only little has been done with the available data --- we are aware
347 only of uses for simple win/loss statistics (TODO: KGS Stats, KGS Analytics,
348 Pro Go Rating) and ''next move'' statistics on a~specific position (TODO:
349 Kombilo, Moyo Go Studio).
351 We present a~more in-depth approach --- from all played moves, we devise
352 a~compact evaluation of each player. We then explore correlations between
353 evaluations of various players in light of externally given information.
354 This way, we can discover similarity between moves characteristics of
355 players with the same playing strength, or discuss the meaning of the
356 "playing style" concept on the assumption that similar playing styles
357 should yield similar moves characteristics.
360 \section{Expert-based knowledge}
361 \label{style-vectors}
362 In order to provide a reference frame for our style analysis,
363 we have gathered some expert-based information about various
364 traditionally perceived style aspects.
365 Three high-level Go players (Alexander Dinerstein 3-pro, Motoki Noguchi
366 7-dan and Vit Brunner 4-dan) have judged style of several Go
367 professionals -- we call them \emph{reference playerse} -- chosen for both
368 being well-known within the community and having large number of played games in our collection.
370 This expert-based knowledge allows us to predict styles of unknown players based on
371 the similarity of their pattern vectors, as well as discover correlations between
372 styles and proportions of played patterns.
374 Experts were asked to assign each of player's style a number
375 on a scale from 1 to 10. These are interpreted
376 as shown in the table below.
378 \vspace{4mm}
379 \noindent
380 %\begin{table}
381 \begin{center}
382 %\caption{Styles}
383 \begin{tabular}{|c|c|c|}
384 \hline
385 \multicolumn{3}{|c|}{Styles} \\ \hline
386 Style & 1 & 10\\ \hline
387 Territoriality & Moyo & Territorial \\
388 Orthodoxity & Classic & Novel \\
389 Aggressivity & Calm & Figting \\
390 Thickness & Safe & Shinogi \\ \hline
391 \end{tabular}
392 \end{center}
393 %\end{table}
394 \vspace{4mm}
396 Averaging this expert based evaluation yields
397 \emph{reference style vector} $\vec s_r$ (of dimension $4$) for each player $r$
398 from the set of \emph{reference players} $R$.
400 %-- each with a \emph{pattern vector} $\vec p_i$ and \emph{style vector} $\vec s_i$.
402 \section{Data Extraction}
403 \label{pattern-vectors}
404 In addition to the explicit expert knowledge, we use the data obtained by...
406 TODO rozvest uvod, nemuze se zacinat jenom As the input...
408 As the input, we assume a~collection of game records\footnote{We
409 use the SGF format (TODO) in our implementation.} organized by player names.
410 We use two collections; the first one is GoGoD Winter 2009 (TODO) containing 42000 (TODO)
411 professional games, dating from the early Go history 1500 years ago to the present.
412 We use this collection for style analysis and detailed correlation analysis
413 of well-known Go professionals.
414 The other source is Go Teaching Ladder reviews (TODO). These include 7600 games
415 of players spanning over all strength levels; we use this collection
416 for finding correlations between moves of players of the same strength rank.
418 In order to generate the required compact description of most frequently played moves,
419 we construct a set of $n$ most occuring patterns (\emph{top patterns})
420 across all players and games from the database\footnote{We use $n=500$ in our analysis.}.
421 For each player, we then count how many times was each of those $n$ patterns played
422 during all his games and finally assign him a~{\em pattern vector} $\vec p$ of dimension $n$, with each
423 dimension corresponding to the relative number of occurences of a given pattern
424 (with respect to player's most played \emph{top pattern}). Using relative numbers of occurences ensures that
425 each dimension of player's \emph{pattern vector} is scaled to range $[0,1]$ and
426 therefore even players with different number of games in the database have comparable \emph{pattern vectors}.
428 \subsection{Pattern Features}
429 We need to define how to compose the patterns we use to describe moves.
430 There are some tradeoffs in play - overly general descriptions carry too few
431 information to discern various player attributes; too specific descriptions
432 gather too few specimen over the games sample and the vector differences are
433 not statistically significant.
