tex: Strength Classifier - clean up

[gostyle.git] / tex / gostyle.tex

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\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 --- DRAFT2p}%
{On Pattern Feature Trends in Large Go Game Corpus --- DRAFT2p}
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\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, computer go, data mining, go theory,
pattern recongition, player strength, playing style,
neural networks, sociomaps, principal component analysis,
naive bayes classifier
\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{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.

\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{PatElo} 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%
\footnote{We do not consider contiguity features in some cases when we are working
on small game samples and need to reduce pattern diversity.}
--- 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
$d(x,y) = |\delta x| + |\delta y| + \max(|\delta x|, |\delta y|)$ produces
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 a 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{Vector Rescaling}

The pattern vector elements can have diverse values since for each object,
we consider different number of games (and thus patterns).
Therefore, we normalize the values to range $[-1,1]$,
the most frequent pattern having the value of $1$ and the least occuring
one being $-1$.
Thus, we obtain vectors describing relative frequency of played patterns
independent on number of gathered patterns.
But there are multiple ways to approach the normalization.

\begin{figure}[!t]
\centering
\includegraphics{patcountdist}
\caption{Log-scaled number of pattern occurences
in the GoGoD games examined in sec. \ref{styleest}.}
\label{fig:patcountdist}
\end{figure}

\subsubsection{Linear Normalization}

One is simply to linearly re-scale the values using:
$$y_i = {x_i - x_{\rm min} \over x_{\rm max}}$$
This is the default approach; we have used data processed by only this
computation unless we note otherwise.
As shown on fig. \ref{fig:patcountdist}, most of the spectrum is covered
by the few most-occuring patterns (describing mostly large-diameter
shapes from the game opening). This means that most patterns will be
always represented by only very small values near the lower bound.

\subsubsection{Extended Normalization}
\label{xnorm}

To alleviate this problem, we have also tried to modify the linear
normalization by applying two steps --- {\em pre-processing}
the raw counts using
$$x_i' = \log (x_i + 1)$$
and {\em post-processing} the re-scaled values by the logistic function:
$$y_i' = {2 \over 1 + e^{-cy_i}}-1$$
However, we have found that this method is not universally beneficial.
In our styles case study (sec. \ref{styleest}), this normalization
produced PCA decomposition with significant dimensions corresponding
better to some of the prior knowledge and more instructive for manual
inspection, but ultimately worsened accuracy of our classifiers;
we conjecture from this that the most frequently occuring patterns are
also most important for classification of major style aspects.

\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{PatElo}}
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 single data set
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 Sociomaps \cite{Sociomaps} creates spatial
representation of the data set elements (e.g. players) based on
similarity of their data set features; we can then project other
information on the map to illutrate its connection to the data set.

Furthermore, we test several \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?? }

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

Another 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.

Finally, a commonly used classifier in statistical inference is
the Naive Bayes classifier; it can infer relative probability of membership
in various classes based on previous evidence (training patterns). \cite{Bayes}

\subsection{Principal Component Analysis}
\label{PCA}
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{Sociomaps}
\label{soc}
Sociomaps are a general mechanism for visualising possibly assymetric
relationships on a 2D plane such that ordering of the maximum possible
object distances in the dataset is preserved in distances on the plane.

In our particular case,%
\footnote{A special case of the {\em Subject-to-Object Relation Mapping (STORM)} indirect sociomap.}
we will consider a dataset $\vec S$ of small-dimensional
vectors $\vec s_i$ and determine projection $\varphi$ of all the $\vec s_i$
to spatial coordinates of an Euclidean plane.
The $\varphi$ projection shall maximize the {\em three-way ordering} criterion:
ordering of any three members in the dataset and on the plane
(by Euclidean metric) must be the same.

