一、PCA算法

PCA算法为主成分分析算法，在数据集中找到“主成分”，可以用于压缩数据维度。

我们将首先通过一个2D数据集进行实验，以获得关于PCA如何工作的直观感受，然后在一个更大的图像数据集上使用它。

PCA算法的好处如下：

1.使得数据集更易使用

2.降低算法的计算开销

3.去除噪声

4.使得结果更易理解

线性回归和神经网络算法，都可以先使用PCA对数据进行降维。

关于PCA算法的理论部分，可以参考我之前的博客：

https://blog.csdn.net/wzk4869/article/details/126017831?spm=1001.2014.3001.5501

二、PCA实战项目

老样子，先放上需要使用的数据集：

链接：https://pan.baidu.com/s/1_8yJo58oCuuo6ZasaC5R3A
提取码：6666

导入需要的包

import matplotlib.pyplot as plt
import matplotlib as mpl
from scipy.io import loadmat
from numpy import *
import pandas as pd

导入数据集

def load_dataset():
    path='./data/ex7data1.mat'
    two_dimension_data=loadmat(path)
    X=two_dimension_data['X']
    return X

我们看一下对应的数据集：

[[3.38156267 3.38911268]
 [4.52787538 5.8541781 ]
 [2.65568187 4.41199472]
 [2.76523467 3.71541365]
 [2.84656011 4.17550645]
 [3.89067196 6.48838087]
 [3.47580524 3.63284876]
 [5.91129845 6.68076853]
 [3.92889397 5.09844661]
 [4.56183537 5.62329929]
 [4.57407171 5.39765069]
 [4.37173356 5.46116549]
 [4.19169388 4.95469359]
 [5.24408518 4.66148767]
 [2.8358402  3.76801716]
 [5.63526969 6.31211438]
 [4.68632968 5.6652411 ]
 [2.85051337 4.62645627]
 [5.1101573  7.36319662]
 [5.18256377 4.64650909]
 [5.70732809 6.68103995]
 [3.57968458 4.80278074]
 [5.63937773 6.12043594]
 [4.26346851 4.68942896]
 [2.53651693 3.88449078]
 [3.22382902 4.94255585]
 [4.92948801 5.95501971]
 [5.79295774 5.10839305]
 [2.81684824 4.81895769]
 [3.88882414 5.10036564]
 [3.34323419 5.89301345]
 [5.87973414 5.52141664]
 [3.10391912 3.85710242]
 [5.33150572 4.68074235]
 [3.37542687 4.56537852]
 [4.77667888 6.25435039]
 [2.6757463  3.73096988]
 [5.50027665 5.67948113]
 [1.79709714 3.24753885]
 [4.3225147  5.11110472]
 [4.42100445 6.02563978]
 [3.17929886 4.43686032]
 [3.03354125 3.97879278]
 [4.6093482  5.879792  ]
 [2.96378859 3.30024835]
 [3.97176248 5.40773735]
 [1.18023321 2.87869409]
 [1.91895045 5.07107848]
 [3.95524687 4.5053271 ]
 [5.11795499 6.08507386]]
(50, 2)

绘制散点图

def plot_scatter(X):
    plt.figure(figsize=(12,8))
    plt.scatter(X[:,0],X[:,1])
    plt.show()

我们看一下可视化后的散点图：

去均值化

我们返回的结果是去均值化后的数据，并生成散点图：

def demean(X):
    X_demean=(X-mean(X,axis=0))
    plt.figure(figsize=(12,8))
    plt.scatter(X_demean[:,0],X_demean[:,1])
    plt.show()
    return X_demean

计算数据的协方差矩阵

def sigma_matrix(X_demean):
    sigma=(X_demean.T @ X_demean)/X_demean.shape[0]
    return sigma

[[1.34852518 0.86535019]
 [0.86535019 1.02641621]]

计算特征值、特征向量

def usv(sigma):
    u,s,v=linalg.svd(sigma)
    return u,s,v

[[-0.76908153 -0.63915068]
 [-0.63915068  0.76908153]]
====================================================================================================
[2.06768062 0.30726078]
====================================================================================================
[[-0.76908153 -0.63915068]
 [-0.63915068  0.76908153]]

对数据进行降维

对数据进行降维，降维后得到Z，在二维数据中Z数据为一条直线点（一维）。

def project_data(X_demean, u, k):
    u_reduced = u[:,:k]
    z=dot(X_demean, u_reduced)
    return z

[[ 1.49876595]
 [-0.95839024]
 [ 1.40325172]
 [ 1.76421694]
 [ 1.40760243]
 [-0.87367998]
 [ 1.27050164]
 [-2.5506712 ]
 [-0.01469839]
 [-0.83694188]
 [-0.70212917]
 [-0.58711016]
 [-0.12493311]
 [-0.74690506]
 [ 1.67629396]
 [-2.10275704]
 [-0.9594953 ]
 [ 1.11633715]
 [-2.37070273]
 [-0.69001651]
 [-2.39397485]
 [ 0.44284714]
 [-1.98340505]
 [-0.01058959]
 [ 1.83205377]
 [ 0.62719172]
 [-1.33171608]
 [-1.4546727 ]
 [ 1.01919098]
 [ 0.01489202]
 [-0.07212622]
 [-1.78539513]
 [ 1.41318051]
 [-0.82644523]
 [ 0.75167377]
 [-1.40551081]
 [ 1.82309802]
 [-1.59458841]
 [ 2.80783613]
 [-0.32551527]
 [-0.98578762]
 [ 0.98465469]
 [ 1.38952836]
 [-1.03742062]
 [ 1.87686597]
 [-0.24535117]
 [ 3.51800218]
 [ 1.54860441]
 [ 0.34412682]
 [-1.55978675]]