435 We have chosen an intuitive and simple approach inspired by pattern features
436 used when computing ELO ratings for candidate patterns in Computer Go play.
437 \cite{ELO} Each pattern is a~combination of several {\em pattern features}
438 (name--value pairs) matched at the position of the played move.
439 We use these features:
441 \begin{itemize}
442 \item capture move flag
443 \item atari move flag
444 \item atari escape flag
445 \item contiguity-to-last flag --- whether the move has been played in one of 8 neighbors of the last move
446 \item contiguity-to-second-last flag
447 \item board edge distance --- only up to distance 4
448 \item spatial pattern --- configuration of stones around the played move
449 \end{itemize}
451 The spatial patterns are normalized (using a dictionary) to be always
452 black-to-play and maintain translational and rotational symmetry.
453 Configurations of radius between 2 and 9 in the gridcular metric%
454 \footnote{The {\em gridcular} metric
455 $d(x,y) = |\delta x| + |\delta y| + \max(|\delta x|, |\delta y|)$ defines
456 a circle-like structure on the Go board square grid. \cite{SpatPat} }
457 are matched.
459 \subsection{Implementation}
461 We have implemented the data extraction by making use of the pattern
462 features matching implementation within the Pachi go-playing program
463 (TODO). We extract information on players by converting the SGF game
464 records to GTP (TODO) stream that feeds Pachi's {\tt patternscan}
465 engine which outputs a~single {\em patternspec} (string representation
466 of the particular pattern features combination) per move.
468 We can then gather all patternspecs played by a~given player and summarize
469 them; the $\vec p$ vector then consists of normalized counts of
470 the given $n$ most frequent patternspecs.
473 \section{Data Mining}
474 \label{data-mining}
476 To assess the properties of gathered \emph{pattern vectors}
477 and their influence on playing styles,
478 we have processes the data using a~few basic data minining techniques.
480 The first two methods ({\em analytic}) rely purely on data gathered
481 from the game collection
482 and serve to show internal structure and correlations within the data set.
483 Principal component analysis finds orthogonal vector components that
484 have the largest variance.
485 Reversing the process then indicates which patterns correlate with each style.
486 Additionally, PCA can be used as a vector-preprocessing for methods
487 that are negatively sensitive to \emph{pattern vector} component correlations.
489 A~second method -- Kohonen maps -- is based on the theory of self-organizing maps of abstract neurons that
490 compete against each other for representation of the input space.
491 Because neurons in the network are organized in a two-dimensional plane,
492 the trained network virtually spreads vectors to the 2D plane,
493 allowing for simple visualization of clusters of players with similar styles.
495 TODO: style vector -> output vector?
497 Furthermore, we have used and compared two \emph{classification} methods
498 that approximate well-defined but unknown \emph{style vector} $\vec S$
499 based on input \emph{pattern vector} $\vec P$.
500 The methods are calibrated based on expert or prior knowledge about
501 training pattern vectors and then their error is measured on a testing
502 set of pattern vectors.
504 One of the methods is $k$-Nearest Neighbor (kNN) classifier:
505 we approximate $\vec S$ by composing the \emph{style vectors} of $k$ nearest \emph{pattern vectors}.
506 The other is based on a multi-layer feed-forward Artificial Neural Network:
507 the neural network can learn correlations between input and output vectors
508 and generalize the ``knowledge'' to unknown vectors; it can be more flexible
509 in the interpretation of different pattern vector elements and discern more
510 complex relations than the kNN classifier, but e.g. requires larger training sample.
512 TODO: Dava ta posledni veta nejaky smysl?!
514 \subsection{Principal Component Analysis}
515 \label{data-mining}
516 We use Principal Component Analysis \emph{PCA} \cite{Jolliffe1986}
517 to reduce the dimensions of the \emph{pattern vectors} while preserving
518 as much information as possible.
520 Briefly, PCA is an eigenvalue decomposition of a~covariance matrix of centered \emph{pattern vectors},
521 producing a~linear mapping $o$ from $n$-dimensional vector space
522 to a~reduced $m$-dimensional vector space.
