Analyse Factorielle des Correspondances Simples (AFC)¶

Mastère ESD - Introduction au Machine Learning¶

Librairies utilisées¶

Données accidents aux Etats-Unis sur la période 1973-1978 à télécharger

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

from prince import CA

Données utilisées¶

In [2]:
df = pandas.read_csv("USAccDeaths.csv", index_col = "Year")
df.columns.rename("Month", inplace = True)
df.index.rename("Year", inplace = True)
df
Out[2]:
Month January February March April May June July August September October November December
Year
1973 9007 8106 8928 9137 10017 10826 11317 10744 9713 9938 9161 8927
1974 7750 6981 8038 8422 8714 9512 10120 9823 8743 9129 8710 8680
1975 8162 7306 8124 7870 9387 9556 10093 9620 8285 8466 8160 8034
1976 7717 7461 7767 7925 8623 8945 10078 9179 8037 8488 7874 8647
1977 7792 6957 7726 8106 8890 9299 10625 9302 8314 8850 8265 8796
1978 7836 6892 7791 8192 9115 9434 10484 9827 9110 9070 8633 9240

Calcul de AFC¶

In [3]:
ca = CA(n_components=6)
ca.fit(df)
Out[3]:
CA(n_components=6)

Valeurs propres¶

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

[0.00037628345587226565, 0.0001856700837173522, 7.578318526200408e-05, 3.768944468224173e-05, 2.457913831520349e-05, 4.207611728329211e-33]
0.0007000053078490677
[0.5375437181019181, 0.2652409655119153, 0.10826087232804832, 0.053841655569800584, 0.03511278848831693, 6.010828319656741e-30]
In [5]:
eig = pandas.DataFrame(
    { 
        "Dimension" : ["Dim" + str(x + 1) for x in range(6)],
        "Valeur propre": ca.eigenvalues_,
        "% variance expliquée": numpy.round(ca.explained_inertia_, 4) * 100,
        "% variance expliquée cumulée": numpy.round(numpy.cumsum(ca.explained_inertia_), 4) * 100,
    }
)
eig
Out[5]:
Dimension Valeur propre % variance expliquée % variance expliquée cumulée
0 Dim1 3.762835e-04 53.75 53.75
1 Dim2 1.856701e-04 26.52 80.28
2 Dim3 7.578319e-05 10.83 91.10
3 Dim4 3.768944e-05 5.38 96.49
4 Dim5 2.457914e-05 3.51 100.00
5 Dim6 4.207612e-33 0.00 100.00

Choix des facteurs¶

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

plt.show()

Représentation des lignes¶

In [7]:
df_row = pandas.DataFrame(ca.row_coordinates(df)).rename(columns = {0: "Dim1", 1: "Dim2"})
df_row
Out[7]:
Dim1 Dim2 2 3 4 5
1973 0.017407 -0.013220 0.004260 0.004290 0.006992 -0.485789
1974 -0.013806 -0.014447 0.011227 -0.001709 -0.006420 -0.485789
1975 0.030026 -0.000779 -0.009876 -0.003899 -0.005081 -0.485789
1976 0.003461 0.024649 0.009397 -0.004836 0.002169 -0.485789
1977 -0.011677 0.010810 -0.004949 0.011591 -0.002571 -0.485789
1978 -0.026633 -0.004477 -0.010296 -0.005888 0.004086 -0.485789
In [8]:
g_row = seaborn.lmplot(x = "Dim1", y = "Dim2", data = df_row, fit_reg = False, 
                       height = 4, aspect = 3)
g_row.fig.suptitle("Modalités en lignes")
for i in df_row.index:
    plt.text(df_row.loc[i].Dim1, df_row.loc[i].Dim2, i, size = "xx-large")
plt.show()

