Extraction de connaissances à partir de données structurées et non structurées

Séance 3 : Analyse en Composantes Principales (ACP)

Utilisation de python

Librairies utilisées

Module sklearn (nommé scikit-learn) très utilisé dans le Machine Learning sous python

• Fonction PCA du module decomposition : réalise les calculs de l'ACP
• Fonction scale du module preprocessing : permet de normaliser les variables (moy = 0, var = 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

Données sur des iris disponibles ici

iris = pandas.read_table("https://fxjollois.github.io/donnees/Iris.txt", sep = "\t")
iris.head()

	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

Suppression de la variable Species

ACP uniquement sur des variables quantitatives

iris2 = iris.drop("Species", axis = 1)
iris2.head()

	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

Ici, scale() permet donc de normaliser chaque variable

pca = PCA(n_components = 4)
pca.fit(scale(iris2))

PCA(n_components=4)

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pca.components_

array([[ 0.52106591, -0.26934744,  0.5804131 ,  0.56485654],
       [ 0.37741762,  0.92329566,  0.02449161,  0.06694199],
       [-0.71956635,  0.24438178,  0.14212637,  0.63427274],
       [-0.26128628,  0.12350962,  0.80144925, -0.52359713]])

pca.singular_values_

array([20.92306556, 11.7091661 ,  4.69185798,  1.76273239])

pca.mean_

array([-4.73695157e-16, -7.81597009e-16, -4.26325641e-16, -4.73695157e-16])

Calcul des valeurs propres

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]

eig = pandas.DataFrame(
    {
        "Dimension" : ["Dim" + str(x + 1) for x in range(len(pca.explained_variance_))], 
        "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

	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

Premier graphique : diagramme des variances expliquées

plt.figure(figsize=(16, 6))

g_eig = seaborn.barplot(x = "Dimension", 
                        y = "% variance expliquée",
                        color = "lightseagreen",
                        data = eig)

# ligne indicatrice du seuil de sélection des dimensions
plt.axhline(y = 25, linewidth = .5, color = "dimgray", linestyle = "--") # 25 = 100 / 4 (nb dimensions)
plt.text(3.25, 26, "25%")

g_eig.set(ylabel = "Variance expliquée (%)")
g_eig.figure.suptitle("Variance expliquée par dimension")

plt.show()

Choix du nombre de facteurs

Deuxième graphique : évolution de la variance expliquée et variance expliqu&e cumulée

plt.figure(figsize=(16, 6))

eig2 = eig.filter(["Dimension", "% variance expliquée", "% cum. var. expliquée"]).melt(id_vars = "Dimension")
g_eig2 = seaborn.lineplot(x = "Dimension", 
                 y = "value",
                 hue = "variable",
                 data = eig2)

g_eig2.set(ylabel = "Variance expliquée (%)")
g_eig2.figure.suptitle("Variance expliquée par dimension")

plt.show()

Visualisation du nuage de points

Récupération des dimensions avec les espèces pour la représentation

iris_pca = pca.transform(scale(iris[iris.columns[:4]]))
iris_pca_df = pandas.DataFrame({
    "Dim1" : iris_pca[:,0], 
    "Dim2" : iris_pca[:,1], 
    "Species" : iris.Species
})
iris_pca_df.head()

	Dim1	Dim2	Species
0	-2.264703	0.480027	setosa
1	-2.080961	-0.674134	setosa
2	-2.364229	-0.341908	setosa
3	-2.299384	-0.597395	setosa
4	-2.389842	0.646835	setosa

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

Obligation de faire un calcul pour avoir les coordonnées des variables

pandas.DataFrame(pca.components_.T, columns=['PC'+str(i) for i in range(1, 5)], index=iris.columns[:4])

	PC1	PC2	PC3	PC4
Sepal Length	0.521066	0.377418	-0.719566	-0.261286
Sepal Width	-0.269347	0.923296	0.244382	0.123510
Petal Length	0.580413	0.024492	0.142126	0.801449
Petal Width	0.564857	0.066942	0.634273	-0.523597

coordvar = pca.components_.T * numpy.sqrt(pca.explained_variance_)
coordvar_df = pandas.DataFrame(coordvar, columns=['PC'+str(i) for i in range(1, 5)], index=iris.columns[:4])
coordvar_df

