16 Données déséquilibrées

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
rng = np.random.default_rng(seed=1234)

Exercice 1 (Critères pour un exemple de données déséquilibrées)  

  1. n = 500
p = 0.05
Y = rng.binomial(1, p=p, size=n)
  2. rng = np.random.default_rng(seed=123)
P1 = rng.binomial(1, p=0.005, size=n)
  3. P2 = np.zeros_like(P1)
for yy in range(n):
    if Y[yy]==0:
        P2[yy] = rng.binomial(1, p=0.10, size=1)[0]
    else:
        P2[yy] = rng.binomial(1, p=0.85, size=1)[0]
  4. from sklearn.metrics import confusion_matrix
print(confusion_matrix(Y, P1))
    [[478   0]
 [ 22   0]]
    print(confusion_matrix(Y, P2))
    [[432  46]
 [  4  18]]
  5. cm = confusion_matrix(Y, P2)
acc = cm.diagonal().sum()/cm.sum()
rec = cm[1,1]/cm[1,:].sum()
prec = cm[1,1]/cm[:,1].sum()
print(acc)
print(rec)
print(prec)
    0.9
0.8181818181818182
0.28125
  6. F1 = 2*(rec*prec)/(rec+prec)
print(F1)
rand = cm[:,0].sum()/n*cm[0,:].sum()/n + cm[:,1].sum()/n*cm[1,:].sum()/n
kappa = (acc-rand)/(1-rand)
print(kappa)
    0.41860465116279066
0.37786183555644093
  7. from sklearn.metrics import accuracy_score, recall_score, precision_score
from sklearn.metrics import f1_score, cohen_kappa_score
print(accuracy_score(Y, P2), "**", accuracy_score(Y, P1))
print(recall_score(Y, P2), "**", recall_score(Y, P1))
print(precision_score(Y, P2), "**", precision_score(Y, P1))
print(f1_score(Y, P2), "**", f1_score(Y, P1))
print(cohen_kappa_score(Y, P2), "**", cohen_kappa_score(Y, P1))
    0.9 ** 0.956
0.8181818181818182 ** 0.0
0.28125 ** 0.0
0.4186046511627907 ** 0.0
0.3778618355564404 ** 0.0

Exercice 2 (Échantillonnage rétrospectif) On remarque d’abord que \(\mathbf P(\tilde y_i=1)=\mathbf P(y_i=1|s_i=1)\). De plus \[ \text{logit}\, p_\beta(x_i)=\log\frac{\mathbf P(y_i=1)}{\mathbf P(y_i=0)}\quad\text{et}\quad \text{logit}\, p_\gamma(x_i)=\log\frac{\mathbf P(y_i=1|s_i=1)}{\mathbf P(y_i=0|s_i=1)}. \] Or \[ \mathbf P(y_i=1|s_i=1)=\frac{\mathbf P(y_i=1,s_i=1)}{\mathbf P(s_i=1)}=\frac{\mathbf P(s_i=1|y_i=1)\mathbf P(y_i=1)}{\mathbf P(s_i=1)} \] et \[ \mathbf P(y_i=0|s_i=1)=\frac{\mathbf P(y_i=0,s_i=1)}{\mathbf P(s_i=1)}=\frac{\mathbf P(s_i=1|y_i=0)\mathbf P(y_i=0)}{\mathbf P(s_i=1)}. \] Donc \[ \text{logit}\, p_\gamma(x_i)=\log\frac{\mathbf P(y_i=1)}{\mathbf P(y_i=0)}+\log\frac{\mathbf P(s_i=1|y_i=1)}{\mathbf P(s_i=1|y_i=0)}=\text{logit}\,p_\beta(x_i)+\log\left(\frac{\tau_{1i}}{\tau_{0i}}\right). \]

Exercice 3 (Rééquilibrage)  

  1. df1 = pd.read_csv("../donnees/dd_exo3_1.csv", header=0, sep=',')
df2 = pd.read_csv("../donnees/dd_exo3_2.csv", header=0, sep=',')
df3 = pd.read_csv("../donnees/dd_exo3_3.csv", header=0, sep=',')
    print(df1.describe())
print(df2.describe())
print(df3.describe())
                    X1           X2            Y
count  1000.000000  1000.000000  1000.000000
mean      0.514433     0.492924     0.441000
std       0.281509     0.291467     0.496755
min       0.000516     0.000613     0.000000
25%       0.284947     0.238695     0.000000
50%       0.518250     0.494121     0.000000
75%       0.753628     0.739679     1.000000
max       0.999567     0.999829     1.000000
                X1           X2            Y
count  1000.000000  1000.000000  1000.000000
mean      0.520809     0.472473     0.308000
std       0.280013     0.283496     0.461898
min       0.002732     0.000890     0.000000
25%       0.296167     0.225272     0.000000
50%       0.521226     0.468858     0.000000
75%       0.764060     0.693746     1.000000
max       0.996044     0.999183     1.000000
                X1           X2            Y
count  1000.000000  1000.000000  1000.000000
mean      0.538032     0.454919     0.158000
std       0.273863     0.271638     0.364924
min       0.004914     0.000613     0.000000
25%       0.322489     0.221447     0.000000
50%       0.545587     0.450438     0.000000
75%       0.781116     0.663637     0.000000
max       0.996044     0.999829     1.000000
    colo = ["C1", "C2"]
mark = ["o", "d"]
for yy in [0, 1]:
    plt.scatter(df1.loc[df1.Y==yy, "X1"], df1.loc[df1.Y==yy, "X2"], color=colo[yy], marker=mark[yy])

    for yy in [0, 1]:
    plt.scatter(df2.loc[df2.Y==yy, "X1"], df2.loc[df2.Y==yy, "X2"], color=colo[yy], marker=mark[yy])

    for yy in [0, 1]:
    plt.scatter(df3.loc[df3.Y==yy, "X1"], df3.loc[df3.Y==yy, "X2"], color=colo[yy], marker=mark[yy])

