In [ ]:
#scree plot
acm.plot_eigenvalues()
#dossier de travail
import os
os.chdir("C:/Users/ricco/Desktop/demo")
#importation des individus actifs
import pandas
ind_actifs = pandas.read_excel("loisirs.xlsx",sheet_name="data")
ind_actifs.info()
<class 'pandas.core.frame.DataFrame'> RangeIndex: 7836 entries, 0 to 7835 Data columns (total 23 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 Reading 7836 non-null object 1 Listening_music 7836 non-null object 2 Cinema 7836 non-null object 3 Show 7836 non-null object 4 Exhibition 7836 non-null object 5 Computer 7836 non-null object 6 Sport 7836 non-null object 7 Walking 7836 non-null object 8 Travelling 7836 non-null object 9 Playing_music 7836 non-null object 10 Collecting 7836 non-null object 11 Volunteering 7836 non-null object 12 Mechanic 7836 non-null object 13 Gardening 7836 non-null object 14 Knitting 7836 non-null object 15 Cooking 7836 non-null object 16 Fishing 7836 non-null object 17 TV 7836 non-null object 18 Sex 7836 non-null object 19 Age 7836 non-null object 20 Marital_status 7836 non-null object 21 Profession 7836 non-null object 22 nb_activitees 7836 non-null int64 dtypes: int64(1), object(22) memory usage: 1.4+ MB
#isoler les variables actives
X_actifs = ind_actifs[ind_actifs.columns[:18]]
X_actifs.columns
Index(['Reading', 'Listening_music', 'Cinema', 'Show', 'Exhibition',
'Computer', 'Sport', 'Walking', 'Travelling', 'Playing_music',
'Collecting', 'Volunteering', 'Mechanic', 'Gardening', 'Knitting',
'Cooking', 'Fishing', 'TV'],
dtype='object')
#nombre d'obs.
n = X_actifs.shape[0]
print(n)
#nombre de var.
p = X_actifs.shape[1]
print(p)
7836 18
#nombre total de modalités
import numpy
M = numpy.sum(X_actifs.apply(axis=0,func=lambda x:x.value_counts().shape[0]))
print(M)
39
#nombre max de facteurs (Livre, page 293)
print(M-p)
21
#inertie totale - Livre, page 299
print(M/p-1)
1.1666666666666665
#installation à la volée du package (éventuellement)
#!pip install fanalysis
#importation du module
from fanalysis.mca import MCA
#instanciation
acm = MCA(row_labels=X_actifs.index,var_labels=X_actifs.columns)
#exécution
acm.fit(X_actifs.values)
#valeurs propres
acm.eig_
array([[1.94203920e-01, 8.17135274e-02, 7.14523138e-02, 6.35737241e-02,
5.88448160e-02, 5.58523058e-02, 5.55688411e-02, 5.32930394e-02,
5.27382461e-02, 4.91374174e-02, 4.67390321e-02, 4.50296758e-02,
4.38053722e-02, 4.35085505e-02, 4.09629230e-02, 3.83005299e-02,
3.72658100e-02, 3.64229806e-02, 3.53180258e-02, 3.23581436e-02,
3.05774725e-02],
[1.66460503e+01, 7.00401663e+00, 6.12448404e+00, 5.44917635e+00,
5.04384137e+00, 4.78734050e+00, 4.76304352e+00, 4.56797480e+00,
4.52042110e+00, 4.21177863e+00, 4.00620275e+00, 3.85968650e+00,
3.75474619e+00, 3.72930433e+00, 3.51110768e+00, 3.28290257e+00,
3.19421229e+00, 3.12196977e+00, 3.02725935e+00, 2.77355517e+00,
2.62092621e+00],
[1.66460503e+01, 2.36500669e+01, 2.97745509e+01, 3.52237273e+01,
