Analyse en composantes principales

# Analyse en composantes principales
## pour visualiser et comprendre des données complexes
### Marie-Pierre Etienne
### <a href="mailto:marie-pierre.etienne@agrocampus-ouest.fr" class="email">marie-pierre.etienne@agrocampus-ouest.fr</a>
### 2021/12/21 (updated: 2023-09-08)

---

# Introduction

## L'exemple des caractéristiques du Doubs

On a mesuré les caractéristiques physico chimiques sur 30  sites différents le long de la rivière Doubs.

```
# A tibble: 30 × 11
    das   alt   pen   deb    pH   dur   pho   nit   amm   oxy   dbo
  <dbl> <int> <dbl> <dbl> <dbl> <int> <dbl> <dbl> <dbl> <dbl> <dbl>
1   0.3   934  48    0.84   7.9    45  0.01  0.2   0     12.2   2.7
2   2.2   932   3    1      8      40  0.02  0.2   0.1   10.3   1.9
3  10.2   914   3.7  1.8    8.3    52  0.05  0.22  0.05  10.5   3.5
4  18.5   854   3.2  2.53   8      72  0.1   0.21  0     11     1.3
5  21.5   849   2.3  2.64   8.1    84  0.38  0.52  0.2    8     6.2
6  32.4   846   3.2  2.86   7.9    60  0.2   0.15  0     10.2   5.3
# … with 24 more rows
```
]

.pull-right[
* das :  distance à al source ( `$km$` ),  
* alt : altitude ( `$m$` )
* pen : la pente (denivelelé pour  1000m)
* deb : le débit () `$m^3.s^{-1}$` )
* pH : le pH de l'eau,
* dur : la concentration en calcium ( `$mg.L^{-1}$` ),
* pho	: concentration en phosphate ( `$mg.L^{-1}$` ),
* nit : concentration en nitrate  ( `$mg.L^{-1}$` ),
* amn	: concentration en ammonium ( `$mg.L^{-1}$` ),
* oxy	: concentration d'oxygène dissous ( `$mg.L^{-1}$` ), 
* dbo	: Demande biologique en oxygène ( `$mg.L^{-1}$` ).
]

--
<p class="question"> Comment visualiser au mieux ces données pour faire apparaître les liens entre variables et identifier des resemblances entre individus ?</p>

---
template: intro

## L'exemple de la morphologie des manchots

On a mesuré les caractéristiques morphologiques de divers manchots :

.pull-left[ Les 6 premières lignes (parmi 333 ) du jeu de données doubs.env
 
 ```
 # A tibble: 333 × 4
   bill_length_mm bill_depth_mm flipper_length_mm body_mass_g
            <dbl>         <dbl>             <int>       <int>
 1           39.1          18.7               181        3750
 2           39.5          17.4               186        3800
 3           40.3          18                 195        3250
 4           36.7          19.3               193        3450
 5           39.3          20.6               190        3650
 6           38.9          17.8               181        3625
 # … with 327 more rows
 ```
]

* bill_length_mm : la longueur du bec,

* bill_depth_mm : l'épaisseur du bec,

* flipper_length_mm : la longueur de la nageoire,

* body_mass_g : le poids du corps.

]

--
<p class="question"> Comment visualiser au mieux ces données pour faire apparaître les liens entre variables et identifier des resemblances entre individus ?</p>

---
template: intro 
## Formalisation

* Un individu statistique `$i$`, on a mesuré `$n$` individus.   
* Pour chaque individu, on a  mesuré `$p$` variables différentes.

Les données sont rangées dans un tableau à `$n$` lignes et `$p$` colonnes.

