PCA 的手动实现会产生错误的图，其中特征向量不是正交的

Question

我需要绘制我这样计算的特征向量：

def fit(self, X):
    
    '''
    fits sorted eigenvalues and eigenvectors to class attributes. same goes for variance and explained variance.
    '''
    
    n_samples = X.shape[0]
    # We center the data and compute the sample covariance matrix.
    X -= np.mean(X, axis=0)
    self.cov_matrix_ = np.dot(X.T, X) / (n_samples-1)
    #test = np.cov(X)
    
    #Negative values are ignored with eigh
    (self.eigvalues_, self.components_) = np.linalg.eigh(self.cov_matrix_)
    
    idx = self.eigvalues_.argsort()[::-1]   
    self.eigvalues_ = self.eigvalues_[idx]
    self.components_ = self.components_[:,idx]
    self.variance_ = np.sum(self.eigvalues_)
    self.explained_variance_ = self.eigvalues_ / self.variance_
    
def transform(self, X):
    #project data onto eigenvectors
    print(self.components_.shape, X.shape)
    self.projected_ = X @ self.components_.T
    return self.projected_

进入我的数据集前 2 个特征的图。

self.components_ 的形状是 100x240 数据集的 240 个特征向量，形状为 240x240。 用最大的特征值绘制我的 2 个特征向量的前两个值后，结果如下：

pca = PCA()

pca.fit(subsample)

#pca.transform(subsample)

plt.scatter(subsample[:,0], subsample[:,1], edgecolor='none', alpha=0.5)
plt.quiver(pca.components_[0,0], pca.components_[0,1], 
       angles='xy', scale_units='xy', scale=1, width=0.002 )
plt.quiver(pca.components_[1,0], pca.components_[1,1], 
       angles='xy', scale_units='xy', scale=1, width=0.002 )

我做错了什么？

Answer 1

您应该按行而不是列对特征向量进行排序

self.components_ = self.components_[:,idx]

应该是

self.components_ = self.components_[idx]

此外，您应该确保以相同的纵横比进行绘图，因为箭袋可能会错位：

plt.gca().set_aspect('equal')

在您的代码中包含一个最低限度的工作示例是一种很好的做法，因此下次请记住 :)。为了获得最小的工作示例，我不得不推断您的其余代码可能是什么。无论如何，这是我建议的代码：

import numpy as np 
from matplotlib import pyplot as plt

class PCA:
    def fit(self, X):
        
        '''
        fits sorted eigenvalues and eigenvectors to class attributes. same goes for variance and explained variance.
        '''
        
        n_samples = X.shape[0]
        # We center the data and compute the sample covariance matrix.
        X -= np.mean(X, axis=0)
        self.cov_matrix_ = np.dot(X.T, X) / (n_samples-1)
        #test = np.cov(X)
        
        #Negative values are ignored with eigh
        (self.eigvalues_, self.components_) = np.linalg.eigh(self.cov_matrix_)
        
        idx = self.eigvalues_.argsort()[::-1]   
        self.eigvalues_ = self.eigvalues_[idx]
        self.components_ = self.components_[idx]
        self.variance_ = np.sum(self.eigvalues_)
        self.explained_variance_ = self.eigvalues_ / self.variance_
        
    def transform(self, X):
        #project data onto eigenvectors
        print(self.components_.shape, X.shape)
        self.projected_ = X @ self.components_.T
        return self.projected_

pca = PCA()

# Generate some dummy data
subsample = np.random.randn(69,2)*0.1 
subsample[:,0] = subsample[:,0]*8 
subsample[:,1] = subsample[:,0]*2 + subsample[:,1] # Add some correlations

pca.fit(subsample)

plt.scatter(subsample[:,0], subsample[:,1], edgecolor='none', alpha=0.5)
plt.quiver(pca.components_[0,0]*2, pca.components_[0,1]*2, # *2 to make arrows larger
       angles='xy', scale_units='xy', scale=1, width=0.006)
plt.quiver(pca.components_[1,0]*2, pca.components_[1,1]*2, 
       angles='xy', scale_units='xy', scale=1, width=0.006)
plt.gca().set_aspect('equal')
plt.show()