Analyse en Composantes Principales (ACP) avec Python¶

Mastère ESD - Introduction au Machine Learning¶

Librairies utilisées¶

Données sur des iris à télécharger

In [1]:
import pandas
import numpy
import matplotlib.pyplot as plt
import seaborn
seaborn.set_style("white")

from sklearn.decomposition import PCA
from sklearn.preprocessing import scale

Données utilisées¶

In [2]:
iris = pandas.read_table("https://fxjollois.github.io/donnees/Iris.txt", sep = "\t")
iris.head()
Out[2]:
Sepal Length Sepal Width Petal Length Petal Width Species
0 5.1 3.5 1.4 0.2 setosa
1 4.9 3.0 1.4 0.2 setosa
2 4.7 3.2 1.3 0.2 setosa
3 4.6 3.1 1.5 0.2 setosa
4 5.0 3.6 1.4 0.2 setosa
In [3]:
iris2 = iris.drop("Species", axis = 1)
iris2.head()
Out[3]:
Sepal Length Sepal Width Petal Length Petal Width
0 5.1 3.5 1.4 0.2
1 4.9 3.0 1.4 0.2
2 4.7 3.2 1.3 0.2
3 4.6 3.1 1.5 0.2
4 5.0 3.6 1.4 0.2

Réalisation de l'ACP¶

In [4]:
pca = PCA(n_components = 4)
pca.fit(scale(iris2))
Out[4]:
PCA(n_components=4)

Calcul des valeurs propres¶

In [5]:
print(pca.explained_variance_)
print(pca.explained_variance_ratio_)

[2.93808505 0.9201649  0.14774182 0.02085386]
[0.72962445 0.22850762 0.03668922 0.00517871]
In [7]:
eig = pandas.DataFrame(
    {
        "Dimension" : ["Dim" + str(x + 1) for x in range(iris2.shape[1])], 
        "Valeur propre" : pca.explained_variance_,
        "% variance expliquée" : numpy.round(pca.explained_variance_ratio_ * 100),
        "% cum. var. expliquée" : numpy.round(numpy.cumsum(pca.explained_variance_ratio_) * 100)
    },
    columns = ["Dimension", "Valeur propre", "% variance expliquée", "% cum. var. expliquée"]
)
eig
Out[7]:
Dimension Valeur propre % variance expliquée % cum. var. expliquée
0 Dim1 2.938085 73.0 73.0
1 Dim2 0.920165 23.0 96.0
2 Dim3 0.147742 4.0 99.0
3 Dim4 0.020854 1.0 100.0

Choix du nombre de facteurs¶

In [8]:
plt.figure(figsize=(16, 6))
g_eig = seaborn.barplot(x = "Dimension", 
                        y = "% variance expliquée",
                        palette = ["lightseagreen"],
                        data = eig)
plt.text(3.25, 26, "25%")
plt.axhline(y = 25, linewidth = .5, color = "dimgray", linestyle = "--") # 25 = 100 / 4 (nb dimensions)
g_eig.set(ylabel = "Variance expliquée (%)")
g_eig.figure.suptitle("Variance expliquée par dimension")

plt.show()
In [9]:
plt.figure(figsize=(16, 6))
g_eig = seaborn.pointplot(x = "Dimension", 
                          y = "% variance expliquée",
                          data = eig)
g_eig.set(ylabel = "Variance expliquée (%)")
g_eig.figure.suptitle("Variance expliquée par dimension")

plt.show()