$$ \min_\varphi \sum_{i\ne j\ne k} \Phi(\varphi, i, j, k) $$
$$ \Phi(\varphi, i, j, k) = \begin{cases}
        1 & \delta(s_i,s_j,s_k) = \delta(\varphi(i),\varphi(j),\varphi(k)) \\
        0 & \hbox{otherwise} \end{cases} $$
$$ \delta(a, b, c) = \begin{cases}
        1  & |a-b| > |a-c| \\
        0  & |a-b| = |a-c| \\
        -1 & |a-b| < |a-c| \end{cases} $$

\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 $\sigma(x)=(1+e^{-(rx+k)})^{-1}$.
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{Naive Bayes Classifier}
\label{naive-bayes}

Naive Bayes Classifier uses existing information to construct
probability model of likelihoods of given {\em feature variables}
based on a discrete-valued {\em class variable}.
Using the Bayes equation, we can then estimate the probability distribution
of class variable for particular values of the feature variables.

In order to approximate player's output vector $\vec O$ based on
pattern vector $\vec P$, we will compute each element of the
output vector separately, covering the output domain by several $k$-sized
discrete intervals (classes).

We will also in fact work on
PCA-represented input $\vec R$ (using the 10 most significant
dimensions), since smaller input dimension is more computationally
feasible and $\vec R$ also better fits the pre-requisites of the
classifier, the dimensions being more independent and
better approximating the normal distribution.

When training the classifier for $\vec O$ element $o_i$
of class $c = \lfloor o_i/k \rfloor$,
we assume the $\vec R$ elements are normally distributed and
feed the classifier information in the form
$$ \vec R \mid c $$
estimating the mean $\mu_c$ and standard deviation $\sigma_c$
of each $\vec R$ element for each encountered $c$.
Then, we can query the built probability model on
$$ \max_c P(c \mid \vec R) $$
obtaining the most probable class $i$ for an arbitrary $\vec R$.
Each probability is obtained using the normal distribution formula:
$$ P(c \mid x) = {1\over \sqrt{2\pi\sigma_c^2}}\exp{-(x-\mu_c)^2\over2\sigma_c^2} $$

\begin{algorithm}
\caption{Training Naive Bayes}
\begin{algorithmic}
\label{alg:tnb}
\REQUIRE{Train set $T = (\mathit{R, c})$}
\FORALL{$(R, c) \in T$}
\STATE $\mathit{RbyC}_c \leftarrow \{\mathit{RbyC}_c, R\}$
\ENDFOR
\FORALL{$c$}
\STATE $\mu_c \leftarrow {1 \over |\mathit{RbyC}_c|} \sum_{R \in \mathit{RbyC}_c} R$
\ENDFOR
\FORALL{$c$}
\STATE $\sigma_c \leftarrow {1 \over |\mathit{RbyC}_c|} \sum_{R \in \mathit{RbyC}_c} R-\mu_c $
\ENDFOR
\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{Python25} programming language.
We use several external libraries, most notably the MDP library \cite{MDP}.
The neural network part of the project is written using the libfann C library\cite{Nissen2003}.

The sociomap has been visualised using the Team Profile Analyzer \cite{TPA}
which is part of the Sociomap suite \cite{SociomapSite}.


\section{Strength Estimation}

\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 {\em rating number} based on tournament games,
and {\em rank} based on their rating.%
\footnote{Elo-type 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 30-kyu (beginner) to 1-kyu (intermediate)
and then follows 1-dan to 7-dan\footnote{9-dan in some systems.} (top-level player).
Multiple independent real-world ranking scales exist
(geographically based), also online servers maintain their own user ranking;
the difference between scales can be up to several ranks and the rank
distributions also differ. \cite{RankComparison}

\subsection{Data used}
As the source game collection, we use Go Teaching Ladder reviews archive%
\footnote{The reviews contain comments and variations --- we consider only the main
variation with the actual played game.}
\cite{GTL} --- this collection contains 7700 games of players with strength ranging
from 30-kyu to 4-dan; we consider only even games with clear rank information.
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 somewhat
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.}

