feat: Add some stuff on generalized linear models
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\includechapters{part1}{2}
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\includechapters{part1}{1}
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\includechapters{part2}{2}
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\includechapters{part2}{2}
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@ -60,12 +60,6 @@ In order to estimate the parameters, we can use penalties (additional terms).
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Lasso regression, Elastic Net, etc.
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Lasso regression, Elastic Net, etc.
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\subsection{Statistical Analysis Workflow}
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\begin{enumerate}[label={\bfseries\color{primary}Step \arabic*.}]
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\item Graphical representation;
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\item ...
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\end{enumerate}
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\[
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\[
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Y = X \beta + \varepsilon,
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Y = X \beta + \varepsilon,
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\]
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\]
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@ -145,21 +139,18 @@ $\Y - \X \hat{\beta} \perp \X \hat{\beta} - \Y \One$ if $\One \in V$, so
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\begin{figure}
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\begin{figure}
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\centering
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\centering
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\includestandalone{figures/schemes/orthogonal_projection}
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\includegraphics{figures/schemes/orthogonal_projection.pdf}
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\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 $\norm{Y - \bar{Y}}^2$.
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\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 $\norm{Y - \bar{Y}}^2$.
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\label{fig:scheme-orthogonal-projection}
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\label{fig:scheme-orthogonal-projection}
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\end{figure}
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\end{figure}
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\begin{figure}
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\begin{figure}
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\centering
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\centering
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\includestandalone{figures/schemes/ordinary_least_squares}
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\includegraphics{figures/schemes/ordinary_least_squares.pdf}
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\caption{Ordinary least squares and regression line with simulated data.}
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\caption{Ordinary least squares and regression line with simulated data.}
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\label{fig:ordinary-least-squares}
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\label{fig:ordinary-least-squares}
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\end{figure}
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\end{figure}
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\begin{definition}[Model dimension]
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\begin{definition}[Model dimension]
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Let $\M$ be a model.
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Let $\M$ be a model.
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The dimension of $\M$ is the dimension of the subspace generated by $\X$, that is the number of parameters in the $\beta$ vector.
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The dimension of $\M$ is the dimension of the subspace generated by $\X$, that is the number of parameters in the $\beta$ vector.
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@ -169,22 +160,21 @@ $\Y - \X \hat{\beta} \perp \X \hat{\beta} - \Y \One$ if $\One \in V$, so
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\section{Gaussian vectors}
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\section{Gaussian vectors}
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\begin{definition}[Normal distribution]
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\begin{definition}[Normal distribution]
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$X \sim \Norm(\mu, \sigma^{2})$, with density function $f$
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\[
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f(x) = \frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^{2}}
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\]
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\end{definition}
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\end{definition}
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\begin{definition}[Gaussian vector]
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\begin{definition}[Gaussian vector]
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A random vector $\Y \in \RR[n]$ is a gaussian vector if every linear combination of its component is ...
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A random vector $\Y \in \RR[n]$ is a gaussian vector if every linear combination of its component is a gaussian random variable.
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\end{definition}
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\end{definition}
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\begin{property}
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\begin{property}
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$m = \EE(Y) = (m_1, \ldots, m_n)^T$, where $m_i = \EE(Y_i)$
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$m = \EE(Y) = (m_1, \ldots, m_n)^T$, where $m_i = \EE(Y_i)$
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...
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\[
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\[
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\Y \sim \Norm_n(m, \Sigma)
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\Y \sim \Norm_n(m, \Sigma)
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\]
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\]
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@ -193,8 +183,6 @@ $\Y - \X \hat{\beta} \perp \X \hat{\beta} - \Y \One$ if $\One \in V$, so
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\Sigma = \E\left[(\Y -m)(\Y - m)^T\right].
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\Sigma = \E\left[(\Y -m)(\Y - m)^T\right].
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\]
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\]
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\end{property}
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\end{property}
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\begin{remark}
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\begin{remark}
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@ -261,24 +249,25 @@ Covariance is really sensitive to scale of variables. For instance, if we measur
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\[
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\[
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\RR[n] = V_1 \overset{\perp}{\oplus} V_2.
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\RR[n] = V_1 \overset{\perp}{\oplus} V_2.
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\]
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\]
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\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 $Z_2 = \Pi_{V_2} (\mathbf{Z}) = \Pi_2 \Y$...
