降维算法实战项目（1）—使用PCA对二维数据降维（Python代码+数据集）

谁借莪１个温暖的怀抱￠ 2023-09-28 17:38 61阅读 0赞

### 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：

![d1360c0c593e47d194a539910c0d8760.png][]  
**去均值化**  
![6900a2b90fbf4e489919ae6c5a88adf0.png][]  
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

![d85064ab17c74b9eb98e6dec4d2ed8f3.png][]  
**计算数据的协方差矩阵**  
![de01fa678ed94345ba59eab8be3275f9.png][]

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()

![e6e7b58661044474841272ea37f2de78.png][]

### 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()

[d1360c0c593e47d194a539910c0d8760.png]: https://img-blog.csdnimg.cn/d1360c0c593e47d194a539910c0d8760.png
[6900a2b90fbf4e489919ae6c5a88adf0.png]: https://img-blog.csdnimg.cn/6900a2b90fbf4e489919ae6c5a88adf0.png
[d85064ab17c74b9eb98e6dec4d2ed8f3.png]: https://img-blog.csdnimg.cn/d85064ab17c74b9eb98e6dec4d2ed8f3.png
[de01fa678ed94345ba59eab8be3275f9.png]: https://img-blog.csdnimg.cn/de01fa678ed94345ba59eab8be3275f9.png
[e6e7b58661044474841272ea37f2de78.png]: https://img-blog.csdnimg.cn/e6e7b58661044474841272ea37f2de78.png