\subsection{PCA analysis}
First, we have created a single pattern vector for each rank, from 30-kyu to 4-dan;
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 quite satisfying value of $r=0.979$
implying extremely strong correlation.
\footnote{Extended vector normalization (sec. \ref{xnorm})
produced noticeably less clear-cut results.}

\subsection{Strength classification}

We have randomly separated $10\%$ of the game database as a testing set,
one $(\vec p, {\rm rank})$ pair per player. We then explore the influence
of game sample size%
\footnote{Arbitrary game numbers are approximated by pattern file sizes,
randomly selecting games of appropriate strength.}
on the accuracy of various classifiers.
In order to reduce the diversity of patterns (negatively impacting accuracy
on small samples), we do not consider the contiguity pattern features.

Using the most significant PCA eigenvector position directly for classification
of players within the test group yields MSE TODO, thus providing
reasonably satisfying accuracy by itself.

Using the $k$-NN classifier, we have achieved the results described
in the table \ref{table-str-class} --- overally obtaining quite impressive
accuracy even on rather small game sample sizes.
The error is listed as MSE (on rank rescaled to $[-1,1]$) and standard deviation
$\sigma$ in percentages (meaning the difference from the real rank on average).

\begin{table}[!t]
% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
\caption{Strength Classifier Performance}
\label{table-str-class}
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
$\sim$ games & MSE & $\sigma \%$ \\ \hline
$85$& $0.007$ & $4\%$   \\
$43$& $0.029$ & $8\%$   \\
$17$& $0.081$ & $14\%$ \\
$9$& $0.131$ & $18\%$ \\
$2$& $0.187$ & $22\%$ \\\hline
\end{tabular}
\end{table}

Finally, we used $8$-fold cross validation on one-file-per-rank files,
yielding a MSE $0.085$ which is equivalent to standard deviation of $15\%$.

\section{Style Estimation}
\label{styleest}

As a~second case study for our pattern analysis,
we investigate pattern vectors $\vec p$ of various well-known players,
their relationships in-between and to prior knowledge
in order to explore the correlation of prior knowledge 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 (table \ref{fig:style_marks});
we have chosen them 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 to use as a prior knowledge.
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 four style aspects of each 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}

We have devised these four style aspects based on our own Go experience
and consultations with other experts.
The used terminology has quite
clear meaning to any experienced Go player and there is not too much
room for confusion, except possibly in the case of ``thickness'' ---
but the concept is not easy to pin-point succintly and we also did not
add extra comments on the style aspects to the questionnaire deliberately
to accurately reflect any diversity in understanding of the terms.

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} $R$.

Throughout our research, we have experimentally found that playing era
is also a major factor differentiating between patterns. Thus, we have
further extended the $\vec s_r$ by median year over all games played
by the player.

\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}{1.4}
\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\tnote{4}&$1.3\pm 0.5$& $6.3 \pm 2.1$ & $7.0 \pm 0.8$ & $1.3 \pm 0.5$ \\
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 received at least two out of three answers are included.
\item [2] Since the playing era column does not fit into 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 eras and also being famous for radical opening experiments throughout the time, and thus featuring especially high diversity in 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 that case 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 and still has this reputation in minds of many players, 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 ten 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. That way, the 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 the first 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}{1.4}
\caption{Covariance Measure of PCA 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||r||r||r||r||r|}
\hline
Eigenval. & $\tau$ & $\omega$ & $\alpha$ & $\theta$ & Year \\
\hline
$0.4473697$ & $\mathbf{-0.530}$ & $ 0.323$ & $ 0.298$ & $\mathbf{-0.554}$ & $ 0.090$ \\
$0.1941057$ & $\mathbf{-0.547}$ & $ 0.215$ & $ 0.249$ & $-0.293$ & $\mathbf{-0.630}$ \\
$0.0463189$ & $ 0.131$ & $-0.002$ & $-0.128$ & $ 0.242$ & $\mathbf{-0.630}$ \\
$0.0280301$ & $-0.011$ & $ 0.225$ & $ 0.186$ & $ 0.131$ & $ 0.067$ \\
$0.0243231$ & $-0.181$ & $ 0.174$ & $-0.032$ & $-0.216$ & $ 0.352$ \\
$0.0180875$ & $-0.364$ & $ 0.226$ & $ 0.339$ & $-0.136$ & $ 0.113$ \\
$0.0138478$ & $-0.194$ & $-0.048$ & $-0.099$ & $-0.333$ & $ 0.055$ \\
$0.0110575$ & $-0.040$ & $-0.254$ & $-0.154$ & $-0.054$ & $-0.089$ \\
$0.0093587$ & $-0.199$ & $-0.115$ & $ 0.358$ & $-0.234$ & $-0.028$ \\
$0.0084930$ & $ 0.046$ & $ 0.190$ & $ 0.305$ & $ 0.176$ & $ 0.089$ \\
\hline
\end{tabular}
\end{table}

