Analyse des Correspondances Multiples (ACM)¶

Mastère ESD - Introduction au Machine Learning¶

Librairies utilisées¶

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

from prince import MCA

Données utilisées¶

Données des passagers du Titanic à télécharger

In [2]:
df = pandas.read_csv("titanic.csv")
df.head()
Out[2]:
Class Sex Age Survived
0 1st Male Adult No
1 3rd Male Adult Yes
2 Crew Male Adult No
3 Crew Male Adult Yes
4 2nd Male Adult No

Calcul de ACM¶

In [3]:
mca = MCA(n_components = 10)
mca.fit(df)
Out[3]:
MCA(n_components=10)

Valeurs propres¶

In [4]:
print(mca.eigenvalues_)
print(mca.total_inertia_)
print(mca.explained_inertia_)

[0.4450794730527672, 0.3050437322075409, 0.25000600109695115, 0.2050373057546917, 0.17851515983455574, 0.1163183280534945, 8.735587841917454e-33, 4.0706080630322065e-33, 2.0357281136133345e-34, 1.990926374845764e-35]
1.5
[0.2967196487018448, 0.20336248813836058, 0.16667066739796743, 0.13669153716979446, 0.1190101065563705, 0.077545552035663, 5.82372522794497e-33, 2.7137387086881377e-33, 1.357152075742223e-34, 1.3272842498971758e-35]
In [5]:
eig = pandas.DataFrame(
    { 
        "Dimension" : ["Dim" + str(x + 1) for x in range(10)],
        "Valeur propre": mca.eigenvalues_,
        "% variance expliquée": numpy.round(mca.explained_inertia_, 4) * 100,
        "% variance expliquée cumulée": numpy.round(numpy.cumsum(mca.explained_inertia_), 4) * 100,
    }
)
eig
Out[5]:
Dimension Valeur propre % variance expliquée % variance expliquée cumulée
0 Dim1 4.450795e-01 29.67 29.67
1 Dim2 3.050437e-01 20.34 50.01
2 Dim3 2.500060e-01 16.67 66.68
3 Dim4 2.050373e-01 13.67 80.34
4 Dim5 1.785152e-01 11.90 92.25
5 Dim6 1.163183e-01 7.75 100.00
6 Dim7 8.735588e-33 0.00 100.00
7 Dim8 4.070608e-33 0.00 100.00
8 Dim9 2.035728e-34 0.00 100.00
9 Dim10 1.990926e-35 0.00 100.00

Représentation des individus¶

In [6]:
df_ind = pandas.DataFrame(mca.row_coordinates(df)).rename(columns = {i: "Dim"+str(i+1) for i in range(10)})
df_ind.head()
Out[6]:
Dim1 Dim2 Dim3 Dim4 Dim5 Dim6 Dim7 Dim8 Dim9 Dim10
0 0.055104 -0.541784 -0.446235 -0.172001 1.057355 -0.000739 -0.581726 0.454156 0.467342 -0.486753
1 0.263386 0.233400 -0.324961 -0.102804 -0.444944 -0.834395 -0.581726 0.454156 0.467342 -0.486753
2 -0.652721 -0.202892 0.039900 0.211090 -0.065702 0.232969 -0.581726 0.454156 0.467342 -0.486753
3 -0.061709 -0.469458 0.045706 0.573659 -0.521356 -0.386500 -0.581726 0.454156 0.467342 -0.486753
4 -0.132518 0.129916 1.194787 -0.423667 0.461580 -0.162475 -0.581726 0.454156 0.467342 -0.486753
In [7]:
g_ind = seaborn.lmplot(x = "Dim1", y = "Dim2", data = df_ind, fit_reg = False, 
                       height = 4, aspect = 3)
g_ind.fig.suptitle("Modalités en lignes")
plt.show()

Représentation des variables¶

In [8]:
df_var = pandas.DataFrame(mca.column_coordinates(df)).rename(columns = {i: "Dim"+str(i+1) for i in range(10)})
df_var
Out[8]:
Dim1 Dim2 Dim3 Dim4 Dim5 Dim6 Dim7 Dim8 Dim9 Dim10
Class_1st 1.151941 -1.231418 -0.890008 -0.049533 1.460940 -0.005902 0.051305 0.077772 -0.013464 0.07928
Class_2nd 0.651259 0.252522 2.392076 -0.505359 0.454055 -0.226544 0.051305 0.077772 -0.013464 0.07928
Class_3rd 0.130599 1.070050 -0.659068 -0.580899 -0.307933 -0.298099 0.051305 0.077772 -0.013464 0.07928
Class_Crew -0.736941 -0.482727 0.082275 0.644339 -0.437074 0.312928 0.051305 0.077772 -0.013464 0.07928
Sex_Female 1.574794 0.008927 -0.009280 -0.338706 -0.501996 0.914393 0.051305 0.077772 -0.013464 0.07928
Sex_Male -0.427587 -0.002424 0.002520 0.091965 0.136302 -0.248275 0.051305 0.077772 -0.013464 0.07928
Age_Adult -0.067828 -0.153321 -0.001242 -0.141832 -0.059028 -0.019826 0.051305 0.077772 -0.013464 0.07928
Age_Child 1.301802 2.942646 0.023828 2.722128 1.132903 0.380510 0.051305 0.077772 -0.013464 0.07928
Survived_No -0.509477 0.190238 -0.003751 -0.212137 0.248761 0.272994 0.051305 0.077772 -0.013464 0.07928
Survived_Yes 1.067680 -0.398669 0.007861 0.444562 -0.521314 -0.572097 0.051305 0.077772 -0.013464 0.07928
In [9]:
g_var = seaborn.lmplot(x = "Dim1", y = "Dim2", data = df_var, fit_reg = False, 
                       height = 4, aspect = 3)
g_var.fig.suptitle("Modalités en colonnes")
for i in df_var.index:
    plt.text(df_var.loc[i].Dim1, df_var.loc[i].Dim2, i, size = "xx-large")
plt.show()

Représentation simultanée¶

In [10]:
mca.plot_coordinates(df, figsize=(16, 8))
plt.show()
In [11]:
fig = plt.figure(figsize = (16,8))
g_simult = seaborn.lmplot(x = "Dim1", y = "Dim2", data = df_ind, fit_reg = False, 
                          height = 4, aspect = 3)

for i in df_var.index:
    plt.scatter(df_var.loc[i].Dim1, df_var.loc[i].Dim2, alpha = .25, c = "black")
    plt.text(df_var.loc[i].Dim1, df_var.loc[i].Dim2, i, size = "xx-large", color = "darkred", ha = "center")

g_simult.fig.suptitle("Représentation conjointe")
plt.show()

<Figure size 1152x576 with 0 Axes>