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\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)
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(\textcolor{red}{look to the slides})
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then:
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\item $z_{1}$, $Z_{2}$ are independent gaussian vectors, $Z_{1} \sim \Norm_{n_{1}} (0_{n}, \Pi_{1})$ and $Z_{2} \sim \Norm(0_{n_{2}}, \Pi_{2})$.
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In particular $\norm{Z_{1}} \sim \chi^{2}(n_{1})$ and $\norm{Z_{2}} \sim \chi^{2}(n_{2})$.
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\end{itemize}
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\end{itemize}
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$Z_2 = \Pi_{V_1}(\Z)$ is the projection of $\Z$ on subspace $V_1$.
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$Z_2 = \Pi_{V_1}(\Z)$ is the projection of $\Z$ on subspace $V_1$.
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\dots
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\dots
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\end{theorem}
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\end{theorem}
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\begin{property}[Estimators properties in the linear model]
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\begin{property}[Estimators properties in the linear model]
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According to \autoref{thm:cochran},
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According to \autoref{thm:cochran},
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\[
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\[
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\hat{m} \text{ is independent from $\hat{\sigma}^2$}
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\hat{m} \text{ is independent from $\hat{\sigma}^2$}
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\]\dots
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\]
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\[
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\[
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\frac{\norm{\Y - \Pi_V(\Y)}^2}{...} \sim
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\norm{\Y - \Pi_V(\Y)}^2 = \norm{\varepsilon - \Pi_{V}(\varepsilon)}^{2} = \norm{\Pi_{V}^{\perp} (\varepsilon)}^{2}
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\]
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\]
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$\hat{m} = \X \hat{\beta}$
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$\hat{m} = \X \hat{\beta}$
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@ -303,40 +292,34 @@ Covariance is really sensitive to scale of variables. For instance, if we measur
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\begin{align*}
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\begin{align*}
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\hat{m} &= \X \hat{\beta} = \X(\X^T\X)^{-1} \X^T \Y \\
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\hat{m} &= \X \hat{\beta} = \X(\X^T\X)^{-1} \X^T \Y \\
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\text{so} \\
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\intertext{so} \\
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&= \Pi_V \Y
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&= \Pi_V \Y
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\end{align*}
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\end{align*}
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According to Cochran theorem, we can deduce that the estimator of the predicted value $\hat{m}$ is independent $\hat{\sigma}^2$
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According to Cochran theorem, we can deduce that the estimator of the predicted value $\hat{m}$ is independent $\hat{\sigma}^2$
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All the sum of squares follows a $\chi^2$ distribution:
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All the sum of squares follows a $\chi^2$ distribution.
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\[
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...
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\]
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\begin{property}
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\end{property}
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\subsection{Estimators properties}
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\subsection{Estimators consistency}
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If $q < n$,
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\begin{itemize}
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\begin{itemize}
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\item $\hat{\sigma}^2 \overunderset{\PP}{n\to\infty} \sigma^{*2}$.
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\item $\hat{m}$ is unbiased and estimator of $m$;
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\item If $(\X^T\X)^{-1}$...
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\item $\EE(\hat{\sigma}^{2}) = \sigma^{2}(n-q)/n$ $\hat{\sigma}^{2}$ is a biased estimator of $\sigma^{2}$.
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\item ...
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\[
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S^{2} = \frac{1}{n-q} \norm{\Y - \Pi_{V}}^{2}
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\]
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is an unbiased estimator of $\sigma²$.
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\end{itemize}
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\end{itemize}
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We can derive statistical test from these properties.
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We can derive statistical test from these properties.
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\section{Statistical tests}
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\section{Statistical tests}
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\subsection{Student $t$-test}
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\subsection{Student $t$-test}
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\[
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\[
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\frac{\hat{\theta}-\theta}{\sqrt{\frac{\widehat{\VVar}(\hat{\theta})}{n}}} \underset{H_0}{\sim} t
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\frac{\hat{\theta}-\theta}{\sqrt{\frac{\widehat{\VVar}(\hat{\theta})}{n}}} \underset{H_0}{\sim} t_{n-q}
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\]
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\]
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where
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where
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@ -627,3 +610,44 @@ Different methods:
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Usually $AIC$ have smaller penalty than $BIC$, thus $AIC$ criterion tends to select models with more variables than $BIC$ criterion.
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Usually $AIC$ have smaller penalty than $BIC$, thus $AIC$ criterion tends to select models with more variables than $BIC$ criterion.
