Calcul matriciel et analyses multivariées pour data scientists
Notes de lecture, Karim Kilani
Matrices avec R et applications
library(tidyverse)## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
## ✔ ggplot2 3.3.6 ✔ purrr 0.3.5
## ✔ tibble 3.1.8 ✔ dplyr 1.0.10
## ✔ tidyr 1.2.1 ✔ stringr 1.4.1
## ✔ readr 2.1.3 ✔ forcats 0.5.2
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
Saisir une matrice
data<-
c(0.617,0.545,0.496,0.493,0.437,0.408,
731,680,621,591,617,615,140,139,143,128,
186,184,3.24,4.13,3.68,4.00,4.80,4.80)
data %>% length()## [1] 24Imprimer avec kable
data%>% matrix(nrow = 6) -> X
X%>%as.data.frame()->DF
rownames(DF)<-c("Tigers","Dragons","BayStars","Swallows","Giants","Carp")
colnames(DF)<-c("Win","Runs","HR","ERA")
DF%>%rownames_to_column(var="Team")->DF
DF%>%knitr::kable("simple")| Team | Win | Runs | HR | ERA |
|---|---|---|---|---|
| Tigers | 0.617 | 731 | 140 | 3.24 |
| Dragons | 0.545 | 680 | 139 | 4.13 |
| BayStars | 0.496 | 621 | 143 | 3.68 |
| Swallows | 0.493 | 591 | 128 | 4.00 |
| Giants | 0.437 | 617 | 186 | 4.80 |
| Carp | 0.408 | 615 | 184 | 4.80 |
Transposer des matrices
X%>%t()## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.617 0.545 0.496 0.493 0.437 0.408
## [2,] 731.000 680.000 621.000 591.000 617.000 615.000
## [3,] 140.000 139.000 143.000 128.000 186.000 184.000
## [4,] 3.240 4.130 3.680 4.000 4.800 4.800print(X)## [,1] [,2] [,3] [,4]
## [1,] 0.617 731 140 3.24
## [2,] 0.545 680 139 4.13
## [3,] 0.496 621 143 3.68
## [4,] 0.493 591 128 4.00
## [5,] 0.437 617 186 4.80
## [6,] 0.408 615 184 4.80knitr::kable(t(X),caption = "**X'**")| 0.617 | 0.545 | 0.496 | 0.493 | 0.437 | 0.408 |
| 731.000 | 680.000 | 621.000 | 591.000 | 617.000 | 615.000 |
| 140.000 | 139.000 | 143.000 | 128.000 | 186.000 | 184.000 |
| 3.240 | 4.130 | 3.680 | 4.000 | 4.800 | 4.800 |
Somme de matrices et multiplication par un scalaire
X<-matrix(c(3,-2,6,8,0,-2),nrow=2,byrow = T)
X%>%knitr::kable(caption = "**X**")| 3 | -2 | 6 |
| 8 | 0 | -2 |
Y<-matrix(c(2,1,-9,-7,2,-3),nrow=2,byrow = T)
Y%>%knitr::kable(caption = "**Y**")| 2 | 1 | -9 |
| -7 | 2 | -3 |
(X+Y)%>%knitr::kable(caption = "**X+Y**")| 5 | -1 | -3 |
| 1 | 2 | -5 |
Z<-matrix(c(8,-2,6,-5,0,-3),nrow=2,byrow = T)
Z%>%knitr::kable(caption = "**Z**")| 8 | -2 | 6 |
| -5 | 0 | -3 |
(-0.1*Z)%>%knitr::kable(caption = "-0.1**Z**")| -0.8 | 0.2 | -0.6 |
| 0.5 | 0.0 | 0.3 |
X<-matrix(c(4,-2,6,8,0,-2),nrow=2,byrow = T)