523 The $m$ eigenvectors of the original vectors' covariance matrix
524 with the largest eigenvalues are used as the base of the reduced vector space;
525 the eigenvectors form the transformation matrix $W$.
527 For each original \emph{pattern vector} $\vec p_i$,
528 we obtain its new representation $\vec r_i$ in the PCA base
529 as shown in the following equation:
530 \begin{equation}
531 \vec r_i = W \cdot \vec p_i
532 \end{equation}
534 The whole process is described in the Algorithm \ref{alg:pca}.
536 \begin{algorithm}
537 \caption{PCA -- Principal Component Analysis}
538 \begin{algorithmic}
539 \label{alg:pca}
540 \REQUIRE{$m > 0$, set of players $R$ with \emph{pattern vectors} $p_r$}
541 \STATE $\vec \mu \leftarrow 1/|R| \cdot \sum_{r \in R}{\vec p_r}$
542 \FOR{ $r \in R$}
543 \STATE $\vec p_r \leftarrow \vec p_r - \vec \mu$
545 \FOR{ $(i,j) \in \{1,... ,n\} \times \{1,... ,n\}$}
546 \STATE $\mathit{Cov}[i,j] \leftarrow 1/|R| \cdot \sum_{r \in R}{\vec p_{ri} \cdot \vec p_{rj}}$
548 \STATE Compute Eigenvalue Decomposition of $\mathit{Cov}$ matrix
549 \STATE Get $m$ largest eigenvalues
550 \STATE Most significant eigenvectors ordered by decreasing eigenvalues form the rows of matrix $W$
551 \FOR{ $r \in R$}
552 \STATE $\vec r_r\leftarrow W \vec p_r$
554 \end{algorithmic}
555 \end{algorithm}
557 \subsection{Kohonen Maps}
558 \label{koh}
559 Kohonen map is a self-organizing network with neurons spread over a two-dimensional plane.
560 Neurons in the map compete for representation of portions of the input vector space.
561 Each neuron $\vec n$ represents a vector
562 and the network is trained so that the neurons that are topologically close
563 tend to represent vectors that are close as well.
565 First, a randomly initialized network is sequentially trained;
566 in each iteration, we choose a random training vector $\vec t$
567 and find the neuron $\vec w$ that is closest to $\vec t$ in Euclidean metric
568 (we call $\vec w$ a \emph{winner neuron}).
570 We then adapt neurons from the neighbourhood of $\vec w$ employing an equation:
571 \begin{equation}
572 \vec n = \vec n + \alpha \cdot \mathit{Influence}(\vec w, \vec n) \cdot (\vec t - \vec n)
573 \end{equation}
574 where $\alpha$ is a learning parameter, usually decreasing in time.
575 $Influence()$ is a function that forces neurons to spread.
576 Such function is usually realised using a mexican hat function or a difference-of-gaussians
577 (see \cite{TODO} for details).
578 The state of the network can be evaluated by calculating mean square difference
579 between each $\vec t \in T$ and its corresponding \emph{winner neuron} $\vec w_t$:
580 \begin{equation}
581 \mathit{Error}(N,T) = \sum_{\vec t \in T}{|\vec w_t - \vec t|}
582 \end{equation}
585 \begin{algorithm}
586 \caption{Kohonen maps -- training}
587 \begin{algorithmic}
588 \label{alg:koh}
589 \REQUIRE{Set of training vectors $T$, input dimension $D$}
590 \REQUIRE{max number of iterations $M$, desired error $E$}
591 \STATE $N \leftarrow \{\vec n | \vec n$ random, $\mathit{dim}(\vec n) = D\}$
593 \STATE $\mathit{It} \leftarrow \mathit{It} + 1$
594 \STATE $\vec t \leftarrow \mathit{PickRandom}(T)$
595 \FORALL{$\vec n \in N$}
596 \STATE $D[\vec n] \leftarrow \mathit{EuclideanDistance}(\vec n, \vec t)$
598 \STATE Find $ \vec w \in N$ so that $D[\vec w] <= D[\vec m], \forall \vec m \in N$
599 \FORALL{$\vec n \in \mathit{TopologicalNeigbors}(N, \vec w)$}
600 \STATE $\vec n \leftarrow \vec n + \alpha(It) \cdot \mathit{Influence}(\vec w, \vec n) \cdot ( \vec t - \vec n ) $
602 \UNTIL{$\mathit{Error}(N, T) < E$ or $ \mathit{It} > M$}
603 \end{algorithmic}
604 \end{algorithm}
607 \subsection{k-nearest Neighbors Classifier}
608 \label{knn}
609 Our goal is to approximate player's \emph{style vector} $\vec S$
610 based on their \emph{pattern vector} $\vec P$.
