Importation et inspection des données

#changement de dossier
import os
os.chdir("C:/Users/ricco/Desktop/demo/candisc")

#importation des données
import pandas
df = pandas.read_excel("Olive_Oil_Candisc.xlsx",sheet_name="dataset")
df.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 569 entries, 0 to 568
Data columns (total 9 columns):
 #   Column       Non-Null Count  Dtype 
---  ------       --------------  ----- 
 0   CLASSE       569 non-null    object
 1   palmitic     569 non-null    int64 
 2   palmitoleic  569 non-null    int64 
 3   stearic      569 non-null    int64 
 4   oleic        569 non-null    int64 
 5   linoleic     569 non-null    int64 
 6   linolenic    569 non-null    int64 
 7   arachidic    569 non-null    int64 
 8   eicosenoic   569 non-null    int64 
dtypes: int64(8), object(1)
memory usage: 40.1+ KB

#première lignes
df.head()

	CLASSE	palmitic	palmitoleic	stearic	oleic	linoleic	linolenic	arachidic	eicosenoic
0	Centre_North	1400	90	270	7420	800	0	20	2
1	Centre_North	990	120	250	7750	870	10	10	2
2	Centre_North	1020	100	220	7530	1030	0	0	3
3	Centre_North	1098	54	202	7945	595	42	32	2
4	Centre_North	1120	80	260	7750	680	30	80	3

#classes - distribution
df.CLASSE.value_counts()

South           322
Centre_North    150
Sardinia         97
Name: CLASSE, dtype: int64

Analyse factorielle discriminante

La procédure CANDISC

#analyse factorielle discriminante
from scientisttools.discriminant_analysis import CANDISC

#instanciation - nombre max de facteurs = min(nb var. descriptives,nb groupes-1)
candisc = CANDISC(n_components=2,
                  target=['CLASSE'],
                  features_labels=list(df.columns[1:]),
                  row_labels=df.index,
                  priors=None)

#entraînement
candisc.fit(df)

CANDISC(features_labels=['palmitic', 'palmitoleic', 'stearic', 'oleic',
                         'linoleic', 'linolenic', 'arachidic', 'eicosenoic'],
        n_components=2, row_labels=RangeIndex(start=0, stop=569, step=1),
        target=['CLASSE'])

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.

#résumé des résultats
from scientisttools.extractfactor import summaryCANDISC
summaryCANDISC(candisc,to_markdown=True)

                     Canonical Discriminant Analysis - Results                     


Summary Information
|       |   Total Sample Size |   Variables |   Classes |   DF Total |   DF Within Classes |   DF Between Classes |
|:------|--------------------:|------------:|----------:|-----------:|--------------------:|---------------------:|
| value |                 569 |           8 |         3 |        568 |                 566 |                    2 |

Class Level information
|                   |   n(k) |     p(k) |
|:------------------|-------:|---------:|
| ('South',)        |    322 | 0.565905 |
| ('Centre_North',) |    150 | 0.26362  |
| ('Sardinia',)     |     97 | 0.170475 |

Importance of components
|                         |    LD1 |     LD2 |
|:------------------------|-------:|--------:|
| Variance                |  8.472 |   2.303 |
| Difference              |  6.168 | nan     |
| % of var.               | 78.623 |  21.377 |
| Cumulative of % of var. | 78.623 | 100     |

Group means:
| CLASSE       |   palmitic |   palmitoleic |   stearic |   oleic |   linoleic |   linolenic |   arachidic |   eicosenoic |
|:-------------|-----------:|--------------:|----------:|--------:|-----------:|------------:|------------:|-------------:|
| Centre_North |    1094.83 |        83.893 |   231.04  | 7791.97 |     727.88 |      21.747 |      37.547 |        1.973 |
| Sardinia     |    1112.06 |        96.351 |   226.351 | 7266.91 |    1197.36 |      27.01  |      73     |        1.928 |
| South        |    1332.37 |       154.888 |   228.708 | 7099.53 |    1034.01 |      38.037 |      63.102 |       27.332 |