X%>%knitr::kable(caption = "**X**")| 4 | -2 | 6 |
| 8 | 0 | -2 |
Y<-matrix(c(2,1,-9,-7,2,-3),nrow=2,byrow = T)
Y%>%knitr::kable(caption = "**Y**")| 2 | 1 | -9 |
| -7 | 2 | -3 |
(0.5*X+(-2)*Y)%>%knitr::kable(caption = "0.5**X**+(-2)**Y**")| -2 | -3 | 21 |
| 18 | -4 | 5 |
Produit de matrices
Produit de deux vecteurs
A<-matrix(c(2,-4,1,7),nrow=2,byrow = T)
A%>%knitr::kable(caption = "**A**")| 2 | -4 |
| 1 | 7 |
B<-matrix(c(-3,1,2,-5),nrow=2,byrow = T)
B%>%knitr::kable(caption = "**B**")| -3 | 1 |
| 2 | -5 |
(A%*%B)%>%knitr::kable(caption = "**AB**")| -14 | 22 |
| 11 | -34 |
X<-matrix(c(2,3,-1,-2,0,4),nrow=2,byrow = T)
X%>%knitr::kable(caption = "**X**")| 2 | 3 | -1 |
| -2 | 0 | 4 |
Y<-matrix(c(3,5,4,-1,0,-2,0,6,0),nrow=3,byrow = T)
Y%>%knitr::kable(caption = "**Y**")| 3 | 5 | 4 |
| -1 | 0 | -2 |
| 0 | 6 | 0 |
(X%*%Y)%>%knitr::kable(caption = "**XY**")| 3 | 4 | 2 |
| -6 | 14 | -8 |
F<-matrix(c(2,-1,-3,0,1,3,-2,-3),nrow=4,byrow = T)
F%>%knitr::kable(caption = "**F**")| 2 | -1 |
| -3 | 0 |
| 1 | 3 |
| -2 | -3 |
A<-matrix(c(-4,1,6,-3,2,5),nrow=3,byrow = T)
A%>%knitr::kable(caption = "**A**")| -4 | 1 |
| 6 | -3 |
| 2 | 5 |
(F%*%t(A)) %>%knitr::kable(caption = "**FA'**")| -9 | 15 | -1 |
| 12 | -18 | -6 |
| -1 | -3 | 17 |
| 5 | -3 | -19 |
A<-matrix(c(-4,1,6,-3,2,5),nrow=3,byrow = T)
A%>%knitr::kable(caption = "**A**")| -4 | 1 |
| 6 | -3 |
| 2 | 5 |
(A%*%t(A)) %>%knitr::kable(caption = "**S=AA'**")| 17 | -27 | -3 |
| -27 | 45 | -3 |
| -3 | -3 | 29 |
(t(A)%*%A) %>%knitr::kable(caption = "**T=A'A**")| 56 | -12 |
| -12 | 35 |
u<-matrix(c(2,-1,3),nrow=3,byrow = T)
u%>%knitr::kable(caption = "**u**")| 2 |
| -1 |
| 3 |
v<-matrix(c(-2,3,-4),nrow=3,byrow = T)
v%>%knitr::kable(caption = "**v**")| -2 |
| 3 |
| -4 |
(t(u)%*%v) %>%knitr::kable(caption = "**u'v**")| -19 |
(u%*%t(v)) %>%knitr::kable(caption = "**uv'**")| -4 | 6 | -8 |
| 2 | -3 | 4 |
| -6 | 9 | -12 |
Représenter des vecteurs en 3D
Avec le package plotly
library(plotly)
X<-data.frame(x=c(1, 4),y=c(2, 5),z=c(3,6))
X%>%knitr::kable("simple")| x | y | z |
|---|---|---|
| 1 | 2 | 3 |
| 4 | 5 | 6 |
X%>%plot_ly(x =~x,y =~y,z =~z)%>%add_markers()%>%
add_trace(x = c(0, X[1,1 ]), y = c(0, X[1,2 ]), z = c(0, X[1,3 ]),type = "scatter3d", mode = "lines", name = "lines", showlegend = FALSE,color='#CCCCCC')%>% add_trace(x = c(0, 4), y = c(0, 5), z = c(0, 6),type = "scatter3d", mode = "lines", name = "lines", showlegend = FALSE,color='#CCCCCC')Avec le package plot3D
