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PCA.py
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PCA.py
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#-*- coding: utf-8 -*-
# Author: Bob
# Date: 2016.11.24
import numpy as np
from matplotlib import pyplot as plt
from scipy import io as spio
from sklearn.decomposition import pca
'''
主成分分析_2维数据降维1维演示函数
'''
def PCA_2D():
data_2d = spio.loadmat("data.mat")
X = data_2d['X']
m = X.shape[0]
plt = plot_data_2d(X,'bo') # 显示二维的数据
plt.show()
X_copy = X.copy()
X_norm,mu,sigma = featureNormalize(X_copy) # 归一化数据
#plot_data_2d(X_norm) # 显示归一化后的数据
#plt.show()
Sigma = np.dot(np.transpose(X_norm),X_norm)/m # 求Sigma
U,S,V = np.linalg.svd(Sigma) # 求Sigma的奇异值分解
plt = plot_data_2d(X,'bo') # 显示原本数据
drawline(plt, mu, mu+S[0]*(U[:,0]), 'r-') # 线，为投影的方向
plt.axis('square')
plt.show()
K = 1 # 定义降维多少维（本来是2维的，这里降维1维）
'''投影之后数据（降维之后）'''
Z = projectData(X_norm,U,K) # 投影
'''恢复数据'''
X_rec = recoverData(Z,U,K) # 恢复
'''作图-----原数据与恢复的数据'''
plt = plot_data_2d(X_norm,'bo')
plot_data_2d(X_rec,'ro')
for i in range(X_norm.shape[0]):
drawline(plt, X_norm[i,:], X_rec[i,:], '--k')
plt.axis('square')
plt.show()
'''主成分分析_PCA图像数据降维'''
def PCA_faceImage():
print (u'加载图像数据.....')
data_image = spio.loadmat('data_faces.mat')
X = data_image['X']
display_imageData(X[0:100,:])
m = X.shape[0] # 数据条数
print (u'运行PCA....')
X_norm,mu,sigma = featureNormalize(X) # 归一化
Sigma = np.dot(np.transpose(X_norm),X_norm)/m # 求Sigma
U,S,V = np.linalg.svd(Sigma) # 奇异值分解
display_imageData(np.transpose(U[:,0:36])) # 显示U的数据
print (u'对face数据降维.....')
K = 100 # 降维100维(原先是32*32=1024维的)
Z = projectData(X_norm, U, K)
print (u'投影之后Z向量的大小：%d %d' %Z.shape)
print (u'显示降维之后的数据......')
X_rec = recoverData(Z, U, K) # 恢复数据
display_imageData(X_rec[0:100,:])
# 可视化二维数据
def plot_data_2d(X,marker):
plt.plot(X[:,0],X[:,1],marker)
return plt
# 归一化数据
def featureNormalize(X):
'''（每一个数据-当前列的均值）/当前列的标准差'''
n = X.shape[1]
mu = np.zeros((1,n));
sigma = np.zeros((1,n))
mu = np.mean(X,axis=0) # axis=0表示列
sigma = np.std(X,axis=0)
for i in range(n):
X[:,i] = (X[:,i]-mu[i])/sigma[i]
return X,mu,sigma
# 映射数据
def projectData(X_norm,U,K):
Z = np.zeros((X_norm.shape[0],K))
U_reduce = U[:,0:K] # 取前K个
Z = np.dot(X_norm,U_reduce)
return Z
# 画一条线
def drawline(plt,p1,p2,line_type):
plt.plot(np.array([p1[0],p2[0]]),np.array([p1[1],p2[1]]),line_type)
# 恢复数据
def recoverData(Z,U,K):
X_rec = np.zeros((Z.shape[0],U.shape[0]))
U_recude = U[:,0:K]
X_rec = np.dot(Z,np.transpose(U_recude)) # 还原数据（近似）
return X_rec
# 显示图片
def display_imageData(imgData):
sum = 0
'''
显示100个数（若是一个一个绘制将会非常慢，可以将要画的图片整理好，放到一个矩阵中，显示这个矩阵即可）
- 初始化一个二维数组
- 将每行的数据调整成图像的矩阵，放进二维数组
- 显示即可
'''
m,n = imgData.shape
width = np.int32(np.round(np.sqrt(n)))
height = np.int32(n/width);
rows_count = np.int32(np.floor(np.sqrt(m)))
cols_count = np.int32(np.ceil(m/rows_count))
pad = 1
display_array = -np.ones((pad+rows_count*(height+pad),pad+cols_count*(width+pad)))
for i in range(rows_count):
for j in range(cols_count):
max_val = np.max(np.abs(imgData[sum,:]))
display_array[pad+i*(height+pad):pad+i*(height+pad)+height,pad+j*(width+pad):pad+j*(width+pad)+width] = imgData[sum,:].reshape(height,width,order="F")/max_val # order=F指定以列优先，在matlab中是这样的，python中需要指定，默认以行
sum += 1
plt.imshow(display_array,cmap='gray') #显示灰度图像
plt.axis('off')
plt.show()
if __name__ == "__main__":
PCA_2D()
PCA_faceImage()