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\subsection{Stepwise}
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\begin{description}
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\item[forward] Add new predictor iteratively, beginning with the most contributing predictors.
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\item[backward] Remove predictors iteratively.
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\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.
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\end{description}
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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.
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In practice, the multiple testing problem is not taken into account in these approaches.
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We can use information criteria or model comparison in these methods.
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\section{Predictions}
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Let $X_i$ the $i$-th row of the matrix $\X$. The observed value $Y_i$ can be estimated by:
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\[
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\hat{Y}_i = (\X \hat{\beta})_i = X_i \hat{\beta}
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\]
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\begin{align*}
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\EE (\hat{Y}_i) &= (\X \beta)_i = X_i \beta \\
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\sigma^{-1} (\X \hat{\beta} - \X \beta) \sim \Norm (0_{p+1}, (\X^T \X)^{-1}), \qquad \text{and} \\
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\Var(\hat{Y}_i) = ... \\
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S^2 = \norm{...}
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\end{align*}
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\paragraph{Prediction Confidence Interval}
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We can build confidence interval for predicted values $(\X \hat{\beta})_i$
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\dots
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\paragraph{Prediction error of $Y$}
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\paragraph{Prediction interval for a new observation $Y_{n+1}$}
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@ -1,4 +1,186 @@
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\chapter{Generalized Linear Model}
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\chapter{Generalized Linear Model}
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\begin{example}
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\begin{description}
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\item[Ex. 1 - Credit Carb Default]
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Let $Y_i$ be a boolean random variable following a Bernoulli distribution.
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\item[Ex. 2 - Horseshoe Crabs]
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Let $Y_i$, be the number of satellites males.
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$Y_i$ can be described as following a Poisson distribution.
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\end{description}
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\end{example}
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\begin{remark}
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A Poisson distribution can be viewed as an approximation of binomial distribution when $n$ is high and $p$ low.
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\end{remark}
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We will consider the following relation:
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\[
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\EE(Y_i) = g^{-1} X_i \beta,
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\]
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equivalently:
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\[
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g(\EE(Y_i)) = X_i \beta.
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\]
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\begin{itemize}
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\item $\beta$ is estimated by the maximum likelihood;
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\item $g$ is called the link function.
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\end{itemize}
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\begin{remark}
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In standard linear model, the OLS estimator is the estimator of maximum of likelihood.
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\end{remark}
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\section{Logistic Regression}
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\section{Logistic Regression}
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\begin{align*}
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& \log(\frac{\Pi}{1 - \Pi}) & = \X \beta \\
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\Leftrightarrow & e^{\ln \frac{\Pi}{1 - \Pi}} = e^{\X \beta} \\
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\Leftrightarrow & \frac{\Pi}{1 - \Pi} = e^{\X \beta} \\
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\Leftrightarrow & \Pi = (1 - \Pi) e^{\X\beta} \\
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\Leftrightarrow & \Pi = e^{\X \beta} - \Pi e^{\X\beta} \\
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\Leftrightarrow & \Pi + \Pi e^{\X\beta} = e^{\X \beta} \\
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\Leftrightarrow & \Pi (1 - e^{\X\beta}) = e^{\X \beta} \\
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\Leftrightarrow & \Pi = \frac{e^{\X\beta}}{1 + e^{\X \beta}}
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\end{align*}
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\section{Maximum Likelihood estimator}
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log-likelihood: the probability to observe what we observe.
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Estimate $\beta$ by $\hat{\beta}$ such that $\forall \beta \in \RR[p+1]$:
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\[
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L_n (\hat{\beta}) \geq L_n (\beta)
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\]
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These estimators are consistent, but not necessarily unbiased.
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\section{Test for each single coordinate}
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\begin{example}[Payment Default]
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Let $Y_i$ be the default value for individual $i$.
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\[
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\log (\frac{\Pi (X)}{1 - \Pi (X)}) = \beta_0 + \beta_1 \text{student} + \beta_2 \text{balance} + \beta_3 \text{income}
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\]
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In this example, only $\beta_0$ and $\beta_2$ are significantly different from 0.
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\end{example}
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\begin{remark}
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We do not add $\varepsilon_i$, because $\log(\frac{\Pi (X)}{1 - \Pi (X)})$ corresponds to the expectation.
