fix: Amend the orthogonal projection scheme

This commit is contained in:
Samuel Ortion 2023-09-30 07:03:23 +02:00
parent b7f323419d
commit 43acae64f3
12 changed files with 205 additions and 16 deletions

6
.gitattributes vendored
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@ -1 +1,7 @@
main.pdf filter=lfs diff=lfs merge=lfs -text
figures/schemes/regression_plan_3D.pdf filter=lfs diff=lfs merge=lfs -text
figures/schemes/vector_orthogonality.pdf filter=lfs diff=lfs merge=lfs -text
figures/schemes/base_plan.pdf filter=lfs diff=lfs merge=lfs -text
figures/schemes/coordinates_systems.pdf filter=lfs diff=lfs merge=lfs -text
figures/schemes/ordinary_least_squares.pdf filter=lfs diff=lfs merge=lfs -text
figures/schemes/orthogonal_projection.pdf filter=lfs diff=lfs merge=lfs -text

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@ -129,6 +129,13 @@ $\X \hat{\beta}$ is the closest point to $\Y$ in the subspace generated by $\X$.
If $H$ is the projection matrix of the subspace generated by $\X$, $X\Y$ is the projection on $\Y$ on this subspace, that corresponds to $\X\hat{\beta}$.
\section{Sum of squares}
$\Y - \X \hat{\beta} \perp \X \hat{\beta} - \Y \One$ if $\One \in V$, so
\[
\underbrace{\norm{\Y - \bar{\Y}\One}}_{\text{Total SS}} = \underbrace{\norm{\Y - \X \hat{\beta}}^2}_{\text{Residual SS}} + \underbrace{\norm{\X \hat{\beta} - \bar{\Y} \One}^2}_{\text{Explicated SS}}
\]
\section{Coefficient of Determination: \texorpdfstring{$R^2$}{R\textsuperscript{2}}}
\begin{definition}[$R^2$]
\[
@ -139,7 +146,7 @@ If $H$ is the projection matrix of the subspace generated by $\X$, $X\Y$ is the
\begin{figure}
\centering
\includestandalone{figures/schemes/orthogonal_projection}
\caption{Orthogonal projection of $\Y$ on plan generated by the base described by $\X$. $\color{blue}a$ corresponds to $\norm{\X\hat{\beta} - \bar{\Y}}^2$ and $\color{blue}b$ corresponds to $\norm{\Y - \hat{\beta}\X}^2$}
\caption{Orthogonal projection of $\Y$ on plan generated by the base described by $\X$. $\color{blue}a$ corresponds to $\norm{\X\hat{\beta} - \bar{\Y}}^2$ and $\color{blue}b$ corresponds to $\hat{\varepsilon} = \norm{\Y - \hat{\beta}\X}^2$} and $\color{blue}c$ corresponds to $\norm{Y - \bar{Y}}^2$.
\label{fig:scheme-orthogonal-projection}
\end{figure}
@ -149,3 +156,165 @@ If $H$ is the projection matrix of the subspace generated by $\X$, $X\Y$ is the
\caption{Ordinary least squares and regression line with simulated data.}
\label{fig:ordinary-least-squares}
\end{figure}
\begin{definition}[Model dimension]
Let $\M$ be a model.
The dimension of $\M$ is the dimension of the subspace generated by $\X$, that is the number of parameters in the $\beta$ vector.
\textit{Nb.} The dimension of the model is not the number of parameter, as $\sigma^2$ is one of the model parameters.
\end{definition}
\section{Gaussian vectors}
\begin{definition}[Normal distribution]
\end{definition}
\begin{definition}[Gaussian vector]
A random vector $\Y \in \RR[n]$ is a gaussian vector if every linear combination of its component is ...
