From 851b3b195ddcd35b16d29de634ef4c50928a40f9 Mon Sep 17 00:00:00 2001 From: AzGeffroy Date: Wed, 20 Mar 2024 22:51:50 +0100 Subject: [PATCH] Corrections inREML and LRT --- rapport.Rnw | 25 +++++++++---------------- rapport.pdf | Bin 1999530 -> 1999979 bytes 2 files changed, 9 insertions(+), 16 deletions(-) diff --git a/rapport.Rnw b/rapport.Rnw index e2d942e..4b22e1b 100644 --- a/rapport.Rnw +++ b/rapport.Rnw @@ -414,42 +414,35 @@ Puis les dérivées secondes: De là on obtient la Hessienne $\begin{pmatrix} \partial_{\sigma^2_{phy}\sigma^2_{phy}}l & \partial_{\sigma^2_{phy}\sigma^2_{err}}l \\ \partial_{\sigma^2_{err}\sigma^2_{phy}}l & \partial_{\sigma^2_{err}\sigma^2_{err}}l \\ -\end{pmatrix}$ puis A en l'inversant, ce qui peut se faire par des méthodes numériques ou analytiques ici car cela signifie inverser une matrice 2 x 2 ce qui est facile. +\end{pmatrix}$ puis A en inversant la matrice de Fisher liée, ce qui peut se faire par des méthodes numériques ou analytiques ici car cela signifie inverser une matrice 2 x 2 ce qui est facile. \end{proof} -% TODO REML voir sujet d'exam corrigée -% Quand estimateur classique on divise par n-p au lieu de diviser par n donc on -% fait sans le dire un REML. -% Au lieu de maximiser la vraisemblance on maximise la vraisemblance restreinte -% Gaussien : effet fixe les betas, pour estimer al variance on projette sur -% l'orthogonal et on estime sigma sur l'orthogonal. - -% Si bayésien on met un prior impropre sur les betas et on intègre apr rapport aux \subsection{REML} -REML, ou Maximum de Vraisemblance Restreint (Restricted Maximum Likelihood en anglais), est une méthode statistique utilisée dans l'estimation des paramètres de modèles linéaires mixtes (ou modèles à effets mixtes) et dans l'analyse de la variance (ANOVA). +Le REML, ou Maximum de Vraisemblance Restreint (Restricted Maximum Likelihood en anglais), est une méthode statistique utilisée dans l'estimation des paramètres de modèles linéaires mixtes (ou modèles à effets mixtes) et dans l'analyse de la variance (ANOVA). Il s'agit d'une approche alternative à la méthode de maximum de vraisemblance (ML) standard, notamment lorsque l'on travaille avec des modèles à effets aléatoires. \newline \newline -L'une des formules pour la vraissemblance restreinte consite à regarder la vraissemblances des observations en intégrant sur les effets fixes, ici $\beta$. +L'une des formules pour la vraissemblance restreinte consite à regarder la vraissemblances des observations en intégrant sur les effets fixes, ici $\beta$ sur lesquels on a mis un prior impropre uniforme entre moins l'infini et plus l'infini. C'est à dire sous l'hypothèse que les estimations des effets fixes sont éliminées (conditionnées) \begin{equation*} - L_{REML} (Y;\theta)=\int _{\beta \in \mathbb{R} ^2} \frac{1}{(2\pi)^{n/2}|V(\theta)|^{1/2}} \exp\left( -\frac{1}{2}(Y - X\beta)^T V(\theta)^{-1} (Y - X\beta) \right), d\beta + L_{REML} (Y;\theta)=\int _{\beta \in \mathbb{R} ^2} \frac{1}{(2\pi)^{n/2}|V(\theta)|^{1/2}} \exp\left( -\frac{1}{2}(Y - X\beta)^T V(\theta)^{-1} (Y - X\beta) \right)d\beta \end{equation*} -Lorsque dans un estimateur on a la division par $(n-p)$ au lieu de n, on fait sans le dire un REML. -TODO: Pourquoi l'utiliser pour Satterthwaite ? +Lorsque dans un estimateur on a la division par $(n-p)$ au lieu de n, on fait sans le dire un REML dans le cas de la régression linéaire classique. +\newline +Satterthewaite maximum likelihood ne fait pas l'astuce de diviser par $n-p$, au contraire de l'anova classique ou phylogénétique. Il est donc particulièrement mauvais. Le REML permet de mieux estimer la variance, et aussi le $\lambda$ (ratio des variances), ce qui est crucial dans la dérivation des degrés de libertés approchés. \subsection{Méthode Likelihood Ratio Test} La méthode du test de rapport de vraisemblance (LRT) est une technique statistique utilisée pour comparer deux modèles statistiques et déterminer s'ils diffèrent significativement en termes d'ajustement aux données. -Le rapport de vraisemblance est calculé comme le rapport des vraisemblances maximales des deux modèles (sous les deux hypothèses): +Le rapport de vraisemblance est calculé comme le rapport des vraisemblances maximales de deux modèles (sous les deux hypothèses) emboîtés : \begin{equation*} \text{LRT} = -2 \log\left(\frac{L(\theta_{\text{H}_0})}{L(\theta_{\text{H}_1})}\right) \end{equation*} Sous l'hypothèse nulle que le modèle plus simple est correct, ce rapport suit approximativement une distribution du chi-deux avec un nombre de degrés de liberté égal à la différence dans le nombre de paramètres entre les deux modèles. Ainsi, en comparant la valeur observée du rapport de vraisemblance à la distribution du chi-deux, on peut décider si l'ajout de paramètres dans le modèle conduit à une amélioration significative de l'ajustement aux données. \newline -Il est à noter que cette méthode est plus coûteuse car il est nécessaire de fitter 2 modèles au lieu d'un seul dans les autres méthodes. +Il est à noter que cette méthode est plus coûteuse car il est nécessaire d'ajuster 2 modèles au lieu d'un seul dans les autres méthodes. \section{Simulations} diff --git a/rapport.pdf b/rapport.pdf index 387d9fafc6ceea801ec565061c92223e8eff11f0..757bfddd26b9ee9f210251eac4607c7597804c9a 100644 GIT binary patch delta 59221 zcmX`SQ+S|Vw5%K3wr#6pCmq|i?c|GX+qT_t$7ZL~v2E-8Ywff5eLe5ptf%IvF{;CS z&Le#q+ap0)Qb4I_Qnn*$0dK&+4@x=_Ls9qNjI#F*0?*0msc*L>365nr?JNVjUgJP0 znwQBCK0h1nZDm%g*Cj}}XTEiy^k8cnIBnK`zRYkgRI53ttGSU#hjd7b9PJQ8=As-7 zcBYG*bo^F{XSnr_+CPNW5!o%aX^e7-xlr{N2C!yJQ%G}k`Df} 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