tex: Data Mining cleanups all over the map

[gostyle.git] / tex / gostyle.tex

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\hyphenation{op-tical net-works semi-conduc-tor know-ledge}


\begin{document}
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% paper title
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\title{On Move Pattern Trends\\in Large Go Games Corpus}

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\author{Petr~Baudi\v{s},~Josef~Moud\v{r}\'{i}k% <-this % stops a space
\thanks{P. Baudi\v{s} 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
\thanks{J. Moud\v{r}\'{i}k is student at the Faculty of Math and Physics, Charles University, Prague, CZ.}}

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\markboth{Transactions on Computational Intelligence and AI in Games}%
{On Pattern Feature Trends in Large Go Game Corpus}
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% make the title area
\maketitle


\begin{abstract}
%\boldmath

We process a~large corpus of game records of the board game of Go and
propose a~way to extract summary information on played moves.
We then apply several basic data-mining methods on the summary
information to identify the most differentiating features within the
summary information, and discuss their correspondence with traditional
Go knowledge. We show mappings of the features to player attributes
like playing strength or informally perceived ``playing style'' (such as
territoriality or aggressivity), and propose applications including
seeding real-work ranks of internet players, aiding in Go study, or
contribution to Go-theoretical discussion on the scope of ``playing
style''.

\end{abstract}
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\begin{IEEEkeywords}
board games, go, data mining, pattern recongition, player strength, playing style,
neural networks, Kohonen maps, principal component analysis
\end{IEEEkeywords}






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\section{Introduction}
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\IEEEPARstart{T}{he} field of Computer Go usually focuses on the problem
of creating a~program to play the game, finding the best move from a~given
board position. \cite{GellySilver2008}
We will make use of one method developed in the course
of such research and apply it to the analysis of existing game records
with the aim of helping humans to play and understand the game better
instead.

Go is a~two-player full-information board game played
on a~square grid (usually $19\times19$ lines) with black and white
stones; the goal of the game is to surround the most territory and
capture enemy stones. We assume basic familiarity with the game.

Many Go players are eager to play using computers (usually over
the internet) and review games played by others on computers as well.
This means that large amounts of game records are collected and digitally
stored, enabling easy processing of such collections. However, so far
only little has been done with the available data --- we are aware
only of uses for simple win/loss statistics \cite{KGSStats} \cite{KGSAnalytics} \cite{ProGoR}
and ``next move'' statistics on a~specific position \cite{Kombilo} \cite{MoyoGo}.

We present a~more in-depth approach --- from all played moves, we devise
a~compact evaluation of each player. We then explore correlations between
evaluations of various players in light of externally given information.
This way, we can discover similarity between moves characteristics of
players with the same playing strength, or discuss the meaning of the
"playing style" concept on the assumption that similar playing styles
should yield similar moves characteristics.


\section{Data Extraction}
\label{pattern-vectors}

As the input of our analysis, we use large collections of game records%
\footnote{We use the SGF format \cite{SGF} in our implementation.}
grouped by the primary object of analysis (player name, player rank, etc.).
We process the games by object, generating a description for each
played move -- a {\em pattern}, being a combination of several
{\em pattern features} described below.

We keep track of the most
occuring patterns, finally composing $n$-dimensional {\em pattern vector}
$\vec p$ of per-pattern counts from the $n$ globally most frequent patterns%
\footnote{We use $n=500$ in our analysis.}
(the mapping from patterns to vector elements is common for all objects).
We can then process and compare just the pattern vectors.

The pattern vector elements can have diverse values since for each object,
we consider different number of games (and thus patterns).
Therefore, we linearly rescale and normalize the values to range $[-1,1]$,
the most frequent pattern having the value of $1$ and the least occuring
one being $-1$.%
\footnote{We did not investigate different methods of re-scaling the vectors;
that might be a good way of improving accuracy of our analysis.}
Thus, we obtain vectors describing relative frequency of played patterns
independent on number of gathered patterns.

\subsection{Pattern Features}
When deciding how to compose the patterns we use to describe moves,
we need to consider a specificity tradeoff --- overly general descriptions carry too few
information to discern various player attributes; too specific descriptions
gather too few specimen over the games sample and the vector differences are
not statistically significant.

We have chosen an intuitive and simple approach inspired by pattern features
used when computing Elo ratings for candidate patterns in Computer Go play.
\cite{Elo} Each pattern is a~combination of several {\em pattern features}
(name--value pairs) matched at the position of the played move.
We use these features:

\begin{itemize}
\item capture move flag
\item atari move flag
\item atari escape flag
\item contiguity-to-last flag --- whether the move has been played in one of 8 neighbors of the last move
\item contiguity-to-second-last flag
\item board edge distance --- only up to distance 4
\item spatial pattern --- configuration of stones around the played move
\end{itemize}

The spatial patterns are normalized (using a dictionary) to be always
black-to-play and maintain translational and rotational symmetry.
Configurations of radius between 2 and 9 in the gridcular metric%
\footnote{The {\em gridcular} metric
$d(x,y) = |\delta x| + |\delta y| + \max(|\delta x|, |\delta y|)$ defines
a circle-like structure on the Go board square grid. \cite{SpatPat} }
are matched.

Pattern vectors representing these features contain information on
played shape as well as basic representation of tactical dynamics
--- threats to capture stones, replying to last move, or ignoring
opponent's move elsewhere to return to an urgent local situation.
The shapes most frequently correspond to opening moves
(either in empty corners and sides, or as part of {\em joseki}
--- commonly played sequences) characteristic for a certain
strategic aim. In the opening, even a single-line difference
in the distance from the border can have dramatic impact on
further local and global development.