611 To achieve this, we require prior knowledge of \emph{reference style vectors}
612 (see section \ref{style-vectors}).
614 In this method, we assume that similarities in players' \emph{pattern vectors}
615 uniformly correlate with similarities in players' \emph{style vectors}.
616 We try to approximate $\vec S$ as a weighted average of \emph{style vectors}
617 $\vec s_i$ of $k$ players with \emph{pattern vectors} $\vec p_i$ closest to $\vec P$.
618 This is illustrated in the Algorithm \ref{alg:knn}.
619 Note that the weight is a function of distance and it is not explicitly defined in Algorithm \ref{alg:knn}.
620 During our research, exponentially decreasing weight has proven to be sufficient.
622 \begin{algorithm}
623 \caption{k-Nearest Neighbors}
624 \begin{algorithmic}
625 \label{alg:knn}
626 \REQUIRE{pattern vector $\vec P$, $k > 0$, set of reference players $R$}
627 \FORALL{$r \in R$ }
628 \STATE $D[r] \leftarrow \mathit{EuclideanDistance}(\vec p_r, \vec P)$
630 \STATE $N \leftarrow \mathit{SelectSmallest}(k, R, D)$
631 \STATE $\vec S \leftarrow \vec 0$
632 \FORALL{$r \in N $}
633 \STATE $\vec S \leftarrow \vec S + \mathit{Weight}(D[r]) \cdot \vec s_r $
635 \end{algorithmic}
636 \end{algorithm}
638 \subsection{Neural Network Classifier}
639 \label{neural-net}
641 As an alternative to the k-Nearest Neigbors algorithm (section \ref{knn}),
642 we have used a classificator based on feed-forward artificial neural networks \cite{TODO}.
643 Neural networks (NN) are known for their ability to generalize
644 and find correlations and patterns between input and output data.
645 Neural network is an adaptive system that must undergo a training
646 period before it can be reasonably used, similarly to the requirement
647 of reference vectors for the k-Nearest Neighbors algorithm above.
649 \subsubsection{Computation and activation of the NN}
650 Technically, neural network is a network of interconnected computational units called neurons.
651 A feedforward neural network has a layered topology;
652 it usually has one \emph{input layer}, one \emph{output layer}
653 and an arbitrary number of \emph{hidden layers} inbetween.
655 Each neuron $i$ is connected to all neurons in the previous layer and each connection has its weight $w_{ij}$
657 The computation proceeds in discrete time steps.
658 In the first step, the neurons in the \emph{input layer}
659 are \emph{activated} according to the \emph{input vector}.
660 Then, we iteratively compute output of each neuron in the next layer
661 until the output layer is reached.
662 The activity of output layer is then presented as the result.
664 The activation $y_i$ of neuron $i$ from the layer $I$ is computed as
665 \begin{equation}
666 y_i = f(\sum_{j \in J}{w_{ij} y_j})
667 \end{equation}
668 where $J$ is the previous layer, while $y_j$ is the activation for neurons from $J$ layer.
669 Function $f()$ is so-called \emph{activation function}
670 and its purpose is to bound the outputs of neurons.