install.packages("plot3D")## Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.2'
## (as 'lib' is unspecified)library("plot3D")## Warning in fun(libname, pkgname): couldn't connect to display ":0"cols <- c("red", "red")
arrows3D(rep(0,2), rep(0,2), rep(0,2), X[,1], X[,2], X[,3], col = cols,
lwd = 2, d = 3,
main = "Représentation des vecteurs en 3D", bty ="g", ticktype = "detailed")
Trace et norme d’une matrice
set.seed(1)
n=2
rnorm(n^2,10,2)%>%matrix(,nrow=n)->X
X%>%data.frame()%>%knitr::kable()| X1 | X2 |
|---|---|
| 8.747092 | 8.328743 |
| 10.367287 | 13.190562 |
(t(X)%*%X)%>%diag()%>%sum()## [1] 427.3511norm(X,type = "F")^2 # Frobenius norm ## [1] 427.3511
Vecteurs composés que de 0 (vecteurs nuls)
- Avec R, créez le vecteur nul avec \(p=5\), et la matrice nulle de dimension \(10\times 5\).
n=10
p=5
matrix(rep(0,p),ncol = 1)## [,1]
## [1,] 0
## [2,] 0
## [3,] 0
## [4,] 0
## [5,] 0matrix(rep(0,n*p),ncol = p)## [,1] [,2] [,3] [,4] [,5]
## [1,] 0 0 0 0 0
## [2,] 0 0 0 0 0
## [3,] 0 0 0 0 0
## [4,] 0 0 0 0 0
## [5,] 0 0 0 0 0
## [6,] 0 0 0 0 0
## [7,] 0 0 0 0 0
## [8,] 0 0 0 0 0
## [9,] 0 0 0 0 0
## [10,] 0 0 0 0 0Vecteurs composés que de 1
- Avec R, créez le vecteur avec \(p=5\) répétitions de 1, et la matrice avec des \(1\) uniquement et de dimension \(10\times 5\) .
matrix(rep(1,5),ncol = 1)## [,1]
## [1,] 1
## [2,] 1
## [3,] 1
## [4,] 1
## [5,] 1matrix(rep(1,10*5),ncol = 5)## [,1] [,2] [,3] [,4] [,5]
## [1,] 1 1 1 1 1
## [2,] 1 1 1 1 1
## [3,] 1 1 1 1 1
## [4,] 1 1 1 1 1
## [5,] 1 1 1 1 1
## [6,] 1 1 1 1 1
## [7,] 1 1 1 1 1
## [8,] 1 1 1 1 1
## [9,] 1 1 1 1 1
## [10,] 1 1 1 1 1matrix(rep(1,10),ncol = 1)%*%t(matrix(rep(1,5),ncol = 1))## [,1] [,2] [,3] [,4] [,5]
## [1,] 1 1 1 1 1
## [2,] 1 1 1 1 1
## [3,] 1 1 1 1 1
## [4,] 1 1 1 1 1
## [5,] 1 1 1 1 1
## [6,] 1 1 1 1 1
## [7,] 1 1 1 1 1
## [8,] 1 1 1 1 1
## [9,] 1 1 1 1 1
## [10,] 1 1 1 1 1Matrices carrées spécifiques
Matrices symmétriques
Définition
### Le produit d’une
matrice par sa transposée est symétrique
Matrice diagonale
Matrice identité
diag(3)%>% knitr::kable()| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
Statistiques avec les matrices
TabA<-data.frame(Participant = 1:6,
Japan = c(82,96,84,90,93,82),
English = c(70,67,54,66,74,60),
Science = c(76,71,65,80,77,89)
)
library(tidyverse)
TabA%>%as_tibble()->TabA
(TabA%>%as.matrix()->MA)## Participant Japan English Science
