feat: Add some stuff on generalized linear models

content/chapters/include.tex

@@ -11,7 +11,7 @@
 		}
 }
 
-\includechapters{part1}{2}
+\includechapters{part1}{1}
 
 \includechapters{part2}{2}
 

content/chapters/part1/1.tex

@@ -60,12 +60,6 @@ In order to estimate the parameters, we can use penalties (additional terms).
 
 Lasso regression, Elastic Net, etc.
 
-\subsection{Statistical Analysis Workflow}
-
-\begin{enumerate}[label={\bfseries\color{primary}Step \arabic*.}]
-    \item Graphical representation;
-    \item ...
-\end{enumerate}
 \[
     Y = X \beta + \varepsilon,
 \]
@@ -145,21 +139,18 @@ $\Y - \X \hat{\beta} \perp \X \hat{\beta} - \Y \One$ if $\One \in V$, so
 
 \begin{figure}
     \centering
-    \includestandalone{figures/schemes/orthogonal_projection}
+    \includegraphics{figures/schemes/orthogonal_projection.pdf}
     \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}
 
 \begin{figure}
     \centering
-    \includestandalone{figures/schemes/ordinary_least_squares}
+    \includegraphics{figures/schemes/ordinary_least_squares.pdf}
     \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.
@@ -169,22 +160,21 @@ $\Y - \X \hat{\beta} \perp \X \hat{\beta} - \Y \One$ if $\One \in V$, so
 
 \section{Gaussian vectors}
 
-
 \begin{definition}[Normal distribution]
-
+  $X \sim \Norm(\mu, \sigma^{2})$, with density function $f$
+  \[
+    f(x) = \frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^{2}}
+  \]
 \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 ...
+    A random vector $\Y \in \RR[n]$ is a gaussian vector if every linear combination of its component is a gaussian random variable.
 \end{definition}
 
 \begin{property}
     $m = \EE(Y) = (m_1, \ldots, m_n)^T$, where $m_i = \EE(Y_i)$
 
-
-    ...
-
     \[
         \Y \sim \Norm_n(m, \Sigma)
     \]
@@ -193,8 +183,6 @@ $\Y - \X \hat{\beta} \perp \X \hat{\beta} - \Y \One$ if $\One \in V$, so
         \Sigma = \E\left[(\Y -m)(\Y - m)^T\right].
     \]
 
-
-
 \end{property}
 
 \begin{remark}
@@ -261,24 +249,25 @@ Covariance is really sensitive to scale of variables. For instance, if we measur
         \[
             \RR[n] = V_1 \overset{\perp}{\oplus} V_2.
         \]
-        \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})
+      \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 $Z_2 = \Pi_{V_2} (\mathbf{Z}) = \Pi_2 \Y$ ($\Pi_{1}$ and $\Pi_{2}$ being projection matrices)
+            then:
+      \item $z_{1}$, $Z_{2}$ are independent gaussian vectors, $Z_{1} \sim \Norm_{n_{1}} (0_{n}, \Pi_{1})$ and $Z_{2} \sim \Norm(0_{n_{2}}, \Pi_{2})$.
+
+            In particular $\norm{Z_{1}} \sim \chi^{2}(n_{1})$ and $\norm{Z_{2}} \sim \chi^{2}(n_{2})$.
     \end{itemize}
 
     $Z_2 = \Pi_{V_1}(\Z)$ is the projection of $\Z$ on subspace $V_1$.
 
     \dots
-
-
 \end{theorem}
 
 \begin{property}[Estimators properties in the linear model]
     According to \autoref{thm:cochran}, 
     \[
         \hat{m} \text{ is independent from $\hat{\sigma}^2$}
-    \]\dots
+    \]
     \[
-        \frac{\norm{\Y - \Pi_V(\Y)}^2}{...} \sim 
+        \norm{\Y - \Pi_V(\Y)}^2 = \norm{\varepsilon - \Pi_{V}(\varepsilon)}^{2} = \norm{\Pi_{V}^{\perp} (\varepsilon)}^{2}
     \]
 
     $\hat{m} = \X \hat{\beta}$
@@ -303,40 +292,34 @@ Covariance is really sensitive to scale of variables. For instance, if we measur
 
 \begin{align*}
     \hat{m} &= \X \hat{\beta} = \X(\X^T\X)^{-1} \X^T \Y \\
-    \text{so} \\
+    \intertext{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:
-\[
-   ...
-\]
+All the sum of squares follows a $\chi^2$ distribution.
 
