tex: Draft text for PCA analysis

[gostyle.git] / tex / gostyle.tex

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


\begin{document}
%
% paper title
% can use linebreaks \\ within to get better formatting as desired
\title{On Move Pattern Trends\\in Large Go Games Corpus}

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\author{Petr~Baudis,~Josef~Moudrik% <-this % stops a space
\thanks{P. Baudis is student at the Faculty of Math and Physics, Charles University, Prague, CZ, and also does some of his Computer Go research as an employee of SUSE Labs Prague, Novell CZ.}% <-this % stops a space
\thanks{J. Moudrik 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 per-player 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 discussion within Go theory 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
\end{IEEEkeywords}






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\section{Introduction}
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% and "HIS" in caps to complete the first word.
\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. 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 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 (TODO: KGS Stats, KGS Analytics,
Pro Go Rating) and ''next move'' statistics on a~specific position (TODO:
Kombilo, Moyo Go Studio).

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{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.
Three high-level Go players (Alexander Dinerstein 3-pro, Motoki Noguchi
7-dan and Vit Brunner 4-dan) have judged style of several Go
professionals -- we call them \emph{reference playerse} -- chosen for both
being well-known within the community and having large number of played games in our collection.

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 assign each of player's style a number
on a scale from 1 to 10. These are interpreted
as shown in the table below.

\vspace{4mm}
\noindent
%\begin{table}
\begin{center}
%\caption{Styles}
\begin{tabular}{|c|c|c|}
\hline
\multicolumn{3}{|c|}{Styles} \\ \hline
Style & 1 & 10\\ \hline
Territoriality & Moyo & Territorial \\
Orthodoxity & Classic & Novel \\
Aggressivity & Calm & Figting \\
Thickness & 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} $R$.

%-- each with a \emph{pattern vector} $\vec p_i$ and \emph{style vector} $\vec s_i$.

\section{Data Extraction}
\label{pattern-vectors}
In addition to the explicit expert knowledge, we use the data obtained by...

TODO rozvest uvod, nemuze se zacinat jenom As the input...

As the input, we assume a~collection of game records\footnote{We
use the SGF format (TODO) in our implementation.} organized by player names.
We use two collections; the first one is GoGoD Winter 2009 (TODO) containing 42000 (TODO)
professional games, dating from the early Go history 1500 years ago to the present.
We use this collection for style analysis and detailed correlation analysis
of well-known Go professionals.
The other source is Go Teaching Ladder reviews (TODO). These include 7600 games
of players spanning over all strength levels; we use this collection
for finding correlations between moves of players of the same strength rank.

In order to generate the required compact description of most frequently played moves,
we construct a set of $n$ most occuring patterns (\emph{top patterns})
across all players and games from the database\footnote{We use $n=500$ in our analysis.}.
For each player, we then count how many times was each of those $n$ patterns played
during all his games and finally assign him a~{\em pattern vector} $\vec p$ of dimension $n$, with each
dimension corresponding to the relative number of occurences of a given pattern
(with respect to player's most played \emph{top pattern}). Using relative numbers of occurences ensures that
each dimension of player's \emph{pattern vector} is scaled to range $[0,1]$ and
therefore even players with different number of games in the database have comparable \emph{pattern vectors}.

\subsection{Pattern Features}
We need to define how to compose the patterns we use to describe moves.
There are some tradeoffs in play - 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.

\subsection{Implementation}

We have implemented the data extraction by making use of the pattern
features matching implementation within the Pachi go-playing program
(TODO). We extract information on players by converting the SGF game
records to GTP (TODO) stream that feeds Pachi's {\tt patternscan}
engine which outputs a~single {\em patternspec} (string representation
of the particular pattern features combination) per move.

We can then gather all patternspecs played by a~given player and summarize
them; the $\vec p$ vector then consists of normalized counts of
the given $n$ most frequent patternspecs.


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

To assess the properties of gathered \emph{pattern vectors}
and their influence on playing styles,
we have processes the data using a~few 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 then indicates which patterns correlate with each style.
Additionally, PCA can be used as a vector-preprocessing for methods
that are negatively sensitive to \emph{pattern vector} component correlations.

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

TODO: style vector -> output vector?

Furthermore, we have used and compared two \emph{classification} methods
that approximate well-defined but unknown \emph{style vector} $\vec S$
based on input \emph{pattern vector} $\vec P$.
The methods are calibrated based on expert or prior knowledge about
training pattern vectors and then their error is measured on a testing
set of pattern vectors.

One of the methods is $k$-Nearest Neighbor (kNN) classifier:
we approximate $\vec S$ by composing the \emph{style vectors} of $k$ nearest \emph{pattern vectors}.
The other is based on 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 e.g. requires larger training sample.

TODO: Dava ta posledni veta nejaky smysl?!