Coefficients of canonical discriminants:
|             |     LD1 |     LD2 |
|:------------|--------:|--------:|
| palmitic    |   0.003 |   0.009 |
| palmitoleic |   0.013 |   0.018 |
| stearic     |  -0.003 |   0.004 |
| oleic       |   0.001 |   0.006 |
| linoleic    |   0.001 |  -0.001 |
| linolenic   |   0.041 |   0.006 |
| arachidic   |  -0.017 |  -0.035 |
| eicosenoic  |   0.163 |   0.01  |
| Intercept   | -13.065 | -56.917 |

Classification functions coefficients:
|             |   Centre_North |   Sardinia |   South |
|:------------|---------------:|-----------:|--------:|
| palmitic    |          0.007 |     -0.034 |   0.007 |
| palmitoleic |         -0.01  |     -0.096 |   0.033 |
| stearic     |          0.017 |     -0.003 |  -0.007 |
| oleic       |          0.009 |     -0.02  |   0.002 |
| linoleic    |         -0.006 |     -0     |   0.003 |
| linolenic   |         -0.126 |     -0.152 |   0.104 |
| arachidic   |         -0.006 |      0.156 |  -0.044 |
| eicosenoic  |         -0.523 |     -0.567 |   0.415 |
| Intercept   |        -70.09  |    194.655 | -37.181 |

Individuals (the 10 first)

|    |    LD1 |   LD2 |
|---:|-------:|------:|
|  0 | -3.245 | 2.894 |
|  1 | -3.058 | 2.107 |
|  2 | -3.184 | 0.602 |
|  3 | -2.746 | 2.63  |
|  4 | -3.695 | 0.503 |
|  5 | -3.568 | 2.404 |
|  6 | -3.185 | 3.136 |
|  7 | -3.803 | 3.569 |
|  8 | -2.244 | 3.061 |
|  9 | -3.767 | 1.808 |

Continues variables

|             |   total.1 |   between.1 |   within.1 |   total.2 |   between.2 |   within.2 |
|:------------|----------:|------------:|-----------:|----------:|------------:|-----------:|
| palmitic    |     0.718 |       0.997 |      0.318 |    -0.038 |      -0.148 |     -0.028 |
| palmitoleic |     0.66  |       0.98  |      0.276 |    -0.09  |      -0.268 |     -0.064 |
| stearic     |    -0.007 |      -0.161 |     -0.002 |     0.049 |       0.996 |      0.027 |
| oleic       |    -0.629 |      -0.667 |     -0.297 |     0.497 |       0.789 |      0.398 |
| linoleic    |     0.264 |       0.288 |      0.116 |    -0.745 |      -0.975 |     -0.553 |
| linolenic   |     0.575 |       0.928 |      0.225 |    -0.155 |      -0.436 |     -0.103 |
| arachidic   |     0.278 |       0.34  |      0.111 |    -0.62  |      -0.962 |     -0.419 |
| eicosenoic  |     0.945 |       1     |      0.683 |     0.004 |      -0.063 |      0.004 |

Tests statistiques

#test MANOVA - les nuages sont-ils séparables ?
#cf. le logiciel SAS
print(candisc.manova_)

                    Multivariate linear model
=================================================================
                                                                 
-----------------------------------------------------------------
         CLASSE          Value   Num DF   Den DF  F Value  Pr > F
-----------------------------------------------------------------
          Wilks' lambda  0.0320 16.0000 1118.0000 320.9837 0.0000
         Pillai's trace  1.5917 16.0000 1120.0000 272.8899 0.0000
 Hotelling-Lawley trace 10.7752 16.0000  911.1473 375.9385 0.0000
    Roy's greatest root  8.4718  8.0000  560.0000 593.0293 0.0000
=================================================================

#autre expression des tests
#cf. TANAGRA
candisc.global_performance_

	Stat	Value	p-value
0	Wilks' Lambda	0.031960	NaN
1	Bartlett -- C(16)	1936.843029	0.000000e+00
2	Rao -- F(16,1118)	320.983726	1.110223e-16

#tests de séparabilité en prenant les variables individuellement
#indépendamment les unes des autres
#triés de manière décroissante selon le carré du rapport de corrélation
candisc.univariate_test_statistis_.sort_values(by='R-squared',ascending=False)