.pull-left[
<img src="./acp_data_pres.png" width="80%" height="30%" style="display: block; margin: auto;" />
]

* `$x_{\bullet k} = \frac{1}{n} \sum_{i=1}^n x_{ik}$` la valeur moyenne de la variable `$k$`,
* `$s_k  = \sqrt{\frac{1}{n} \sum_{i=1}^n (x_{ik}-x_{\bullet k})^2}$` l'écart type de la variable  `$k$`,

]

---
template: intro

## Même question dans des domaines très variés

* Analyse sensorielle : note du descripteur `$k$` pour le produit `$i$`

* Economie : valeur de l’indicateur `$k$` pour l’année `$i$`

* Génomique : expression du gène `$k$` pour l'e patient'échantilon  `$i$`

* Marketing : valeur d’indice de satisfaction `$k$` pour la marque `$i$`

etc ...

On a `$p$` variables mesurées sur `$n$` individus et on souhaite visualiser ces données pour comprendre les liens entre variables et les proximités entre individus.

---
template: intro

## Voir c'est comprendre : comment représenter l'information contenue dans ce tableau ?

Idée 1 : on représente les liens des variables 2 à 2

---
template: intro

## Voir c'est comprendre : comment représenter l'information contenue dans ce tableau ?

Idée 1 : on représente les liens des variables 2 à 2

On perd l'information sur les autres axes

---
template: intro

## Voir c'est comprendre : comment représenter l'information contenue dans ce tableau ?

Idée 1 : on représente les liens des variables 2 à 2

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---
template: intro

* Représenter sans perdre trop d'information 
* idéalement des individus éloignés dans le nuage initial, restent éloignés dans la représentation.

* quantifier l'information perdue dans la représentation 
* construire la représentation qui perd le moins d'information possible

* pour rendre comparable des variables exprimées dans des unités différentes, on les centre et on les réduit

on définit les mesures centrées réduites

`$$x_{ik}' = \frac{x_{ik} - x_{\bullet k}}{s_k}$$`

dans la suite, on omet le `$\prime$` pour alléger l'écriture.

---

# Une mesure de la quantité d'information

## L'inertie du nuage de points

Les données étant centrées, le point `$O$` de coordonnées `$(0, \ldots,0)$` est le centre du nuage de points.

.pull-left[
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]

`$$I =\frac{1}{n}  \sum_{i=1}^n d^2(x_{i}, 0) = \frac{1}{n}  \sum_{i=1}^n  \sum_{k=1}^p x_{ik}^2$$`
.orange[Remarque :]

`$$I =\sum_{k=1}^p  \left( \frac{1}{n}  \sum_{i=1}^n  x_{ik}^2 \right) = \sum_{k=1}^p  Var(x_{.k})$$`
Si les variables sont réduites

`$$I = p$$`
]

---

## L'inertie portée par rapport à un axe `$\Delta$`  passant par `$O$`.

---

## L'inertie portée par rapport à un axe `$\Delta$`  passant par `$O$`.

L'inertie `$I_{\Delta}$` quantifie l'nformation préservée sur la droite  et `$I_{\Delta^\perp}= I - I_{\Delta}$` 
l'information perdue (merci Pythagore)

`$$I_{\Delta} = \frac{1}{n} \sum_{i=1}^n d^2(\class{rouge}{x^{\Delta}_{i}}, 0)$$`
`$$I_{\Delta^\perp} =\frac{1}{n}  \sum_{i=1}^n d^2( \class{rouge}{x^{\Delta}_{i}}, x_{i})$$`

---

# Déterminer les meilleurs axes
 
--

## L'intuition

### Situation simplifiée à 2 variables

]
--

`$$\class{jaune}{I_{\Delta} =  1}$$`

`$$\class{vert}{I_{\Delta} =  1.89}$$`

]
L'inertie mesure la quatité d'information contenue dans les données

* Si on ne s'intéresse qu'à la variable poids du corps, on a 1 d'inertie.

* Si on ne s'intéresse qu'à la variable longueur des ailes, on a 1 d'inertie.

