* green/doc : New documentation directory. * green/doc/formulae : New specific directory. * green/doc/formulae/formulae.tex : New recipe of 3d formulae. --- trunk/milena/sandbox/ChangeLog | 8 + .../milena/sandbox/green/doc/formulae/formulae.tex | 1157 ++++++++++++++++++++ 2 files changed, 1165 insertions(+), 0 deletions(-) create mode 100644 trunk/milena/sandbox/green/doc/formulae/formulae.tex diff --git a/trunk/milena/sandbox/ChangeLog b/trunk/milena/sandbox/ChangeLog index 5a96357..b50e807 100644 --- a/trunk/milena/sandbox/ChangeLog +++ b/trunk/milena/sandbox/ChangeLog @@ -1,3 +1,11 @@ +2009-09-10 Yann Jacquelet &lt;jacquelet(a)lrde.epita.fr&gt; + + Write down 3d currently used formulaes. + + * green/doc : New documentation directory. + * green/doc/formulae : New specific directory. + * green/doc/formulae/formulae.tex : New recipe of 3d formulae. + 2009-09-09 Yann Jacquelet &lt;jacquelet(a)lrde.epita.fr&gt; Fix bugs an compilation problem on histo3d_rgb source code. diff --git a/trunk/milena/sandbox/green/doc/formulae/formulae.tex b/trunk/milena/sandbox/green/doc/formulae/formulae.tex new file mode 100644 index 0000000..52af7a7 --- /dev/null +++ b/trunk/milena/sandbox/green/doc/formulae/formulae.tex @@ -0,0 +1,1157 @@ +%% Copyright (C) 2009 EPITA Research and Development Laboratory (LRDE) +%% +%% This file is part of Olena. +%% +%% Olena is free software: you can redistribute it and/or modify it under +%% the terms of the GNU General Public License as published by the Free +%% Software Foundation, version 2 of the License. +%% +%% Olena is distributed in the hope that it will be useful, +%% but WITHOUT ANY WARRANTY; without even the implied warranty of +%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU +%% General Public License for more details. +%% +%% You should have received a copy of the GNU General Public License +%% along with Olena. If not, see <http://www.gnu.org/licenses/>. + +\documentclass{article} + +%\usepackage{hevea} + +\usepackage{html} +\usepackage{hyperref} +\usepackage{graphicx} +\usepackage{makeidx} +\usepackage{xcolor} +\usepackage{color} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{amssymb} + + +%% \bm +%% input : the mathematical text to put in bold +%% +\newcommand{\bm}[1]{\mbox{\boldmath$#1$}} + +%% \fm +%% input : the mathematical text to put in mathematic font set +%% +\newcommand{\fm}[1]{\mathbb{#1}} + +%% \df +%% input : the set which the dim we are intering in +%% +\newcommand{\df}[1]{|{\mathbb{#1}}|} + +\title{Recipe in statistics and linear algebra\\ + \large{A few formulas in $\fm{R}^3$ space} } +\author{LRDE} +\date{} +\makeindex + +\begin{document} + +\maketitle + +%################################################################# +\section{Introduction} +The goal of this document is to keep in mind some statictic formulas which are +not so evident to represent. They help us to keep some background mathematics +in the development. + +%################################################################# +\section{Notations} + +\begin{itemize} + \item