ACP_main.Rmd

---
title: "ACP à la main"
output: pdf_document
---

# ACP à la main

```{r}
X <- means.variables
library(FactoMineR)
res.pca <- PCA(X, graph = F)
F <- res.pca$ind$coord
V <- res.pca$var$coord

# Centrer réduire
m <- apply(X,2,mean) #mean
s <- apply(X,2,sd) #standard deviation
Y <- sweep(X,2,m,"-")
apply(Y,2,mean)
apply(Y,2,sd)
Z <- sweep(Y,2,s,"/")
apply(Z,2,sd)

# Calcul vectoriel
R <- t(as.matrix(Z))%*%as.matrix(Z)/3
e <- eigen(R)
names(e)
dim(e$vectors)
q <- 3
V <- e$vectors[,1:q]
Fbis <- as.matrix(Z)%*%V
head(Fbis)
head(F)

# SVD triplet
n <- nrow(X)
#first compute Z with the non corrected standard deviation
s <- apply(Y,2,sd)*sqrt((n-1)/n)
Z <- sweep(Y,2,s,"/")
e_svd <- svd(Z)
names(e_svd)
(e_svd$d^2)[1:q]/n #division by n because of the metric
U <- e_svd$u[,1:q] #matrix of the q first left singular vectors
Fbisbis <- U%*%diag(e_svd$d[1:q]) #principal components
head(Fbisbis)
head(F)
e$values
res.pca$eig
```*