	PC1	PC2	PC3	PC4
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

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(coordvar_df.shape[0]):
    axes.text(coordvar_df["PC1"].iloc[j],
              coordvar_df["PC2"].iloc[j], 
              coordvar_df.index[j], 
              size = 25)
    axes.plot([0,coordvar_df["PC1"].iloc[j]], 
              [0,coordvar_df["PC2"].iloc[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)

g_pca = seaborn.lmplot(x = "Dim1", y = "Dim2", 
                       data = iris_pca_df, fit_reg = False, 
                       height = 6, aspect = 2)

g_pca.set(xlabel = "Dimension 1 (73%)", ylabel = "Dimension 2 (23 %)")
g_pca.fig.suptitle("Premier plan factoriel")

axes = g_pca.axes[0,0]
for j in range(coordvar_df.shape[0]):
    axes.text(3 * coordvar_df["PC1"].iloc[j],
              3 * coordvar_df["PC2"].iloc[j], 
              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

Gros intérêt de l'ACP : représenter une variable qualitative sur le plan factoriel

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()

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()

Exercice - Wine

Nous allons travailler sur des données concernant 3 types de vin. Elles sont disponibles sur cette page de l'UCI MLR. Il s'agit de 178 vins, réparties en 3 classes donc, et décrit par 13 variables quantitatives (lire la description dans le fichier wine.names pour plus d'informations).

Le code suivant permet de charger les données, et de nommer correctement les variables.

url = "https://archive.ics.uci.edu/ml/machine-learning-databases/wine/wine.data"
wine = pandas.read_csv(url, header = None, sep = ",")
wine.columns = ["class", "Alcohol", "Malic acid", "Ash", "Alcalinity of ash", "Magnesium", 
                "Total phenols", "Flavanoids", "Nonflavanoid phenols", "Proanthocyanins", 
                "Color intensity", "Hue", "OD280/OD315 of diluted wines", "Proline"]
wine

	class	Alcohol	Malic acid	Ash	Alcalinity of ash	Magnesium	Total phenols	Flavanoids	Nonflavanoid phenols	Proanthocyanins	Color intensity	Hue	OD280/OD315 of diluted wines	Proline
0	1	14.23	1.71	2.43	15.6	127	2.80	3.06	0.28	2.29	5.64	1.04	3.92	1065
1	1	13.20	1.78	2.14	11.2	100	2.65	2.76	0.26	1.28	4.38	1.05	3.40	1050
2	1	13.16	2.36	2.67	18.6	101	2.80	3.24	0.30	2.81	5.68	1.03	3.17	1185
3	1	14.37	1.95	2.50	16.8	113	3.85	3.49	0.24	2.18	7.80	0.86	3.45	1480
4	1	13.24	2.59	2.87	21.0	118	2.80	2.69	0.39	1.82	4.32	1.04	2.93	735
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
173	3	13.71	5.65	2.45	20.5	95	1.68	0.61	0.52	1.06	7.70	0.64	1.74	740
174	3	13.40	3.91	2.48	23.0	102	1.80	0.75	0.43	1.41	7.30	0.70	1.56	750
175	3	13.27	4.28	2.26	20.0	120	1.59	0.69	0.43	1.35	10.20	0.59	1.56	835
176	3	13.17	2.59	2.37	20.0	120	1.65	0.68	0.53	1.46	9.30	0.60	1.62	840
177	3	14.13	4.10	2.74	24.5	96	2.05	0.76	0.56	1.35	9.20	0.61	1.60	560

178 rows × 14 columns

Travail à faire

Vous devez donc réaliser les étapes suivantes :

• Décrire les données
• Réaliser une ACP centrée ou normée (choix à justifier)
• Produire les graphiques nécessaires à l’interprétation
• Identifier les classes sur le plan factoriel
• Que peut-on dire globalement ?

