From a3224ef3d41347b1351d55dedd5da577708d1035 Mon Sep 17 00:00:00 2001
From: Petr Baudis <pasky@ucw.cz>
Date: Wed, 10 Mar 2010 13:50:33 +0100
Subject: [PATCH] tex: Chi-square intro

---
 tex/gostyle.bib | 12 ++++++++++++
 tex/gostyle.tex | 18 +++++++++++++++---
 2 files changed, 27 insertions(+), 3 deletions(-)

diff --git a/tex/gostyle.bib b/tex/gostyle.bib
index 6c177d3..44ae081 100644
--- a/tex/gostyle.bib
+++ b/tex/gostyle.bib
@@ -73,6 +73,18 @@
   timestamp = {2010.03.05}
 }
 
+@article{Pearson,
+	author = {R. L. Plackett},
+	title = {Karl Pearson and the Chi-Squared Test},
+	journal = {International Statistical Review},
+	year = {1983},
+	volume = {51},
+	number = {1},
+	pages = {59--72},
+	issn = {03067734},
+	owner = {pasky}
+}
+
 @misc{GoR,
   author = {Ales Cieply and others},
   title = {EGF ratings system --- Sys\-tem des\-crip\-tion},
diff --git a/tex/gostyle.tex b/tex/gostyle.tex
index dace37b..c584205 100644
--- a/tex/gostyle.tex
+++ b/tex/gostyle.tex
@@ -505,6 +505,15 @@ 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.
+
+TODO: Chi-square computation.
+
 \subsection{Kohonen Maps}
 \label{koh}
 Kohonen map is a self-organizing network with neurons spread over a two-dimensional plane.
@@ -731,8 +740,11 @@ 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 \ref{fig:strength_pca}.)
+(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.
 Using the eigenvector position directly for classification
 of players within the test group yields MSE TODO, thus providing
 reasonably satisfying accuracy.
@@ -859,7 +871,7 @@ Chen Yaoye           & $6.0 \pm 1.0$ & $4.0 \pm 1.0$ & $6.0 \pm 1.0$ & $5.5 \pm
 
 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
+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
-- 
2.11.4.GIT