4.02675686e+01, 4.50549091e+01, 4.98179527e+01, 5.43859275e+01,
5.89063486e+01, 6.31181272e+01, 6.71243299e+01, 7.09840164e+01,
7.47387626e+01, 7.84680670e+01, 8.19791746e+01, 8.52620772e+01,
8.84562895e+01, 9.15782593e+01, 9.46055186e+01, 9.73790738e+01,
1.00000000e+02]])
#petite vérif. inertie totale = somme des val.p
print(numpy.sum(acm.eig_[0]))
1.1666666666666672
#scree plot
acm.plot_eigenvalues()
#pourcentage de variance cumulée
acm.plot_eigenvalues("cumulative")
#infos
info_lig = acm.row_topandas()
info_lig.columns
Index(['row_coord_dim1', 'row_coord_dim2', 'row_coord_dim3', 'row_coord_dim4',
'row_coord_dim5', 'row_coord_dim6', 'row_coord_dim7', 'row_coord_dim8',
'row_coord_dim9', 'row_coord_dim10', 'row_coord_dim11',
'row_coord_dim12', 'row_coord_dim13', 'row_coord_dim14',
'row_coord_dim15', 'row_coord_dim16', 'row_coord_dim17',
'row_coord_dim18', 'row_coord_dim19', 'row_coord_dim20',
'row_coord_dim21', 'row_contrib_dim1', 'row_contrib_dim2',
'row_contrib_dim3', 'row_contrib_dim4', 'row_contrib_dim5',
'row_contrib_dim6', 'row_contrib_dim7', 'row_contrib_dim8',
'row_contrib_dim9', 'row_contrib_dim10', 'row_contrib_dim11',
'row_contrib_dim12', 'row_contrib_dim13', 'row_contrib_dim14',
'row_contrib_dim15', 'row_contrib_dim16', 'row_contrib_dim17',
'row_contrib_dim18', 'row_contrib_dim19', 'row_contrib_dim20',
'row_contrib_dim21', 'row_cos2_dim1', 'row_cos2_dim2', 'row_cos2_dim3',
'row_cos2_dim4', 'row_cos2_dim5', 'row_cos2_dim6', 'row_cos2_dim7',
'row_cos2_dim8', 'row_cos2_dim9', 'row_cos2_dim10', 'row_cos2_dim11',
'row_cos2_dim12', 'row_cos2_dim13', 'row_cos2_dim14', 'row_cos2_dim15',
'row_cos2_dim16', 'row_cos2_dim17', 'row_cos2_dim18', 'row_cos2_dim19',
'row_cos2_dim20', 'row_cos2_dim21'],
dtype='object')
#nuage de points - premier plan factoriel
#points "anonymes" ici, peu intéressant affichage des index
#mais possible - cf. https://github.com/OlivierGarciaDev/fanalysis/blob/master/doc/mca_tutorial.ipynb
#acm.mapping_row(num_x_axis=1,num_y_axis=2)
#coordonnées et infos
info_col = acm.col_topandas()
info_col.columns
Index(['col_coord_dim1', 'col_coord_dim2', 'col_coord_dim3', 'col_coord_dim4',
'col_coord_dim5', 'col_coord_dim6', 'col_coord_dim7', 'col_coord_dim8',
'col_coord_dim9', 'col_coord_dim10', 'col_coord_dim11',
'col_coord_dim12', 'col_coord_dim13', 'col_coord_dim14',
'col_coord_dim15', 'col_coord_dim16', 'col_coord_dim17',
'col_coord_dim18', 'col_coord_dim19', 'col_coord_dim20',
'col_coord_dim21', 'col_contrib_dim1', 'col_contrib_dim2',
'col_contrib_dim3', 'col_contrib_dim4', 'col_contrib_dim5',
'col_contrib_dim6', 'col_contrib_dim7', 'col_contrib_dim8',
'col_contrib_dim9', 'col_contrib_dim10', 'col_contrib_dim11',
'col_contrib_dim12', 'col_contrib_dim13', 'col_contrib_dim14',
'col_contrib_dim15', 'col_contrib_dim16', 'col_contrib_dim17',
'col_contrib_dim18', 'col_contrib_dim19', 'col_contrib_dim20',
'col_contrib_dim21', 'col_cos2_dim1', 'col_cos2_dim2', 'col_cos2_dim3',