Worldwide Governance Indicators

La banque mondiale fournit un grand nombre de données, dont des indicateurs de gouvernance au niveau mondial (voir ici). Le code ci-dessous importe les données 2019 présentes dans le fichier WGI_Data.csv (que vous pouvez donc télécharger). Les informations concernant la définition des indicateurs sont les suivantes :

• CC : Control of Corruption
  • Control of Corruption captures perceptions of the extent to which public power is exercised for private gain, including both petty and grand forms of corruption, as well as capture" of the state by elites and private interests. Estimate gives the country's score on the aggregate indicator
• GE : Government Effectiveness
  • Government Effectiveness captures perceptions of the quality of public services, the quality of the civil service and the degree of its independence from political pressures, the quality of policy formulation and implementation, and the credibility of the government's commitment to such policies. Estimate gives the country's score on the aggregate indicator, in units of a standard normal distribution, i.e. ranging from approximately -2.5 to 2.5.
• PV :Political Stability and Absence of Violence/Terrorism
  • Political Stability and Absence of Violence/Terrorism measures perceptions of the likelihood of political instability and/or politically-motivated violence, including terrorism. Estimate gives the country's score on the aggregate indicator, in units of a standard normal distribution, i.e. ranging from approximately -2.5 to 2.5.
• RQ : Regulatory Quality
  • Regulatory Quality captures perceptions of the ability of the government to formulate and implement sound policies and regulations that permit and promote private sector development. Estimate gives the country's score on the aggregate indicator, in units of a standard normal distribution, i.e. ranging from approximately -2.5 to 2.5.
• RL : Rule of Law
  • Rule of Law captures perceptions of the extent to which agents have confidence in and abide by the rules of society, and in particular the quality of contract enforcement, property rights, the police, and the courts, as well as the likelihood of crime and violence. Estimate gives the country's score on the aggregate indicator, in units of a standard normal distribution, i.e. ranging from approximately -2.5 to 2.5.
• VA : Voice and Accountability
  • Voice and Accountability captures perceptions of the extent to which a country's citizens are able to participate in selecting their government, as well as freedom of expression, freedom of association, and a free media. Estimate gives the country's score on the aggregate indicator, in units of a standard normal distribution, i.e. ranging from approximately -2.5 to 2.5.

wgi = pandas.read_csv("https://fxjollois.github.io/donnees/WGI/wgi2019.csv")
wgi

	Country	Code	Voice and Accountability	Political Stability and Absence of Violence/Terrorism	Government Effectiveness	Regulatory Quality	Rule of Law	Control of Corruption
0	Aruba	ABW	1.294189	1.357372	1.029933	0.857360	1.263128	1.217238
1	Andorra	ADO	1.139154	1.615139	1.908749	1.228176	1.579939	1.234392
2	Afghanistan	AFG	-0.988032	-2.649407	-1.463875	-1.120555	-1.713527	-1.401076
3	Angola	AGO	-0.777283	-0.311101	-1.117144	-0.893871	-1.054343	-1.054683
4	Anguilla	AIA	NaN	1.367357	0.815824	0.846231	0.355737	1.234392
...	...	...	...	...	...	...	...	...
209	Serbia	SRB	0.026626	-0.091665	0.019079	0.113867	-0.119070	-0.445551
210	South Africa	ZAF	0.670388	-0.217931	0.367380	0.156172	-0.076408	0.084924
211	Congo, Dem. Rep.	ZAR	-1.365966	-1.808007	-1.627429	-1.509667	-1.786088	-1.538931
212	Zambia	ZMB	-0.286199	-0.102216	-0.675215	-0.554269	-0.462069	-0.640345
213	Zimbabwe	ZWE	-1.141875	-0.920179	-1.205337	-1.463199	-1.257009	-1.238796

214 rows × 8 columns

Travail à faire

Vous devez donc réaliser les étapes suivantes :

• Décrire rapidement les données
• Réaliser une ACP centrée ou normée (choix à justifier), sur les données
• Produire les graphiques nécessaires à l’interprétation
• Identifier les pays les plus représentatifs de chaque axe (en se basant sur leurs coordonnées par exempe)
• Que peut-on dire globalement ?