## [1,] 1 82 70 76
## [2,] 2 96 67 71
## [3,] 3 84 54 65
## [4,] 4 90 66 80
## [5,] 5 93 74 77
## [6,] 6 82 60 89
data<-
c(0.617,0.545,0.496,0.493,0.437,0.408,
731,680,621,591,617,615,140,139,143,128,
186,184,3.24,4.13,3.68,4.00,4.80,4.80)
data%>% matrix(nrow = 6) -> MC
MC%>%as.data.frame()->TabC
rownames(TabC)<-c("Tigers","Dragons","BayStars","Swallows","Giants","Carp")
colnames(TabC)<-c("Win","Runs","HR","ERA")
TabC%>%rownames_to_column(var="Team")->TabC
TabC%>%knitr::kable("simple")| Team | Win | Runs | HR | ERA |
|---|---|---|---|---|
| Tigers | 0.617 | 731 | 140 | 3.24 |
| Dragons | 0.545 | 680 | 139 | 4.13 |
| BayStars | 0.496 | 621 | 143 | 3.68 |
| Swallows | 0.493 | 591 | 128 | 4.00 |
| Giants | 0.437 | 617 | 186 | 4.80 |
| Carp | 0.408 | 615 | 184 | 4.80 |
MC## [,1] [,2] [,3] [,4]
## [1,] 0.617 731 140 3.24
## [2,] 0.545 680 139 4.13
## [3,] 0.496 621 143 3.68
## [4,] 0.493 591 128 4.00
## [5,] 0.437 617 186 4.80
## [6,] 0.408 615 184 4.80
TabAp<-data.frame(Participant = 1:6,
History = c(66,72,44,58,70,56),
Mathematics = c(74,98,62,88,56,84)
)
library(tidyverse)
TabAp%>%as_tibble()->TabAp
(TabAp%>%as.matrix()->MAp)## Participant History Mathematics
## [1,] 1 66 74
## [2,] 2 72 98
## [3,] 3 44 62
## [4,] 4 58 88
## [5,] 5 70 56
## [6,] 6 56 84
library(ggplot2)
TabAp%>%
ggplot(aes(x=History, y=0,label =History))+geom_point(size = 2,shape=25)+ scale_color_manual(values = unname(colours)) +geom_text(vjust = -1,size=2)+geom_hline(yintercept=0,linetype="dashed",
color = "red", size=1)+xlim(0,100)
TabAp%>%
ggplot(aes(x=History, y=0,label =History))+ geom_boxplot(alpha = 0.80) +
geom_point(size = 2)+geom_text(vjust = -1)+
theme(text = element_text(size = 10),
axis.title.x = element_blank(),
axis.title.y = element_blank(),
panel.grid.minor.x = element_blank(),
panel.grid.major.x = element_blank(),
legend.position = "none")+xlim(0,100)
Moyenne
TabAp%>%summarise(mean = mean(History), n = n())matrix(rep(1,6),ncol = 1)->u
TabAp%>%select(History)%>%as.matrix(ncol = 1)->x
t(u)%*%x/6## History
## [1,] 61Scores centrés
matrix(rep(1,6),ncol = 1)->u
TabAp%>%select(History)%>%as.matrix(ncol = 1)->x
x-mean(x)## History
## [1,] 5
## [2,] 11
## [3,] -17
## [4,] -3
## [5,] 9
## [6,] -5x-mean(x)*u## History
## [1,] 5
## [2,] 11
## [3,] -17
## [4,] -3
## [5,] 9
## [6,] -5J<-diag(6)-(1/6)*u%*%t(u)
J%*%x## History
