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05_b_svd_pca.pdf
05_b_svd_pca_cache/
05_b_svd_pca_files/
05_b_svd_pca_slides.pdf
05_b_svd_pca_slides_cache/
05_b_svd_pca_slides_files/
06_matrice_definie_non_negative.pdf
07_multiplicateurs_de_lagrange.pdf
08_derivation_svd.pdf
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14_geometrie_ridge_svd.pdf
14_geometrie_ridge_svd_slides.pdf
15_loocv.pdf
15_loocv_slides.pdf
16_biais_variance_estimateur.pdf
17_biais_variance_ridge.pdf
18_kernel_ridge_regression.pdf

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---
title: "05-b SVD et Analyse en Composantes Principales"
author: Pierre-Edouard Portier
date: mars 2022
output:
beamer_presentation:
incremental: false
---
# Projection sur les axes principaux
>- $\mathbf{X}\in\mathbb{R}^{n \times p} \quad ; \quad \mathbf{X} = \sum_{\alpha}\sqrt{\lambda_\alpha}\mathbf{u_\alpha}\mathbf{v_\alpha}^T \quad\equiv\quad \mathbf{X} = \mathbf{U} \mathbf{D} \mathbf{V}^T$
>- Coordonnées (facteurs) des observations sur les axes principaux $\mathbf{F} \triangleq \mathbf{X}\mathbf{V} = \mathbf{U} \mathbf{D} \mathbf{V}^T\mathbf{V} = \mathbf{U} \mathbf{D}$
>- Covariance des facteurs $\left(\mathbf{U} \mathbf{D}\right)^T\left(\mathbf{U} \mathbf{D}\right) = \mathbf{D}^2$
>- Les facteurs sont orthogonaux (par construction)
>- $\lambda_\alpha = \sum_i f_{i,\alpha}^2$ est la variance expliquée par l'axe $\mathbf{v_\alpha}$
>- Contribution des observations aux axes $CTR_{i,\alpha} = \frac{f_{i,\alpha}^2}{\sum_i f_{i,\alpha}^2} = \frac{f_{i,\alpha}^2}{\lambda_\alpha}$
>- $\sum_i CTR_{i,\alpha} = 1$ et $\mathbf{x_i}$ pèse sur $\mathbf{v_\alpha}$ quand $CTR_{i,\alpha} \geq 1/n$
>- Contribution des axes aux observations $COS2_{i,\alpha} = \frac{f_{i,\alpha}^2}{\sum_\alpha f_{i,\alpha}^2} = \frac{f_{i,\alpha}^2}{d_{i,g}^2}$
>- $d_{i,g}^2 = \sum_j \left(x_{i,j}-g_j\right)^2$ ($\sum_j x_{i,j}^2$ si données centrées)
>- $\sum_i d_{i,g}^2 = \mathcal{I} = \sum_\alpha \lambda_\alpha$
>- Contribution des variables aux axes $VARCTR_{j,\alpha} = \left(\frac{\lambda_\alpha}{\sqrt{n-1}}\mathbf{v_\alpha}\right)^2$
# Exemple `abalone`
## Récupération du jeu de données
```{r cache=TRUE}
abalone.cols = c("sex", "length", "diameter", "height",
"whole.wt", "shucked.wt", "viscera.wt",
"shell.wt", "rings")
url <- 'http://archive.ics.uci.edu/ml/machine-learning-databases/abalone/abalone.data'
abalone <- read.table(url, sep=",", row.names=NULL,
col.names=abalone.cols,
nrows=4177)
abalone <- subset(abalone, height!=0)
```
## Sélection des variables explicatives numériques
```{r}
X <- abalone[,c("length", "diameter", "height", "whole.wt",
"shucked.wt", "viscera.wt", "shell.wt")]
```
# Exemple `abalone`
## Standardisation
```{r}
standard <- function(X) {
meanX <- apply(X, 2, mean);
sdX <- apply(X, 2, sd);
Y <- sweep(X, 2, meanX, FUN = "-");
Y <- sweep(Y, 2, sdX, FUN = "/");
res <- list("X" = Y, "meanX" = meanX, "sdX" = sdX)
}
```
# Exemple `abalone`
## Clustering
```{r warning=FALSE}
set.seed(1123)
std.X <- standard(X)
A <- std.X$X
nbClust <- 100
clst <- kmeans(A, nbClust, nstart = 25)
```
# Exemple `abalone`
## SVD des centres des clusters
```{r}
std.centers <- standard(clst$centers)
C <- std.centers$X
svd.C <- svd(C)
```