\end{definition}
\begin{property}
$m = \EE(Y) = (m_1, \ldots, m_n)^T$, where $m_i = \EE(Y_i)$
...
\[
\Y \sim \Norm_n(m, \Sigma)
\]
where $\Sigma$ is the variance-covariance matrix!
\[
\Sigma = \E\left[(\Y -m)(\Y - m)^T\right].
\]
\end{property}
\begin{remark}
\[
\Cov(Y_i, Y_i) = \Var(Y_i)
\]
\end{remark}
\begin{definition}[Covariance]
\[
\Cov(Y_i, Y_j) = \EE\left((Y_i-\EE(Y_j))(Y_j-\EE(Y_j))\right)
\]
\end{definition}
When two variable are linked, the covariance is large.
If two variables $X, Y$ are independent, $\Cov(X, Y) = 0$.
\begin{definition}[Correlation coefficient]
\[
\Cor(Y_i, Y_j) = \frac{\EE\left((Y_i-\EE(Y_j))(Y_j-\EE(Y_j))\right)}{\sqrt{\EE(Y_i - \EE(Y_i)) \cdot \EE(Y_j - \EE(Y_j))}}
\]
\end{definition}
Covariance is really sensitive to scale of variables. For instance, if we measure distance in millimeters, the covariance would be larger than in the case of a measure expressed in metters. Thus the correlation coefficient, which is a sort of normalized covariance is useful, to be able to compare the values.
\begin{remark}
\begin{align*}
\Cov(Y_i, Y_i) &= \EE((Y_i - \EE(Y_i)) (Y_i - \EE(Y_i))) \\
&= \EE((Y_i - \EE(Y_i))^2) \\
&= \Var(Y_i)
\end{align*}
\end{remark}
\begin{equation}
\Sigma = \begin{pNiceMatrix}
\VVar(Y_1) & & & &\\
& \Ddots & & & \\
& \Cov(Y_i, Y_j) & \VVar(Y_i) & & \\
& & & \Ddots & \\
& & & & \VVar(Y_n)
\end{pNiceMatrix}
\end{equation}
\begin{definition}[Identity matrix]
\[
\mathcal{I}_n = \begin{pNiceMatrix}
1 & 0 & 0 \\
0 & \Ddots & 0\\
0 & 0 & 1
\end{pNiceMatrix}
\]
\end{definition}
\begin{theorem}[Cochran Theorem (Consequence)]
Let $\mathbf{Z}$ be a gaussian vector: $\mathbf{Z} \sim \Norm_n(0_n, I_n)$.
\begin{itemize}
\item If $V_1, V_n$ are orthogonal subspaces of $\RR[n]$ with dimensions $n_1, n_2$ such that
\[
\RR[n] = V_1 \overset{\perp}{\oplus} V_2.
\]
\item If $Z_1, Z_2$ are orthogonal of $\mathbf{Z}$ on $V_1$ and $V_2$ i.e. $Z_1 = \Pi_{V_1}(\mathbf{Z}) = \Pi_1 \Y$ and $Z_2 = \Pi_{V_2} (\mathbf{Z}) = \Pi_2 \Y$...
(\textcolor{red}{look to the slides})
\end{itemize}
\end{theorem}
\begin{definition}[Chi 2 distribution]
If $X_1, \ldots, X_n$ i.i.d. $\sim \Norm(0, 1)$, then;,
\[
X_1^2 + \ldots X_n^2 \sim \chi_n^2
\]
\end{definition}
\subsection{Estimator's properties}
\[
\Pi_V = \X(\X^T\X)^{-1} \X^T
\]
\begin{align*}
\hat{m} &= \X \hat{\beta} = \X(\X^T\X)^{-1} \X^T \Y \\
\text{so} \\
&= \Pi_V \Y
\end{align*}
According to Cochran theorem, we can deduce that the estimator of the predicted value $\hat{m}$ is independent $\hat{\sigma}^2$
All the sum of squares follows a $\chi^2$ distribution:
\[
...
\]
\begin{property}
\end{property}
\subsection{Estimators consistency}
If $q < n$,
\begin{itemize}
\item $\hat{\sigma}^2 \overunderset{\PP}{n\to\infty} \sigma^{*2}$.
\item If $(\X^T\X)^{-1}$...
\item ...
\end{itemize}
We can derive statistical test from these properties.
\section{Statistical tests}
\subsection{Student $t$-test}
\[
\frac{\hat{\theta}-\theta}{\sqrt{\frac{\widehat{\VVar}(\hat{\theta})}{n}}} \underset{H_0}{\sim} t
\]
where