671 A typical example of an activation function is the sigmoid function.%
672 \footnote{A special case of the logistic function, defined by the formula
673 $\sigma(x)=\frac{1}{1+e^{-(rx+k)}}$; parameters control the growth rate ($r$)
674 and the x-position ($k$).}
676 \subsubsection{Training}
677 The training of the feed-forward neural network usually involves some
678 modification of supervised Backpropagation learning algorithm. \cite{TODO}
679 We use first-order optimization algorithm called RPROP \cite{Riedmiller1993}.
681 Because the \emph{reference set} is usually not very large,
682 we have devised a simple method for its extension.
683 This enhancement is based upon adding random linear combinations
684 of \emph{style and pattern vectors} to the training set.
686 TODO: Tohle je puvodni napad?
688 As outlined above, the training set consists of pairs of
689 input vectors (\emph{pattern vectors}) and
690 desired output vectors (\emph{style vectors}).
691 The training set $T$ is then extended by adding the linear combinations:
692 \begin{equation}
693 T_\mathit{base} = \{(\vec p_r, \vec s_r) | r \in R\}\\
694 \end{equation}
695 \begin{equation}
696 T_\mathit{ext} = \{(\vec p, \vec s) | \exists D \subseteq R : \vec p = \sum_{d \in D}{g_d \vec p_d}, \vec s = \sum_{d \in D}{g_d \vec s_d}\}
697 \end{equation}
698 TODO zabudovat $g_d$ dovnitr?
699 where $g_d, d \in D$ are random coeficients, so that $\sum_{d \in D}{g_d} = 1$.
700 The training set is then constructed as:
701 \begin{equation}
702 T = T_\mathit{base} \cup \mathit{SomeFiniteSubset}(T_\mathit{ext})
703 \end{equation}
705 The network is trained as shown in Algorithm \ref{alg:tnn}.
707 \begin{algorithm}
708 \caption{Training Neural Network}
709 \begin{algorithmic}
710 \label{alg:tnn}
711 \REQUIRE{Train set $T$, desired error $e$, max iterations $M$}
712 \STATE $N \leftarrow \mathit{RandomlyInitializedNetwork}()$
713 \STATE $\mathit{It} \leftarrow 0$
715 \STATE $\mathit{It} \leftarrow \mathit{It} + 1$
716 \STATE $\Delta \vec w \leftarrow \vec 0$
717 \STATE $\mathit{TotalError} \leftarrow 0$
718 %\FORALL{$(\overrightarrow{Input}, \overrightarrow{DesiredOutput}) \in T$}
719 %\STATE $\overrightarrow{Output} \leftarrow Result(N, \overrightarrow{Input})$
720 %\STATE $E \leftarrow |\overrightarrow{DesiredOutput} - \overrightarrow{Output}|$
721 \FORALL{$(\mathit{Input}, \mathit{DesiredOutput}) \in T$}
722 \STATE $\mathit{Output} \leftarrow \mathit{Result}(N, \mathit{Input})$
723 \STATE $\mathit{Error} \leftarrow |\mathit{DesiredOutput} - \mathit{Output}|$
724 \STATE $\Delta \vec w \leftarrow \Delta \vec w + \mathit{WeightUpdate}(N,\mathit{Error})$
725 \STATE $\mathit{TotalError} \leftarrow \mathit{TotalError} + \mathit{Error}$
727 \STATE $N \leftarrow \mathit{ModifyWeights}(N, \Delta \vec w)$
728 \UNTIL{$\mathit{TotalError} < e$ or $ \mathit{It} > M$}
729 \end{algorithmic}
730 \end{algorithm}
733 \subsubsection{Architecture details}
734 TODO num layers, num neurons, ..
737 \subsection{Implementation}
739 We have implemented the data mining methods as an open-source framework ``gostyle'' \cite{TODO},
740 made available under the GNU GPL licence.
741 We use python for the basic processing and most of the analysis;
742 the MDP library \cite{MDP} is used for PCA analysis, Kohonen library \cite{KohonenPy} for Kohonen maps.
