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)
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
MCA(n_components=10)

Valeurs propres¶

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

[4.45079473e-01 3.05043732e-01 2.50006001e-01 2.05037306e-01
 1.78515160e-01 1.16318328e-01 4.85494016e-33 1.42400212e-33
 3.71815520e-34]
1.5000000000000373
[2.96719649e+01 2.03362488e+01 1.66670667e+01 1.36691537e+01
 1.19010107e+01 7.75455520e+00 3.23662678e-31 9.49334747e-32
 2.47877013e-32]
In [5]:
eig = pandas.DataFrame(
    { 
        "Dimension" : ["Dim" + str(x + 1) for x in range(9)],
        "Valeur propre": mca.eigenvalues_,
        "% variance expliquée": numpy.round(mca.percentage_of_variance_, 4),
        "% variance expliquée cumulée": numpy.round(numpy.cumsum(mca.percentage_of_variance_), 4),
    }
)
eig
Out[5]:
Dimension Valeur propre % variance expliquée % variance expliquée cumulée
0 Dim1 4.450795e-01 29.6720 29.6720
1 Dim2 3.050437e-01 20.3362 50.0082
2 Dim3 2.500060e-01 16.6671 66.6753
3 Dim4 2.050373e-01 13.6692 80.3444
4 Dim5 1.785152e-01 11.9010 92.2454
5 Dim6 1.163183e-01 7.7546 100.0000
6 Dim7 4.854940e-33 0.0000 100.0000
7 Dim8 1.424002e-33 0.0000 100.0000
8 Dim9 3.718155e-34 0.0000 100.0000
In [6]:
plt.figure(figsize=(16, 6))
g_eig = seaborn.barplot(x = "Dimension", 
                        y = "% variance expliquée",
                        color = "lightseagreen",
                        data = eig)
g_eig.set(ylabel = "Variance expliquée (%)")
g_eig.figure.suptitle("Variance expliquée par dimension")

plt.show()

Représentation des individus¶

In [7]:
df_ind = pandas.DataFrame(mca.row_coordinates(df)).rename(columns = {i: "Dim"+str(i+1) for i in range(10)})
df_ind.head()
Out[7]:
Dim1 Dim2 Dim3 Dim4 Dim5 Dim6 Dim7 Dim8 Dim9
0 0.055104 -0.541784 -0.446235 -0.172001 1.057355 -0.000739 0.004029 -0.30581 -0.723675
1 0.263386 0.233400 -0.324961 -0.102804 -0.444944 -0.834395 0.004029 -0.30581 -0.723675
2 -0.652721 -0.202892 0.039900 0.211090 -0.065702 0.232969 0.004029 -0.30581 -0.723675
3 -0.061709 -0.469458 0.045706 0.573659 -0.521356 -0.386500 0.004029 -0.30581 -0.723675
4 -0.132518 0.129916 1.194787 -0.423667 0.461580 -0.162475 0.004029 -0.30581 -0.723675
In [8]:
g_ind = seaborn.lmplot(x = "Dim1", y = "Dim2", data = df_ind, fit_reg = False, 
                       height = 6, aspect = 2)
g_ind.fig.suptitle("Modalités en lignes")
plt.show()
In [9]:
df_ind_bis = df_ind.filter(["Dim1", "Dim2"]).groupby(["Dim1", "Dim2"]).agg(Nb = ("Dim1", "count")).reset_index()
df_ind_bis.head()
Out[9]:
Dim1 Dim2 Nb
0 -0.652721 -0.202892 670
1 -0.327626 0.499967 387
2 -0.132518 0.129916 154
3 -0.061709 -0.469458 192
4 0.055104 -0.541784 118
In [10]:
plt.figure(figsize=(16, 6))
g_ind = seaborn.scatterplot(x = "Dim1", y = "Dim2", size = "Nb", data = df_ind_bis)
g_ind.figure.suptitle("Modalités en lignes")
plt.show()

Représentation des variables¶

In [11]:
df_var = pandas.DataFrame(mca.column_coordinates(df)).rename(columns = {i: "Dim"+str(i+1) for i in range(10)})
df_var

/usr/local/lib/python3.9/site-packages/prince/ca.py:206: FutureWarning: is_sparse is deprecated and will be removed in a future version. Check `isinstance(dtype, pd.SparseDtype)` instead.
  is_sparse = X.dtypes.apply(pd.api.types.is_sparse).all()
Out[11]:
Dim1 Dim2 Dim3 Dim4 Dim5 Dim6 Dim7 Dim8 Dim9
Class_1st 1.151941 -1.231418 -0.890008 -0.049533 1.460940 -0.005902 0.395783 0.006324 -0.171133
Class_2nd 0.651259 0.252522 2.392076 -0.505359 0.454055 -0.226544 0.395783 0.006324 -0.171133
Class_3rd 0.130599 1.070050 -0.659068 -0.580899 -0.307933 -0.298099 0.395783 0.006324 -0.171133
Class_Crew -0.736941 -0.482727 0.082275 0.644339 -0.437074 0.312928 0.395783 0.006324 -0.171133
Sex_Female 1.574794 0.008927 -0.009280 -0.338706 -0.501996 0.914393 0.395783 0.006324 -0.171133
Sex_Male -0.427587 -0.002424 0.002520 0.091965 0.136302 -0.248275 0.395783 0.006324 -0.171133
Age_Adult -0.067828 -0.153321 -0.001242 -0.141832 -0.059028 -0.019826 0.395783 0.006324 -0.171133
Age_Child 1.301802 2.942646 0.023828 2.722128 1.132903 0.380510 0.395783 0.006324 -0.171133
Survived_No -0.509477 0.190238 -0.003751 -0.212137 0.248761 0.272994 0.395783 0.006324 -0.171133
Survived_Yes 1.067680 -0.398669 0.007861 0.444562 -0.521314 -0.572097 0.395783 0.006324 -0.171133
In [12]:
g_var = seaborn.lmplot(x = "Dim1", y = "Dim2", data = df_var, fit_reg = False, 
                       height = 6, aspect = 2)
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 [13]:
plt.figure(figsize=(16, 8))

g_simult = seaborn.scatterplot(x = "Dim1", y = "Dim2", size = "Nb", data = df_ind_bis)

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.figure.suptitle("Représentation conjointe")

plt.show()