\subsection{Principal Component Analysis}
\label{data-mining}
We use Principal Component Analysis \emph{PCA} \cite{Jolliffe1986}
to reduce the dimensions of the \emph{pattern vectors} while preserving
as much information as possible.

Briefly, PCA is an eigenvalue decomposition of a~covariance matrix of centered \emph{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 the transformation matrix $W$.

For each original \emph{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 \emph{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}

\subsection{Kohonen Maps}
\label{koh}
Kohonen map is a self-organizing network with neurons spread over a two-dimensional plane.
Neurons in the map compete for representation of portions of the input vector space.
Each neuron $\vec n$ represents a vector
and the network is trained so that the neurons that are topologically close
tend to represent vectors that are close as well.

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

We then adapt neurons from the neighbourhood of $\vec w$ employing an 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
(see \cite{TODO} for details).
The state of the network can be evaluated by calculating mean square difference
between each $\vec t \in T$ and its corresponding \emph{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 \emph{style vector} $\vec S$
based on their \emph{pattern vector} $\vec P$.
To achieve this, we require prior knowledge of \emph{reference style vectors}
(see section \ref{style-vectors}).

In this method, we assume that similarities in players' \emph{pattern vectors}
uniformly correlate with similarities in players' \emph{style vectors}.
We try to approximate $\vec S$ as a weighted average of \emph{style vectors}
$\vec s_i$ 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 it 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 $\vec S \leftarrow \vec 0$
\FORALL{$r \in N $}
\STATE $\vec S \leftarrow \vec S + \mathit{Weight}(D[r]) \cdot \vec s_r $
\ENDFOR
\end{algorithmic}
\end{algorithm}

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

As an alternative to the k-Nearest Neigbors algorithm (section \ref{knn}),
we have used a classificator based on feed-forward artificial neural networks \cite{TODO}.
Neural networks (NN) are known for their ability to generalize
and find correlations and patterns between input and output data.
Neural network is an adaptive system that must undergo a training
period before it can be reasonably used, similarly to the requirement
of reference vectors for the k-Nearest Neighbors algorithm above.

\subsubsection{Computation and activation of the NN}
Technically, 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} inbetween.

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(\sum_{j \in J}{w_{ij} y_j})
\end{equation}
where $J$ is the previous layer, while $y_j$ is the activation for neurons from $J$ layer.
Function $f()$ is 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}
The training of the feed-forward neural network usually involves some
modification of supervised Backpropagation learning algorithm. \cite{TODO}
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.

TODO: Tohle je puvodni napad?

As outlined above, the training set consists of pairs of
input vectors (\emph{pattern vectors}) and
desired output vectors (\emph{style vectors}).
The training set $T$ is then extended by adding the linear combinations:
\begin{equation}
T_\mathit{base} = \{(\vec p_r, \vec s_r) | r \in R\}\\
\end{equation}
\begin{equation}
T_\mathit{ext} = \{(\vec p, \vec s) | \exists D \subseteq R : \vec p = \sum_{d \in D}{g_d \vec p_d}, \vec s = \sum_{d \in D}{g_d \vec s_d}\}
\end{equation}
TODO zabudovat $g_d$ dovnitr?
where $g_d, d \in D$ are random coeficients, so that $\sum_{d \in D}{g_d} = 1$.
The training set is then constructed as:
\begin{equation}
T = T_\mathit{base} \cup \mathit{SomeFiniteSubset}(T_\mathit{ext})
\end{equation}

The network is trained as 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}


\subsubsection{Architecture details}
TODO num layers, num neurons, ..


\subsection{Implementation}

We have implemented the data mining methods as an open-source framework ``gostyle'' \cite{TODO},
made available under the GNU GPL licence.
We use python for the basic processing and most of the analysis;
the MDP library \cite{MDP} is used for PCA analysis, Kohonen library \cite{KohonenPy} for Kohonen maps.
The neuron network classifier is using the libfann C library. \cite{TODO}


\section{Strength Estimation Analysis}

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 white 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}

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 orders of magnitude smaller, with no discernable
structure revealed within the lower-order eigenvectors.}
(chi-square test TODO).
(Figure TODO.) 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 Components Analysis}

PCA analysis yielded X, chi-square test...

We then tried to apply the NN classifier with linear output function on the dataset
and that yielded Y (see fig. Z), with MSE abcd.


\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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\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:
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% or
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%\appendices
%\section{Proof of the First Zonklar Equation}
%Appendix one text goes here.
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%\section{}
%Appendix two text goes here.


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


We would like to thank X for reviewing our paper.
We appreciate helpful comments on our general methodology
by John Fairbairn, T. M. Hall, Robert Jasiek
and several GoDiscussions.com users. \cite{GoDiscThread}
Finally, we are very grateful for ranking of go styles of selected professionals
by Alexander Dinerstein 3-pro, Motoki Noguchi 7-dan and Vit Brunner 4-dan.


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