'col_cos2_dim4', 'col_cos2_dim5', 'col_cos2_dim6', 'col_cos2_dim7',
'col_cos2_dim8', 'col_cos2_dim9', 'col_cos2_dim10', 'col_cos2_dim11',
'col_cos2_dim12', 'col_cos2_dim13', 'col_cos2_dim14', 'col_cos2_dim15',
'col_cos2_dim16', 'col_cos2_dim17', 'col_cos2_dim18', 'col_cos2_dim19',
'col_cos2_dim20', 'col_cos2_dim21'],
dtype='object')
#coordoonées dans le premier plan factoriel
coord_col = info_col[['col_coord_dim1', 'col_coord_dim2']]
print(coord_col)
col_coord_dim1 col_coord_dim2 Reading_n 0.733412 -0.041586 Reading_y -0.362301 0.020543 Listening_music_n 0.839848 0.254103 Listening_music_y -0.318176 -0.096267 Cinema_n 0.527020 0.307260 Cinema_y -0.718375 -0.418822 Show_n 0.405560 0.112424 Show_y -0.943322 -0.261496 Exhibition_n 0.431890 -0.007079 Exhibition_y -0.929998 0.015243 Computer_n 0.455623 0.200060 Computer_y -0.681404 -0.299199 Sport_n 0.415821 0.181959 Sport_y -0.653901 -0.286140 Walking_n 0.412714 -0.338255 Walking_y -0.397413 0.325715 Travelling_n 0.496057 0.007750 Travelling_y -0.711117 -0.011110 Playing_music_n 0.214572 0.029402 Playing_music_y -0.972008 -0.133189 Collecting_n 0.067469 -0.057665 Collecting_y -0.559681 0.478349 Volunteering_n 0.139880 -0.045265 Volunteering_y -0.751258 0.243108 Mechanic_n 0.307265 -0.345452 Mechanic_y -0.394287 0.443290 Gardening_n 0.177713 -0.553866 Gardening_y -0.261996 0.816546 Knitting_n 0.050650 -0.179588 Knitting_y -0.252092 0.893832 Cooking_n 0.311962 -0.342517 Cooking_y -0.378000 0.415023 Fishing_n -0.003218 -0.106732 Fishing_y 0.024103 0.799389 TV_n0 0.473323 -0.317982 TV_n1 -0.271869 -0.137644 TV_n2 -0.175706 0.109968 TV_n3 -0.028438 0.218662 TV_n4 0.132535 -0.060858
#MASS = poids relatif des modalités
#Livre, page 297
#Qui constitue aussi le profil moyen : Livre, page 290
col_mass = acm.c_/(n*p)
print(col_mass)
[[0.01836963 0.03718592 0.01526431 0.04029125 0.03204583 0.02350973 0.03885202 0.01670353 0.03793744 0.01761812 0.03329363 0.02226193 0.03396007 0.02159549 0.02725313 0.02830242 0.03272645 0.02282911 0.04550933 0.01004623 0.04957887 0.00597669 0.04683512 0.00872044 0.03122341 0.02433214 0.03310221 0.02245335 0.04626085 0.00929471 0.03043645 0.02511911 0.04901168 0.00654387 0.00674239 0.00820288 0.01446316 0.01171232 0.0144348 ]]
#facteurs sont centrés - vérifions avec moyenne pondérée des modalités
#sur les 2 premiers facteurs
coord_col.apply(axis=0,func=lambda x: numpy.sum(col_mass[0]*x))
col_coord_dim1 -5.551115e-17 col_coord_dim2 -4.336809e-18 dtype: float64
#disperion = valeurs propres ==> oui, Livre Page 298
coord_col.apply(axis=0,func=lambda x: numpy.sum(col_mass[0]*(x**2)))
col_coord_dim1 0.194204 col_coord_dim2 0.081714 dtype: float64
#carte des modalités
acm.mapping_col(num_x_axis=1,num_y_axis=2,short_labels=False)
#pour mieux rendre compte des dispersions
#affichage dans le premier plan factoriel
import matplotlib.pyplot as plt
fig, ax = plt.subplots(figsize=(7,7))
ax.axis([-1.0,+1.0,-1.0,+1.0])
ax.plot([-1.0,+1.0],[0,0],color='silver',linestyle='--')