\subsection{Implementation}

We have implemented the data extraction by making use of the pattern
features matching implementation%
\footnote{The pattern features matching was developed according
to the Elo-rating playing scheme. \cite{Elo}}
within the Pachi go-playing program \cite{Pachi}.
We extract information on players by converting the SGF game
records to GTP stream \cite{GTP} that feeds Pachi's {\tt patternscan}
engine, outputting a~single {\em patternspec} (string representation
of the particular pattern features combination) per move. Of course,
only moves played by the appropriate color in the game are collected.

\section{Data Mining}
\label{data-mining}

To assess the properties of gathered pattern vectors
and their influence on playing styles,
we process the data by several basic data minining techniques.

The first two methods {\em (analytic)} rely purely on data gathered
from the game collection
and serve to show internal structure and correlations within the data set.

Principal Component Analysis finds orthogonal vector components that
have the largest variance.
Reversing the process can indicate which patterns correlate with each component.
Additionally, PCA can be used as vector preprocessing for methods
that are negatively sensitive to pattern vector component correlations.

The~second method of Kohonen Maps
is based on the theory of self-organizing maps of abstract units (neurons) that
compete against each other for the representation of the input space.
Because neurons in the network are organized in a two-dimensional plane,
the trained network spreads the vectors on a 2D plane,
allowing for visualization of clusters of players with similar properties.


Furthermore, we use two \emph{classification} methods that assign
each pattern vector $\vec P$ an \emph{output vector $\vec O$,
representing e.g.~information about styles, player's strength or even
meta-information like the player's era or a country of origin.
Initially, the methods must be calibrated (trained) on some prior knowledge,
usually in the form of \emph{reference pairs} of pattern vectors
and the associated output vectors.

Moreover, the reference set can be divided into training and testing pairs
and the methods can be compared by the mean square error on testing data set
(difference of output vectors approximated by the method and their real desired value).

%\footnote{However, note that dicrete characteristics such as country of origin are
%not very feasible to use here, since WHAT??? is that even true?? }

The $k$-Nearest Neighbors \cite{CoverHart1967} classifier
approximates $\vec O$ by composing the output vectors
of $k$ reference pattern vectors closest to $\vec P$.

The other classifier is a~multi-layer feed-forward Artificial Neural Network:
the neural network can learn correlations between input and output vectors
and generalize the ``knowledge'' to unknown vectors; it can be more flexible
in the interpretation of different pattern vector elements and discern more
complex relations than the kNN classifier,
but may not be as stable and requires larger training sample.

\subsection{Principal Component Analysis}
\label{data-mining}
We use Principal Component Analysis \emph{PCA} \cite{Jolliffe1986}
to reduce the dimensions of the pattern vectors while preserving
as much information as possible, assuming inter-dependencies between
pattern vector dimensions are linear.

Briefly, PCA is an eigenvalue decomposition of a~covariance matrix of centered pattern vectors,
producing a~linear mapping $o$ from $n$-dimensional vector space
to a~reduced $m$-dimensional vector space.
The $m$ eigenvectors of the original vectors' covariance matrix
with the largest eigenvalues are used as the base of the reduced vector space;
the eigenvectors form projection matrix $W$.

For each original pattern vector $\vec p_i$,
we obtain its new representation $\vec r_i$ in the PCA base
as shown in the following equation:
\begin{equation}
\vec r_i = W \cdot \vec p_i
\end{equation}

The whole process is described in the Algorithm \ref{alg:pca}.

\begin{algorithm}
\caption{PCA -- Principal Component Analysis}
\begin{algorithmic}
\label{alg:pca}
\REQUIRE{$m > 0$, set of players $R$ with pattern vectors $p_r$}
\STATE $\vec \mu \leftarrow 1/|R| \cdot \sum_{r \in R}{\vec p_r}$
\FOR{ $r \in R$}
\STATE $\vec p_r \leftarrow \vec p_r - \vec \mu$
\ENDFOR
\FOR{ $(i,j) \in \{1,... ,n\} \times \{1,... ,n\}$}
\STATE $\mathit{Cov}[i,j] \leftarrow 1/|R| \cdot \sum_{r \in R}{\vec p_{ri} \cdot \vec p_{rj}}$
\ENDFOR
\STATE Compute Eigenvalue Decomposition of $\mathit{Cov}$ matrix
\STATE Get $m$ largest eigenvalues
\STATE Most significant eigenvectors ordered by decreasing eigenvalues form the rows of matrix $W$
\FOR{ $r \in R$}
\STATE $\vec r_r\leftarrow W \vec p_r$
\ENDFOR
\end{algorithmic}
\end{algorithm}

\label{pearson}
We want to find correlations between PCA dimensions and
some prior knowledge (player rank, style vector).
For this purpose, we compute the well-known
{\em Pearson product-moment correlation coefficient} \cite{Pearson},
measuring the strength of the linear dependence%
\footnote{A desirable property of PMCC is that it is invariant to translations and rescaling
of the vectors.}
between the dimensions:

$$ r_{X,Y} = {{\rm cov}(X,Y) \over \sigma_X \sigma_Y} $$

\subsection{Kohonen Maps}
\label{koh}
Kohonen map is a self-organizing network with neurons spread evenly over a~two-dimensional plane.
Neurons $\vec n$ in the map compete for representation of portions of the input vector space,
each vector being represented by some neuron.
The network is trained so that the neurons
that are topologically close tend to represent vectors that are close in suitable metric as well.

First, a~randomly initialized network is sequentially trained;
in each iteration, we choose a~random training vector $\vec t$
and find the {\em winner neuron} $\vec w$ that is closest to $\vec t$ in Euclidean metric.

