From 4c9915c294d766294207f8373b8e77c5e031d94f Mon Sep 17 00:00:00 2001
From: Petr Baudis <pasky@ucw.cz>
Date: Wed, 10 Mar 2010 22:01:27 +0100
Subject: [PATCH] tex: Chi-squared -> Pearson r

---
 tex/gostyle.bib | 15 ++++++++-------
 tex/gostyle.tex | 41 ++++++++++++++++++++++-------------------
 2 files changed, 30 insertions(+), 26 deletions(-)

diff --git a/tex/gostyle.bib b/tex/gostyle.bib
index 34acba7..5e74177 100644
--- a/tex/gostyle.bib
+++ b/tex/gostyle.bib
@@ -55,14 +55,15 @@
 }
 
 @ARTICLE{Pearson,
-  author = {R. L. Plackett},
-  title = {Karl Pearson and the Chi-Squared Test},
-  journal = {International Statistical Review},
-  year = {1983},
-  volume = {51},
-  pages = {59--72},
+  author = {J. L. Rodgers and W. A. Nicewander},
+  title = {Thirteen ways to look at the correlation coefficient},
+  journal = {The American Statistician},
+  year = {1988},
+  month = {Feb},
+  volume = {42},
   number = {1},
-  issn = {03067734},
+  pages = {59--66},
+  issn = {},
   owner = {pasky}
 }
 
diff --git a/tex/gostyle.tex b/tex/gostyle.tex
index 9c4cafd..409a31d 100644
--- a/tex/gostyle.tex
+++ b/tex/gostyle.tex
@@ -511,14 +511,16 @@ The whole process is described in the Algorithm \ref{alg:pca}.
 \end{algorithmic}
 \end{algorithm}
 
-We will want to find dependencies between PCA dimensions and dimensions
-of some prior knowledge (player rank, style vector). For this, we use
-the well-known {\em Pearson's $\chi^2$ test} \cite{Pearson}; the test
-yields the probability of a null hypothesis that two distributions
-are statistically independent, we will instead use the probability
-of the alternative hypothesis that they are in fact dependent.
+\label{pearson}
+We will want to find correlations between PCA dimensions and
+some prior knowledge (player rank, style vector).
+We compute the well-known {\em Pearson product-moment correlation coefficient} \cite{Pearson}
+values for this purpose, 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:
 
-TODO: Chi-square computation.
+$$ r_{X,Y} = {{\rm cov}(X,Y) \over \sigma_X \sigma_Y} $$
 
 \subsection{Kohonen Maps}
 \label{koh}
@@ -747,9 +749,8 @@ second dimension was four orders of magnitude smaller, with no discernable
 structure revealed within the lower-order eigenvectors.}
 (figure \ref{fig:strength_pca}).
 
-In order to measure the accuracy of approximation of strength by the first dimension,
-we have used the $\chi^2$ test, yielding probability $p=TODO$ that it is dependent
-on the player strength.
+We measure the accuracy of strength approximation by the first dimension
+using Pearson's $r$ (see \ref{pearson}), yielding satisfying value $r=TODO$.
 Using the eigenvector position directly for classification
 of players within the test group yields MSE TODO, thus providing
 reasonably satisfying accuracy.
@@ -874,16 +875,14 @@ Chen Yaoye           & $6.0 \pm 1.0$ & $4.0 \pm 1.0$ & $6.0 \pm 1.0$ & $5.5 \pm
 \label{fig:style_pca}
 \end{figure}
 
-We have looked at the three most significant dimensions of the pattern data
-yielded by the PCA analysis (fig. \ref{fig:style_pca}). We have again
-performend $\chi^2$--test between the three most significant PCA dimensions
-and dimensions of the prior knowledge style vectors to find correlations;
-the found correlations are presented in table \ref{fig:style_chisq}.
-We also list the characteristic spatial patterns of the PCA dimension
-extremes (table \ref{fig:style_patterns}).
+We have looked at the five most significant dimensions of the pattern data
+yielded by the PCA analysis (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.
 
 It is immediately
-obvious that by far the most significant vector corresponds very well
+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 just below $-0.5$ on the first dimension.}
@@ -893,7 +892,11 @@ most obvious in the played shapes and sequences
 (that can obviously aim directly at taking secure territory
 or building center-oriented framework).
 
-The other PCA dimensions are far less obvious --- TODO.
+Other PCA dimensions are far less 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_chisq}.
+We also list the characteristic spatial patterns of the PCA dimension
+extremes (table \ref{fig:style_patterns}).
 
 Kohonen map view.
 
-- 
2.11.4.GIT