ax.plot([0,0],[-1.0,+1.0],color='silver',linestyle='--')
ax.set_xlabel("Dim.1")
ax.set_ylabel("Dim.2")
plt.title("Carte des modalités")
for i in range(coord_col.shape[0]):
ax.text(coord_col.iloc[i,0],coord_col.iloc[i,1],coord_col.index[i])
plt.show()
#contributions au premier facteur
acm.plot_col_contrib(num_axis=1,nb_values=7,short_labels=False,figsize=(5,3))
#contributions au second facteur
acm.plot_col_contrib(num_axis=2,nb_values=7,short_labels=False,figsize=(5,3))
#représentation des individus et des modalités possible
#dans le même repère en ACM
#acm.mapping(num_x_axis=1,num_y_axis=2)
#mais pas pertinent ici parce qu'individus nombreux et anonymes
#cf. tuto -- https://github.com/OlivierGarciaDev/fanalysis/blob/master/doc/mca_tutorial.ipynb
#variables illustratives qualitatives
X_illus_quali = ind_actifs[ind_actifs.columns[18:-1]]
X_illus_quali.columns
Index(['Sex', 'Age', 'Marital_status', 'Profession'], dtype='object')
#positionnement de profession par ex.
coord = info_lig[['row_coord_dim1', 'row_coord_dim2']].copy()
coord["Profession"] = X_illus_quali["Profession"]
#moyennes conditionnelles - Livre, page 341
coord_fact = pandas.pivot_table(data=coord,values=['row_coord_dim1', 'row_coord_dim2'],index="Profession",aggfunc='mean')
#corrigés par la racine carrée des valeurs propres
coord_fact = coord_fact/numpy.sqrt(acm.eig_[0][:2])
print(coord_fact)
row_coord_dim1 row_coord_dim2 Profession Employee 0.028047 0.034695 Foreman -0.375970 0.012330 Management -0.688832 -0.177703 Manual_labourer 0.398133 0.237908 NotAvailable 0.049242 -0.191748 Other -0.141590 0.035449 Technician -0.120254 -0.018906 Unskilled_worker 0.615680 0.115093
#placer les modalités dans le repère factoriel
fig, ax = plt.subplots(figsize=(7,7))
ax.axis([-1.0,+1.0,-1.0,+1.0])
ax.plot([-1.0,+1.0],[0,0],color='silver',linestyle='--')
ax.plot([0,0],[-1.0,+1.0],color='silver',linestyle='--')
ax.set_xlabel("Dim.1")
ax.set_ylabel("Dim.2")
plt.title("Carte des modalités + Modalités 'Profession'")
#modalités actives
for i in range(coord_col.shape[0]):
ax.text(coord_col.iloc[i,0],coord_col.iloc[i,1],coord_col.index[i],color="grey")
#modalités illustratives (celles de Profession)
for i in range(coord_fact.shape[0]):
ax.text(coord_fact.iloc[i,0],coord_fact.iloc[i,1],coord_fact.index[i],color="blue")
plt.show()
#corrélation avec les facteurs - Livre, page 339
#on raisonne en "directions"
corr_illus = info_lig[['row_coord_dim1', 'row_coord_dim2']].corrwith(ind_actifs.nb_activitees)
print(corr_illus)
row_coord_dim1 -0.973648 row_coord_dim2 0.202817 dtype: float64
#un cercle des corrélations pour les illustratives quantitatives
#surtout sintéressant sur plusieurs variables illustratives quantitatives
fig, ax = plt.subplots(figsize=(5,5))
ax.axis([-1.0,+1.0,-1.0,+1.0])
ax.plot([-1.0,+1.0],[0,0],color='silver',linestyle='--')
ax.plot([0,0],[-1.0,+1.0],color='silver',linestyle='--')
ax.set_xlabel("Dim.1")
ax.set_ylabel("Dim.2")