We then adapt neurons $n$ from the neighborhood of $\vec w$ employing the equation
\begin{equation}
\vec n = \vec n + \alpha \cdot \mathit{Influence}(\vec w, \vec n) \cdot (\vec t - \vec n) 
\end{equation}
where $\alpha$ is a learning parameter, usually decreasing in time.
$Influence()$ is a function that forces neurons to spread.
Such function is usually realised using a mexican hat function or a difference-of-gaussians
\cite{TODO}.
The state of the network can be evaluated by calculating mean square difference
between each $\vec t \in T$ and its corresponding winner neuron $\vec w_t$:
\begin{equation}
\mathit{Error}(N,T) = \sum_{\vec t \in T}{|\vec w_t - \vec t|}
\end{equation}


\begin{algorithm}
\caption{Kohonen maps -- training}
\begin{algorithmic}
\label{alg:koh}
\REQUIRE{Set of training vectors $T$, input dimension $D$}
\REQUIRE{max number of iterations $M$, desired error $E$}
\STATE $N \leftarrow \{\vec n | \vec n$ random, $\mathit{dim}(\vec n) = D\}$
\REPEAT
\STATE $\mathit{It} \leftarrow \mathit{It} + 1$
\STATE $\vec t \leftarrow \mathit{PickRandom}(T)$
\FORALL{$\vec n \in N$}
\STATE $D[\vec n] \leftarrow \mathit{EuclideanDistance}(\vec n, \vec t)$
\ENDFOR
\STATE Find $ \vec w \in N$ so that $D[\vec w] <= D[\vec m], \forall \vec m \in N$
\FORALL{$\vec n \in \mathit{TopologicalNeigbors}(N, \vec w)$}
\STATE $\vec n \leftarrow \vec n + \alpha(It) \cdot \mathit{Influence}(\vec w, \vec n) \cdot ( \vec t - \vec n ) $
\ENDFOR
\UNTIL{$\mathit{Error}(N, T) < E$ or $ \mathit{It} > M$}
\end{algorithmic}
\end{algorithm}


\subsection{k-nearest Neighbors Classifier}
\label{knn}
Our goal is to approximate player's output vector $\vec O$;
we know his pattern vector $\vec P$.
We further assume that similarities in players' pattern vectors
uniformly correlate with similarities in players' output vectors.

We require a set of reference players $R$ with known \emph{pattern vectors} $\vec p_r$
and \emph{output vectors} $\vec o_r$.

$\vec O$ is approximated as a~weighted average of \emph{output vectors}
$\vec o_i$ of $k$ players with \emph{pattern vectors} $\vec p_i$ closest to $\vec P$.
This is illustrated in the Algorithm \ref{alg:knn}.
Note that the weight is a function of distance and is not explicitly defined in Algorithm \ref{alg:knn}.
During our research, exponentially decreasing weight has proven to be sufficient.

\begin{algorithm}
\caption{k-Nearest Neighbors}
\begin{algorithmic}
\label{alg:knn}
\REQUIRE{pattern vector $\vec P$, $k > 0$, set of reference players $R$}
\FORALL{$r \in R$ }
\STATE $D[r] \leftarrow \mathit{EuclideanDistance}(\vec p_r, \vec P)$
\ENDFOR
\STATE $N \leftarrow \mathit{SelectSmallest}(k, R, D)$
\STATE $\vec O \leftarrow \vec 0$
\FORALL{$r \in N $}
\STATE $\vec O \leftarrow \vec O + \mathit{Weight}(D[r]) \cdot \vec o_r $
\ENDFOR
\end{algorithmic}
\end{algorithm}

\subsection{Neural Network Classifier}
\label{neural-net}

Feed-forward neural networks \cite{ANN} are known for their ability to generalize
and find correlations between input patterns and output classifications.
Before use, the network is iteratively trained on the training data
until the error on the training set is reasonably small.

%Neural network is an adaptive system that must undergo a training
%period  similarly to the requirement
%of reference vectors for the k-Nearest Neighbors algorithm above.

\subsubsection{Computation and activation of the NN}
Technically, the neural network is a network of interconnected
computational units called neurons.
A feedforward neural network has a layered topology;
it usually has one \emph{input layer}, one \emph{output layer}
and an arbitrary number of \emph{hidden layers} between.

Each neuron $i$ is connected to all neurons in the previous layer and each connection has its weight $w_{ij}$

The computation proceeds in discrete time steps.
In the first step, the neurons in the \emph{input layer}
are \emph{activated} according to the \emph{input vector}.
Then, we iteratively compute output of each neuron in the next layer
until the output layer is reached.
The activity of output layer is then presented as the result.

The activation $y_i$ of neuron $i$ from the layer $I$ is computed as
\begin{equation}
y_i = f\left(\sum_{j \in J}{w_{ij} y_j}\right)
\end{equation}
where $J$ is the previous layer, while $y_j$ is the activation for neurons from $J$ layer.
Function $f()$ is a~so-called \emph{activation function}
and its purpose is to bound the outputs of neurons.
A typical example of an activation function is the sigmoid function.%
\footnote{A special case of the logistic function, defined by the formula
$\sigma(x)=\frac{1}{1+e^{-(rx+k)}}$; parameters control the growth rate ($r$)
and the x-position ($k$).}

\subsubsection{Training}
Training of the feed-forward neural network usually involves some
modification of supervised Backpropagation learning algorithm.
We use first-order optimization algorithm called RPROP. \cite{Riedmiller1993}

%Because the \emph{reference set} is usually not very large,
%we have devised a simple method for its extension.
%This enhancement is based upon adding random linear combinations
%of \emph{style and pattern vectors} to the training set.