## [1,] 5
## [2,] 11
## [3,] -17
## [4,] -3
## [5,] 9
## [6,] -5Variance et écart-type
## Variance selon les différentes formules
matrix(rep(1,6),ncol = 1)->u
TabAp%>%select(History)%>%as.matrix(ncol = 1)->x
x-mean(x)## History
## [1,] 5
## [2,] 11
## [3,] -17
## [4,] -3
## [5,] 9
## [6,] -5x-mean(x)*u## History
## [1,] 5
## [2,] 11
## [3,] -17
## [4,] -3
## [5,] 9
## [6,] -5J<-diag(6)-(1/6)*u%*%t(u)
(1/6)*t(x-mean(x)*u)%*%(x-mean(x)*u)## History
## History 91.66667(1/6)*t(x)%*%J%*%x## History
## History 91.66667(1/6)*(norm(J%*%x, type="2"))^2## [1] 91.66667(5/6)*var(x)## History
## History 91.66667## Ecart-type
sqrt(1/6)*(norm(J%*%x, type="2"))## [1] 9.574271sqrt(5/6)*sd(x)## [1] 9.574271Standard Scores
matrix(rep(1,6),ncol = 1)->u
TabAp%>%select(History)%>%as.matrix(ncol = 1)->x
(x-mean(x))/(sd(x))## History
## [1,] 0.4767313
## [2,] 1.0488088
## [3,] -1.6208864
## [4,] -0.2860388
## [5,] 0.8581163
## [6,] -0.4767313scale(x)## History
## [1,] 0.4767313
## [2,] 1.0488088
## [3,] -1.6208864
## [4,] -0.2860388
## [5,] 0.8581163
## [6,] -0.4767313
## attr(,"scaled:center")
## History
## 61
## attr(,"scaled:scale")
## History
## 10.48809scale(x,scale = FALSE)## History
## [1,] 5
## [2,] 11
## [3,] -17
## [4,] -3
## [5,] 9
## [6,] -5
## attr(,"scaled:center")
## History
## 61x-mean(x)## History
## [1,] 5
## [2,] 11
## [3,] -17
## [4,] -3
## [5,] 9
## [6,] -5Que se passe-t-il lorsqu’on standardise ?
Représentation matricielle
`
Exercices
Variabilité inter-variables
X<-data.frame(Food=1:10,
Sweet=c(32,28,20,34,25,35,25,30,34,22),
Spice=c(10,20,19,21,16,14,20,18,13,26),
Sales=c(62,83,34,91,53,70,62,73,84,63))
## Reproduire ces trois graphiques avec un
panel ggplot2
Covariance
X[,2:4]%>% as.matrix()->M
n<-dim(M)[1]
t(M[,1]-mean(M[,1]))%*%(M[,2]-mean(M[,2]))/n## [,1]
## [1,] -12.65u<-matrix(1,nrow = n)
J<-diag(n)-u%*%t(u)/n
t(M[,1])%*%J%*%M[,2]/n## [,1]
## [1,] -12.65((n-1)/n)*cov(M)## Sweet Spice Sales
## Sweet 25.65 -12.65 60.15
## Spice -12.65 19.01 0.15
## Sales 60.15 0.15 251.45Coefficient de corrélation
X[,2:4]%>% as.matrix()->M
n<-dim(M)[1]
res<-t(J%*%M[,1])%*%J%*%M[,2]/(norm(J%*%M[,1],type="2")*norm(J%*%M[,2],type="2"))
round(res,2)## [,1]
## [1,] -0.57cor(M)%>%round(2)## Sweet Spice Sales
## Sweet 1.00 -0.57 0.75
## Spice -0.57 1.00 0.00
## Sales 0.75 0.00 1.00Vecteurs et corrélations
angles<-cor(M)%>%acos()
round(angles*360/(2*pi),0)## Sweet Spice Sales
## Sweet 0 125 41
## Spice 125 0 90
## Sales 41 90 0