ax.add_artist(plt.Circle(xy=(0,0),radius=1,fill=False,color="gray",linestyle="--"))
ax.arrow(0,0,corr_illus[0],corr_illus[1],head_width=0.05,length_includes_head=True)
ax.text(corr_illus[0]-0.02,corr_illus[1]+0.05,"Nb_Activitees",color="blue")
plt.show()
#chargement
ind_supp = pandas.read_excel("loisirs.xlsx",sheet_name="seniors")
ind_supp.info()
<class 'pandas.core.frame.DataFrame'> RangeIndex: 567 entries, 0 to 566 Data columns (total 23 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 Reading 567 non-null object 1 Listening_music 567 non-null object 2 Cinema 567 non-null object 3 Show 567 non-null object 4 Exhibition 567 non-null object 5 Computer 567 non-null object 6 Sport 567 non-null object 7 Walking 567 non-null object 8 Travelling 567 non-null object 9 Playing_music 567 non-null object 10 Collecting 567 non-null object 11 Volunteering 567 non-null object 12 Mechanic 567 non-null object 13 Gardening 567 non-null object 14 Knitting 567 non-null object 15 Cooking 567 non-null object 16 Fishing 567 non-null object 17 TV 567 non-null object 18 Sex 567 non-null object 19 Age 567 non-null object 20 Marital_status 567 non-null object 21 Profession 567 non-null object 22 nb_activitees 567 non-null int64 dtypes: int64(1), object(22) memory usage: 102.0+ KB
#ditribution - variable Age (qui est illustrative en réalité)
ind_supp.Age.value_counts()
(75,85] 482 (85,100] 85 Name: Age, dtype: int64
#récupération des variables actives
X_supp = ind_supp[ind_supp.columns[:18]]
X_supp.columns
Index(['Reading', 'Listening_music', 'Cinema', 'Show', 'Exhibition',
'Computer', 'Sport', 'Walking', 'Travelling', 'Playing_music',
'Collecting', 'Volunteering', 'Mechanic', 'Gardening', 'Knitting',
'Cooking', 'Fishing', 'TV'],
dtype='object')
#calcul des coordonnées factorielles - Livre, page 335
coord_supp = acm.transform(X_supp.values)
#affichage des 2 premiers facteurs
coord_supp[:,:2]
array([[ 0.12327656, -0.28475875],
[-0.30491255, -0.29849733],
[ 0.58200401, -0.19445755],
...,
[ 0.40001713, -0.10040159],
[ 0.81407973, -0.24158056],
[ 0.5941269 , -0.14633458]])
#superposition des individus supplémentaires
#dans la représentation factorielle des individus
fig, ax = plt.subplots(figsize=(7,7))
ax.axis([-1.2,+1.2,-1.2,+1.2])
ax.plot([-1.2,+1.2],[0,0],color='silver',linestyle='--')
ax.plot([0,0],[-1.2,+1.2],color='silver',linestyle='--')
ax.set_xlabel("Dim.1")
ax.set_ylabel("Dim.2")
plt.scatter(info_lig.iloc[:,0],info_lig.iloc[:,1],color='grey',marker='.',s=0.3)
plt.scatter(coord_supp[:,0],coord_supp[:,1],color='blue',marker='o',s=5)
plt.show()
#comparaison du nombre d'activités
#chez les individus initiaux
print(ind_actifs.nb_activitees.mean())
#chez les seniors (individus supplémentaires ici)
print(ind_supp.nb_activitees.mean())
7.048749361919347 4.340388007054674
#voir le package "ca" pour R
#https://cran.r-project.org/package=ca
#ne conserver que les val.p. supérieures à la moyenne des val.p.