743 The neuron network classifier is using the libfann C library. \cite{TODO}
746 \section{Strength Estimation Analysis}
748 First, we have used our framework to analyse correlations of pattern vectors
749 and playing strength. Like in other competitively played board games, Go players
750 receive real-world rating based on tournament games, and rank based on their
751 rating.\footnote{Elo-like rating system \cite{GoR} is usually used,
752 corresponding to even win chances for game of two players with the same rank,
753 and about 2:3 win chance for white in case of one rank difference.}%
754 \footnote{Professional ranks and dan ranks in some Asia countries may
755 be assigned differently.} The amateur ranks range from 30kyu (beginner) to
756 1kyu (intermediate) and then follows 1dan to 7dan (9dan in some systems;
757 top-level player). Multiple independent real-world ranking scales exist
758 (geographically based) and online servers maintain their own user ranking;
759 the difference can be up to several stones.
761 As the source game collection, we use Go Teaching Ladder
762 reviews\footnote{The reviews contain comments and variations --- we consider only the actual played game.}
763 \cite{GTL} --- this collection contains 7700 games of players with strength ranging
764 from 30k to 4d; we consider only even games with clear rank information, and then
765 randomly separate 770 games as a testing set. Since the rank information is provided
766 by the users and may not be consistent, we are forced to take a simplified look
767 at the ranks, discarding the differences between various systems and thus increasing
768 error in our model.\footnote{Since
769 our results seem satisfying, we did not pursue to try another collection}
771 First, we have created a single pattern vector for each rank, from 30k to 4d;
772 we have performed PCA analysis on the pattern vectors, achieving near-perfect
773 rank correspondence in the first PCA dimension\footnote{The eigenvalue of the
774 second dimension was four orders of magnitude smaller, with no discernable
775 structure revealed within the lower-order eigenvectors.}
776 (chi-square test TODO).
777 (Figure TODO.) Using the eigenvector position directly for classification
778 of players within the test group yields MSE TODO, thus providing
779 reasonably satisfying accuracy.
781 To further enhance the strength estimator accuracy,
782 we have tried to train a NN classifier on our train set, consisting
783 of one $(\vec p, {\rm rank})$ pair per player --- we use the pattern vector
784 for activation of input neurons and rank number as result of the output
785 neuron. We then proceeded to test the NN on per-player pattern vectors built
786 from the games in the test set, yielding MSE of TODO with TODO games per player
787 on average.
790 \section{Style Components Analysis}
792 PCA analysis yielded X, chi-square test...
794 We then tried to apply the NN classifier with linear output function on the dataset
795 and that yielded Y (see fig. Z), with MSE abcd.
798 \section{Proposed Applications}
800 We believe that our findings might be useful for many applications
801 in the area of Go support software as well as Go-playing computer engines.
803 The style analysis can be an excellent teaching aid --- classifying style
804 dimensions based on player's pattern vector, many study recommendations
805 can be given, e.g. about the professional games to replay, the goal being
806 balancing understanding of various styles to achieve well-rounded skill set.
807 This was also our original aim when starting the research and a user-friendly
808 tool based on our work is now being created.
810 We hope that more strong players will look into the style dimensions found
811 by our statistical analysis --- analysis of most played patterns of prospective
812 opponents might prepare for the game, but we especially hope that new insights
813 on strategic purposes of various shapes and general human understanding
814 of the game might be achieved by investigating the style-specific patterns.
816 Classifying playing strength of a pattern vector of a player can be used
817 e.g. to help determine initial real-world rating of a player before their
818 first tournament based on games played on the internet; some players especially
819 in less populated areas could get fairly strong before playing their first
820 real tournament.
822 Analysis of pattern vectors extracted from games of Go-playing programs
823 in light of the shown strength and style distributions might help to
824 highlight some weaknesses and room for improvements. (However, since
825 correlation does not imply causation, simply optimizing Go-playing programs
826 according to these vectors is unlikely to yield good results.)
827 Another interesting applications in Go-playing programs might be strength
828 adjustment; the program can classify the player's level based on the pattern
829 vector from its previous games and auto-adjust its difficulty settings
830 accordingly to provide more even games for beginners.
833 % An example of a floating figure using the graphicx package.