## Pourcentage de variance expliquée par chaque axe principal
```{r}
round((svd.C$d^2 / sum(svd.C$d^2))*100, 2)
```
# Exemple `abalone`
## Facteurs et des contributions des observations aux axes
- $\mathbf{F} \triangleq \mathbf{X}\mathbf{V} = \mathbf{U} \mathbf{D} \mathbf{V}^T\mathbf{V} = \mathbf{U} \mathbf{D}$
- $CTR_{i,\alpha} = \frac{f_{i,\alpha}^2}{\sum_i f_{i,\alpha}^2} = \frac{f_{i,\alpha}^2}{\lambda_\alpha}$
```{r}
fact <- svd.C$u %*% diag(svd.C$d)
fact2 <- fact^2
ctr <- sweep(fact2, 2, colSums(fact2), "/")
```
# Exemple `abalone`
## Affichage des observations importantes pour les deux premiers axes principaux
```{r, eval=FALSE}
n <- dim(fact)[1]
d1BestCtr <- which(ctr[,1] > 1/n)
d2BestCtr <- which(ctr[,2] > 1/n)
d1d2BestCtr <- union(d1BestCtr,d2BestCtr)
plot(fact[d1d2BestCtr,1], fact[d1d2BestCtr,2], pch="")
text(fact[d1d2BestCtr,1], fact[d1d2BestCtr,2],
d1d2BestCtr, cex=0.8)
```
# Exemple `abalone`
```{r, echo=FALSE}
n <- dim(fact)[1]
d1BestCtr <- which(ctr[,1] > 1/n)
d2BestCtr <- which(ctr[,2] > 1/n)
d1d2BestCtr <- union(d1BestCtr,d2BestCtr)
plot(fact[d1d2BestCtr,1], fact[d1d2BestCtr,2], pch="")
text(fact[d1d2BestCtr,1], fact[d1d2BestCtr,2],
d1d2BestCtr, cex=1)
```
# Exemple `abalone`
```{r}
f2.max <- which.max(abs(fact[,2]))
f2.max.size <- clst$size[f2.max]
f2.max.name <- names(clst$cluster[clst$cluster==f2.max])
f2.max.ctr <- round(ctr[f2.max,2]*100, 2)
```
- Le cluster `r f2.max` explique `r f2.max.ctr` % de la variance du second axe.
- Il est composé de `r f2.max.size` élément
```{r}
abalone[f2.max.name,]
```
# Exemple `abalone`
## Nouvelle analyse
```{r, warning=FALSE}
abalone <- abalone[!(row.names(abalone) %in% f2.max.name),]
X <- abalone[,c("length", "diameter", "height", "whole.wt","shucked.wt", "viscera.wt",
"shell.wt")]
std.X <- standard(X)
A <- std.X$X
nbClust <- 100
clst <- kmeans(A, nbClust, nstart = 25)
std.centers <- standard(clst$centers)
C <- std.centers$X
svd.C <- svd(C)
round((svd.C$d^2 / sum(svd.C$d^2))*100, 2)
```
# Exemple `abalone`
## Nouvelle analyse (suite) et affichage des observations sur les deux premiers axes
```{r}
fact <- svd.C$u %*% diag(svd.C$d)
fact2 <- fact^2
ctr <- sweep(fact2, 2, colSums(fact2), "/")
n <- dim(fact)[1]
d1BestCtr <- which(ctr[,1] > 1/n)
d2BestCtr <- which(ctr[,2] > 1/n)
d1d2BestCtr <- union(d1BestCtr,d2BestCtr)
```
```{r, eval=FALSE}
plot(fact[d1d2BestCtr,1], fact[d1d2BestCtr,2], pch="")
text(fact[d1d2BestCtr,1], fact[d1d2BestCtr,2],
d1d2BestCtr, cex=0.8)
```
# Exemple `abalone`
```{r, echo=FALSE}
plot(fact[d1d2BestCtr,1], fact[d1d2BestCtr,2], pch="")
text(fact[d1d2BestCtr,1], fact[d1d2BestCtr,2], d1d2BestCtr, cex=0.8)
```
# Exemple `abalone`
## Un cluster intrigant...
```{r}
intrig<-43
intrig.name <- names(clst$cluster[clst$cluster==intrig])
abalone[intrig.name,]
```
# Exemple `abalone`
## Un cluster intrigant...
```{r}
boxplot(abalone$height)
```
# Exemple `abalone`
## Un cluster intrigant...
- Contributions des axes aux écarts des observations au centre d'inertie $COS2_{i,\alpha} = \frac{f_{i,\alpha}^2}{\sum_\alpha f_{i,\alpha}^2} = \frac{f_{i,\alpha}^2}{d_{i,g}^2}$
```{r}
cos2 <- sweep(fact2, 1, rowSums(fact2), "/")
round(cos2[intrig,]*100,2)
```
- Observation intrigante toute expliquée par les deux premiers axes.