\begin{table}[!t]
% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.6}
\begin{threeparttable}
\caption{Characteristic Patterns of PCA$_{1,2}$ Dimensions \tnote{1}}
\label{fig:style_patterns}
\centering
% Some packages, such as MDW tools, offer better commands for making tables
% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{|p{2.3cm}p{2.4cm}p{2.4cm}p{0cm}|}
% The virtual last column is here because otherwise we get random syntax errors.

\hline \multicolumn{4}{|c|}{PCA$_1$ --- Moyo-oriented, thin-playing player} \\
\centering \begin{psgopartialboard*}{(8,1)(12,6)}
\stone[\marktr]{black}{k}{4}
\end{psgopartialboard*} &
\centering \begin{psgopartialboard*}{(1,2)(5,6)}
\stone{white}{d}{3}
\stone[\marktr]{black}{d}{5}
\end{psgopartialboard*} &
\centering \begin{psgopartialboard*}{(5,1)(10,6)}
\stone{white}{f}{3}
\stone[\marktr]{black}{j}{4}
\end{psgopartialboard*} & \\
\centering $0.274$ & \centering $0.086$ & \centering $0.083$ & \\
\centering high corner/side opening \tnote{2} & \centering high corner approach & \centering high distant pincer & \\

\hline \multicolumn{4}{|c|}{PCA$_1$ --- Territorial, thick-playing player} \\
\centering \begin{psgopartialboard*}{(3,1)(7,6)}
\stone{white}{d}{4}
\stone[\marktr]{black}{f}{3}
\end{psgopartialboard*} &
\centering \begin{psgopartialboard*}{(3,1)(7,6)}
\stone{white}{c}{6}
\stone{black}{d}{4}
\stone[\marktr]{black}{f}{3}
\end{psgopartialboard*} &
\centering \begin{psgopartialboard*}{(3,1)(7,6)}
\stone{black}{d}{4}
\stone[\marktr]{black}{f}{3}
\end{psgopartialboard*} & \\
\centering $-0.399$ & \centering $-0.399$ & \centering $-0.177$ & \\
\centering low corner approach & \centering low corner reply & \centering low corner enclosure & \\

\hline \multicolumn{4}{|c|}{PCA$_2$ --- Territorial, current player \tnote{3}} \\
\centering \begin{psgopartialboard*}{(3,1)(7,6)}
\stone{white}{c}{6}
\stone{black}{d}{4}
\stone[\marktr]{black}{f}{3}
\end{psgopartialboard*} &
\centering \begin{psgopartialboard*}{(3,1)(8,6)}
\stone{white}{d}{4}
\stone[\marktr]{black}{g}{4}
\end{psgopartialboard*} &
\centering \begin{psgopartialboard*}{(4,1)(9,6)}
\stone{black}{d}{4}
\stone{white}{f}{3}
\stone[\marktr]{black}{h}{3}
\end{psgopartialboard*} & \\
\centering $-0.193$ & \centering $-0.139$ & \centering $-0.135$ & \\
\centering low corner reply \tnote{4} & \centering high distant approach/pincer & \centering near low pincer & \\

\hline
\end{tabular}
\begin{tablenotes}
\item [1] We present the patterns in a simplified compact form;
in reality, they are usually somewhat larger and always circle-shaped
(centered on the triangled move).
We omit only pattern segments that are entirely empty.
\item [2] We give some textual interpretation of the patterns, especially
since some of them may not be obvious unless seen in game context; we choose
the descriptions based on the most frequently observer contexts, but of course
the pattern can be also matched in other positions and situations.
\item [3] In the second PCA dimension, we find no correlated patterns;
only uncorrelated and anti-correlated ones.
\item [4] As the second most significant pattern,
we skip a slide follow-up pattern to this move.
\end{tablenotes}
\end{threeparttable}
\end{table}

\begin{table}[!t]
% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.8}
\begin{threeparttable}
\caption{Characteristic Patterns of PCA$_3$ Dimension \tnote{1}}
\label{fig:style_patterns3}
\centering
% Some packages, such as MDW tools, offer better commands for making tables
% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{|p{2.4cm}p{2.4cm}p{2.4cm}p{0cm}|}
% The virtual last column is here because otherwise we get random syntax errors.

\hline \multicolumn{4}{|c|}{PCA$_3$ --- Old-time player} \\
\centering \begin{psgopartialboard*}{(1,3)(5,7)}
\stone{white}{d}{4}
\stone[\marktr]{black}{c}{6}
\end{psgopartialboard*} &
\centering \begin{psgopartialboard*}{(8,1)(12,5)}
\stone[\marktr]{black}{k}{3}
\end{psgopartialboard*} &
\centering \begin{psgopartialboard*}{(1,1)(5,5)}
\stone[\marktr]{black}{c}{3}
\end{psgopartialboard*} & \\
\centering $0.515$ & \centering $0.264$ & \centering $0.258$ & \\
\centering low corner approach & \centering low side or mokuhazushi opening & \centering san-san opening & \\

\hline \multicolumn{4}{|c|}{PCA$_3$ --- Current player} \\
\centering \begin{psgopartialboard*}{(3,1)(7,5)}
\stone{black}{d}{4}
\stone[\marktr]{black}{f}{3}
\end{psgopartialboard*} &
\centering \begin{psgopartialboard*}{(1,1)(5,5)}
\stone[\marktr]{black}{c}{4}
\end{psgopartialboard*} &
\centering \begin{psgopartialboard*}{(1,2)(5,6)}
\stone{black}{d}{3}
\stone{white}{d}{5}
\stone[\marktr]{black}{c}{5}
\end{psgopartialboard*} & \\
\centering $-0.276$ & \centering $-0.273$ & \centering $-0.116$ & \\
\centering low corner enclosure & \centering 3-4 corner opening \tnote{2} & \centering high approach reply & \\

\hline
\end{tabular}
\begin{tablenotes}
\item [1] We cannot use terms ``classic'' and ''modern'' in case of PCA$_3$
since the current patterns are commonplace in games of past centuries
(not included in our training set) and many would call a lot of the old-time patterns
modern inventions. Perhaps we can infer that the latest 21th-century play trends abandon
many of the 20th-century experiments (lower echelon of our by-year samples)
to return to the more ordinary but effective classic patterns.
\item [2] At this point, we skip two patterns already shown elsewhere:
{\em high side/corner opening} and {\em low corner reply}.
\end{tablenotes}
\end{threeparttable}
\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 territoriality of the players,%
\footnote{Cho Chikun, perhaps the best-known
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 are already
correlated in the prior data.