* Si on s'intéresse à la nouvelle variable (compromis entre le poids du corps et la longueur des ailes), on a 1.89. AVec une seule variable bien choisie, on représente presque toute l'information du jeu de données

---

---

Les .rouge[composantes principales] sont des combinaisons linéaires des variables initiales

Bien sûr ça n'a pas d'intérêt dans le cas de deux variables

---

## ACP sur l'exemple des manchots

```r
library(FactoMineR)
dta <- penguins %>% select(bill_length_mm, bill_depth_mm, flipper_length_mm, body_mass_g) 
penguins.PCA <- PCA(X = dta, ncp = 4, graph = FALSE)
penguins.PCA$eig
```

```
       eigenvalue percentage of variance cumulative percentage of variance
comp 1  2.7453557              68.633893                          68.63389
comp 2  0.7781172              19.452929                          88.08682
comp 3  0.3686425               9.216063                          97.30289
comp 4  0.1078846               2.697115                         100.00000
```

---

## ACP sur l'exemple des manchots

Il s'agit de la représentation en 2 dimensions qui déforment le moins possible le nuage global des points

A l'aide de 2 dimensions seulement, on représente 68.63 + 19.45 de l'inertie totale.

---
name: interprétation

# Interprétation

## Quelle information peut on extraire ?

### Sur l'exemple des manchots simplifiés (2var)

Comment représenter le lien entre les anciennes et les nouvelles variables ?

---
template: interprétation

## Qualité de la représentation des variables

On regarde l'angle entre les composantes principales et les variables

Les `$cos^2$` des variables :

```
      Dim.1        Dim.2   Plan.12
1 0.8294881 4.549411e-03 0.8340375
2 0.5652466 2.803042e-01 0.8455508
3 0.9134542 2.606919e-05 0.9134803
4 0.4371669 4.932375e-01 0.9304044
```
]

Une représentation graphique : Le cercle des corrélations

```r
plot(penguins.PCA, choix = "var")
```

---
template: interprétation

## Qualité de la représentation des individus

.pull-lefts[
<img src="./cos2_ind.png" width="40%" height="30%" style="display: block; margin: auto;" />
]

--
.pull-rightb[

* Le cosinus (au carré pour éviter les questions de signe) de l'angle entre la composante principale est le vecteur `$\vec{Ox_{i}}$` nous informe de la qualité de la représentation de l'individu.

* Pour mesurer la qualité de la représentation d'un individu sur le plan CP1, CP2, on additionne la qualité de la représentation sur chaque axe.

* On ne peut pas discuter des individus mal représentés

]

---
template: interprétation

## Visualisation conjointe individus et variables sur le premier plan principal  1-2

---
template: interprétation

## Visualisation conjointe individus et variables sur le premier plan principal 3-4

---
name: exemple

# Une analyse complète : les caractéristiques du Doubs

## La projection sur le premier plan principal

```r
doubs.PCA <- PCA(doubs.env, graph = FALSE, ncp = 10)
p1 <- plot(doubs.PCA, axes = c(1,2))
p2 <- plot(doubs.PCA, choix = "var", axes = c(1, 2) )
ggpubr::ggarrange(nrow = 1, plotlist = list(p1,p2), ncol = 2)
```

---
template: exemple

##  Faut-il aller voir les autres composantes principales

---
name: fin

# Pour finir en beauté

```r
library(Factoshiny)
PCAshiny(penguins)
```

---

# Ce qu'il faut retenir du cours

* L'inertie mesure la quantité d'information.

* La valeur propre associée à une composante principale est l'inertie du nuage de points limités à cet axe (nuage projeté sur l'axe).

* L'ACP est un changement de repère pour projeter de façon pertinente donc les composantes principales sont des combinaisons linéaires des variables d'origine.

* On perd de l'information en projetant, malgré toutes nos précautions.

* Vérifier la qualité de la représentation des variables et des individus avant toute interprétation.

* Deux variables orthogonales, sont des variables ayant un coefficient de corrélation de 0.

* On peut interpréter plus que 2 axes.