As outlined above, the training set $T$ consists of
$(\vec p_i, \vec o_i)$ pairs.
The training algorithm is shown in Algorithm \ref{alg:tnn}.

\begin{algorithm}
\caption{Training Neural Network}
\begin{algorithmic}
\label{alg:tnn}
\REQUIRE{Train set $T$, desired error $e$, max iterations $M$}
\STATE $N \leftarrow \mathit{RandomlyInitializedNetwork}()$
\STATE $\mathit{It} \leftarrow 0$
\REPEAT
\STATE $\mathit{It} \leftarrow \mathit{It} + 1$
\STATE $\Delta \vec w \leftarrow \vec 0$
\STATE $\mathit{TotalError} \leftarrow 0$
%\FORALL{$(\overrightarrow{Input}, \overrightarrow{DesiredOutput}) \in T$}
%\STATE $\overrightarrow{Output} \leftarrow Result(N, \overrightarrow{Input})$
%\STATE $E \leftarrow |\overrightarrow{DesiredOutput} - \overrightarrow{Output}|$
\FORALL{$(\mathit{Input}, \mathit{DesiredOutput}) \in T$}
\STATE $\mathit{Output} \leftarrow \mathit{Result}(N, \mathit{Input})$
\STATE $\mathit{Error} \leftarrow |\mathit{DesiredOutput} - \mathit{Output}|$
\STATE $\Delta \vec w \leftarrow \Delta \vec w + \mathit{WeightUpdate}(N,\mathit{Error})$
\STATE $\mathit{TotalError} \leftarrow \mathit{TotalError} + \mathit{Error}$
\ENDFOR
\STATE $N \leftarrow \mathit{ModifyWeights}(N, \Delta \vec w)$
\UNTIL{$\mathit{TotalError} < e$ or $ \mathit{It} > M$}
\end{algorithmic}
\end{algorithm}

\subsection{Implementation}

We have implemented the data mining methods as the
``gostyle'' open-source framework \cite{GoStyle},
made available under the GNU GPL licence.

The majority of our basic processing and the analysis parts
are implemented in the Python \cite{Python2005} programming language.
We use several external libraries, most notably the MDP library \cite{MDP} (used for PCA analysis)
and Kohonen library \cite{KohonenPy}.
The neural network part of the project is written using the libfann C library\cite{Nissen2003}.


\section{Strength Estimator}

\begin{figure*}[!t]
\centering
\includegraphics[width=7in]{strength-pca}
\caption{PCA of by-strength vectors}
\label{fig:strength_pca}
\end{figure*}

First, we have used our framework to analyse correlations of pattern vectors
and playing strength. Like in other competitively played board games, Go players
receive real-world rating based on tournament games, and rank based on their
rating.\footnote{Elo-like rating system \cite{GoR} is usually used,
corresponding to even win chances for game of two players with the same rank,
and about 2:3 win chance for stronger in case of one rank difference.}%
\footnote{Professional ranks and dan ranks in some Asia countries may
be assigned differently.} The amateur ranks range from 30kyu (beginner) to
1kyu (intermediate) and then follows 1dan to 7dan (9dan in some systems;
top-level player). Multiple independent real-world ranking scales exist
(geographically based) and online servers maintain their own user ranking;
the difference can be up to several stones.

As the source game collection, we use Go Teaching Ladder
reviews\footnote{The reviews contain comments and variations --- we consider only the actual played game.}
\cite{GTL} --- this collection contains 7700 games of players with strength ranging
from 30k to 4d; we consider only even games with clear rank information, and then
randomly separate 770 games as a testing set. Since the rank information is provided
by the users and may not be consistent, we are forced to take a simplified look
at the ranks, discarding the differences between various systems and thus increasing
error in our model.\footnote{Since
our results seem satisfying, we did not pursue to try another collection;
one could e.g. look at game archives of some Go server.}

First, we have created a single pattern vector for each rank, from 30k to 4d;
we have performed PCA analysis on the pattern vectors, achieving near-perfect
rank correspondence in the first PCA dimension%
\footnote{The eigenvalue of the second dimension was four times smaller,
with no discernable structure revealed within the lower-order eigenvectors.}
(figure \ref{fig:strength_pca}).

We measure the accuracy of strength approximation by the first dimension
using Pearson's $r$ (see \ref{pearson}), yielding satisfying value $r=0.979$.
Using the eigenvector position directly for classification
of players within the test group yields MSE TODO, thus providing
reasonably satisfying accuracy.

To further enhance the strength estimator accuracy,
we have tried to train a NN classifier on our train set, consisting
of one $(\vec p, {\rm rank})$ pair per player --- we use the pattern vector
for activation of input neurons and rank number as result of the output
neuron. We then proceeded to test the NN on per-player pattern vectors built
from the games in the test set, yielding MSE of TODO with TODO games per player
on average.


\section{Style Estimator}

As a~second case study for our pattern analysis, we investigate pattern vectors $\vec p$
of various well-known players, their relationships and correlations to prior
knowledge to explore its correlaction with extracted patterns. We look for
relationships between pattern vectors and perceived ``playing style'' and
attempt to use our classifiers to transform pattern vector $\vec p$ to style vector $\vec s$.

The source game collection is GoGoD Winter 2008 \cite{GoGoD} containing 55000
professional games, dating from the early Go history 1500 years ago to the present.
We consider only games of a small subset of players (fig. \ref{fig:style_marks});
we have chosen these for being well-known within the players community,
having large number of played games in our collection and not playing too long
ago.\footnote{Over time, many commonly used sequences get altered, adopted and
dismissed; usual playing conditions can also differ significantly.}

\subsection{Expert-based knowledge}
\label{style-vectors}
In order to provide a reference frame for our style analysis,
we have gathered some expert-based information about various
traditionally perceived style aspects.
This expert-based knowledge allows us to predict styles of unknown players based on
the similarity of their pattern vectors, as well as discover correlations between
styles and proportions of played patterns.