还原数据

def recover_data(z, u, k):
    u_reduced = u[:,:k]
    X_recover=dot(z, u_reduced.T)
    return X_recover

[[-1.15267321 -0.95793728]
 [ 0.73708023  0.61255577]
 [-1.07921498 -0.89688929]
 [-1.35682667 -1.12760046]
 [-1.08256103 -0.89967005]
 [ 0.67193114  0.55841316]
 [-0.97711935 -0.81204199]
 [ 1.96167412  1.63026324]
 [ 0.01130426  0.00939449]
 [ 0.64367655  0.53493198]
 [ 0.53999458  0.44876634]
 [ 0.45153559  0.37525186]
 [ 0.09608375  0.07985108]
 [ 0.57443089  0.47738488]
 [-1.28920673 -1.07140443]
 [ 1.61719161  1.34397859]
 [ 0.73793012  0.61326208]
 [-0.85855429 -0.71350765]
 [ 1.82326369  1.51523626]
 [ 0.53067896  0.44102452]
 [ 1.84116185  1.53011066]
 [-0.34058556 -0.28304605]
 [ 1.5254002   1.26769469]
 [ 0.00814426  0.00676834]
 [-1.40899873 -1.17095842]
 [-0.48236157 -0.40087002]
 [ 1.02419824  0.85116724]
 [ 1.11876191  0.92975505]
 [-0.78384096 -0.65141661]
 [-0.01145318 -0.00951825]
 [ 0.05547094  0.04609952]
 [ 1.37311442  1.14113651]
 [-1.08685103 -0.90323529]
 [ 0.63560376  0.52822303]
 [-0.57809841 -0.4804328 ]
 [ 1.08095241  0.89833319]
 [-1.40211102 -1.16523434]
 [ 1.2263685   1.01918227]
 [-2.15945492 -1.79463038]
 [ 0.25034779  0.20805331]
 [ 0.75815106  0.63006683]
 [-0.75727974 -0.62934272]
 [-1.0686606  -0.888118  ]
 [ 0.79786104  0.6630681 ]
 [-1.44346296 -1.19960017]
 [ 0.18869505  0.15681637]
 [-2.70563051 -2.24853349]
 [-1.19100306 -0.98979157]
 [-0.26466158 -0.21994889]
 [ 1.19960319  0.99693877]]

数据可视化

fig, ax = plt.subplots(figsize=(12, 8))
ax.scatter(X_demean[:,0],X_demean[:,1])
ax.scatter(list(X_recover[:, 0]), list(X_recover[:, 1]),c='r')
ax.plot([X_demean[:,0],list(X_recover[:, 0])]，[X_demean[:,1],list(X_recover[:, 1])])
plt.show()

完整版代码如下：

import matplotlib.pyplot as plt
import matplotlib as mpl
from scipy.io import loadmat
from numpy import *
import pandas as pd
"""
PCA对二维数据进行降维
"""

def load_dataset():
    path='./data/ex7data1.mat'
    two_dimension_data=loadmat(path)
    X=two_dimension_data['X']
    return X

def plot_scatter(X):
    plt.figure(figsize=(12,8))
    cm=mpl.colors.ListedColormap(['blue'])
    plt.scatter(X[:,0],X[:,1],cmap=cm)
    plt.show()

"""
对X去均值,并可视化图像
"""
def demean(X):
    X_demean=(X-mean(X,axis=0))
    plt.figure(figsize=(12,8))
    plt.scatter(X_demean[:,0],X_demean[:,1])
    plt.show()
    return X_demean

"""
计算协方差矩阵
"""
def sigma_matrix(X_demean):
    sigma=(X_demean.T @ X_demean)/X_demean.shape[0]
    return sigma

"""
计算特征值、特征向量
"""
def usv(sigma):
    u,s,v=linalg.svd(sigma)
    return u,s,v

def project_data(X_demean, u, k):
    u_reduced = u[:,:k]
    z=dot(X_demean, u_reduced)
    return z

def recover_data(z, u, k):
    u_reduced = u[:,:k]
    X_recover=dot(z, u_reduced.T)
    return X_recover

if __name__=='__main__':
    X=load_dataset()
    print(X)
    print('=='*50)
    print(X.shape)
    print('=='*50)
    plot_scatter(X)
    X_demean=demean(X)
    sigma=sigma_matrix(X_demean)
    print(sigma)
    print('=='*50)
    u, s, v=usv(sigma)
    print(u)
    print('=='*50)
    print(s)
    print('=='*50)
    print(v)
    print('=='*50)
    z = project_data(X_demean, u, 1)
    print(z)
    print('=='*50)
    X_recover = recover_data(z, u, 1)
    print(X_recover)
    print('=='*50)
    fig, ax = plt.subplots(figsize=(12, 8))
    ax.scatter(X_demean[:,0],X_demean[:,1])
    ax.scatter(list(X_recover[:, 0]), list(X_recover[:, 1]),c='r')
    ax.plot([X_demean[:,0],list(X_recover[:, 0])],[X_demean[:,1],list(X_recover[:, 1])])
    plt.show()