-\begin{property}
 
-\end{property}
+\subsection{Estimators properties}
 
-\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 ...
+  \item $\hat{m}$ is unbiased and estimator of $m$;
+  \item $\EE(\hat{\sigma}^{2}) = \sigma^{2}(n-q)/n$ $\hat{\sigma}^{2}$ is a biased estimator of $\sigma^{2}$.
+        \[
+        S^{2} = \frac{1}{n-q} \norm{\Y - \Pi_{V}}^{2}
+        \]
+        is an unbiased estimator of $\sigma²$.
 \end{itemize}
 
 We can derive statistical test from these properties.
 
-
 \section{Statistical tests}
 
 \subsection{Student $t$-test}
 
-
 \[
-    \frac{\hat{\theta}-\theta}{\sqrt{\frac{\widehat{\VVar}(\hat{\theta})}{n}}} \underset{H_0}{\sim} t
+    \frac{\hat{\theta}-\theta}{\sqrt{\frac{\widehat{\VVar}(\hat{\theta})}{n}}} \underset{H_0}{\sim} t_{n-q}
 \]
 
 where 
@@ -627,3 +610,44 @@ Different methods:
 
 Usually $AIC$ have smaller penalty than $BIC$, thus $AIC$ criterion tends to select models with more variables than $BIC$ criterion.
 
+\subsection{Stepwise}
+
+\begin{description}
+    \item[forward] Add new predictor iteratively, beginning with the most contributing predictors. 
+    \item[backward] Remove predictors iteratively.
+    \item[stepwise] Combination of forward and backward selection. We start by no predictors. We add predictor. Before adding the predictor, we check whether all previously predictors remain meaningful. 
+\end{description}
+
+The problem with this iterative regression, is that at each step we make a test. We have to reduce the confidence level for multiple test.
+
+In practice, the multiple testing problem is not taken into account in these approaches.
+
+We can use information criteria or model comparison in these methods.
+
+\section{Predictions}
+
+Let $X_i$ the $i$-th row of the matrix $\X$. The observed value $Y_i$ can be estimated by:
+\[
+    \hat{Y}_i = (\X \hat{\beta})_i = X_i \hat{\beta}
+\]
+
+\begin{align*}
+    \EE (\hat{Y}_i) &= (\X \beta)_i = X_i \beta \\
+    \sigma^{-1} (\X \hat{\beta} - \X \beta) \sim \Norm (0_{p+1}, (\X^T \X)^{-1}), \qquad \text{and} \\
+    \Var(\hat{Y}_i) = ... \\
+    S^2 = \norm{...}
+\end{align*}
+
+
+\paragraph{Prediction Confidence Interval}
+We can build confidence interval for predicted values $(\X \hat{\beta})_i$
+
+\dots
+
+\paragraph{Prediction error of $Y$}
+
+
+\paragraph{Prediction interval for a new observation $Y_{n+1}$}
+
+
+

content/chapters/part1/2.tex

@@ -1,4 +1,186 @@
 \chapter{Generalized Linear Model}
 
+\begin{example}
+
+  \begin{description}
+    \item[Ex. 1 - Credit Carb Default]
+      Let $Y_i$ be a boolean random variable following a Bernoulli distribution.
+    \item[Ex. 2 - Horseshoe Crabs]
+      Let $Y_i$, be the number of satellites males.
+
+      $Y_i$ can be described as following a Poisson distribution.
+  \end{description}
+\end{example}
+
+\begin{remark}
+  A Poisson distribution can be viewed as an approximation of binomial distribution when $n$ is high and $p$ low.
+\end{remark}
+
+
+We will consider the following relation:
+\[
+  \EE(Y_i) = g^{-1} X_i \beta,
+\]
+equivalently:
+\[
+  g(\EE(Y_i)) = X_i \beta.
+\]
+
+\begin{itemize}
+  \item $\beta$ is estimated by the maximum likelihood;
+  \item $g$ is called the link function.
+\end{itemize}
+
+\begin{remark}
+  In standard linear model, the OLS estimator is the estimator of maximum of likelihood.
+\end{remark}
+
 \section{Logistic Regression}
 