Use lowercase and normal font for the scalar variables. + \item Use lowercase and bold font for the vector variables. + \item Use uppercase and normal font for the matrix variables. + \item Use uppercase and double font for the set variables. +\end {itemize} + +%================================================================= +\subsection{Sets} + +There is three particular sets that we use every time: +\begin{itemize} + \item The color space $\fm{C}$ in which the pixels take their value. + \item The dataset $\fm{P}$ which contains every pixel we take care. + \item The group set $\fm{G}$ that define any splitting of the dataset. +\end{itemize} + +\begin{tabular}{|c|l|l|c|c|} + \hline + Sets & Contents & (in sample) & + Dimension & (in sample) \\ + \hline + $\fm{C}$ & $\fm{R}^q$ & $\fm{R}^3$ & + $\df{C} = q$ & $q = 3$ \\ + $\fm{P}$ & $\{\bm{p_i}\}$ &$\{\bm{a},\bm{b},\bm{c},\bm{d}\}$ & + $\df{P} = r$ & $r = 4$ \\ + $\fm{G}$ & $\{\fm{G}_i\}$ &$\{\fm{F}, \fm{H}\}$ & + $\df{G} = k$ & $k = 2$ \\ + \hline +\end{tabular} + +%================================================================= +\subsection{Color space} + +We use the euclidian distance. + +$$ +d(a,b) = +\sqrt{\sum_{i=0}^q (a_i - b_i)^2} = +\sqrt{(a_x - b_x)^2 + (a_y - b_y)^2 + (a_z - b_z)^2} +$$ + +$$ +d(a,b)^2 = +\left[\begin{array}{ccc} + a_x - b_x & a_y - b_y & a_z - b_z +\end{array}\right] +\times +\left[\begin{array}{c} + a_x - b_x \\ + a_y - b_y \\ + a_y - b_z +\end{array}\right] +$$ + +%================================================================= +\subsection{Data points in $\mathbb{R}^3$} +We present the four points of the dataset with their vector representation. + +$$ +\bm{a} = +\left[\begin{array}{c} + a_x \\ + a_y \\ + a_z +\end{array}\right] += +\left[\begin{array}{c} + p_{1x} \\ + p_{1y} \\ + p_{1z} +\end{array}\right] += +\left[\begin{array}{c} + p_{11} \\ + p_{12} \\ + p_{13} +\end{array}\right] += \bm{p}_1 +$$ + +$$ +\bm{b} = +\left[\begin{array}{c} + b_x \\ + b_y \\ + b_z +\end{array}\right] += +\left[\begin{array}{c} + p_{2x} \\ + p_{2y} \\ + p_{2z} +\end{array}\right] += +\left[\begin{array}{c} + p_{21} \\ + p_{22} \\ + p_{23} +\end{array}\right] += \bm{p}_2 +$$ + +$$ +\bm{c} = +\left[\begin{array}{c} + c_x \\ + c_y \\ + c_z +\end{array}\right] += +\left[\begin{array}{c} + p_{3x} \\ + p_{3y} \\ + p_{3z} +\end{array}\right] += +\left[\begin{array}{c} + p_{31} \\ + p_{32} \\ + p_{33} +\end{array}\right] += \bm{p}_3 +$$ + +$$ +\mbox{\boldmath$d$} = +\left[\begin{array}{c} + d_x \\ + d_y \\ + d_z +\end{array}\right] += +\left[\begin{array}{c} + p_{4x} \\ + p_{4y} \\ + p_{4z} +\end{array}\right] += +\left[\begin{array}{c} + p_{41} \\ + p_{42} \\ + p_{43} +\end{array}\right] += \bm{p}_4 +$$ + +One may group the four points in one matrix P : + +$$ +P = +\left[\begin{array}{c} + \bm{a}^t \\ + \bm{b}^t \\ + \bm{c}^t \\ + \bm{d}^t +\end{array}\right] += +\left[\begin{array}{ccc} + a_x & a_y & a_z \\ + b_x & b_y & b_z \\ + c_x & c_y & c_z \\ + d_x & d_y & d_z +\end{array}\right] += +\left[\begin{array}{ccc} + p_{1x} & p_{1y} & p_{1z} \\ + p_{2x} & p_{2y} & p_{2z} \\ + p_{3x} & p_{3y} & p_{3z} \\ + p_{4x} & p_{4y} & p_{4z} +\end{array}\right] += +\left[\begin{array}{ccc} + p_{11} & p_{22} & p_{13} \\ + p_{21} & p_{22} & p_{23} \\ + p_{31} & p_{32} & p_{33} \\ + p_{41} & p_{42} & p_{43} +\end{array}\right] += +\left[\begin{array}{c} + \bm{p}_1^t \\ + \bm{p}_2^t \\ + \bm{p}_3^t \\ + \bm{p}_4^t +\end{array}\right] +$$ + +%================================================================= +\subsection{The group} + +We can define the group set $\fm{G}$ in thow context: +\begin{itemize} + \item First, the group set is a partition. + \item Second, the group set if a fuzzy set. +\end{itemize} + +$\df{P} = \sum_{i=1}^{k} |\fm{G}_i| = \df{F} + \df{H} = 4$ + +$\dim F = \dim H = \df{P} = 4$ + +$ +\bm{f} = +\left[\begin{array}{c} + f_a \\ + f_b \\ + f_c \\ + f_d +\end{array}\right] += +\frac{1}{2} +\left[\begin{array}{c} + 1 \\ + 1 \\ + 0 \\ + 0 +\end{array}\right] +$ + +$\sum_{i=1}^{k} f_i = 1$ + +$ +\bm{h} = +\left[\begin{array}{c} + h_a \\ + h_b \\ + h_c \\ + h_d +\end{array}\right] += +\frac{1}{2} +\left[\begin{array}{c} + 0 \\ + 0 \\ + 1 \\ + 1 +\end{array}\right] +$ + +$\sum_{i=1}^{k} h_i = 1$ + +The appartenance degree of $\bm{a}$ to the group $\fm{H}$ is $h_a$ such that: + +$ h_a \in +\begin{cases} + \{0,1\} \text{if $\fm{G}$ is a partition.} \\ + [0,1] \text{if $\fm{G}$ is a fuzzy set.} +\end{cases} +$ + +%################################################################# +\section{Momentum} + +Let's have a look to the first three moments. + +%================================================================= +\subsection{The mean} + +$$ + \mbox{\boldmath$m$} = +\left[\begin{array}{c} + m_x \\ + m_y \\ + m_z +\end{array}\right] += +\left[\begin{array}{c} + m_1 \\ + m_2 \\ + m_3 +\end{array}\right] += +\frac{1}{4} +\left[\begin{array}{c} + a_x + b_x + c_x + d_x \\ + a_y + b_y + c_y + d_y \\ + a_z + b_z + c_z + d_z +\end{array}\right] += +\frac{1}{4} +\left[\begin{array}{c} + \sum_{i=1}^{4}p_{ix} \\ + \sum_{i=1}^{4}p_{iy} \\ + \sum_{i=1}^{4}p_{iz} +\end{array}\right] += +\frac{1}{4}(\mbox{\boldmath$a$} + \mbox{\boldmath$b$} + \mbox{\boldmath$c$} + + \mbox{\boldmath$d$}) += +\frac{1}{4}\sum_{i=1}^{4}\mbox{\boldmath$p$}_i +$$ +$$ + \mbox{\boldmath$m$} = +\frac{1}{4} +\left[\begin{array}{cccc} + a_x & b_x & c_x & d_x \\ + a_y & b_y & c_y & d_y \\ + a_z & b_z & c_z & d_z +\end{array}\right] +\left[\begin{array}{c} + 1 \\ + 1 \\ + 1 \\ + 1 +\end{array}\right] += +\frac{1}{4} +\left[\begin{array}{cccc} + \mbox{\boldmath$p$}_1^t & \mbox{\boldmath$p$}_2^t & + \mbox{\boldmath$p$}_3^t & \mbox{\boldmath$p$}_4^t +\end{array}\right] +\left[\begin{array}{c} + 1 \\ + 1 \\ + 1 \\ + 1 +\end{array}\right] += +\frac{1}{4} P^t \mbox{\boldmath$ 1$} +$$ +The mean matrix: + +$$ +M = +\left[\begin{array}{c} + \mbox{\boldmath$m$}^t \\ + \mbox{\boldmath$m$}^t \\ + \mbox{\boldmath$m$}^t \\ + \mbox{\boldmath$m$}^t +\end{array}\right] += +\left[\begin{array}{ccc} + m_x & m_y & m_z \\ + m_x & m_y & m_z \\ + m_x & m_y & m_z \\ + m_x & m_y & m_z +\end{array}\right] += +\left[\begin{array}{ccc} + m_1 & m_2 & m_3 \\ + m_1 & m_2 & m_3 \\ + m_1 & m_2 & m_3 \\ + m_1 & m_2 & m_3 +\end{array}\right] +$$ + +We define the difference between a point $\mbox{\boldmath$p$}_i$ and the mean: +$$ +(\mbox{\boldmath$p$}_i - \mbox{\boldmath$m$}) = +\left[\begin{array}{c} + p_{ix} - m_x \\ + p_{iy} - m_y \\ + p_{iz} - m_z +\end{array}\right] += +\left[\begin{array}{c} + p_{i1} - m_1 \\ + p_{i2} - m_2 \\ + p_{i3} - m_3 +\end{array}\right] +$$ + +And for all the dataset: + +$$ +(P - M) = +\left[\begin{array}{ccc} + a_x & a_y & a_z \\ + b_x & b_y & b_z \\ + c_x & c_y & c_z \\ + d_x & d_y & d_z +\end{array}\right] +- +\left[\begin{array}{ccc} + m_x & m_y & m_z \\ + m_x & m_y & m_z \\ + m_x & m_y & m_z \\ + m_x & m_y & m_z +\end{array}\right] += +\left[\begin{array}{ccc} + a_x - m_x & a_y - m_y & a_z - m_z \\ + b_x - m_x & b_y - m_y & b_z - m_z \\ + c_x - m_x & c_y - m_y & c_z - m_z \\ + d_x - m_x & d_y - m_y & d_z - m_z +\end{array}\right] +$$ + +%================================================================= +\subsection{The variance} + +$$ +\begin{array}{lcl} +V & = & +\left[\begin{array}{ccc} + v_{xx} & v_{xy} & v_{xz} \\ + v_{yx} & v_{yy} & v_{yz} \\ + v_{zx} & v_{zy} & v_{zz} +\end{array}\right] += +\left[\begin{array}{ccc} + v_{11} & v_{12} & v_{13} \\ + v_{21} & v_{22} & v_{23} \\ + v_{31} & v_{32} & v_{33} +\end{array}\right] +\\ +V & = & +\frac{1}{4} +\left[\begin{array}{ccc} + \sum_{i=1}^4 (p_{ix} - m_x)(p_{ix} - m_x) & + \sum_{i=1}^4 (p_{ix} - m_x)(p_{iy} - m_y) & + \sum_{i=1}^4 (p_{ix} - m_x)(p_{iz} - m_z) \\ + \sum_{i=1}^4 (p_{iy} - m_y)(p_{ix} - m_x) & + \sum_{i=1}^4 (p_{iy} - m_y)(p_{iy} - m_y) & + \sum_{i=1}^4 (p_{iy} - m_y)(p_{iz} - m_z) \\ + \sum_{i=1}^4 (p_{iz} - m_z)(p_{ix} - m_x) & + \sum_{i=1}^4 (p_{iz} - m_z)(p_{iy} - m_y) & + \sum_{i=1}^4 (p_{iz} - m_z)(p_{iz} - m_z) +\end{array}\right] +\\ +V & = & +\frac{1}{4} +\left[\begin{array}{cccc} + a_x - m_x & b_x - m_x & c_x - m_x & d_x - m_x \\ + a_y - m_y & b_y - m_y & c_y - m_y & d_y - m_y \\ + a_z - m_z & b_z - m_z & c_z - m_z & d_z - m_z +\end{array}\right] +\left[\begin{array}{ccc} + a_x - m_x & a_y - m_y & a_z - m_z \\ + b_x - m_x & b_y - m_y & b_z - m_z \\ + c_x - m_x & c_y - m_y & c_z - m_z \\ + d_x - m_x & d_y - m_y & d_z - m_z +\end{array}\right] +\\ +V & = & +\frac{1}{4} +(P - M)^t (P - M) +\end{array} +$$ + +%################################################################# +\section{Splitting into groups} +When we study some mixed population, the total variance can be splitted in the +the variance between the groups and in the variance whithin each group. + +We define two groups in the population. Each group owns its moments of the +second order. + +%================================================================= +\subsection{Decomposing the count} + +$$ +\begin{array}{lcl} +n_t & = & n_1 + n_2 \\ + & = & \sum_{i=1}^{2} n_i +\end{array} +$$ + + +%================================================================= +\subsection{Decomposing the mean} + +$$ +\begin{array}{lcl} +\mbox{\boldmath$m_t$} & = & + \frac{1}{n_t}(n_1 \mbox{\boldmath$m_1$} + n_2 \mbox{\boldmath$m_2$}) \\ + & = & + \frac{1}{n_t}\sum_{i=1}^{2} n_i \mbox{\boldmath$m_i$} +\end{array} +$$ + + +%================================================================= +\subsection{Decomposing the variance} +When we study some mixed population, the total variance can be splitted in the +the variance between the groups and in the variance whithin each group. + +$$ +V_t = V_i + V_b +$$ + +$$ +\begin{array}{lcl} +V_i & = & \frac{1}{n_t}(n_1 V_1 + n_2 V_2) \\ + & = & \frac{1}{n_t}\sum_{i=1}^2 n_i V_i +\end{array} +$$ + +$$ +\begin{array}{lcl} +V_b & = & \frac{1}{n_t}(n_1 (\mbox{\boldmath$m_1$} - \mbox{\boldmath$m_t$})^2 + + n_2 (\mbox{\boldmath$m_2$} - \mbox{\boldmath$m_t$})^2 \\ + & = & \frac{1}{n_t} \sum_{i=1}^2 n_i + (\mbox{\boldmath$m_i$} - \mbox{\boldmath$m_t$})^2 +\end{array} +$$ + +%################################################################# +\section{Basis} + +%%================================================================= +\subsection{Determinant of a square matrix 3x3} + +$$ +\det{V} = +\left|\begin{array}{ccc} + v_{11} & v_{12} & v_{13} \\ + v_{21} & v_{22} & v_{23} \\ + v_{31} & v_{32} & v_{33} +\end{array}\right| += +v_{11}(v_{22}v_{33} - v_{32}v_{23}) +- v_{12}(v_{21}v_{33} - v_{31}v_{23}) ++ v_{13}(v_{21}v_{32} - v_{31}v_{22}) +$$ + +%%================================================================= +\subsection{Transpose} + +$ +V^t = +\left[\begin{array}{ccc} + v_{11} & v_{12} & v_{13} \\ + v_{21} & v_{22} & v_{23} \\ + v_{31} & v_{32} & v_{33} +\end{array}\right]^t += +\left[\begin{array}{ccc} + v_{11} & v_{21} & v_{31} \\ + v_{12} & v_{22} & v_{32} \\ + v_{13} & v_{23} & v_{33} +\end{array}\right] +$ + +%%================================================================= +\subsection{Inverse of a square matrix 3x3} + +\begin{tabular}{c c} +$ +minor(v_{11}) += +\left|\begin{array}{cc} + v_{22} & v_{23} \\ + v_{32} & v_{33} +\end{array}\right| += +v_{22}v_{33} - v_{32}v_{23} +$ +& +$ +minor(v_{21}) += +\left|\begin{array}{cc} + v_{12} & v_{13} \\ + v_{31} & v_{33} +\end{array}\right| += +v_{12}v_{33} - v_{31}v_{31} +$ +\\ +\\ +$ +minor(v_{31}) += +\left|\begin{array}{cc} + v_{12} & v_{13} \\ + v_{22} & v_{23} +\end{array}\right| += +v_{12}v_{23} - v_{22}v_{13} +$ +& +$ +minor(v_{12}) += +\left|\begin{array}{cc} + v_{21} & v_{23} \\ + v_{31} & v_{33} +\end{array}\right| += +v_{21}v_{33} - v_{31}v_{23} +$ +\\ +\\ +$ +minor(v_{22}) += +\left|\begin{array}{cc} + v_{11} & v_{13} \\ + v_{31} & v_{33} +\end{array}\right| += +v_{11}v_{33} - v_{31}v_{13} +$ +& +$ +minor(v_{32}) += +\left|\begin{array}{cc} + v_{11} & v_{13} \\ + v_{21} & v_{23} +\end{array}\right| += +v_{11}v_{23} - v_{21}v_{13} +$ +\\ +\\ +$ +minor(v_{13}) += +\left|\begin{array}{cc} + v_{21} & v_{22} \\ + v_{31} & v_{32} +\end{array}\right| += +v_{21}v_{32} - v_{31}v_{22} +$ +& +$ +minor(v_{23}) += +\left|\begin{array}{cc} + v_{11} & v_{12} \\ + v_{31} & v_{32} +\end{array}\right| += +v_{11}v_{32} - v_{31}v_{12} +$ +\\ +\\ +$ +minor(v_{33}) += +\left|\begin{array}{cc} + v_{11} & v_{12} \\ + v_{21} & v_{22} +\end{array}\right| += +v_{11}v_{22} - v_{21}v_{12} +$ +\end{tabular} + + +$$ +minor(V) = +minor( +\left[\begin{array}{ccc} + v_{11} & v_{12} & v_{13} \\ + v_{21} & v_{22} & v_{23} \\ + v_{31} & v_{32} & v_{33} +\end{array}\right]) += +\left[\begin{array}{ccc} + minor(v_{11}) & minor(v_{12}) & minor(v_{13}) \\ + minor(v_{21}) & minor(v_{22}) & minor(v_{23}) \\ + minor(v_{31}) & minor(v_{32}) & minor(v_{33}) +\end{array}\right] +$$ + +$$ +cofactor(V) += +\left[\begin{array}{ccc} + (+)minor(v_{11}) & (-)minor(v_{12}) & (+)minor(v_{13}) \\ + (-)minor(v_{21}) & (+)minor(v_{22}) & (-)minor(v_{23}) \\ + (+)minor(v_{31}) & (-)minor(v_{32}) & (+)minor(v_{33}) +\end{array}\right] +$$ + + +$$ +adj(V) = cofactor(V)^t += +\left[\begin{array}{ccc} + v_{22}v_{33} - v_{32}v_{23} & + v_{31}v_{13} - v_{12}v_{33} & + v_{12}v_{23} - v_{22}v_{13} \\ + v_{31}v_{23} - v_{21}v_{33} & + v_{11}v_{33} - v_{31}v_{13} & + v_{21}v_{13} - v_{11}v_{23} \\ + v_{21}v_{32} - v_{31}v_{22} & + v_{21}v_{13} - v_{11}v_{23} & + v_{11}v_{22} - v_{21}v_{12} +\end{array}\right] +$$ + +$$ +V^{-1} += +\frac{adj(V)}{\det V} +$$ + + +%%================================================================= +\subsection{Eigenvalues and eigenvectors} + +We assume that we work on variance/covariance matrix which is real and symetric. +In this case, all the three eigen values are real. + +$$ +V \bm{x} = \lambda \bm{x} +$$ + +$$ +\det V - \lambda I = 0 +$$ + +\begin{tabular}{lcl} +$\det V - \lambda I$ & +$=$ & +$\left|\begin{array}{ccc} + v_{11} - \lambda & v_{21} & v_{31} \\ + v_{12} & v_{22} - \lambda & v_{32} \\ + v_{13} & v_{23} & v_{33} - \lambda +\end{array}\right|$ +\\ +\\ +& +$=$ & +$(v_{11} - \lambda)((v_{22} - \lambda)(v_{33} - \lambda) - v_{32}v_{23})$ +\\ +& +& +$- v_{12}(v_{21}(v_{33} - \lambda) - v_{31}v_{23})$ +\\ +& +& +$- v_{13}(v_{21}v_{32} - v_{31}(v_{22} - \lambda))$ +\\ +\\ +& += & +$- \lambda^3 + (v_{11} + v_{22} + v_{33})\lambda^2$ +\\ +& +& +$ -(v_{12}v_{21} + v_{13}v_{31} + v_{23}v_{32} + - v_{11}v_{22} - v_{11}v_{33} - v_{22}v_{33})\lambda$ +\\ +& +& +$+ v_{11}(v_{22}v_{33} - v_{32}v_{23}) - +v_{12}(v_{21}v_{33} - v_{31}v_{23}) - +v_{13}(v_{21}v_{32} - v_{31}v_{22})$ +\\ +\\ +& $=$ & +$-\lambda^3 + tr(V) \lambda^2 + \frac{1}{2}[tr(A^2) - tr(A)^2]\lambda + \det A$ +\\ +\\ +\end{tabular} + + +\begin{tabular}{ccccccc} +\\ +$A$ +& $=$ & +$ +\left[\begin{array}{ccc} + a & d & e \\ + d & b & f \\ + e & f & c +\end{array}\right] +$ +\\ +\\ +& $=$ & +$P$ & $\times$ & $D$ & $\times$ & $P^t$ +\\ +\\ +& $=$ & +$ +\left[\begin{array}{ccc} + u_1 & v_1 & w_1 \\ + u_2 & v_2 & w_2 \\ + u_3 & v_3 & w_3 +\end{array}\right] +$ +& $\times$ & +$ +\left[\begin{array}{ccc} + \lambda_1 & 0 & 0 \\ + 0 & \lambda_2 & 0 \\ + 0 & 0 & \lambda_3 +\end{array}\right] +$ +& $\times$ & +$ +\left[\begin{array}{ccc} + u_1 & u_2 & u_3 \\ + v_1 & v_2 & v_3 \\ + w_1 & w_2 & w_3 +\end{array}\right] +$ +\\ +\\ +& $=$ & +$ +\lambda_1 +\left[\begin{array}{ccc} + u_1^2 & u_1u_2 & u_1u_3 \\ + u_2u_1 & u_2^2 & u_2u_3 \\ + u_3u_1 & u_3u_2 & u_3^2 \\ +\end{array}\right] +$ +& $+$ & +$ +\lambda_2 +\left[\begin{array}{ccc} + v_1^2 & v_1v_2 & v_1v_3 \\ + v_2v_1 & v_2^2 & v_2v_3 \\ + v_3v_1 & v_3v_2 & v_3^2 \\ +\end{array}\right] +$ +& $+$ & +$ +\lambda_3 +\left[\begin{array}{ccc} + w_1^2 & w_1w_2 & w_1w_3 \\ + w_2w_1 & w_2^2 & w_2w_3 \\ + w_3w_1 & w_3w_2 & w_3^2 \\ +\end{array}\right] +$ +\\ +\\ +& $=$ & +$\lambda_1\bm{uu^t}$ +& $+$ & +$\lambda_2\bm{vv^t}$ +& $+$ & +$\lambda_3\bm{ww^t}$ +\\ +\\ +$I$ +& $=$ & +$PP^t$ +& $=$ & +$P^tP$ +\\ +\\ +$0$ +& $=$ & +$A\bm{u}-\lambda_1\bm{u}$ +& $=$ & +$A\bm{v}-\lambda_2\bm{v}$ +& $=$ & +$A\bm{w}-\lambda_3\bm{w}$ +\\ +\\ +$|P|$ +& $=$ & +$ +\left|\begin{array}{ccc} + u_1 & u_2 & u_3 \\ + v_1 & v_2 & v_3 \\ + w_1 & w_2 & w_3 +\end{array}\right| +$ +& $=$ & +1 +\end{tabular} + + +%% sample + +\begin{tabular}{ccccccc} +\\ +$ +\left[\begin{array}{ccc} + \frac{3}{2} & \frac{1}{2} & 0 \\ +\\ + \frac{1}{2} & \frac{3}{2} & 0 \\ +\\ + 0 & 0 & 2 +\end{array}\right] +$ +& $=$ & +$ +\left[\begin{array}{ccc} + \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{1}} & -\frac{1}{\sqrt{2}} \\ +\\ + \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{2}} \\ +\\ + \frac{1}{\sqrt{3}} & -\frac{2}{\sqrt{6}} & 0 +\end{array}\right] +$ +& $\times$ & +$ +\left[\begin{array}{ccc} + 2 & 0 & 0 \\ +\\ + 0 & 2 & 0 \\ +\\ + 0 & 0 & 1 +\end{array}\right] +$ +& $\times$ & +$ +\left[\begin{array}{ccc} + \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ +\\ + \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & -\frac{2}{\sqrt{6}} \\ +\\ + -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 +\end{array}\right] +$ +\\ +\\ +& $=$ & +$ +2 +\left[\begin{array}{ccc} + \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ +\\ + \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ +\\ + \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ +\end{array}\right] +$ +& $+$ & +$ +2 +\left[\begin{array}{ccc} + \frac{1}{6} & \frac{1}{6} & -\frac{2}{6} \\ +\\ + \frac{1}{6} & \frac{1}{6} & -\frac{2}{6} \\ +\\ + -\frac{2}{6} & -\frac{2}{6} & \frac{4}{3} \\ +\end{array}\right] +$ +& $+$ & +$ +1 +\left[\begin{array}{ccc} + \frac{1}{2} & -\frac{1}{2} & 0 \\ +\\ + -\frac{1}{2} & \frac{1}{2} & 0 \\ +\\ + 0 & 0 & 0 \\ +\end{array}\right] +$ +\\ +\\ +$ +\left[\begin{array}{ccc} + \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ +\\ + \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ +\\ + \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ +\end{array}\right] +$ +& $=$ & +$ +\left[\begin{array}{c} + \frac{1}{\sqrt{3}} \\ + \frac{1}{\sqrt{3}} \\ + \frac{1}{\sqrt{3}} \\ +\end{array}\right] +$ +& $\times$ & +$ +\left[\begin{array}{ccc} + \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ +\end{array}\right] +$ +\\ +\\ +$ +\left[\begin{array}{ccc} + \frac{1}{6} & \frac{1}{6} & -\frac{2}{6} \\ +\\ + \frac{1}{6} & \frac{1}{6} & -\frac{2}{6} \\ +\\ + -\frac{2}{6} & -\frac{2}{6} & \frac{4}{3} \\ +\end{array}\right] +$ +& $=$ & +$ +\left[\begin{array}{c} + \frac{1}{\sqrt{6}} \\ + \frac{1}{\sqrt{6}} \\ + -\frac{2}{\sqrt{6}} \\ +\end{array}\right] +$ +& $\times$ & +$ +\left[\begin{array}{ccc} + \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & -\frac{2}{\sqrt{6}} \\ +\end{array}\right] +$ +\\ +\\ +$ +\left[\begin{array}{ccc} + \frac{1}{2} & -\frac{1}{2} & 0 \\ +\\ + -\frac{1}{2} & \frac{1}{2} & 0 \\ +\\ + 0 & 0 & 0 \\ +\end{array}\right] +$ +& $=$ & +$ +\left[\begin{array}{c} + -\frac{1}{\sqrt{2}} \\ + \frac{1}{\sqrt{2}} \\ + 0 \\ +\end{array}\right] +$ +& $\times$ & +$ +\left[\begin{array}{ccc} + -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ +\end{array}\right] +$ +\end{tabular} + +%%================================================================= +\subsection{MCO for eigenvalues} + +In 3d, it's difficult to extract the cubic roots from the characteristic +polynomia. The difficulties disappear when we find one of the three roots. +A planar regression allows us to reach the equation of the plane. From the +equation, we can determine its normal vector $\bm{w}$. It satisfies the +following equation $A\bm{w} = \lambda_3\bm{w}$. Thus we know $\lambda_3$. +By the way, as far as $trace(A) = \lambda_1 + \lambda_2 + \lambda_3$ and +$det(A) = \lambda_1 \lambda_2 \lambda_3$, then we can access to the value of +the two others eigenvalues (just solve $\lambda^2 - (trace(A) - \lambda_3) +\lambda + \frac{det(A)}{\lambda_3} = 0$). The knowledge of $\lambda_1$ and +$\lambda_2$ let us find $\bm{u}$ and $\bm{v}$ by the equations $A\bm{u} = +\lambda_1\bm{u}$ and $A\bm{v} = \lambda_2\bm{v}$. + +Let's center the points by susbtracting their center of mass. Now, we have +three equivalent ways to estimate the coefficients of the plane: +\begin{itemize} + \item if not $c = 0$, then $\frac{a}{c}x + \frac{b}{c}y + z = 0$, + \item if not $b = 0$, then $\frac{a}{b}x + y + \frac{c}{b}z = 0$, + \item if not $a = 0$, then $x + \frac{b}{a}y + \frac{c}{a}z = 0$. +\end{itemize} +As we cannot decide which way is the best, may be we have to test the three +ones. + +Let's choose the linear model of the major inertia plane +($ax + by + cz + d = 0$). With not $c = 0$. + +\begin{tabular}{lcl} +$\bm{y}$ & $=$ & +$\left[\begin{array}{c} + z_1 \\ + \vdots \\ + z_r \\ +\end{array}\right]$ +\\ +\\ +$\bm{\theta}$ & $=$ & +$\left[\begin{array}{c} + a \\ + b \\ + d \\ +\end{array}\right]$ +\\ +\\ +$J$ & $=$ & +$\left[\begin{array}{ccc} + x_1 & y_1 & 1 \\ + \vdots & \vdots & \vdots\\ + x_r & y_r & 1\\ +\end{array}\right]$ +\\ +\\ +$W$ & $=$ & +$\left[\begin{array}{ccc} + 1 & \ldots & 0 \\ + \vdots & \ddots & \vdots\\ + 0 & \ldots & 1\\ +\end{array}\right]$ +\\ +\\ +$\chi(\bm{\theta})^2$ & $=$ & +$(J\theta - \bm{y})^t W (J\theta - \bm{y})$ +\\ +\\ +$grad \chi(\bm{\theta})^2$ & $=$ & +$2J^tWJ\bm{\theta} - 2J^tW\bm{y}$ +\\ +\\ +$\bm{\theta_{min}}$ & $=$ & +$(J^tWJ)^{-1} J^tWy$ +\end{tabular} + +\end{document} -- 1.5.6.5