- Pas de risque à considérer comme significatif son écart à l'origine.
# Exemple `abalone`
## Contributions des variables aux axes principaux
- $VARCTR_{j,\alpha} = \left(\frac{\lambda_\alpha}{\sqrt{n-1}}\mathbf{v_\alpha}\right)^2$
```{r}
varctr <- ((svd.C$v %*% diag(svd.C$d))/sqrt(n-1))^2
rownames(varctr) <- colnames(X)
round(varctr,2)
```
- Variables très corrélées qui pèsente toutes fortement sur la variance expliquée par l'axe 1

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---
title: "15 Validation croisée un contre tous"
author: Pierre-Edouard Portier
date: mars 2022
output:
beamer_presentation:
incremental: false
---
```{r, include=FALSE}
source("01_intro.R", local = knitr::knit_global())
source("04_validation_croisee.R", local = knitr::knit_global())
source("15_loocv.R", local = knitr::knit_global())
```
# Validation croisée un contre tous (Leave-One-Out Cross-Validation)
>- Validation croisée à $n$-plis !
>- $\hat{\boldsymbol\beta}_\lambda^{(-i)} = \left( \mathbf{X^{(-i)}}^T\mathbf{X^{(-i)}} + \lambda\mathbf{I} \right)^{-1} \mathbf{X^{(-i)}}^T \mathbf{y^{(-i)}}$
>- $LOO_\lambda = \frac{1}{n} \sum_{i=1}^{n} \left( y_i - \mathbf{x_i}^T \hat{\boldsymbol\beta}_\lambda^{(-i)} \right)^2$
>- $\mathbf{X^{(-i)}}^T\mathbf{X^{(-i)}} + \lambda\mathbf{I} = \mathbf{X^T}\mathbf{X} + \lambda\mathbf{I} - \mathbf{x_i}\mathbf{x_i}^T$
# Matrice chapeau
>- $\mathbf{\hat{y}_\lambda} = \mathbf{X}\hat{\boldsymbol\beta}_\lambda = \mathbf{X} \left( \mathbf{X}^T \mathbf{X} + \lambda \mathbf{I}\right)^{-1} \mathbf{X}^T \mathbf{y} = \mathbf{H} \mathbf{y}$
>- $h_{ij}$ est l'influence qu'exerce $y_j$ sur la prédiction $\hat{y}_i$
>- $h_{ii}$, l'influence de $y_i$ sur $\hat{y}_i$ identifie les observations à l'influence prépondérante
>- $\mathbf{x_i}^T \left( \mathbf{X}^T \mathbf{X} + \lambda \mathbf{I}\right)^{-1} \mathbf{x_i} = h_{ii} \triangleq h_i$
>- $\mathbf{\hat{y}_\lambda} = \sum_{d_j>0} \mathbf{u_j} \frac{d_j^2}{d_j^2 + \lambda} \mathbf{u_j}^T\mathbf{y}$
>- $h_i = \sum_{d_j>0} \frac{d_j^2}{d_j^2 + \lambda} u_{ij}$
>- $tr(\mathbf{H(\lambda)}) = \sum_i h_i = \sum_{d_j>0} \frac{d_j^2}{d_j^2 + \lambda}$
>- $tr(\mathbf{H(0)}) = rang(\mathbf{X})$ ou \emph{degré de liberté} de la régression
# Formule de Morrison et calcul efficace de LOOCV
>- $\left( \mathbf{A} + \mathbf{u}\mathbf{v}^T \right)^{-1} = \mathbf{A}^{-1} - \frac{\mathbf{A}^{-1}\mathbf{u}\mathbf{v}^T\mathbf{A}^{-1}}{1 + \mathbf{v}^T\mathbf{A}^{-1}\mathbf{u}}$
>- Permet de simplifier $\left( \mathbf{X^{(-i)}}^T\mathbf{X^{(-i)}} + \lambda\mathbf{I} \right)^{-1}$ pour obtenir...