Représentation des colonnes¶

In [9]:
df_col = pandas.DataFrame(ca.column_coordinates(df)).rename(columns = {0: "Dim1", 1: "Dim2"})
df_col
Out[9]:
Dim1 Dim2 2 3 4 5
January 0.023375 0.003002 -0.004639 0.001429 0.001994 1.0
February 0.029632 0.024436 0.014401 -0.006230 0.005528 1.0
March 0.018686 -0.000595 0.006133 -0.003373 -0.006826 1.0
April -0.006568 -0.003396 0.013345 0.004948 -0.001407 1.0
May 0.018069 0.002971 -0.018906 -0.002541 -0.001863 1.0
June 0.013585 -0.011738 -0.001110 0.005372 0.000419 1.0
July -0.006641 0.014813 -0.006718 0.011894 0.000836 1.0
August 0.001480 -0.010823 0.000180 -0.007366 -0.002311 1.0
September -0.014577 -0.019645 -0.003744 -0.005379 0.011479 1.0
October -0.013734 -0.007586 0.008094 0.006472 0.002180 1.0
November -0.015175 -0.012299 0.000979 -0.001664 -0.008809 1.0
December -0.041467 0.024106 -0.002788 -0.006655 -0.000887 1.0
In [10]:
g_col = seaborn.lmplot(x = "Dim1", y = "Dim2", data = df_col, fit_reg = False, 
                       height = 4, aspect = 3)
g_col.fig.suptitle("Modalités en colonnes")
for i in df_col.index:
    plt.text(df_col.loc[i].Dim1, df_col.loc[i].Dim2, i, size = "xx-large")
plt.show()

Représentation simultanée¶

In [11]:
ca.plot_coordinates(df,
                    figsize=(16, 8));
In [12]:
fig = plt.figure(figsize = (16,8))
plt.xlim(-.05, .05)
plt.ylim(-.05, .05)

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

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

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

Importance de chaque modalité¶

In [13]:
plt.figure(figsize = (16, 4))
plt.bar(df.index, ca.row_masses_, color = "darkblue")
plt.show()
In [14]:
plt.figure(figsize = (16, 4))
plt.bar(df.columns, ca.col_masses_, color = "darkred")
plt.show()

AFC - Profils lignes¶

In [15]:
p_row = df.div(df.sum(axis=1), axis=0).round(4) * 100
p_row
Out[15]:
Month January February March April May June July August September October November December
Year
1973 7.78 7.00 7.71 7.89 8.65 9.35 9.77 9.28 8.39 8.58 7.91 7.71
1974 7.41 6.67 7.68 8.05 8.33 9.09 9.67 9.39 8.36 8.73 8.33 8.30
1975 7.92 7.09 7.88 7.64 9.11 9.27 9.79 9.33 8.04 8.21 7.92 7.80
1976 7.66 7.41 7.71 7.87 8.56 8.88 10.00 9.11 7.98 8.43 7.82 8.58
1977 7.57 6.76 7.51 7.88 8.64 9.03 10.32 9.04 8.08 8.60 8.03 8.55
1978 7.42 6.53 7.38 7.76 8.63 8.93 9.93 9.30 8.62 8.59 8.17 8.75
In [16]:
fig = plt.figure(figsize = (16, 8))
base = numpy.zeros(6)
for m in p_row.columns:
    plt.bar(p_row.index, p_row[m], bottom = base, label = m)
    base = base + p_row[m]
plt.margins(0.05, 0.15)
plt.legend(ncol = 6, loc = 9)
plt.show()

AFC - Profils colonnes¶

In [17]:
p_col = df.div(df.sum(axis=0), axis=1).round(4) * 100
p_col
Out[17]:
Month January February March April May June July August September October November December
Year
1973 18.66 18.55 18.46 18.40 18.30 18.80 18.04 18.37 18.61 18.42 18.03 17.06
1974 16.06 15.97 16.62 16.96 15.92 16.52 16.14 16.79 16.75 16.92 17.14 16.59
1975 16.91 16.72 16.79 15.85 17.15 16.60 16.09 16.45 15.87 15.69 16.06 15.35
1976 15.99 17.07 16.06 15.96 15.75 15.54 16.07 15.69 15.40 15.74 15.50 16.53
1977 16.14 15.92 15.97 16.33 16.24 16.15 16.94 15.90 15.93 16.41 16.27 16.81
1978 16.24 15.77 16.11 16.50 16.65 16.39 16.72 16.80 17.45 16.81 16.99 17.66
In [18]:
fig = plt.figure(figsize = (16, 8))
base = numpy.zeros(12)
for m in p_col.index:
    plt.bar(p_col.columns, p_col.loc[m], bottom = base, label = m)
    base = base + p_col.loc[m]
plt.margins(0.05, 0.15)
plt.legend(ncol = 6, loc = 9)
plt.show()