	Std. Dev.	R-squared	Rsq/(1-Rsq)	F-statistic	Prob (F-statistic)
eicosenoic	14.092942	0.797929	3.948758	1117.498522	2.867939e-197
oleic	405.966522	0.526521	1.112025	314.703134	1.288711e-92
palmitic	168.715167	0.461491	0.856978	242.524854	8.465810e-77
linoleic	242.802985	0.449629	0.816955	231.198312	4.032628e-74
palmitoleic	52.558263	0.395095	0.653152	184.841942	1.650063e-62
arachidic	22.033442	0.337110	0.508547	143.918675	2.936859e-51
linolenic	12.985725	0.312278	0.454076	128.503464	9.724383e-47
stearic	36.795195	0.001728	0.001731	0.489942	6.129213e-01

Représentation factorielle

#représentation graphique dans le plan
from scientisttools.pyplot import plotCANDISC
import matplotlib.pyplot as plt 

fig, axe =plt.subplots(figsize=(8,5))
plotCANDISC(candisc,color=['skyblue','chartreuse','orange'],marker=['.','.','.'],ax=axe)
plt.show()

#coefficients des équations de projection
from scientisttools.extractfactor import get_candisc_coef
coef = get_candisc_coef(candisc,choice="absolute")
coef

	LD1	LD2
palmitic	0.002760	0.008854
palmitoleic	0.013092	0.018422
stearic	-0.002794	0.004253
oleic	0.000626	0.006239
linoleic	0.001131	-0.001260
linolenic	0.041077	0.005822
arachidic	-0.017345	-0.034675
eicosenoic	0.163051	0.010059
Intercept	-13.064616	-56.916913

#test des facteurs
#c.-à-d. le facteur concerné et les suivants
#présentent-ils des rapports de corrélation nuls ?
candisc.likelihood_test_

	statistic	DDL num.	DDL den.	Pr>F
0	320.983726	16.0	1118.0	1.110223e-16
1	184.272085	7.0	560.0	1.110223e-16

#barycentres conditionnels
candisc.gcenter_

	LD1	LD2
CLASSE
Centre_North	-3.322164	1.843764
Sardinia	-3.302678	-2.860687
South	2.542498	0.002863

#faire apparaître les moyennes conditionnelles dans le graphique
fig, axe =plt.subplots(figsize=(8,5))
plotCANDISC(candisc,color=['skyblue','chartreuse','orange'],marker=['.','.','.'],ax=axe)
plt.scatter(candisc.gcenter_.LD1,candisc.gcenter_.LD2,c=['blue','green','red'],marker='X',s=90)
plt.show()

#distance (au carré) entre les centres de classes
candisc.squared_mdist_

	Centre_North	Sardinia	South
Centre_North	0.000000	22.132240	37.783184
Sardinia	22.132240	0.000000	42.366007
South	37.783184	42.366007	0.000000

Interprétation des facteurs

#corrélation (totales) des variables avec les facteurs
candisc.tcorr_

	LD1	LD2
palmitic	0.717538	-0.037627
palmitoleic	0.659883	-0.089794
stearic	-0.007083	0.049135
oleic	-0.629206	0.497264
linoleic	0.264460	-0.745062
linolenic	0.574829	-0.154918
arachidic	0.278118	-0.619875
eicosenoic	0.944513	0.003556

#cercle des corrélations
#direction des variables par rapport aux facteurs
fig, axe =plt.subplots(figsize=(6,6))
axe.axis([-1.0,+1.0,-1.0,+1.0])
axe.plot([-1.0,+1.0],[0,0],color='gainsboro',linestyle='--')
axe.plot([0,0],[-1.0,+1.0],color='gainsboro',linestyle='--')
axe.add_artist(plt.Circle((0,0),radius=1.0,
                facecolor='khaki',edgecolor='silver',fill=False))
plt.scatter(candisc.tcorr_.LD1,candisc.tcorr_.LD2,alpha=0.2)
for i in range(candisc.tcorr_.shape[0]):
    axe.text(candisc.tcorr_.LD1[i],candisc.tcorr_.LD2[i],candisc.tcorr_.index[i],
            horizontalalignment='center',verticalalignment='center',fontsize=7)
    plt.arrow(0,0,candisc.tcorr_.LD1[i]*0.85,candisc.tcorr_.LD2[i]*0.85,width=0.0005,head_width=0.02,color='grey')
plt.show()