834 % Note that \label must occur AFTER (or within) \caption.
835 % For figures, \caption should occur after the \includegraphics.
836 % Note that IEEEtran v1.7 and later has special internal code that
837 % is designed to preserve the operation of \label within \caption
838 % even when the captionsoff option is in effect. However, because
839 % of issues like this, it may be the safest practice to put all your
840 % \label just after \caption rather than within \caption{}.
842 % Reminder: the "draftcls" or "draftclsnofoot", not "draft", class
843 % option should be used if it is desired that the figures are to be
844 % displayed while in draft mode.
846 %\begin{figure}[!t]
847 %\centering
848 %\includegraphics[width=2.5in]{myfigure}
849 % where an .eps filename suffix will be assumed under latex,
850 % and a .pdf suffix will be assumed for pdflatex; or what has been declared
851 % via \DeclareGraphicsExtensions.
852 %\caption{Simulation Results}
853 %\label{fig_sim}
854 %\end{figure}
856 % Note that IEEE typically puts floats only at the top, even when this
857 % results in a large percentage of a column being occupied by floats.
860 % An example of a double column floating figure using two subfigures.
861 % (The subfig.sty package must be loaded for this to work.)
862 % The subfigure \label commands are set within each subfloat command, the
863 % \label for the overall figure must come after \caption.
864 % \hfil must be used as a separator to get equal spacing.
865 % The subfigure.sty package works much the same way, except \subfigure is
866 % used instead of \subfloat.
868 %\begin{figure*}[!t]
869 %\centerline{\subfloat[Case I]\includegraphics[width=2.5in]{subfigcase1}%
870 %\label{fig_first_case}}
871 %\hfil
872 %\subfloat[Case II]{\includegraphics[width=2.5in]{subfigcase2}%
873 %\label{fig_second_case}}}
874 %\caption{Simulation results}
875 %\label{fig_sim}
876 %\end{figure*}
878 % Note that often IEEE papers with subfigures do not employ subfigure
879 % captions (using the optional argument to \subfloat), but instead will
880 % reference/describe all of them (a), (b), etc., within the main caption.
883 % An example of a floating table. Note that, for IEEE style tables, the
884 % \caption command should come BEFORE the table. Table text will default to
885 % \footnotesize as IEEE normally uses this smaller font for tables.
886 % The \label must come after \caption as always.
888 %\begin{table}[!t]
889 %% increase table row spacing, adjust to taste
890 %\renewcommand{\arraystretch}{1.3}
891 % if using array.sty, it might be a good idea to tweak the value of
892 % \extrarowheight as needed to properly center the text within the cells
893 %\caption{An Example of a Table}
894 %\label{table_example}
895 %\centering
896 %% Some packages, such as MDW tools, offer better commands for making tables
897 %% than the plain LaTeX2e tabular which is used here.
898 %\begin{tabular}{|c||c|}
899 %\hline
900 %One & Two\\
901 %\hline
902 %Three & Four\\
903 %\hline
904 %\end{tabular}
905 %\end{table}
908 % Note that IEEE does not put floats in the very first column - or typically
909 % anywhere on the first page for that matter. Also, in-text middle ("here")
910 % positioning is not used. Most IEEE journals use top floats exclusively.
911 % Note that, LaTeX2e, unlike IEEE journals, places footnotes above bottom
912 % floats. This can be corrected via the \fnbelowfloat command of the
913 % stfloats package.
917 \section{Conclusion}
918 The conclusion goes here.
919 We have shown brm and proposed brm.
921 Since we are not aware of any previous research on this topic and we
922 are limited by space and time constraints, plenty of research remains
923 to be done. There is plenty of room for further research in all parts
924 of our analysis --- different methods of generating the $\vec p$ vectors
925 can be explored; other data mining methods could be tried.
926 It can be argued that many players adjust their style by game conditions
927 (Go development era, handicap, komi and color, time limits, opponent)
928 or styles might express differently in various game stages.
929 More professional players could be consulted on the findings
930 and for style scales calibration. Impact of handicap games on by-strength
931 $\vec p$ distribution should be investigated.