The other PCA dimensions are somewhat harder to interpret, but there
certainly is significant influence of the styles on the patterns;
the found correlations are all 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 (tables \ref{fig:style_patterns}, \ref{fig:style_patterns3}), determined by their coefficients
in the PCA projection matrix --- however, such naive approach
has limited reliability, 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 do not show the other pattern features since they carry no useful
information in the opening stage.%
\footnote{The board distance feature can be useful in some cases,
but here all the spatial patterns are big enough to reach to the edge
on their own.}

\begin{table}[!t]
% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.4}
\caption{Covariance Measure of Externed-Normalization PCA and~Prior Information}
\label{fig:style_normr}
\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||r||r||r||r||r|}
\hline
Eigenval. & $\tau$ & $\omega$ & $\alpha$ & $\theta$ & Year \\
\hline
$6.3774289$ & $ \mathbf{0.436}$ & $-0.220$ & $-0.289$ & $ \mathbf{0.404}$ & $\mathbf{-0.576}$ \\
$1.7269775$ & $\mathbf{-0.690}$ & $ 0.340$ & $ 0.315$ & $\mathbf{-0.445}$ & $\mathbf{-0.639}$ \\
$1.1747101$ & $-0.185$ & $ 0.156$ & $ 0.107$ & $-0.315$ & $ 0.320$ \\
$0.8452797$ & $ 0.064$ & $-0.102$ & $-0.189$ & $ 0.032$ & $ 0.182$ \\
$0.8038992$ & $-0.185$ & $ 0.261$ & $ \mathbf{0.620}$ & $ 0.120$ & $ 0.056$ \\
$0.6679533$ & $-0.027$ & $ 0.055$ & $ 0.147$ & $-0.198$ & $ 0.155$ \\
$0.5790000$ & $ 0.079$ & $ \mathbf{0.509}$ & $ 0.167$ & $ 0.294$ & $-0.019$ \\
$0.4971474$ & $ 0.026$ & $-0.119$ & $-0.071$ & $ 0.049$ & $ 0.043$ \\
$0.4938777$ & $-0.061$ & $ 0.061$ & $ 0.104$ & $-0.168$ & $ 0.015$ \\
$0.4888848$ & $ 0.203$ & $-0.283$ & $-0.120$ & $ 0.083$ & $-0.220$ \\
\hline
\end{tabular}
\end{table}

The PCA results presented above do not show much correlation between
the significant PCA dimensions and the $\omega$ and $\alpha$ style dimensions.
However, when we applied the extended vector normalization
(sec. \ref{xnorm}; see table \ref{fig:style_normr}),
some less significant PCA dimensions exhibited clear correlations.%
\footnote{We have found that $c=6$ in the post-processing logistic function
produces the most instructive PCA output on our particular game collection.}
It appears that less-frequent patterns that appear only in the middle-game
phase\footnote{In the middle game, the board is much more filled and thus
particular specific-shape patterns repeat less often.} are defining
for these dimensions, and these are not represented in the pattern vectors
as well as the common opening patterns.
However, we do not use the extended normalization results since
they produced noticeably less accurate classifiers in all dimensions,
including $\omega$ and $\alpha$.

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

\begin{figure}[!t]
\centering
\includegraphics[width=3.5in,angle=-90]{sociomap}
\caption{Sociomap visualisation. The spatial positioning of players
is based on the expert knowledge, while the node heights (depicted by
contour lines) represent the pattern vectors.%
%The light lines denote coherence-based hierarchical clusters.
}
\label{fig:sociomap}
\end{figure}

Fig. \ref{fig:sociomap} shows the Sociomap visualisation
as an alternate view of the player relationships and similarity,
as well as correlation between the expert-given style marks
and the PCA decomposition. The four-dimensional style vectors
are used as input for the Sociomap renderer and determine the
spatial positions of players. The height of a node is then
determined using first two PCA dimensions $R_1,R_2$ and their
eigenvalues $\lambda_1,\lambda_2$ as their linear combination:
$$ h=\lambda_1R_1 + \lambda_2R_2 $$

We can observe that the terrain of the sociomap is reasonably
``smooth'', again demonstrating some level of connection between
the style vectors and data-mined information. High countour density
indicates some discrepancy; in case of Takemiya Masaki and Yi Ch'ang-ho,
this seems to be merely an issue of scale,
while the Rui Naiwei --- Gu Li cliff suggests a genuine problem;
we cannot say now whether it is because of imprecise prior information
or bad approximation abilities of our model.

\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 tested the style inference ability
of neural network (sec. \ref{neural-net}), $k$-NN (sec. \ref{knn}) and Bayes (sec. \ref{naive-bayes}) classifers.

\subsubsection{Reference (Training) Data}
As the~reference data, we use expert-based knowledge presented in section \ref{style-vectors}.
For each reference player, that gives $4$-dimensional \emph{style vector} (each component in the
range of $[1,10]$).\footnote{Since the neural network has activation function with range $[-1,1]$, we
have linearly rescaled the \emph{style vectors} from interval $[1,10]$ to $[-1,1]$ before using the training
data. The network's output was afterwards rescaled back to allow for MSE comparison.}

All input (pattern) vectors were preprocessed using PCA, reducing the input dimension from $400$ to $23$.

\subsubsection{Cross-validation}
\label{crossval}
To compare and evaluate all methods, we have performed $5$-fold cross validation
and compared each method's performance with a~random classifier.
In the $5$-fold cross-validation, we randomly divide the training set
(organized by players) 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, in turn yielding mean square error.

\subsubsection{Results}
The results are shown in the table \ref{crossval-cmp}. Second to fifth columns in the table represent
mean square error of the examined styles, $\mathit{Mean}$ is the
mean square error across the styles and finally, the last column $\mathit{Cmp}$
represents $\mathit{Mean}(\mathit{Random classifier}) / \mathit{Mean}(\mathit{X})$ -- comparison of mean square error
of method $\mathit{X}$ with the random classifier. To minimize the
effect of random variables, all numbers were taken as an average of $200$ runs of the cross validation.

Analysis of the performance of $k$-NN classifier for different $k$-values showed that different
$k$-values are suitable to approximate different styles. Combining the $k$-NN classifiers with the 
neural network (so that each style is approximated by the method with lowest MSE in that style)
results in \emph{Joint classifier}, which outperforms all other methods. (Table \ref{crossval-cmp})
The \emph{Joint classifier} has outstanding MSE $3.960$, which is equivalent to standard deviation
of $\sigma = 1.99$ per style.

\begin{table}[!t]
\renewcommand{\arraystretch}{1.4}
\begin{threeparttable}
\caption{Comparison of style classifiers}
\label{crossval-cmp}
\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 Cmp}\\ \hline
Joint classifier\tnote{1} & 4.04 & {\bf 5.25} & {\bf 3.52} & {\bf 3.05} & {\bf 3.960}& 2.97 \\\hline
Neural network   & {\bf 4.03} & 6.15       & {\bf 3.58} & 3.79       & 4.388      & 2.68 \\ 
$k$-NN ($k=2$)   & 4.08       & 5.40       & 4.77       & 3.37       & 4.405      & 2.67 \\
$k$-NN ($k=3$)   & 4.05       & 5.58       & 5.06       & 3.41       & 4.524      & 2.60 \\
$k$-NN ($k=1$)   & 4.52       & {\bf 5.26} & 5.36       & {\bf 3.09} & 4.553      & 2.59 \\
$k$-NN ($k=4$)   & 4.10       & 5.88       & 5.16       & 3.60       & 4.684      & 2.51 \\
Naive Bayes      & 4.48       & 6.90       & 5.48       & 3.70       & 5.143      & 2.29 \\
Random class.    & 12.26      & 12.33      & 12.40      & 10.11      & 11.776     & 1.00 \\\hline

