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 = 5) # minimum du nombre de modalités - 1
ca.fit(df)
Out[3]:
<prince.ca.CA at 0x10ca5cbb0>

Valeurs propres¶

In [4]:
print(ca.eigenvalues_)
print(ca.total_inertia_)
print(ca.percentage_of_variance_) # CHANGEMENT ICI

[3.76283456e-04 1.85670084e-04 7.57831853e-05 3.76894447e-05
 2.45791383e-05]
0.0007000053078490678
[53.75437181 26.52409655 10.82608723  5.38416556  3.51127885]
In [5]:
eig = pandas.DataFrame(
    { 
        "Dimension" : ["Dim" + str(x + 1) for x in range(5)],
        "% variance expliquée": numpy.round(ca.percentage_of_variance_, 4) * 100,
        "% variance expliquée cumulée": numpy.round(numpy.cumsum(ca.percentage_of_variance_), 4) * 100,
    }
)
eig
Out[5]:
Dimension % variance expliquée % variance expliquée cumulée
0 Dim1 5375.44 5375.44
1 Dim2 2652.41 8027.85
2 Dim3 1082.61 9110.46
3 Dim4 538.42 9648.87
4 Dim5 351.13 10000.00

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()

/usr/local/lib/python3.9/site-packages/seaborn/_core.py:1218: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead
  if pd.api.types.is_categorical_dtype(vector):
/usr/local/lib/python3.9/site-packages/seaborn/_core.py:1218: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead
  if pd.api.types.is_categorical_dtype(vector):
/usr/local/lib/python3.9/site-packages/seaborn/_core.py:1218: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead
  if pd.api.types.is_categorical_dtype(vector):

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
Year
1973 0.017407 -0.013220 0.004260 0.004290 0.006992
1974 -0.013806 -0.014447 0.011227 -0.001709 -0.006420
1975 0.030026 -0.000779 -0.009876 -0.003899 -0.005081
1976 0.003461 0.024649 0.009397 -0.004836 0.002169
1977 -0.011677 0.010810 -0.004949 0.011591 -0.002571
1978 -0.026633 -0.004477 -0.010296 -0.005888 0.004086
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

/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[9]:
Dim1 Dim2 2 3 4
Month
January 0.023375 0.003002 -0.004639 0.001429 0.001994
February 0.029632 0.024436 0.014401 -0.006230 0.005528
March 0.018686 -0.000595 0.006133 -0.003373 -0.006826
April -0.006568 -0.003396 0.013345 0.004948 -0.001407
May 0.018069 0.002971 -0.018906 -0.002541 -0.001863
June 0.013585 -0.011738 -0.001110 0.005372 0.000419
July -0.006641 0.014813 -0.006718 0.011894 0.000836
August 0.001480 -0.010823 0.000180 -0.007366 -0.002311
September -0.014577 -0.019645 -0.003744 -0.005379 0.011479
October -0.013734 -0.007586 0.008094 0.006472 0.002180
November -0.015175 -0.012299 0.000979 -0.001664 -0.008809
December -0.041467 0.024106 -0.002788 -0.006655 -0.000887
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]:
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 [12]:
plt.figure(figsize = (16, 4))
plt.bar(df.index, ca.row_masses_, color = "darkblue")
plt.show()
In [13]:
plt.figure(figsize = (16, 4))
plt.bar(df.columns, ca.col_masses_, color = "darkred")
plt.show()

AFC - Profils lignes¶

In [14]:
p_row = df.div(df.sum(axis=1), axis=0).round(4) * 100
p_row
Out[14]:
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 [15]:
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 [16]:
p_col = df.div(df.sum(axis=0), axis=1).round(4) * 100
p_col
Out[16]:
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 [17]:
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()