Manual Implementation of PCA produces a wrong plot, where eigenvectors are not orthogonal

Question

I need to plot my eigenvectors that I calculated like this:

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_

Into the plot of the first 2 features of my dataset.

The shape of my self.components_ which are my 240 eigenvectors of my 100x240 dataset, have shape 240x240. After plotting the first two values of my 2 eigenvectors with the largest eigenvalue, it comes out like this:

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 )

What am I doing wrong?

Answer 1

Your should sort your eigenvectors by the rows, not the columns, that is

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

should be

self.components_ = self.components_[idx]

Also, you should ensure that you plot with equal aspect ratio, as the quivers may be misaligned:

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

It is good practice to include a minimum working example in your code, so remember that next time :). I had to infer what the rest of your code could be in order to get a minimum working example. Anyways, here is my proposed code:

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()