+\begin{align*}
+                  & \log(\frac{\Pi}{1 - \Pi})                   & = \X \beta \\
+  \Leftrightarrow & e^{\ln \frac{\Pi}{1 - \Pi}} =  e^{\X \beta}              \\
+  \Leftrightarrow & \frac{\Pi}{1 - \Pi} = e^{\X \beta}                       \\
+  \Leftrightarrow & \Pi = (1 - \Pi) e^{\X\beta}                              \\
+  \Leftrightarrow & \Pi = e^{\X \beta} - \Pi e^{\X\beta}                     \\
+  \Leftrightarrow & \Pi + \Pi e^{\X\beta} = e^{\X \beta}                     \\
+  \Leftrightarrow & \Pi (1 - e^{\X\beta}) = e^{\X \beta}                     \\
+  \Leftrightarrow & \Pi = \frac{e^{\X\beta}}{1 + e^{\X \beta}}
+\end{align*}
+
+
+\section{Maximum Likelihood estimator}
+
+log-likelihood: the probability to observe what we observe.
+
+Estimate $\beta$ by $\hat{\beta}$ such that $\forall \beta \in \RR[p+1]$:
+\[
+  L_n (\hat{\beta}) \geq L_n (\beta)
+\]
+
+These estimators are consistent, but not necessarily unbiased.
+
+
+\section{Test for each single coordinate}
+
+
+
+\begin{example}[Payment Default]
+  Let $Y_i$ be the default value for individual $i$.
+
+  \[
+    \log (\frac{\Pi (X)}{1 - \Pi (X)}) = \beta_0 + \beta_1 \text{student} + \beta_2 \text{balance} + \beta_3 \text{income}
+  \]
+
+  In this example, only $\beta_0$ and $\beta_2$ are significantly different from 0.
+\end{example}
+
+\begin{remark}
+  We do not add $\varepsilon_i$, because $\log(\frac{\Pi (X)}{1 - \Pi (X)})$ corresponds to the expectation.
+\end{remark}
+
+\subsection{Comparison of nested models}
+
+To test $H_0:\: \beta_0 = \ldots = \beta_p = 0$, we use the likelihood ratio test:
+\[
+  T_n = -2 \log (\mathcal{L}^{\texttt{null}}) + 2 \log (\mathcal{L}(\hat{\beta})) \underset{H_0}{\overunderset{\mathcal{L}}{n \to \infty}{\longrightarrow}} \chi^2(p).
+\]
+
+\begin{remark}[Family of Tests]
+  \begin{itemize}
+    \item Comparison of estimated values and values under the null hypothesis;
+    \item Likelihood ratio test;
+    \item Based on the slope on the derivative.
+  \end{itemize}
+\end{remark}
+
+\section{Relative risk}
+
+$RR_i$ is the probably to have the disease, conditional to the predictor $X_{i1}$ over the probability of having the disease, conditional to the predictor $X_{i2}$.
+
+\[
+  RR(j) = \frac{\Prob(Y_{i_1} = 1 \: | \: X_{i_1})}{\Prob(Y_{i_2} = 1) \: | \: X_{i_2}} = \frac{\EE(Y_{i_1})}{\EE(Y_{i_2})}.
+\]
+
+$\pi(X_i)$ is the probability of having the disease, according to $X_i$.
+
+The relative risk can be written as\dots
+
+\section{Odds}
+
+Quantity providing a measure of the likelihood of a particular outcome:
+\[
+  odd = \frac{\pi(X_i)}{1 - \pi(X_i)}
+\]
+
+\[
+  odds = \exp(X_i \beta)
+\]
+odds is the ratio of people having the disease, if Y represent the disease, over the people not having the disease.
+
+\section{Odds Ratio}
+
+\begin{align*}
+  OR(j) =\frac{odds(X_{i_1})}{odds(X_{i_2})} & =  \frac{\frac{\pi{X_{i_1}}}{1 - \pi(X_{i_1})}}{\frac{\pi{X_{i_2}}}{1 - \pi(X_{i_2})}}
+\end{align*}
+
+The OR can be written as:
+\[
+  OR(j) = \exp(\beta_j)
+\]
+
+\begin{exercise}
+  Show that $OR(j) = \exp(\beta_j)$.
+\end{exercise}
+
+\begin{align*}
+  OR(j) & = \frac{odds(X_{i_1})}{odds(X_{i_2})}             \\
+        & = \frac{\exp(X_{i_1} \beta)}{\exp(X_{i_2} \beta)} \\
+\end{align*}
+
+  \[
+    \log \left(
+    \frac{\Prob(Y=1 \: |\: X_{i_1})}{1 - \Prob(Y=1 \: |\: X_{i_1})}\right)
+    = \beta_0 + \beta_1 X_1^{(1)} + \beta_2 X_2^{(1)} + \ldots + \beta_p X_p^{(1)}
+\]
+    Similarly
+\[
+    \log \left(
+      \frac{\Prob(Y=1 \: |\: X_{i_2})}{1 - \Prob(Y=1 \: |\: X_{i_2})}\right)
+    = \beta_0 + \beta_1 X_1^{(2)} + \beta_2 X_2^{(2)} + \ldots + \beta_p X_p^{(2)}
+\]
+  We substract both equations:
+
+    \begin{align*}
+      &\log \left(
+        \frac{\Prob(Y=1 \: |\: X_{i_1})}{1 - \Prob(Y=1 \: |\: X_{i_1})} \right) - \log \left(\frac{\Prob(Y=1 \: |\: X_{i_2})}{1 - \Prob(Y=1 \: |\: X_{i_2})}\right) \\
+      & = \beta_0 + \beta_1 X_1^{(1)} + \beta_2 X_2^{(1)} + \ldots + \beta_p X_p^{(1)} - \beta_0 + \beta_1 X_1^{(2)} + \beta_2 X_2^{(2)} + \ldots + \beta_p X_p^{(2)} \\
+      & = \log OR(j)  \\
+      & = \cancel{(\beta_0 - \beta_0)} + \beta_1 \cancel{(X_1^{(1)} - X_1^{(2)})} + \beta_2 \cancel{(X_2^{(1)} - X_2^{(2)})} + \ldots + \beta_j \cancelto{1}{(X_j^{(1)} - X_j^{(2)})} + \ldots + \beta_p \cancel{(X_p^{(1)} - X_p^{(2)})} \\
+      &\Leftrightarrow \log (OR_j) = \beta_j \\
+      &\Leftrightarrow OR(j) = \exp(\beta_j)
+    \end{align*}
+
+OR is not equal to RR, except in the particular case of probability (?)
+
+If OR is significantly different from 1, the $\exp(\beta_j)$ is significantly different from 1, thus $\beta_j$ is significantly different from 0.
+
+If we have more than two classes, we do not know what means $X_{i_1} - X_{i_2} = 0$. We will have to take a reference class, and compare successively each class with the reference class.
+
+$\hat{\pi}(X_{+}) = \hat{\Prob(X=1 \: | X_{i1})}$ for a new individual.
+
+
+\section{Poisson model}
+
+Let $Y_{i} \sim \mathcal{P}(\lambda_{i})$, corresponding to a counting.
+
+\begin{align*}
+	\EE(Y_{i}) & = g^{-1}(X_{i} \beta) \\
+	\Leftrightarrow g(\EE(Y_{i})) = X_{i} \beta
+\end{align*}
+
+where $g(x) = \ln(x)$, and $g^{-1}(x) = e^{x}$.
+
+\[
+	\lambda_{i} = \EE(Y_{i}) = \Var(Y_{i})
+\]