Experts were asked to mark each style aspect of the given players
on the scale from 1 to 10. The style aspects are defined as shown:

%\vspace{4mm}
%\noindent
\begin{table}
\begin{center}
\caption{Styles}
\begin{tabular}{|c|c|c|}
\hline
Style & 1 & 10\\ \hline
Territoriality $\tau$ & Moyo & Territory \\
Orthodoxity $\omega$ & Classic & Novel \\
Aggressivity $\alpha$ & Calm & Figting \\
Thickness $\theta$ & Safe & Shinogi \\ \hline
\end{tabular}
\end{center}
\end{table}
%\vspace{4mm}

Averaging this expert based evaluation yields
\emph{reference style vector} $\vec s_r$ (of dimension $4$) for each player $r$
from the set of \emph{reference players} $R$.

\begin{table}[!t]
% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
\caption{Covariance Measure of Prior Information Dimensions}
\label{fig:style_marks_r}
\centering
% Some packages, such as MDW tools, offer better commands for making tables
% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{|r||r||r||r||r||r|}
\hline
  & $\tau$ & $\omega$ & $\alpha$ & $\theta$ & year \\
\hline
$\tau$  &$1.000$&$-0.438$&$-0.581$&$ 0.721$&$ 0.108$\\
$\omega$&       &$ 1.000$&$ 0.682$&$ 0.014$&$-0.021$\\
$\alpha$&       &        &$ 1.000$&$-0.081$&$ 0.030$\\
$\theta$&       &\multicolumn{1}{c||}{---}
                         &        &$ 1.000$&$-0.073$\\
y.      &       &        &        &        &$ 1.000$\\
\hline
\end{tabular}
\end{table}

Three high-level Go players (Alexander Dinerstein 3-pro, Motoki Noguchi
7-dan and V\'{i}t Brunner 4-dan) have judged style of the reference
players.
The complete list of answers is in table \ref{fig:style_marks}.
Mean standard deviation of the answers is 0.952,
making the data reasonably reliable,
though much larger sample would of course be more desirable.
We have also found significant correlation between the various
style aspects, as shown by the Pearson's $r$ values
in table \ref{fig:style_marks_r}.

\begin{table}[!t]
% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
\begin{threeparttable}
\caption{Expert-Based Style Aspects of Selected Professionals\tnote{1} \tnote{2}}
\label{fig:style_marks}
\centering
% Some packages, such as MDW tools, offer better commands for making tables
% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{|c||c||c||c||c|}
\hline
{Player} & $\tau$ & $\omega$ & $\alpha$ & $\theta$ \\
\hline
Go Seigen\tnote{3}   & $6.0 \pm 2.0$ & $9.0 \pm 1.0$ & $8.0 \pm 1.0$ & $5.0 \pm 1.0$ \\
Ishida Yoshio\tnote{4}&$8.0 \pm 1.4$ & $5.0 \pm 1.4$ & $3.3 \pm 1.2$ & $5.3 \pm 0.5$ \\
Miyazawa Goro        & $1.5 \pm 0.5$ & $10  \pm 0  $ & $9.5 \pm 0.5$ & $4.0 \pm 1.0$ \\
Yi Ch'ang-ho\tnote{5}& $7.0 \pm 0.8$ & $5.0 \pm 1.4$ & $2.6 \pm 0.9$ & $2.6 \pm 1.2$ \\
Sakata Eio           & $7.6 \pm 1.7$ & $4.6 \pm 0.5$ & $7.3 \pm 0.9$ & $8.0 \pm 1.6$ \\
Fujisawa Hideyuki    & $3.5 \pm 0.5$ & $9.0 \pm 1.0$ & $7.0 \pm 0.0$ & $4.0 \pm 0.0$ \\
Otake Hideo          & $4.3 \pm 0.5$ & $3.0 \pm 0.0$ & $4.6 \pm 1.2$ & $3.6 \pm 0.9$ \\
Kato Masao           & $2.5 \pm 0.5$ & $4.5 \pm 1.5$ & $9.5 \pm 0.5$ & $4.0 \pm 0.0$ \\
Takemiya Masaki      & $1.3 \pm 0.5$ & $6.3 \pm 2.1$ & $7.0 \pm 0.8$ & $1.3 \pm 0.5$ \\
Kobayashi Koichi     & $9.0 \pm 1.0$ & $2.5 \pm 0.5$ & $2.5 \pm 0.5$ & $5.5 \pm 0.5$ \\
Cho Chikun           & $9.0 \pm 0.8$ & $7.6 \pm 0.9$ & $6.6 \pm 1.2$ & $9.0 \pm 0.8$ \\
Ma Xiaochun          & $8.0 \pm 2.2$ & $6.3 \pm 0.5$ & $5.6 \pm 1.9$ & $8.0 \pm 0.8$ \\
Yoda Norimoto        & $6.3 \pm 1.7$ & $4.3 \pm 2.1$ & $4.3 \pm 2.1$ & $3.3 \pm 1.2$ \\
Luo Xihe             & $7.3 \pm 0.9$ & $7.3 \pm 2.5$ & $7.6 \pm 0.9$ & $6.0 \pm 1.4$ \\
O Meien              & $2.6 \pm 1.2$ & $9.6 \pm 0.5$ & $8.3 \pm 1.7$ & $3.6 \pm 1.2$ \\
Rui Naiwei           & $4.6 \pm 1.2$ & $5.6 \pm 0.5$ & $9.0 \pm 0.8$ & $3.3 \pm 1.2$ \\
Yuki Satoshi         & $3.0 \pm 1.0$ & $8.5 \pm 0.5$ & $9.0 \pm 1.0$ & $4.5 \pm 0.5$ \\
Hane Naoki           & $7.5 \pm 0.5$ & $2.5 \pm 0.5$ & $4.0 \pm 0.0$ & $4.5 \pm 1.5$ \\
Takao Shinji         & $5.0 \pm 1.0$ & $3.5 \pm 0.5$ & $5.5 \pm 1.5$ & $4.5 \pm 0.5$ \\
Yi Se-tol            & $5.3 \pm 0.5$ & $6.6 \pm 2.5$ & $9.3 \pm 0.5$ & $6.6 \pm 1.2$ \\
Yamashita Keigo\tnote{4}&$2.0\pm 0.0$& $9.0 \pm 1.0$ & $9.5 \pm 0.5$ & $3.0 \pm 1.0$ \\
Cho U                & $7.3 \pm 2.4$ & $6.0 \pm 0.8$ & $5.3 \pm 1.7$ & $6.3 \pm 1.7$ \\
Gu Li                & $5.6 \pm 0.9$ & $7.0 \pm 0.8$ & $9.0 \pm 0.8$ & $4.0 \pm 0.8$ \\
Chen Yaoye           & $6.0 \pm 1.0$ & $4.0 \pm 1.0$ & $6.0 \pm 1.0$ & $5.5 \pm 0.5$ \\
\hline
\end{tabular}
\begin{tablenotes}
\item [1] Including standard deviation. Only players where we got at least two out of tree answers are included.
\item [2] We consider era as one of factors when finding correlations with pattern vectors; we quantify era by taking median year over all games played by the player. Since this quantity does not fit to the table, we at least sort the players ascending by their median year.
\item [3] We do not consider games of Go Seigen due to him playing across several distinct Go-playing eras and thus specifically high diversity of patterns.
\item [4] We do not consider games of Ishida Yoshio and Yamashita Keigo for the PCA analysis since they are significant outliers, making high-order dimensions much like purely ``similarity to this player''. Takemiya Masaki has the similar effect for the first dimension, but this corresponds to common knowledge of him being an extreme proponent of anti-territorial (``moyo'') style.
\item [5] We consider games only up to year 2004, since Yi Ch'ang-ho was prominent representative of a balanced, careful player until then, but is regarded to have altered his style significantly afterwards.
\end{tablenotes}
\end{threeparttable}
\end{table}