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@ -0,0 +1,6 @@
\DeclareMathOperator{\VVar}{\mathbb{V}} % variance
\DeclareMathOperator{\One}{\mathbf{1}}
\DeclareMathOperator{\Cor}{\mathrm{Cor}}
\newcommand{\M}[1][]{\ensuremath{\ifstrempty{#1}{\mathcal{M}}{\mathbb{M}_{#1}}}}
\newcommand{\X}{\ensuremath{\mathbf{X}}}
\newcommand{\Y}{\ensuremath{\mathbf{Y}}}

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@ -12,21 +12,31 @@
\tkzDefParallelogram(A,B,C)
\tkzGetPoint{D}
\tkzDrawPolygon[fill=gray!25!white](A,B,C,D)
\draw[decorate,decoration={brace,
amplitude=8pt},xshift=0pt,very thin,gray] (2,0) -- ++(-1,-0.5) node [black,midway,xshift=0.5em,yshift=-1em] {\color{blue}$a$};
\end{scope}
\begin{scope}[canvas is xz plane at y=0]
% Draw the rectangle triangle scheme
\begin{scope}[canvas is xz plane at y=1]
\draw[thick,fill=white,fill opacity=0.7,nodes={opacity=1}]
(2,0) node[bullet,label=below right:{$\mathbf{X}$}] {}
-- (0,0) node[bullet] {}
-- (0,3) node[bullet,label=above:{$\mathbf{Y}$}] {} -- cycle;
\draw (0.25,0) -- (0.25,0.25) -- (0,0.25);
(2,0) node[bullet,label=right:{$\bar{\mathbf{Y}}$}] (Y_bar) {}
-- (0,-0.5) node (B) {}
-- (0,3) node[label=above:{$\mathbf{Y}$}] (Y) {} -- cycle;
% Right angle annotation
\tkzPicRightAngle[draw,
angle eccentricity=.5,angle radius=2mm](Y,B,Y_bar)
% epsilon: Y - X \hat{\beta} curly brackets annotations
\draw[decorate,decoration={brace,
amplitude=8pt},xshift=0pt,very thin,gray] (0,0) -- (0,3) node [black,midway,xshift=-1.25em,yshift=0em] {\color{blue}$b$};
amplitude=8pt},xshift=0pt,very thin,gray] (B) -- (Y) node [black,midway,xshift=-1.25em,yshift=0em] {\color{blue}$b$};
% X\hat{\beta} - \hat{Y}
\draw[decorate,decoration={brace,
amplitude=8pt},xshift=0pt,very thin,gray] (Y_bar) -- (B) node [black,midway,xshift=0.5em,yshift=-1em] {\color{blue}$a$};
%
\draw[decorate,decoration={brace,
amplitude=8pt},xshift=0pt,very thin,gray] (Y) -- (Y_bar) node [black,midway,xshift=1em,yshift=1em] {\color{blue}$c$};
\end{scope}
% Coordinate system
\begin{scope}[canvas is xy plane at z=0]
\draw[->] (2,0) -- ++(-0.75,0.75) node [left] {$\mathbf{1}$};
\draw[->] (2,0) -- ++(-1,-0.5);
\draw[->] (2,1) -- node [above] {$\mathbf{1}$} ++(-1,0) ;
\draw[->] (2,1) -- ++(-0.45,-1) node [right] {$X_1$};
\end{scope}
\end{tikzpicture}
\end{document}

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main.pdf (Stored with Git LFS)

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@ -28,9 +28,6 @@
\definecolor{myblue}{HTML}{5654fa}
\colorlet{primary}{myblue}
\input{definitions}
\input{preamble}
\hypersetup{
pdftitle={Course - Multivariate Statistics},
pdfauthor={Samuel Ortion},
@ -51,6 +48,7 @@
\input{glossary}
\input{definitions}
\makeindex%
\makeglossary%
\begin{document}