二十二、降维算法-PCA主成分分析

df.columns=[\'sepal_len\', \'sepal_wid\', \'petal_len\', \'petal_wid\', \'class\']
df.head()

 

本节内容：

• PCA降维概述
• PCA要优化的目标
• PCA求解
• PCA实例

 1、PCA降维概述

 

 

 

 

2、PCA要优化的目标

 

 

 

 

 

 

 

 3、PCA求解

 

 

 

 

 

 

 

 

 

 

 

4、PCA实例

import numpy as np
import pandas as pd
df = pd.read_csv(\'iris.data\')
df.head()

df.columns=[\'sepal_len\', \'sepal_wid\', \'petal_len\', \'petal_wid\', \'class\']
df.head()

# split data table into data X and class labels y

X = df.ix[:,0:4].values
y = df.ix[:,4].values

 

from matplotlib import pyplot as plt
import math

label_dict = {1: \'Iris-Setosa\',
              2: \'Iris-Versicolor\',
              3: \'Iris-Virgnica\'}

feature_dict = {0: \'sepal length [cm]\',
                1: \'sepal width [cm]\',
                2: \'petal length [cm]\',
                3: \'petal width [cm]\'}


plt.figure(figsize=(8, 6))
for cnt in range(4):
    plt.subplot(2, 2, cnt+1)
    for lab in (\'Iris-setosa\', \'Iris-versicolor\', \'Iris-virginica\'):
        plt.hist(X[y==lab, cnt],
                     label=lab,
                     bins=10,
                     alpha=0.3,)
    plt.xlabel(feature_dict[cnt])
    plt.legend(loc=\'upper right\', fancybox=True, fontsize=8)

plt.tight_layout()
plt.show()

from sklearn.preprocessing import StandardScaler
X_std = StandardScaler().fit_transform(X)
print (X_std)

mean_vec = np.mean(X_std, axis=0)
cov_mat = (X_std - mean_vec).T.dot((X_std - mean_vec)) / (X_std.shape[0]-1)
print(\'Covariance matrix \n%s\' %cov_mat)

 

Covariance matrix 
[[ 1.00675676 -0.10448539  0.87716999  0.82249094]
 [-0.10448539  1.00675676 -0.41802325 -0.35310295]
 [ 0.87716999 -0.41802325  1.00675676  0.96881642]
 [ 0.82249094 -0.35310295  0.96881642  1.00675676]]

print(\'NumPy covariance matrix: \n%s\' %np.cov(X_std.T))

 

NumPy covariance matrix: 
[[ 1.00675676 -0.10448539  0.87716999  0.82249094]
 [-0.10448539  1.00675676 -0.41802325 -0.35310295]
 [ 0.87716999 -0.41802325  1.00675676  0.96881642]
 [ 0.82249094 -0.35310295  0.96881642  1.00675676]]

cov_mat = np.cov(X_std.T)

eig_vals, eig_vecs = np.linalg.eig(cov_mat)

print(\'Eigenvectors \n%s\' %eig_vecs)
print(\'\nEigenvalues \n%s\' %eig_vals)

Eigenvectors 特征向量
[[ 0.52308496 -0.36956962 -0.72154279  0.26301409]
[-0.25956935 -0.92681168  0.2411952  -0.12437342]
[ 0.58184289 -0.01912775  0.13962963 -0.80099722]
[ 0.56609604 -0.06381646  0.63380158  0.52321917]]

Eigenvalues 特征值
[ 2.92442837  0.93215233  0.14946373  0.02098259]

# Make a list of (eigenvalue, eigenvector) tuples
eig_pairs = [(np.abs(eig_vals[i]), eig_vecs[:,i]) for i in range(len(eig_vals))]
print (eig_pairs)
print (\'----------\')
# Sort the (eigenvalue, eigenvector) tuples from high to low
eig_pairs.sort(key=lambda x: x[0], reverse=True)

# Visually confirm that the list is correctly sorted by decreasing eigenvalues
print(\'Eigenvalues in descending order:\')
for i in eig_pairs:
    print(i[0])

 

[(2.9244283691111144, array([ 0.52308496, -0.25956935,  0.58184289,  0.56609604])), (0.93215233025350641, array([-0.36956962, -0.92681168, -0.01912775, -0.06381646])), (0.14946373489813314, array([-0.72154279,  0.2411952 ,  0.13962963,  0.63380158])), (0.020982592764270606, array([ 0.26301409, -0.12437342, -0.80099722,  0.52321917]))]
----------
Eigenvalues in descending order:
2.92442836911
0.932152330254
0.149463734898
0.0209825927643

tot = sum(eig_vals)
var_exp = [(i / tot)*100 for i in sorted(eig_vals, reverse=True)]
print (var_exp)
cum_var_exp = np.cumsum(var_exp)
cum_var_exp

[72.620033326920336, 23.147406858644135, 3.7115155645845164, 0.52104424985101538]

Out[49]:

array([  72.62003333,   95.76744019,   99.47895575,  100.        ])

a = np.array([1,2,3,4])
print (a)
print (\'-----------\')
print (np.cumsum(a))

 

[1 2 3 4]
-----------
[ 1  3  6 10]

 

plt.figure(figsize=(6, 4))

plt.bar(range(4), var_exp, alpha=0.5, align=\'center\',
            label=\'individual explained variance\')
plt.step(range(4), cum_var_exp, where=\'mid\',
             label=\'cumulative explained variance\')
plt.ylabel(\'Explained variance ratio\')
plt.xlabel(\'Principal components\')
plt.legend(loc=\'best\')
plt.tight_layout()
plt.show()

matrix_w = np.hstack((eig_pairs[0][1].reshape(4,1),
                      eig_pairs[1][1].reshape(4,1)))

print(\'Matrix W:\n\', matrix_w)

 

Matrix W:
 [[ 0.52308496 -0.36956962]
 [-0.25956935 -0.92681168]
 [ 0.58184289 -0.01912775]
 [ 0.56609604 -0.06381646]]

Y = X_std.dot(matrix_w)#150*4经过降维（乘上一个4*2的矩阵）后，得到一个150*2的矩阵
Y

 

plt.figure(figsize=(6, 4))
for lab, col in zip((\'Iris-setosa\', \'Iris-versicolor\', \'Iris-virginica\'),
                        (\'blue\', \'red\', \'green\')):
     plt.scatter(X[y==lab, 0],
                X[y==lab, 1],
                label=lab,
                c=col)
plt.xlabel(\'sepal_len\')
plt.ylabel(\'sepal_wid\')
plt.legend(loc=\'best\')
plt.tight_layout()
plt.show()

plt.figure(figsize=(6, 4))
for lab, col in zip((\'Iris-setosa\', \'Iris-versicolor\', \'Iris-virginica\'),
                        (\'blue\', \'red\', \'green\')):
     plt.scatter(Y[y==lab, 0],
                Y[y==lab, 1],
                label=lab,
                c=col)
plt.xlabel(\'Principal Component 1\')
plt.ylabel(\'Principal Component 2\')
plt.legend(loc=\'lower center\')
plt.tight_layout()
plt.show()

 

详见：http://www.pianshen.com/article/110236350/