\subsection{Style Components Analysis}

\begin{figure}[!t]
\centering
\includegraphics[width=3.75in]{style-pca}
\caption{PCA of per-player vectors}
\label{fig:style_pca}
\end{figure}

We have looked at the five most significant dimensions of the pattern data
yielded by the PCA analysis of the reference player set%
\footnote{We also tried to observe PCA effect of removing outlying Takemiya
Masaki. This second dimension strongly
correlated to territoriality and third dimension strongly correlacted to era,
however the first dimension remained mysteriously uncorrelated and with no
obvious interpretation.}
(fig. \ref{fig:style_pca} shows three).
We have again computed the Pearson's $r$ for all combinations of PCA dimensions
and dimensions of the prior knowledge style vectors to find correlations.

\begin{table}[!t]
% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
\caption{Covariance Measure of Patterns and Prior Information}
\label{fig:style_r}
\centering
% Some packages, such as MDW tools, offer better commands for making tables
% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{|c||c||c||c||c||c|}
\hline
Eigenval. & $\tau$ & $\omega$ & $\alpha$ & $\theta$ & Year \\
\hline
0.447 & {\bf -0.530} &  0.323 &  0.298 & {\bf -0.554} &  0.090 \\
0.194 & {\bf -0.547} &  0.215 &  0.249 & -0.293 & {\bf -0.630} \\
0.046 &  0.131 & -0.002 & -0.128 &  0.242 & {\bf -0.630} \\
0.028 & -0.011 &  0.225 &  0.186 &  0.131 &  0.067 \\
0.024 & -0.181 &  0.174 & -0.032 & -0.216 &  0.352 \\
\hline
\end{tabular}
\end{table}

\begin{table}[!t]
% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
\caption{Characteristic Patterns of PCA Dimensions}
\label{fig:style_ptterns}
\centering
% Some packages, such as MDW tools, offer better commands for making tables
% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{|cccc|}
\hline
PCA1 top &
\begin{psgopartialboard*}{(8,1)(12,6)}
\stone[\marktr]{black}{k}{4}
\end{psgopartialboard*} &
\begin{psgopartialboard*}{(3,1)(5,6)}
\stone{white}{d}{3}
\stone[\marktr]{black}{d}{5}
\end{psgopartialboard*} &
\begin{psgopartialboard*}{(5,1)(10,6)}
\stone{white}{f}{3}
\stone[\marktr]{black}{j}{4}
\end{psgopartialboard*} \\
$0.447 \cdot$ & $0.274$ & $0.086$ & $0.083$ \\
& side extension or \par 4--4 corner opening & high corner approach & high distant pincer \\
PCA1 bot. &
\begin{psgopartialboard*}{(3,1)(7,6)}
\stone{white}{d}{4}
\stone[\marktr]{black}{f}{3}
\end{psgopartialboard*} &
\begin{psgopartialboard*}{(3,1)(7,6)}
\stone{white}{c}{6}
\stone{black}{d}{4}
\stone[\marktr]{black}{f}{3}
\end{psgopartialboard*} &
\begin{psgopartialboard*}{(3,1)(7,6)}
\stone{black}{d}{4}
\stone[\marktr]{black}{f}{3}
\end{psgopartialboard*} \\
$0.447 \cdot$ & $-0.399$ & $-0.399$ & $-0.177$ \\
& low corner approach & low corner reply & low corner enclosure \\
\hline
\end{tabular}
\end{table}

It is immediately
obvious both from the measured $r$ and visual observation
that by far the most significant vector corresponds very well
to the player territoriality,\footnote{Cho Chikun, perhaps the best-known
super-territorial player, is not well visible in the cluster, but he is
positioned around $-0.8$ on the first dimension.}
confirming the intuitive notion that this aspect of style
is the one easiest to pin-point and also
most obvious in the played shapes and sequences
(that can obviously aim directly at taking secure territory
or building center-oriented framework). Thick (solid) play also plays
a role, but these two style dimensions have been already shown
to be correlated in prior data.

