Dimensionality Reduction with Eigenvector / Eigenvalues and Correlation Matrix (PCA)

In [1]:

    

%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns
import pandas as pd
import numpy as np
from numpy import linalg as LA

from sklearn import datasets


    

In [2]:

    

iris = datasets.load_iris()


    

In [3]:

    

df = pd.DataFrame(iris.data, columns=iris.feature_names)
corr = df.corr()
df.corr()


    

    
Out[3]:






  

    

      

      
sepal length (cm)
      
sepal width (cm)
      
petal length (cm)
      
petal width (cm)
    
  
  

    

      
sepal length (cm)
      
1.000000
      
-0.109369
      
0.871754
      
0.817954
    
    

      
sepal width (cm)
      
-0.109369
      
1.000000
      
-0.420516
      
-0.356544
    
    

      
petal length (cm)
      
0.871754
      
-0.420516
      
1.000000
      
0.962757
    
    

      
petal width (cm)
      
0.817954
      
-0.356544
      
0.962757
      
1.000000
    
  

In [4]:

    

sns.heatmap(corr)


    

    
Out[4]:





<matplotlib.axes._subplots.AxesSubplot at 0x10a8c8f28>






    

In [5]:

    

eig_vals, eig_vecs = LA.eig(corr)
eig_pairs = [(np.abs(eig_vals[i]), eig_vecs[:,i]) for i in range(len(eig_vals))]

eig_pairs.sort(key=lambda x: x[0], reverse=True)


    

In [6]:

    

pd.DataFrame([eig_vals])


    

    
Out[6]:






  

    

      

      
0
      
1
      
2
      
3
    
  
  

    

      
0
      
2.910818
      
0.921221
      
0.147353
      
0.020608
    
  

In [7]:

    

pd.DataFrame(eig_vecs)


    

    
Out[7]:






  

    

      

      
0
      
1
      
2
      
3
    
  
  

    

      
0
      
0.522372
      
-0.372318
      
-0.721017
      
0.261996
    
    

      
1
      
-0.263355
      
-0.925556
      
0.242033
      
-0.124135
    
    

      
2
      
0.581254
      
-0.021095
      
0.140892
      
-0.801154
    
    

      
3
      
0.565611
      
-0.065416
      
0.633801
      
0.523546
    
  

In [8]:

    

matrix_w = np.hstack((eig_pairs[0][1].reshape(len(corr),1),
                      eig_pairs[1][1].reshape(len(corr),1)))


pd.DataFrame(matrix_w, columns=['PC1', 'PC2'])


    

    
Out[8]:






  

    

      

      
PC1
      
PC2
    
  
  

    

      
0
      
0.522372
      
-0.372318
    
    

      
1
      
-0.263355
      
-0.925556
    
    

      
2
      
0.581254
      
-0.021095
    
    

      
3
      
0.565611
      
-0.065416
    
  

In [9]:

    

new_dim = np.dot(np.array(iris.data), matrix_w)

df = pd.DataFrame(new_dim, columns=['X', 'Y'])
df['label'] = iris.target
df.head()


    

    
Out[9]:






  

    

      

      
X
      
Y
      
label
    
  
  

    

      
0
      
2.669231
      
-5.180887
      
0
    
    

      
1
      
2.696434
      
-4.643645
      
0
    
    

      
2
      
2.481163
      
-4.752183
      
0
    
    

      
3
      
2.571512
      
-4.626615
      
0
    
    

      
4
      
2.590658
      
-5.236211
      
0
    
  

In [10]:

    

fig = plt.figure()
fig.suptitle('PCA with Eigenvector', fontsize=14, fontweight='bold')
ax = fig.add_subplot(111)

plt.scatter(df[df.label == 0].X, df[df.label == 0].Y, color='red', label=iris.target_names[0])
plt.scatter(df[df.label == 1].X, df[df.label == 1].Y, color='blue', label=iris.target_names[1])
plt.scatter(df[df.label == 2].X, df[df.label == 2].Y, color='green', label=iris.target_names[2])

plt.legend(bbox_to_anchor=(1.25, 1))


    

    
Out[10]:





<matplotlib.legend.Legend at 0x10ab8bbe0>