content/chapters/part1/3.tex

@@ -0,0 +1,25 @@
+\chapter{Tests Reminders}
+
+\section{$\chi^2$ test of independence}
+
+
+\section{$\chi^2$ test of goodness of fit}
+
+Check if the observations is in adequation with a particular distribution.
+
+\begin{example}[Mendel experiments]
+	Let $AB$, $Ab$, $aB$, $ab$ be the four possible genotypes of peas: colors and grain shape.
+	\begin{tabular}
+		\toprule
+		AB  & Ab  & aB  & ab \\
+		\midrule
+		315 & 108 & 101 & 32 \\
+		\bottomrule
+	\end{tabular}
+\end{example}
+
+The test statistics is:
+\[
+	D_{k,n} = \sum_{i=1}^{k} \frac{(N_i - np_i)^2}{np_i} \underoverset{H_0}{\mathcal{L}} \chi^2_{(n-1)(q-1)??}
+\]
+

content/chapters/part2/1.tex

@@ -68,7 +68,7 @@ Let $u = \begin{pmatrix}
 
 \begin{figure}
     \centering
-    \includestandalone{figures/schemes/vector_orthogonality}
+    \includegraphics{figures/schemes/vector_orthogonality.pdf}
     \caption{Scalar product of two orthogonal vectors.}
     \label{fig:scheme-orthogonal-scalar-product}
 \end{figure}
@@ -215,6 +215,6 @@ The number of columns has to be the same as the dimension of the vector to which
 
 \begin{figure}
     \centering
-    \includestandalone{figures/schemes/coordinates_systems}
+    \includegraphics{figures/schemes/coordinates_systems.pdf}
     \caption{Coordinate systems}
 \end{figure}

figures/plots/logistic_curve.tex

@@ -0,0 +1,23 @@
+\documentclass[margin=0.5cm]{standalone}
+\usepackage{tikz}
+\usepackage{pgfplots}
+\pgfplotsset{compat=1.18} 
+
+\begin{document}
+\begin{tikzpicture}
+  \begin{axis}[
+    title={Logit function},
+    xlabel={$x$},
+    ylabel={$y$},
+    domain=-5:5,
+    samples=200,
+    legend style={at={(0.95,0.05)},anchor=south east}
+]
+\newcommand{\Lvar}{1}
+\newcommand{\kvar}{1}
+\newcommand{\xvar}{0}
+\addplot [blue] {\Lvar / (1 + exp(-\kvar*(x-\xvar)))};
+\addlegendentry{$L = \Lvar, k=\kvar, x_0=\xvar$};
+\end{axis}
+\end{tikzpicture}
+\end{document}

preamble.tex

@@ -4,3 +4,4 @@
 \usepackage{tikz-3dplot}
 \usepackage{tkz-euclide}
 \usepackage{nicematrix}
+\usepackage{luacode}