In other PCA dimensions correspond well to to identify and name, but there
certainly is some influence of the styles on the patterns;
the found correlations are presented in table \ref{fig:style_r}.
(Larger absolute value means better linear correspondence.)

We also list the characteristic spatial patterns of the PCA dimension
extremes (table \ref{fig:style_patterns}) --- however, naive inference
of characteristic patterns based on projection matrix coefficients
does not work well, better methods will have to be researched.%
\footnote{For example, as one of highly ranked ``Takemiya's'' PCA1 patterns,
3,3 corner opening was generated, completely inappropriately;
it reflects some weak ordering in bottom half of the dimension,
not global ordering within the dimension.}

We have not found significant correspondence for the style aspects
representing aggressiveness and novelty of play; this means either
these are not as well defined, the prior information do not represent
them accurately, or we cannot capture them well with our chosen pattern
extraction techniques.

We believe that the next step
in interpreting our results will be more refined prior information input
and precise analysis by Go experts.

Kohonen map view.

\subsection{Style Classification}

%TODO vsude zcheckovat jestli pouzivame stejny cas "we perform, we apply" X "we have performed, ..." 

Apart from the PCA-based analysis, we applied neural network (sec. \ref{neural-net})
and $k$-NN classifiers (sec. \ref{knn}).

To compare and evaluate both methods, we have performed $5$-fold cross validation \cite{TODO} and
compared them with a~random classificator.
In the $5$-fold cross-validation, we randomly divide the training set into $5$ distinct parts with comparable
sizes and then iteratively use each part as a~testing set (yielding square error value), while
the rest (remaining $4$ parts) is taken as a~training set. The square errors across all $5$ iterations are
averaged, yielding mean square error.

The results are shown in table \ref{crossval-cmp}. Second to fifth columns in the table represent
mean square error of different styles (see \ref{style vectors}), $\mathit{Mean}$ is the
mean square error across the styles and finally, the last column $\mathit{Comp}$
represents $\mathit{Mean}_\mathit{RND} / \mathit{X}$ -- comparison of mean square error (across styles)
with random classificator. To minimize the
effect of random variables, all numbers were taken as an average of $30$ runs of the cross validation.

\begin{table}[!t]
\label{crossval-cmp}
\begin{center}
\caption{Comparison of style classificators}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
%Classifier & $\sigma_\tau$ & $\sigma_\omega$ & $\sigma_\alpha$ & $\sigma_\theta$ & Tot $\sigma$ & $\mathit{RndC}$\\ \hline
%Neural network & 0.420 & 0.488 & 0.365 & 0.371 & 0.414 & 1.82 \\
%$k$-NN ($k=4$) & 0.394 & 0.507 & 0.457 & 0.341 & 0.429 & 1.76 \\
%Random classifier & 0.790 & 0.773 & 0.776 & 0.677 & 0.755 & 1.00 \\ \hline
&\multicolumn{5}{|c|}{MSE}& \\ \hline
{Classifier} & $\tau$ & $\omega$ & $\alpha$ & $\theta$ & {\bf Mean} & {\bf Comp}\\ \hline
Neural network & 0.173 & 0.236 & 0.136 & 0.143 & 0.172 & 3.3 \\
$k$-NN ($k=4$) & 0.156 & 0.257 & 0.209 & 0.116 & 0.184 & 3.1\\
Random classifier & 0.544 & 0.640 & 0.647 & 0.458 & 0.572 & 1.0 \\ \hline
\end{tabular}
\end{center}
\end{table}

\subsubsection{Reference (Training) Data}
As a~reference data, we use expert based knowledge presented in section \ref{style-vectors}.
For both methods to yield comparable errors, we have rescaled style vectors to interval $[-1,1]$
(since neural network's activation function has such range).

% TODO presunout konkretni parametry do Appendixu?  (neni jich tolik, mozna ne)
\subsubsection{$k$-NN parameters}
$k=4$, Weight function is $0.8^{(10*EuclideanDistance)}$

\subsubsection{Neural network's parameters}
$3$ layers, $23 - 30 - 4$ architecture


\section{Proposed Applications}

We believe that our findings might be useful for many applications
in the area of Go support software as well as Go-playing computer engines.

The style analysis can be an excellent teaching aid --- classifying style
dimensions based on player's pattern vector, many study recommendations
can be given, e.g. about the professional games to replay, the goal being
balancing understanding of various styles to achieve well-rounded skill set.
This was also our original aim when starting the research and a user-friendly
tool based on our work is now being created.

We hope that more strong players will look into the style dimensions found
by our statistical analysis --- analysis of most played patterns of prospective
opponents might prepare for the game, but we especially hope that new insights
on strategic purposes of various shapes and general human understanding
of the game might be achieved by investigating the style-specific patterns.

Classifying playing strength of a pattern vector of a player can be used
e.g. to help determine initial real-world rating of a player before their
first tournament based on games played on the internet; some players especially
in less populated areas could get fairly strong before playing their first
real tournament.

Analysis of pattern vectors extracted from games of Go-playing programs
in light of the shown strength and style distributions might help to
highlight some weaknesses and room for improvements. (However, since
correlation does not imply causation, simply optimizing Go-playing programs
according to these vectors is unlikely to yield good results.)
Another interesting applications in Go-playing programs might be strength
adjustment; the program can classify the player's level based on the pattern
vector from its previous games and auto-adjust its difficulty settings
accordingly to provide more even games for beginners.