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\end{remark}
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\subsection{Comparison of nested models}
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To test $H_0:\: \beta_0 = \ldots = \beta_p = 0$, we use the likelihood ratio test:
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\[
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T_n = -2 \log (\mathcal{L}^{\texttt{null}}) + 2 \log (\mathcal{L}(\hat{\beta})) \underset{H_0}{\overunderset{\mathcal{L}}{n \to \infty}{\longrightarrow}} \chi^2(p).
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\]
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\begin{remark}[Family of Tests]
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\begin{itemize}
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\item Comparison of estimated values and values under the null hypothesis;
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\item Likelihood ratio test;
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\item Based on the slope on the derivative.
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\end{itemize}
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\end{remark}
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\section{Relative risk}
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$RR_i$ is the probably to have the disease, conditional to the predictor $X_{i1}$ over the probability of having the disease, conditional to the predictor $X_{i2}$.
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\[
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RR(j) = \frac{\Prob(Y_{i_1} = 1 \: | \: X_{i_1})}{\Prob(Y_{i_2} = 1) \: | \: X_{i_2}} = \frac{\EE(Y_{i_1})}{\EE(Y_{i_2})}.
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\]
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$\pi(X_i)$ is the probability of having the disease, according to $X_i$.
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The relative risk can be written as\dots
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\section{Odds}
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Quantity providing a measure of the likelihood of a particular outcome:
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\[
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odd = \frac{\pi(X_i)}{1 - \pi(X_i)}
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\]
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\[
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odds = \exp(X_i \beta)
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\]
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odds is the ratio of people having the disease, if Y represent the disease, over the people not having the disease.
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\section{Odds Ratio}
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\begin{align*}
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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})}}
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\end{align*}
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The OR can be written as:
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\[
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OR(j) = \exp(\beta_j)
|
||||||
|
\]
|
||||||
|
|
||||||
|
\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 $g^{-1}(x) = e^{x}$.
|
||||||
|
|
||||||
|
\[
|
||||||
|
\lambda_{i} = \EE(Y_{i}) = \Var(Y_{i})
|
||||||
|
\]
|
||||||
|
|
|
@ -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:
|
||||||
|
\[
|
||||||
|
D_{k,n} = \sum_{i=1}^{k} \frac{(N_i - np_i)^2}{np_i} \underoverset{H_0}{\mathcal{L}} \chi^2_{(n-1)(q-1)??}
|
||||||
|
\]
|
||||||
|
|
|
@ -68,7 +68,7 @@ Let $u = \begin{pmatrix}
|
||||||
|
|
||||||
\begin{figure}
|
\begin{figure}
|
||||||
\centering
|
\centering
|
||||||
\includestandalone{figures/schemes/vector_orthogonality}
|
\includegraphics{figures/schemes/vector_orthogonality.pdf}
|
||||||
\caption{Scalar product of two orthogonal vectors.}
|
\caption{Scalar product of two orthogonal vectors.}
|
||||||
\label{fig:scheme-orthogonal-scalar-product}
|
\label{fig:scheme-orthogonal-scalar-product}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
@ -215,6 +215,6 @@ The number of columns has to be the same as the dimension of the vector to which
|
||||||
|
|
||||||
\begin{figure}
|
\begin{figure}
|
||||||
\centering
|
\centering
|
||||||
\includestandalone{figures/schemes/coordinates_systems}
|
\includegraphics{figures/schemes/coordinates_systems.pdf}
|
||||||
\caption{Coordinate systems}
|
\caption{Coordinate systems}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
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|
@ -0,0 +1,23 @@
|
||||||
|
\documentclass[margin=0.5cm]{standalone}
|
||||||
|
\usepackage{tikz}
|
||||||
|
\usepackage{pgfplots}
|
||||||
|
\pgfplotsset{compat=1.18}
|
||||||
|
|
||||||
|
\begin{document}
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\begin{axis}[
|
||||||
|
title={Logit function},
|
||||||
|
xlabel={$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}
|
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|
@ -3,4 +3,5 @@
|
||||||
\usepackage{standalone}
|
\usepackage{standalone}
|
||||||
\usepackage{tikz-3dplot}
|
\usepackage{tikz-3dplot}
|
||||||
\usepackage{tkz-euclide}
|
\usepackage{tkz-euclide}
|
||||||
\usepackage{nicematrix}
|
\usepackage{nicematrix}
|
||||||
|
\usepackage{luacode}
|
Loading…
Reference in New Issue