Visualisation du nuage de points¶

In [22]:
iris_pca_df = pandas.DataFrame(pca.transform(iris2), 
                               columns = ["Dim" + str(i+1) for i in range(iris2.shape[1])]) \
                    .assign(Species = iris.Species)
iris_pca_df.head()
Out[22]:
Dim1 Dim2 Dim3 Dim4 Species
0 2.640270 5.204041 -2.488621 0.117033 setosa
1 2.670730 4.666910 -2.466898 0.107536 setosa
2 2.454606 4.773636 -2.288321 0.104350 setosa
3 2.545517 4.648463 -2.212378 0.278417 setosa
4 2.561228 5.258629 -2.392226 0.155513 setosa
In [23]:
g_pca = seaborn.lmplot(x = "Dim1", y = "Dim2", data = iris_pca_df, fit_reg = False, 
                       height = 4, aspect = 3)
g_pca.set(xlabel = "Dimension 1 (73%)", ylabel = "Dimension 2 (23 %)")
g_pca.fig.suptitle("Premier plan factoriel")

plt.show()

Représentation des variables¶

In [24]:
coordvar = pca.components_.T * numpy.sqrt(pca.explained_variance_)
coordvar_df = pandas.DataFrame(coordvar, 
                               columns = ["Dim" + str(i+1) for i in range(iris2.shape[1])], 
                               index = iris.columns[:4])
coordvar_df
Out[24]:
Dim1 Dim2 Dim3 Dim4
Sepal Length 0.893151 0.362039 -0.276581 -0.037732
Sepal Width -0.461684 0.885673 0.093934 0.017836
Petal Length 0.994877 0.023494 0.054629 0.115736
Petal Width 0.968212 0.064214 0.243797 -0.075612
In [25]:
fig, axes = plt.subplots(figsize = (10, 10))
fig.suptitle("Cercle des corrélations")
axes.set_xlim(-1, 1)
axes.set_ylim(-1, 1)
axes.axvline(x = 0, color = 'lightgray', linestyle = '--', linewidth = 1)
axes.axhline(y = 0, color = 'lightgray', linestyle = '--', linewidth = 1)
for j in range(4):
    axes.text(coordvar_df["Dim1"][j],coordvar_df["Dim2"][j], coordvar_df.index[j], size = 25)
    axes.plot([0,coordvar_df["Dim1"][j]], [0,coordvar_df["Dim2"][j]], color = "gray", linestyle = 'dashed')
plt.gca().add_artist(plt.Circle((0,0),1,color='blue',fill=False))

plt.show()

Représentation simultanée¶

Analyse conjointe des deux nuages (individus et variables)

In [26]:
g_pca = seaborn.lmplot(x = "Dim1", y = "Dim2", data = iris_pca_df, fit_reg = False, 
                       height = 4, aspect = 3)
g_pca.set(xlabel = "Dimension 1 (73%)", ylabel = "Dimension 2 (23 %)")
axes = g_pca.axes[0,0]
for j in range(4):
    axes.text(coordvar_df["Dim1"][j] * 3 + iris_pca_df.Dim1.mean(),
              coordvar_df["Dim2"][j] * 1.5 + iris_pca_df.Dim2.mean(), 
              coordvar_df.index[j], size = 25)
plt.axvline(x = iris_pca_df.Dim1.mean(), linewidth = .5, color = "dimgray", linestyle = "--")
plt.axhline(y = iris_pca_df.Dim2.mean(), linewidth = .5, color = "dimgray", linestyle = "--")

plt.show()

Visualisation des espèces sur le premier plan factoriel¶

In [27]:
g_pca = seaborn.lmplot(x = "Dim1", y = "Dim2", hue = "Species", data = iris_pca_df, fit_reg = False, 
                       height = 4, aspect = 3)
g_pca.set(xlabel = "Dimension 1 (73%)", ylabel = "Dimension 2 (23 %)")
g_pca.fig.suptitle("Premier plan factoriel")

plt.show()
In [28]:
g_pca2 = seaborn.lmplot(x = "Dim1", y = "Dim2", hue = "Species", col = "Species", 
                        data = iris_pca_df, fit_reg = False,
                        height = 4, aspect = 1.1)
g_pca2.set(xlabel = "Dimension 1 (73%)", ylabel = "Dimension 2 (23 %)")
g_pca.fig.suptitle("Premier plan factoriel")

plt.show()
In [ ]: