Bk5_Ch18_01

佚名 / 2024-10-13 / 原文 

import numpy as np
import statsmodels.api as sm
import matplotlib.pyplot as plt 
import pandas as pd  
from sklearn.datasets import load_iris

plt.close('all')

iris = load_iris()
# A copy from Sklearn

X_1_to_4 = iris.data
y = iris.target

feature_names = ['Sepal length, $X_1$','Sepal width, $X_2$',
                 'Petal length, $X_3$','Petal width, $X_4$']

X_df = pd.DataFrame(X_1_to_4, columns=feature_names)
y_df = pd.DataFrame(y, columns=['label'])

y_df[y_df==0] = 'C_1'
y_df[y_df==1] = 'C_2'
y_df[y_df==2] = 'C_3'

X1_df = X_df['Sepal length, $X_1$']

#%% likelihood PDF, given class Y

# given C1 (y = 0)

x1 = np.linspace(4,8,161)

fig, ax = plt.subplots()

KDE_C1 = sm.nonparametric.KDEUnivariate(X1_df[y==0])
KDE_C1.fit(bw=0.1)

f_x1_given_C1 = KDE_C1.evaluate(x1)

ax.fill_between(x1, f_x1_given_C1, facecolor = '#FF3300',alpha = 0.2)
ax.plot(x1, f_x1_given_C1,color = '#FF3300', 
        label = '$f_{X1|Y}(x_1|C_1)$, likelihood')

ax.autoscale(enable=True, axis='x', tight=True)
ax.autoscale(enable=True, axis='y', tight=True)
ax.set_ylim([0,1.5])
ax.set_yticks([0, 0.5, 1, 1.5])
ax.set_xlim([4,8])
ax.set_ylabel('PDF')
ax.set_xlabel('Sepal length, $x_1$')
ax.legend()

# given C2 (y = 1)

fig, ax = plt.subplots()

KDE_C2 = sm.nonparametric.KDEUnivariate(X1_df[y==1])
KDE_C2.fit(bw=0.1)

f_x1_given_C2 = KDE_C2.evaluate(x1)

ax.fill_between(x1, f_x1_given_C2, facecolor = '#0099FF',alpha = 0.2)
ax.plot(x1, f_x1_given_C2,color = '#0099FF',
        label = '$f_{X1|Y}(x_1|C_2)$, likelihood')

ax.autoscale(enable=True, axis='x', tight=True)
ax.autoscale(enable=True, axis='y', tight=True)
ax.set_ylim([0,1.5])
ax.set_yticks([0, 0.5, 1, 1.5])
ax.set_xlim([4,8])
ax.set_ylabel('PDF')
ax.set_xlabel('Sepal length, $x_1$')
ax.legend()

# given C3 (y = 2)

fig, ax = plt.subplots()

KDE_C3 = sm.nonparametric.KDEUnivariate(X1_df[y==2])
KDE_C3.fit(bw=0.1)

f_x1_given_C3 = KDE_C3.evaluate(x1)

ax.fill_between(x1, f_x1_given_C3, facecolor = '#8A8A8A',alpha = 0.2)
ax.plot(x1, f_x1_given_C3,color = '#8A8A8A',
        label = '$f_{X1|Y}(x_1|C_3)$, likelihood')

ax.autoscale(enable=True, axis='x', tight=True)
ax.autoscale(enable=True, axis='y', tight=True)
ax.set_ylim([0,1.5])
ax.set_yticks([0, 0.5, 1, 1.5])
ax.set_xlim([4,8])
ax.set_ylabel('PDF')
ax.set_xlabel('Sepal length, $x_1$')
ax.legend()

#%% compare three likelihood curves


fig, ax = plt.subplots()

ax.fill_between(x1, f_x1_given_C1, facecolor = '#FF3300',alpha = 0.2)
ax.plot(x1, f_x1_given_C1,color = '#FF3300', label = '$f_{X1|Y}(x_1|C_1)$')

ax.fill_between(x1, f_x1_given_C2, facecolor = '#0099FF',alpha = 0.2)
ax.plot(x1, f_x1_given_C2,color = '#0099FF', label = '$f_{X1|Y}(x_1|C_2)$')

ax.fill_between(x1, f_x1_given_C3, facecolor = '#8A8A8A',alpha = 0.2)
ax.plot(x1, f_x1_given_C3,color = '#8A8A8A', label = '$f_{X1|Y}(x_1|C_3)$')

ax.autoscale(enable=True, axis='x', tight=True)
ax.autoscale(enable=True, axis='y', tight=True)
ax.set_ylim([0,1.5])
ax.set_yticks([0, 0.5, 1, 1.5])
ax.set_xlim([4,8])
ax.set_ylabel('Likelihood PDF')
ax.set_xlabel('Sepal length, $x_1$')
ax.legend()

#%% prior probability

y_counts = y_df['label'].value_counts()

#Plot the data:
my_colors = ['#FF3300', '#0099FF', '#8A8A8A']

fig, ax = plt.subplots()

y_counts.plot.bar(color=my_colors)

plt.show()

y_prob = y_counts/y_df.count().values[0]
plt.ylabel('Count')

fig, ax = plt.subplots()

y_prob.plot.bar(color=my_colors)
plt.ylabel('Prior probability')

#%% Joint PDF

f_x1_joint_C1 = f_x1_given_C1*y_prob['C_1']
f_x1_joint_C2 = f_x1_given_C2*y_prob['C_2']
f_x1_joint_C3 = f_x1_given_C3*y_prob['C_3']

# C1

fig, ax = plt.subplots()

# Conditional likelihood
ax.plot(x1, f_x1_given_C1,color = '#FF3300', linestyle = '--',
        label = '$f_{X1|Y}(x_1|C_1)$, likelihood')

# Joint
ax.fill_between(x1, f_x1_joint_C1, facecolor = '#FF3300',alpha = 0.2)
ax.plot(x1, f_x1_joint_C1,color = '#FF3300', 
        label = '$f_{X1,Y}(x_1,C_1)$, joint')

ax.autoscale(enable=True, axis='x', tight=True)
ax.autoscale(enable=True, axis='y', tight=True)
ax.set_ylim([0,1.5])
ax.set_yticks([0, 0.5, 1, 1.5])
ax.set_xlim([4,8])
ax.set_ylabel('$PDF$')
ax.set_xlabel('Sepal length, $x_1$')
ax.legend()

# C2

fig, ax = plt.subplots()

# Conditional likelihood
ax.plot(x1, f_x1_given_C2,color = '#0099FF', linestyle = '--', 
        label = '$f_{X1|Y}(x_1|C_2)$, likelihood')

# Joint
ax.fill_between(x1, f_x1_joint_C2, facecolor = '#0099FF',alpha = 0.2)
ax.plot(x1, f_x1_joint_C2,color = '#0099FF', 
        label = '$f_{X1,Y}(x_1,C_2)$, joint')

ax.autoscale(enable=True, axis='x', tight=True)
ax.autoscale(enable=True, axis='y', tight=True)
ax.set_ylim([0,1.5])
ax.set_yticks([0, 0.5, 1, 1.5])
ax.set_xlim([4,8])
ax.set_ylabel('$PDF$')
ax.set_xlabel('Sepal length, $x_1$')
ax.legend()

# C3

fig, ax = plt.subplots()

# Conditional likelihood
ax.plot(x1, f_x1_given_C3,color = '#8A8A8A', linestyle = '--', 
        label = '$f_{X1|Y}(x_1|C_3)$, likelihood')

# Joint
ax.fill_between(x1, f_x1_joint_C3, facecolor = '#8A8A8A',alpha = 0.2)
ax.plot(x1, f_x1_joint_C3,color = '#8A8A8A', 
        label = '$f_{X1,Y}(x_1,C_3)$, joint')

ax.autoscale(enable=True, axis='x', tight=True)
ax.autoscale(enable=True, axis='y', tight=True)
ax.set_ylim([0,1.5])
ax.set_yticks([0, 0.5, 1, 1.5])
ax.set_xlim([4,8])
ax.set_ylabel('$PDF$')
ax.set_xlabel('Sepal length, $x_1$')
ax.legend()

# compare joint 

fig, ax = plt.subplots()

# Joint
ax.fill_between(x1, f_x1_joint_C1, facecolor = '#FF3300',alpha = 0.2)
ax.plot(x1, f_x1_joint_C1,color = '#FF3300', 
        label = '$f_{X1,Y}(x_1,C_1)$')

ax.fill_between(x1, f_x1_joint_C2, facecolor = '#0099FF',alpha = 0.2)
ax.plot(x1, f_x1_joint_C2,color = '#0099FF', 
        label = '$f_{X1,Y}(x_1,C_2)$')

ax.fill_between(x1, f_x1_joint_C3, facecolor = '#8A8A8A',alpha = 0.2)
ax.plot(x1, f_x1_joint_C3,color = '#8A8A8A', 
        label = '$f_{X1,Y}(x_1,C_3)$')

ax.autoscale(enable=True, axis='x', tight=True)
ax.autoscale(enable=True, axis='y', tight=True)
ax.set_ylim([0,1.5])
ax.set_yticks([0, 0.5, 1, 1.5])
ax.set_xlim([4,8])
ax.set_ylabel('Conditional PDF')
ax.set_xlabel('Sepal length, $x_1$')
ax.legend()

#%% Evidence fX_1(x_1)

f_x1 = f_x1_joint_C1 + f_x1_joint_C2 + f_x1_joint_C3

fig, ax = plt.subplots()

ax.plot(x1, f_x1,color = '#00448A', 
        label = '$f_{X1}(x_1)$, evidence (marginal)')
ax.fill_between(x1, f_x1, facecolor = '#00448A',alpha = 0.1)

ax.plot(x1, f_x1_joint_C1,color = '#FF3300', 
        label = '$f_{X1,Y}(x_1,C_1)$')
ax.plot(x1, f_x1_joint_C2,color = '#0099FF', 
        label = '$f_{X1,Y}(x_1,C_2)$')
ax.plot(x1, f_x1_joint_C3,color = '#8A8A8A', 
        label = '$f_{X1,Y}(x_1,C_3)$')

ax.autoscale(enable=True, axis='x', tight=True)
ax.autoscale(enable=True, axis='y', tight=True)
ax.set_ylim([0,0.75])
ax.set_yticks([0, 0.5])
ax.set_xlim([4,8])
ax.set_ylabel('PDF')
ax.set_xlabel('Sepal length, $x_1$')
ax.legend()


#%%

#%% Posterior

f_C1_given_x1 = f_x1_joint_C1/f_x1
f_C2_given_x1 = f_x1_joint_C2/f_x1
f_C3_given_x1 = f_x1_joint_C3/f_x1

# C1

fig, (ax1, ax2, ax3) = plt.subplots(3)

# joint C1
ax1.plot(x1, f_x1_joint_C1,color = '#FF3300',
         label = '$f_{x1, Y}(x_1, C_1)$, joint')
ax1.fill_between(x1, f_x1_joint_C1, facecolor = '#FF3300',alpha = 0.2)
ax1.set_ylim([0,1])
ax1.set_yticks([0, 0.5, 1])
ax1.set_xlim([4,8])
ax1.set_xticks([])
ax1.legend()

# marginal, x1
ax2.plot(x1, f_x1, color = '#00448A',
                 label = '$f_{X1}(x_1)$, marginal')
ax2.fill_between(x1, f_x1, facecolor = '#00448A',alpha = 0.1)
ax2.set_ylim([0,1])
ax2.set_yticks([0, 0.5, 1])
ax2.set_xlim([4,8])
ax2.set_xticks([])
ax2.legend()

# given x1, probability of C1
ax3.fill_between(x1, f_C1_given_x1, facecolor = '#FF3300',alpha = 0.2)
ax3.plot(x1, f_C1_given_x1,color = '#FF3300',
         label = '$f_{Y|X1}(C_1|x_1)$, posterior')

ax3.autoscale(enable=True, axis='x', tight=True)
ax3.autoscale(enable=True, axis='y', tight=True)
ax3.set_ylim([0,1])
ax3.set_yticks([0, 0.5, 1])
ax3.set_xlim([4,8])
ax3.set_xlabel('Sepal length, $x_1$')
ax3.legend()

# C2

fig, (ax1, ax2, ax3) = plt.subplots(3)

# joint C2
ax1.plot(x1, f_x1_joint_C2,color = '#0099FF',
         label = '$f_{x1, Y}(x_1, C_2)$, joint')
ax1.fill_between(x1, f_x1_joint_C2, facecolor = '#0099FF',alpha = 0.2)
ax1.set_ylim([0,1])
ax1.set_yticks([0, 0.5, 1])
ax1.set_xlim([4,8])
ax1.set_xticks([])
ax1.legend()

# marginal, x1
ax2.plot(x1, f_x1, color = '#00448A',
                 label = '$f_{X1}(x_1)$, marginal')
ax2.fill_between(x1, f_x1, facecolor = '#00448A',alpha = 0.1)
ax2.set_ylim([0,1])
ax2.set_yticks([0, 0.5, 1])
ax2.set_xlim([4,8])
ax2.set_xticks([])
ax2.legend()

# given x1, probability of C2
ax3.fill_between(x1, f_C2_given_x1, facecolor = '#0099FF',alpha = 0.2)
ax3.plot(x1, f_C2_given_x1,color = '#0099FF',
         label = '$f_{Y|X1}(C_2|x_1)$, posterior')

ax3.autoscale(enable=True, axis='x', tight=True)
ax3.autoscale(enable=True, axis='y', tight=True)
ax3.set_ylim([0,1])
ax3.set_yticks([0, 0.5, 1])
ax3.set_xlim([4,8])
ax3.set_xlabel('Sepal length, $x_1$')
ax3.legend()


# C3

fig, (ax1, ax2, ax3) = plt.subplots(3)

# joint C3
ax1.plot(x1, f_x1_joint_C3,color = '#8A8A8A',
         label = '$f_{x1, Y}(x_1, C_3)$, joint')
ax1.fill_between(x1, f_x1_joint_C3, facecolor = '#8A8A8A',alpha = 0.2)
ax1.set_ylim([0,1])
ax1.set_yticks([0, 0.5, 1])
ax1.set_xlim([4,8])
ax1.set_xticks([])
ax1.legend()

# marginal, x1
ax2.plot(x1, f_x1, color = '#00448A',
                 label = '$f_{X1}(x_1)$, marginal')
ax2.fill_between(x1, f_x1, facecolor = '#00448A',alpha = 0.1)
ax2.set_ylim([0,1])
ax2.set_yticks([0, 0.5, 1])
ax2.set_xlim([4,8])
ax2.set_xticks([])
ax2.legend()

# given x1, probability of C3
ax3.fill_between(x1, f_C3_given_x1, facecolor = '#8A8A8A',alpha = 0.2)
ax3.plot(x1, f_C3_given_x1,color = '#8A8A8A',
         label = '$f_{Y|X1}(C_3|x_1)$, posterior')

ax3.autoscale(enable=True, axis='x', tight=True)
ax3.autoscale(enable=True, axis='y', tight=True)
ax3.set_ylim([0,1])
ax3.set_yticks([0, 0.5, 1])
ax3.set_xlim([4,8])
ax3.set_xlabel('Sepal length, $x_1$')
ax3.legend()

#%% compare three Posterior curves


fig, ax = plt.subplots()

ax.fill_between(x1, f_C1_given_x1, facecolor = '#FF3300',alpha = 0.2)
ax.plot(x1, f_C1_given_x1,color = '#FF3300', label = '$f_{Y|X1}(C_1|x_1)$')

ax.fill_between(x1, f_C2_given_x1, facecolor = '#0099FF',alpha = 0.2)
ax.plot(x1, f_C2_given_x1,color = '#0099FF', label = '$f_{Y|X1}(C_2|x_1)$')

ax.fill_between(x1, f_C3_given_x1, facecolor = '#8A8A8A',alpha = 0.2)
ax.plot(x1, f_C3_given_x1,color = '#8A8A8A', label = '$f_{Y|X1}(C_3|x_1)$')

ax.axhline(y = 1, color = 'k', linestyle = '--')

ax.autoscale(enable=True, axis='x', tight=True)
ax.autoscale(enable=True, axis='y', tight=True)
ax.set_ylim([0,1])
ax.set_yticks([0, 0.5, 1])
ax.set_xlim([4,8])
ax.set_ylabel('Posterior probability')
ax.set_xlabel('Sepal length, $x_1$')
ax.legend()

#%%

#%% compare posterior, likelihood, marginal (evidence), and joint

fig, ax = plt.subplots()

# posterior
ax.plot(x1, f_C1_given_x1,color = 'r',label = '$f_{Y|X1}(C_1|x_1)$, posterior')

# likelihood
ax.plot(x1, f_x1_given_C1,color = '#0099FF',label = '$f_{X1|Y}(x_1|C_1)$, likelihood')
ax.fill_between(x1, f_x1_given_C1,alpha = 0.2,color = '#0099FF')

# marginal (evidence)
ax.plot(x1, f_x1, color = '#00448A',label = '$f_{X1}(x_1)$, evidence (marginal)')

# joint
ax.plot(x1, f_x1_joint_C1, color = '#92D050', label = '$f_{X1,Y}(x_1,C_1)$, joint')
ax.fill_between(x1,f_x1_joint_C1, 
                edgecolor = 'k',
                hatch='///',
                facecolor="none")

ax.set_ylim([0,1.5])
ax.set_yticks([0, 0.5, 1, 1.5])
ax.set_xlim([4,8])

ax.set_xlabel('Sepal length, $x_1$')
ax.legend()

# C2

fig, ax = plt.subplots()

# posterior
ax.plot(x1, f_C2_given_x1,color = 'r',label = '$f_{Y|X1}(C_2|x_1)$, posterior')

# likelihood
ax.plot(x1, f_x1_given_C2,color = '#0099FF',label = '$f_{X1|Y}(x_1|C_2)$, likelihood')
ax.fill_between(x1, f_x1_given_C2, alpha = 0.2,color = '#0099FF')

# marginal (evidence)
ax.plot(x1, f_x1, color = '#00448A',label = '$f_{X1}(x_1)$, evidence (marginal)')

# joint
ax.plot(x1, f_x1_joint_C2, color = '#92D050', label = '$f_{X1,Y}(x_1,C_2)$, joint')
ax.fill_between(x1,f_x1_joint_C2, 
                edgecolor = 'k',
                hatch='///',
                facecolor="none")

ax.set_ylim([0,1.5])
ax.set_yticks([0, 0.5, 1, 1.5])
ax.set_xlim([4,8])

ax.set_xlabel('Sepal length, $x_1$')
ax.legend()

# C3

fig, ax = plt.subplots()

# posterior
ax.plot(x1, f_C3_given_x1,color = 'r',label = '$f_{Y|X1}(C_3|x_1)$, posterior')

# likelihood
ax.plot(x1, f_x1_given_C3,color = '#0099FF',label = '$f_{X1|Y}(x_1|C_3)$, likelihood')
ax.fill_between(x1, f_x1_given_C3, alpha = 0.2,color = '#0099FF')

# marginal (evidence)
ax.plot(x1, f_x1, color = '#00448A', label = '$f_{X1}(x_1)$, evidence (marginal)')

# joint
ax.plot(x1, f_x1_joint_C3, color = '#92D050', label = '$f_{X1,Y}(x_1,C_3)$, joint')
ax.fill_between(x1,f_x1_joint_C3, 
                edgecolor = 'k',
                hatch='///',
                facecolor="none")

ax.set_ylim([0,1.5])
ax.set_yticks([0, 0.5, 1, 1.5])
ax.set_xlim([4,8])

ax.set_xlabel('Sepal length, $x_1$')
ax.legend()

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