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%\hfil
%\subfloat[Case II]{\includegraphics[width=2.5in]{subfigcase2}%
%\label{fig_second_case}}}
%\caption{Simulation results}
%\label{fig_sim}
%\end{figure*}
%
% Note that often IEEE papers with subfigures do not employ subfigure
% captions (using the optional argument to \subfloat), but instead will
% reference/describe all of them (a), (b), etc., within the main caption.


% An example of a floating table. Note that, for IEEE style tables, the 
% \caption command should come BEFORE the table. Table text will default to
% \footnotesize as IEEE normally uses this smaller font for tables.
% The \label must come after \caption as always.
%
%\begin{table}[!t]
%% increase table row spacing, adjust to taste
%\renewcommand{\arraystretch}{1.3}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
%\caption{An Example of a Table}
%\label{table_example}
%\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
%\begin{tabular}{|c||c|}
%\hline
%One & Two\\
%\hline
%Three & Four\\
%\hline
%\end{tabular}
%\end{table}


% Note that IEEE does not put floats in the very first column - or typically
% anywhere on the first page for that matter. Also, in-text middle ("here")
% positioning is not used. Most IEEE journals use top floats exclusively.
% Note that, LaTeX2e, unlike IEEE journals, places footnotes above bottom
% floats. This can be corrected via the \fnbelowfloat command of the
% stfloats package.



\section{Conclusion}
The conclusion goes here.
We have shown brm and proposed brm.

Since we are not aware of any previous research on this topic and we
are limited by space and time constraints, plenty of research remains
to be done. There is plenty of room for further research in all parts
of our analysis --- different methods of generating the $\vec p$ vectors
can be explored; other data mining methods could be tried.
It can be argued that many players adjust their style by game conditions
(Go development era, handicap, komi and color, time limits, opponent)
or styles might express differently in various game stages.
More professional players could be consulted on the findings
and for style scales calibration. Impact of handicap games on by-strength
$\vec p$ distribution should be investigated.

TODO: Future research --- Sparse PCA




% if have a single appendix:
%\appendix[Proof of the Zonklar Equations]
% or
%\appendix  % for no appendix heading
% do not use \section anymore after \appendix, only \section*
% is possibly needed

% use appendices with more than one appendix
% then use \section to start each appendix
% you must declare a \section before using any
% \subsection or using \label (\appendices by itself
% starts a section numbered zero.)
%


%\appendices
%\section{Proof of the First Zonklar Equation}
%Appendix one text goes here.
%
%% you can choose not to have a title for an appendix
%% if you want by leaving the argument blank
%\section{}
%Appendix two text goes here.


% use section* for acknowledgement
\section*{Acknowledgment}
\label{acknowledgement}

We would like to thank Radka ``chidori'' Hane\v{c}kov\'{a} for the original research idea
and X for reviewing our paper.
We appreciate helpful comments on our general methodology
by John Fairbairn, T. M. Hall, Cyril H\"oschl, Robert Jasiek, Franti\v{s}ek Mr\'{a}z
and several GoDiscussions.com users. \cite{GoDiscThread}
Finally, we are very grateful for detailed input on specific go styles
by Alexander Dinerstein, Motoki Noguchi and V\'{i}t Brunner.


% Can use something like this to put references on a page
% by themselves when using endfloat and the captionsoff option.
\ifCLASSOPTIONcaptionsoff
  \newpage
\fi



% trigger a \newpage just before the given reference
% number - used to balance the columns on the last page
% adjust value as needed - may need to be readjusted if
% the document is modified later
%\IEEEtriggeratref{8}
% The "triggered" command can be changed if desired:
%\IEEEtriggercmd{\enlargethispage{-5in}}

% references section

% can use a bibliography generated by BibTeX as a .bbl file
% BibTeX documentation can be easily obtained at:
% http://www.ctan.org/tex-archive/biblio/bibtex/contrib/doc/
% The IEEEtran BibTeX style support page is at:
% http://www.michaelshell.org/tex/ieeetran/bibtex/
\bibliographystyle{IEEEtran}
% argument is your BibTeX string definitions and bibliography database(s)
\bibliography{gostyle}
%
% <OR> manually copy in the resultant .bbl file
% set second argument of \begin to the number of references
% (used to reserve space for the reference number labels box)
%\begin{thebibliography}{1}
%
%\bibitem{MasterMCTS}
%
%\end{thebibliography}

% biography section
% 
% If you have an EPS/PDF photo (graphicx package needed) extra braces are
% needed around the contents of the optional argument to biography to prevent
% the LaTeX parser from getting confused when it sees the complicated
% \includegraphics command within an optional argument. (You could create
% your own custom macro containing the \includegraphics command to make things
% simpler here.)
%\begin{biography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{mshell}}]{Michael Shell}
% or if you just want to reserve a space for a photo:

\begin{IEEEbiography}{Michael Shell}
Biography text here.
\end{IEEEbiography}

% if you will not have a photo at all:
\begin{IEEEbiographynophoto}{John Doe}
Biography text here.
\end{IEEEbiographynophoto}

% insert where needed to balance the two columns on the last page with
% biographies
%\newpage

\begin{IEEEbiographynophoto}{Jane Doe}
Biography text here.
\end{IEEEbiographynophoto}

% You can push biographies down or up by placing
% a \vfill before or after them. The appropriate
% use of \vfill depends on what kind of text is
% on the last page and whether or not the columns
% are being equalized.

%\vfill

% Can be used to pull up biographies so that the bottom of the last one
% is flush with the other column.
%\enlargethispage{-5in}



% that's all folks
\end{document}