lambada = acm.eig_[0]
lambada = lambada[acm.eig_[0] > (1/p)]
print(lambada)
[0.19420392 0.08171353 0.07145231 0.06357372 0.05884482 0.05585231 0.05556884]
#correction de Benzécri - Livre, section 5.3.2
lambada_prim = ((p/(p-1)*(lambada-1/p)))**2
print(lambada_prim)
[2.15514585e-02 7.67105863e-04 2.83311566e-04 7.20771375e-05 1.21295220e-05 9.87255227e-08 1.97880577e-10]
#screeplot
plt.title("Screeplot après correction de Benzécri")
plt.plot(numpy.arange(1,lambada_prim.shape[0]+1,1),lambada_prim)
plt.show()
#pourcentage d'information véhiculée -- correction de Greenacre - Livre, section 5.3.3
#somme corrigée de l'info dispo dans les données
S2nd = p/(p-1)*(numpy.sum(acm.eig_[0]**2)-(M-p)/(p**2))
print(S2nd)
0.024929277937850762
#application à la correction de Benzécri pour avoir les pourcentages
percent_2nd = lambada_prim/S2nd
print(percent_2nd*100)
[8.64503920e+01 3.07712829e+00 1.13646118e+00 2.89126455e-01 4.86557295e-02 3.96022392e-04 7.93767783e-07]
#graphique
plt.title("Part d'info cumulée sur les axes")
plt.plot(numpy.arange(0,lambada_prim.shape[0]+1,1),numpy.append([0],numpy.cumsum(percent_2nd)))
plt.show()
#correction des coordonnées des modalités
#pour rendre compte de la dispersion corrigée
coord_prim = coord_col/numpy.sqrt(acm.eig_[0][:2])*numpy.sqrt(lambada_prim[:2])
print(coord_prim)
col_coord_dim1 col_coord_dim2 Reading_n 0.244319 -0.004029 Reading_y -0.120692 0.001990 Listening_music_n 0.279776 0.024620 Listening_music_y -0.105993 -0.009327 Cinema_n 0.175565 0.029771 Cinema_y -0.239310 -0.040580 Show_n 0.135103 0.010893 Show_y -0.314246 -0.025336 Exhibition_n 0.143874 -0.000686 Exhibition_y -0.309807 0.001477 Computer_n 0.151780 0.019384 Computer_y -0.226994 -0.028989 Sport_n 0.138521 0.017630 Sport_y -0.217832 -0.027724 Walking_n 0.137486 -0.032774 Walking_y -0.132389 0.031559 Travelling_n 0.165250 0.000751 Travelling_y -0.236892 -0.001076 Playing_music_n 0.071479 0.002849 Playing_music_y -0.323802 -0.012905 Collecting_n 0.022476 -0.005587 Collecting_y -0.186445 0.046347 Volunteering_n 0.046598 -0.004386 Volunteering_y -0.250264 0.023555 Mechanic_n 0.102358 -0.033471 Mechanic_y -0.131348 0.042950 Gardening_n 0.059201 -0.053664 Gardening_y -0.087278 0.079115 Knitting_n 0.016873 -0.017400 Knitting_y -0.083979 0.086604 Cooking_n 0.103923 -0.033187 Cooking_y -0.125922 0.040212 Fishing_n -0.001072 -0.010341 Fishing_y 0.008029 0.077453 TV_n0 0.157676 -0.030809 TV_n1 -0.090567 -0.013336 TV_n2 -0.058532 0.010655 TV_n3 -0.009473 0.021186 TV_n4 0.044151 -0.005897
#vérifions les dispersions == valeurs propres corrigées ?? OUI
coord_prim.apply(axis=0,func=lambda x: numpy.sum(col_mass[0]*(x**2)))
col_coord_dim1 0.021551 col_coord_dim2 0.000767 dtype: float64
#représentation dans le plan factoriel
fig, ax = plt.subplots(figsize=(8,8))
ax.axis([-0.35,+0.35,-0.35,+0.35])
ax.plot([-0.35,+0.35],[0,0],color='silver',linestyle='--')
ax.plot([0,0],[-0.35,+0.35],color='silver',linestyle='--')
ax.set_xlabel("Dim.1 (86.5%)")
ax.set_ylabel("Dim.2 (3.1%)")
plt.title("Carte des modalités")
for i in range(coord_prim.shape[0]):
ax.text(coord_prim.iloc[i,0],coord_prim.iloc[i,1],coord_prim.index[i],fontsize=7)
plt.show()