>- $$y_i - \hat{y}^{(-i)}_{\lambda i} = \frac{y_i - \hat{y}_{\lambda i}}{1 - h_i}$$
>- $$
\begin{aligned}
&LOO_\lambda = \frac{1}{n} \sum_{i=1}^{n} \left( \frac{y_i - \hat{y}_{\lambda i}}{1 - h_i} \right)^2 \\
&\text{avec } h_i = \sum_{d_j>0} \frac{d_j^2}{d_j^2 + \lambda} u_{ij}
\end{aligned}
$$
>- Cette mesure peut être instable pour des $h_i$ proches de $1$...
# Validation croisée généralisée (GCV)
$$
\begin{aligned}
&GCV_\lambda = \frac{1}{n} \sum_{i=1}^{n} \left( \frac{y_i - \hat{y}_{\lambda i}}{1 - \frac{1}{n} tr(\mathbf{H}(\lambda))} \right)^2 \\
&\text{avec } tr(\mathbf{H}(\lambda)) = \sum_{d_j>0} \frac{d_j^2}{d_j^2 + \lambda}
\end{aligned}
$$
# Implémentation de GCV
\small
```{r, eval=FALSE}
ridge.svd <- function(data, degre) {
X <- data$X
n <- nrow(X)
A <- scale(outer(c(X), 1:degre, "^"))
Ym <- mean(data$Y)
Y <- data$Y - Ym
As <- svd(A)
d <- As$d
function(lambda) {
coef <- c(As$v %*% ((d / (d^2 + lambda)) * (t(As$u) %*% Y)))
coef <- coef / attr(A,"scaled:scale")
inter <- Ym - coef %*% attr(A,"scaled:center")
coef <- c(inter, coef)
trace.H <- sum(d^2 / (d^2 + lambda))
pred <- polyeval(coef, X)
gcv <- sum( ((Y - pred) / (1 - (trace.H / n))) ^ 2 ) / n
list(coef = coef, gcv = gcv)
}
}
```
\normalsize
# Implémentation de GCV
\small
```{r, eval=FALSE}
ridge.loocv <- function(data, deg, lambdas) {
errs <- double(length(lambdas))
coefs <- matrix(data = NA, nrow = length(lambdas), ncol = 1+deg)
ridge <- ridge.svd(data, deg)
idx <- 1
for(lambda in lambdas) {
res <- ridge(lambda)
coefs[idx,] <- res$coef
errs[idx] <- res$gcv
idx <- idx + 1
}
err.min <- min(errs)
lambda.best <- lambdas[which(errs == err.min)]
coef.best <- coefs[which(errs == err.min),]
pred <- polyeval(coef.best, data$X)
mae <- mean(abs(pred - data$Y))
list(coef = coef.best, lambda = lambda.best, mae = mae)
}
```
\normalsize
# Exemple
\small
```{r, eval=FALSE}
set.seed(1123)
n <- 100
deg <- 8
data = gendat(n,0.2)
splitres <- splitdata(data,0.8)
entr <- splitres$entr
test <- splitres$test
lambdas <- c(1E-8, 1E-7, 1E-6, 1E-5, 1E-4, 1E-3, 1E-2, 1E-1, 1)
ridge.res <- ridge.loocv(entr, deg, lambdas)
plt(entr,f)
pltpoly(ridge.res$coef)
```
\normalsize
# Exemple - jeu d'entraînement
```{r, echo=FALSE}
set.seed(1123)
n <- 100
deg <- 8
data = gendat(n,0.2)
splitres <- splitdata(data,0.8)
entr <- splitres$entr
test <- splitres$test
lambdas <- c(1E-8, 1E-7, 1E-6, 1E-5, 1E-4, 1E-3, 1E-2, 1E-1, 1)
ridge.res <- ridge.loocv(entr, deg, lambdas)
plt(entr,f)
pltpoly(ridge.res$coef)
```
```{r, echo=FALSE}
bestLambda <- ridge.res$lambda
maeTrain <- ridge.res$mae
```
- $\lambda$ : `r bestLambda`
- Erreur absolue moyenne sur le jeu d'entraînement : `r round(maeTrain,3)`
# Exemple - jeu de test
```{r, echo=FALSE}
plt(test,f)
pltpoly(ridge.res$coef)
```
# Exemple - jeu de test
```{r}
testpred <- polyeval(ridge.res$coef, test$X)
testmae <- mean(abs(testpred - test$Y))
```
- erreur absolue moyenne de `r round(testmae,3)` sur le jeu de test