  2. from sklearn.model_selection import train_test_split
## separation en matrice X, Y (et creation du produit=interaction)
T1 = df1.drop(columns="Y")
X1 = T1.assign(inter= T1.X1 * T1.X2).to_numpy()
y1 = df1.Y.to_numpy()
T2 = df2.drop(columns="Y")
X2 = T2.assign(inter= T2.X1 * T2.X2).to_numpy()
y2 = df2.Y.to_numpy()
T3 = df3.drop(columns="Y")
X3 = T3.assign(inter= T3.X1 * T3.X2).to_numpy()
y3 = df3.Y.to_numpy()
## separation apprentissage/validation
X1_app, X1_valid, y1_app, y1_valid = train_test_split(
    X1, y1, test_size=0.33, random_state=1234)
X2_app, X2_valid, y2_app, y2_valid = train_test_split(
    X2, y2, test_size=0.33, random_state=1234)
X3_app, X3_valid, y3_app, y3_valid = train_test_split(
    X3, y3, test_size=0.33, random_state=1234)
  3. from sklearn.linear_model import LogisticRegression
mod1 = LogisticRegression(penalty=None, solver="newton-cholesky").fit(X1_app, y1_app)
mod2 = LogisticRegression(penalty=None, solver="newton-cholesky").fit(X2_app, y2_app)
mod3 = LogisticRegression(penalty=None, solver="newton-cholesky").fit(X3_app, y3_app)
    from sklearn.metrics import accuracy_score, balanced_accuracy_score, f1_score, cohen_kappa_score
P1 = mod1.predict(X1_valid)
P2 = mod1.predict(X2_valid)
P3 = mod1.predict(X3_valid)
s1 = pd.DataFrame({"crit": ["acc", "bal_acc", "F1", "Kappa"]})
s2 = pd.DataFrame({"crit": ["acc", "bal_acc", "F1", "Kappa"]})
s3 = pd.DataFrame({"crit": ["acc", "bal_acc", "F1", "Kappa"]})
print("--- donnees 1 ---")
s1 = s1.assign(brut=0.0)
s1.iloc[0,1] = accuracy_score(y1_valid, P1)
s1.iloc[1,1] = balanced_accuracy_score(y1_valid, P1)
s1.iloc[2,1] = f1_score(y1_valid, P1)
s1.iloc[3,1] = cohen_kappa_score(y1_valid, P1)
print(s1)
print("--- donnees 2 ---")
s2 = s2.assign(brut=0.0)
s2.iloc[0,1] = accuracy_score(y2_valid, P2)
s2.iloc[1,1] = balanced_accuracy_score(y2_valid, P2)
s2.iloc[2,1] = f1_score(y2_valid, P2)
s2.iloc[3,1] = cohen_kappa_score(y2_valid, P2)
print(s2)
print("--- donnees 3 ---")
s3 = s3.assign(brut=0.0)
s3.iloc[0,1] = accuracy_score(y3_valid, P3)
s3.iloc[1,1] = balanced_accuracy_score(y3_valid, P3)
s3.iloc[2,1] = f1_score(y3_valid, P3)
s3.iloc[3,1] = cohen_kappa_score(y3_valid, P3)
print(s3)
    --- donnees 1 ---
      crit      brut
0      acc  0.657576
1  bal_acc  0.655825
2       F1  0.622074
3    Kappa  0.309572
--- donnees 2 ---
      crit      brut
0      acc  0.724242
1  bal_acc  0.740841
2       F1  0.637450
3    Kappa  0.427280
--- donnees 3 ---
      crit      brut
0      acc  0.693939
1  bal_acc  0.729286
2       F1  0.435754
3    Kappa  0.278103
  4. from imblearn.under_sampling import RandomUnderSampler
from imblearn.under_sampling import TomekLinks
from imblearn.over_sampling import RandomOverSampler
from imblearn.over_sampling import SMOTE
## RandomOverSampler
ros3 = RandomOverSampler(random_state=123)
X3_app_reech, y3_app_reech = ros3.fit_resample(X3_app, y3_app)
mod3_ros = LogisticRegression(penalty=None, solver="newton-cholesky").fit(X3_app_reech, y3_app_reech)
## Smote
sm = RandomOverSampler(random_state=123)
X3_app_reech, y3_app_reech = sm.fit_resample(X3_app, y3_app)
mod3_sm = LogisticRegression(penalty=None, solver="newton-cholesky").fit(X3_app_reech, y3_app_reech)
## RandomUnderSampler
rus3 = RandomUnderSampler(random_state=123)
X3_app_reech, y3_app_reech = rus3.fit_resample(X3_app, y3_app)
mod3_rus = LogisticRegression(penalty=None, solver="newton-cholesky").fit(X3_app_reech, y3_app_reech)
## Tomek
tl = TomekLinks(sampling_strategy='all')
X3_app_reech, y3_app_reech = tl.fit_resample(X3_app, y3_app)
mod3_tl = LogisticRegression(penalty=None, solver="newton-cholesky").fit(X3_app_reech, y3_app_reech)
    P3_ros = mod3_ros.predict(X3_valid)
P3_sm = mod3_sm.predict(X3_valid)
P3_rus = mod3_rus.predict(X3_valid)
P3_tl = mod3_tl.predict(X3_valid)
    s3 = s3.assign(ros=[accuracy_score(y3_valid, P3_ros),
balanced_accuracy_score(y3_valid, P3_ros),
f1_score(y3_valid, P3_ros),
cohen_kappa_score(y3_valid, P3_ros)])
s3 = s3.assign(sm=[accuracy_score(y3_valid, P3_sm),
balanced_accuracy_score(y3_valid, P3_sm),
f1_score(y3_valid, P3_sm),
cohen_kappa_score(y3_valid, P3_sm)])
s3 = s3.assign(rus=[accuracy_score(y3_valid, P3_rus),
balanced_accuracy_score(y3_valid, P3_rus),
f1_score(y3_valid, P3_rus),
cohen_kappa_score(y3_valid, P3_rus)])
s3 = s3.assign(tl=[accuracy_score(y3_valid, P3_tl),
balanced_accuracy_score(y3_valid, P3_tl),
f1_score(y3_valid, P3_tl),
cohen_kappa_score(y3_valid, P3_tl)])
print(s3)
          crit      brut       ros        sm       rus        tl
0      acc  0.693939  0.603030  0.603030  0.612121  0.854545
1  bal_acc  0.729286  0.683929  0.683929  0.689286  0.520000
2       F1  0.435754  0.379147  0.379147  0.384615  0.076923
3    Kappa  0.278103  0.192415  0.192415  0.200606  0.066038
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