#vérification des moyennes conditionnelles
gb = df.groupby('CLASSE')
gb.mean()

	palmitic	palmitoleic	stearic	oleic	linoleic	linolenic	arachidic	eicosenoic
CLASSE
Centre_North	1094.833333	83.893333	231.040000	7791.973333	727.880000	21.746667	37.546667	1.973333
Sardinia	1112.061856	96.350515	226.350515	7266.907216	1197.360825	27.010309	73.000000	1.927835
South	1332.369565	154.888199	228.708075	7099.531056	1034.009317	38.037267	63.102484	27.332298

#corrélation (inter-groupes) des variables avec les facteurs
#indique le rôle que joue la variable dans l'écartement des groupes
#pour le facteur étudié
candisc.bcorr_

	LD1	LD2
palmitic	0.996819	-0.148078
palmitoleic	0.979502	-0.268340
stearic	-0.160870	0.995704
oleic	-0.666629	0.789483
linoleic	0.287779	-0.975225
linolenic	0.927928	-0.435710
arachidic	0.340070	-0.961566
eicosenoic	0.999983	-0.063032

Traitement des individus supplémentaires

Importation et calcul des coordonnées factorielles

#importation
XSupp = pandas.read_excel("Olive_Oil_Candisc.xlsx",sheet_name="supplementaires")
XSupp

	palmitic	palmitoleic	stearic	oleic	linoleic	linolenic	arachidic	eicosenoic
0	1306	127	250	7554	869	47	68	16

#calcul des coordonnées factorielles
coordSupp = candisc.transform(XSupp)
coordSupp

	LD1	LD2
0	0.578707	2.157809

#rajouter dans le graphique
fig, axe =plt.subplots(figsize=(8,5))
plotCANDISC(candisc,color=['skyblue','chartreuse','orange'],marker=['.','.','.'],ax=axe)
plt.scatter(candisc.gcenter_.LD1,candisc.gcenter_.LD2,c=['blue','green','red'],marker='X',s=90)
plt.scatter(coordSupp.LD1,coordSupp.LD2,marker='o',c='black',s=30)
#matéraliser l'écartement par rapport aux centroïdes
candisc.gcenter_.apply(axis=1,
                       func=lambda x:plt.plot([coordSupp.values[0,0],x.values[0]],[coordSupp.values[0,1],x.values[1]],
                                              c='grey',linestyle=':'))
plt.show()

Affectation à la classe la plus proche

Calcul de la distance aux barycentres.

#la plus proche au sens du barycentre conditionnel
#rappel - affichage des moyennes conditionnelles
candisc.gcenter_

	LD1	LD2
CLASSE
Centre_North	-3.322164	1.843764
Sardinia	-3.302678	-2.860687
South	2.542498	0.002863

#distance aux barycentres conditionnels
import numpy
dist2 = candisc.gcenter_.apply(axis=1,func=lambda x:numpy.sum((x.values-coordSupp.values[0,:])**2))
numpy.sqrt(dist2)

CLASSE
Centre_North    3.913492
Sardinia        6.344325
South           2.915522
dtype: float64

Conversion en probabilité d'affectation.

#fréquence des classes
candisc.class_level_information_[['p(k)']].sort_index()

	p(k)
CLASSE
Centre_North	0.263620
Sardinia	0.170475
South	0.565905

#distance carrée généralisée
distGen = dist2.values - 2.0 * numpy.log(candisc.class_level_information_[['p(k)']].sort_index().values.reshape(-1))
distGen

array([17.98191075, 43.78879377,  9.63892755])

#proba d'affectation - softmax
proba = numpy.exp(-0.5*distGen)/numpy.sum(numpy.exp(-0.5*distGen))
proba

array([1.51947843e-02, 3.78269298e-08, 9.84805178e-01])

Vérification avec la probabilité fournie par "scientisttools".

#proba forunie par scientisttols
#complètement cohérente !!!
candisc.predict_proba(XSupp)

	Centre_North	Sardinia	South
0	0.015195	3.782693e-08	0.984805