\end{tabular}
\begin{tablenotes}
\item [1] Note that these measurements have a certain variance. The Joint classifier measurements were taken independently and
they can differ from the according methods.
\end{tablenotes}
\end{threeparttable}
\end{table}

% TODO presunout konkretni parametry do Appendixu?  (neni jich tolik, mozna ne)
\subsubsection{$k$-NN parameters}
All three variants of $k$-NN classifier ($k=2,3,4$) we have used and compared had the following weight function
\begin{equation}
\mathit{Weight}(\vec x) = 0.8^{10*\mathit{Distance}(\vec x)}
\end{equation}
The parameters were chosen empirically to minimize the MSE.

\subsubsection{Neural network's parameters}
The neural network classifier had $3$-layered architecture (one hidden layer) with following numbers of
neurons:
\vspace{4mm}
\noindent
%\begin{table}
\begin{center}
%\caption{Styles}
\begin{tabular}{|c|c|c|}
\hline
\multicolumn{3}{|c|}{Layer} \\\hline
Input & Hidden & Output \\ \hline
 23 & 30 & 4 \\ \hline
\end{tabular}
\end{center}
%\end{table}
\vspace{4mm}

The network was trained until the square error on the training set was smaller than $0.0003$.
Due to a small number of input vectors, this only took $20$ iterations of RPROP learning algorithm on average.

\subsubsection{Naive Bayes parameters}

We have chosen $k = 10/7$ as our discretization parameter;
ideally, we would use $k = 1$ to fully cover the style marks
domain, however our training sample is probably too small for
that.

\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.
Time by time, new historical game records are still being discovered;
pattern-based classification might help to suggest origin of the games
in Go Archeology.

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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\section{Future Research}

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, in all parts of our analysis --- we have already noted
many in the text above. Most significantly, different methods of generating
and normalizing the $\vec p$ vectors can be explored
and other data mining methods could be investigated.
Better ways of visualising the relationships would be desirable,
together with thorough dissemination of internal structure
of the player pattern vectors space.

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 also investigated.

% TODO: Future research --- Sparse PCA

\section{Conclusion}
We have proposed a way to extract summary pattern information from
game collections and combined this with various data mining methods
to show correspondence of our pattern summaries with various player
meta-information like playing strength, era of play or playing style
as ranked by expert players. We have implemented and measured our
proposals in two case studies: per-rank characteristics of amateur
players and per-player style/era characteristics of well-known
professionals.

While many details remain to be worked out,
we have demonstrated that many significant correlations do exist and
it is practically viable to infer the player meta-information from
extracted pattern summaries. We proposed wide range of applications
for such inference. Finally, we outlined some of the many possible
directions of future work in this newly staked research field
on the boundary of Computer Go, Data Mining and Go Theory.


% if have a single appendix:
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%\section{Proof of the First Zonklar Equation}
%Appendix one text goes here.
%
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%Appendix two text goes here.


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

Foremostly, we are very grateful for detailed input on specific go styles
by Alexander Dinerstein, Motoki Noguchi and V\'{i}t Brunner.
We appreciate X reviewing our paper, and 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 would like to thank Radka ``chidori'' Hane\v{c}kov\'{a}
for the original research idea and acknowledge major inspiration
by R\'{e}mi Coulom's paper \cite{PatElo} on the extraction of pattern information.


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Received BSc degree in Informatics at Charles University, Prague in 2009,
currently a graduate student.
Doing research in the fields of Computer Go, Monte Carlo Methods
and Version Control Systems.
Plays Go with the rank of 2-kyu on European tournaments
and 2-dan on the KGS Go Server.
\end{IEEEbiographynophoto}

\begin{IEEEbiographynophoto}{Josef Moud\v{r}\'{i}k}
Received BSc degree in Informatics at Charles University, Prague in 2009,
currently a graduate student.
Doing research in the fields of Neural Networks and Cognitive Sciences.
His Go skills are not worth mentioning.
\end{IEEEbiographynophoto}

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