933 TODO: Future research --- Sparse PCA
938 % if have a single appendix:
939 %\appendix[Proof of the Zonklar Equations]
940 % or
941 %\appendix % for no appendix heading
942 % do not use \section anymore after \appendix, only \section*
943 % is possibly needed
945 % use appendices with more than one appendix
946 % then use \section to start each appendix
947 % you must declare a \section before using any
948 % \subsection or using \label (\appendices by itself
949 % starts a section numbered zero.)
953 %\appendices
954 %\section{Proof of the First Zonklar Equation}
955 %Appendix one text goes here.
957 %% you can choose not to have a title for an appendix
958 %% if you want by leaving the argument blank
959 %\section{}
960 %Appendix two text goes here.
963 % use section* for acknowledgement
964 \section*{Acknowledgment}
965 \label{acknowledgement}
968 We would like to thank X for reviewing our paper.
969 We appreciate helpful comments on our general methodology
970 by John Fairbairn, T. M. Hall, Robert Jasiek
971 and several GoDiscussions.com users. \cite{GoDiscThread}
972 Finally, we are very grateful for ranking of go styles of selected professionals
973 by Alexander Dinerstein 3-pro, Motoki Noguchi 7-dan and Vit Brunner 4-dan.
976 % Can use something like this to put references on a page
977 % by themselves when using endfloat and the captionsoff option.
978 \ifCLASSOPTIONcaptionsoff
979 \newpage
984 % trigger a \newpage just before the given reference
985 % number - used to balance the columns on the last page
986 % adjust value as needed - may need to be readjusted if
987 % the document is modified later
988 %\IEEEtriggeratref{8}
989 % The "triggered" command can be changed if desired:
990 %\IEEEtriggercmd{\enlargethispage{-5in}}
992 % references section
994 % can use a bibliography generated by BibTeX as a .bbl file
995 % BibTeX documentation can be easily obtained at:
996 % http://www.ctan.org/tex-archive/biblio/bibtex/contrib/doc/
997 % The IEEEtran BibTeX style support page is at:
998 % http://www.michaelshell.org/tex/ieeetran/bibtex/
999 \bibliographystyle{IEEEtran}
1000 % argument is your BibTeX string definitions and bibliography database(s)
1001 \bibliography{gostyle}
1003 % <OR> manually copy in the resultant .bbl file
1004 % set second argument of \begin to the number of references
1005 % (used to reserve space for the reference number labels box)
1006 %\begin{thebibliography}{1}
1008 %\bibitem{MasterMCTS}
1010 %\end{thebibliography}
1012 % biography section
1014 % If you have an EPS/PDF photo (graphicx package needed) extra braces are
1015 % needed around the contents of the optional argument to biography to prevent
1016 % the LaTeX parser from getting confused when it sees the complicated
1017 % \includegraphics command within an optional argument. (You could create
1018 % your own custom macro containing the \includegraphics command to make things
1019 % simpler here.)
1020 %\begin{biography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{mshell}}]{Michael Shell}
1021 % or if you just want to reserve a space for a photo:
1023 \begin{IEEEbiography}{Michael Shell}
1024 Biography text here.
1025 \end{IEEEbiography}
1027 % if you will not have a photo at all:
1028 \begin{IEEEbiographynophoto}{John Doe}
1029 Biography text here.
1030 \end{IEEEbiographynophoto}
1032 % insert where needed to balance the two columns on the last page with
1033 % biographies
1034 %\newpage
1036 \begin{IEEEbiographynophoto}{Jane Doe}
1037 Biography text here.
1038 \end{IEEEbiographynophoto}
1040 % You can push biographies down or up by placing
1041 % a \vfill before or after them. The appropriate
1042 % use of \vfill depends on what kind of text is
1043 % on the last page and whether or not the columns
1044 % are being equalized.
1046 %\vfill
1048 % Can be used to pull up biographies so that the bottom of the last one
1049 % is flush with the other column.
1050 %\enlargethispage{-5in}
1054 % that's all folks
1055 \end{document}