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PRINCIPLE COMPONENT ANALYSIS

Authors

Ndèye Gagnessiry Ndiaye and Christin Seifert

License

This work is licensed under the Creative Commons Attribution 3.0 Unported License https://creativecommons.org/licenses/by/3.0/

This notebook:

  • creates PCA projections of Iris dataset




In [7]:



    


import pandas as pd
import numpy as np
import pylab as plt
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA

The Iris dataset represents 3 kind of Iris flowers (Setosa, Versicolour and Virginica) with 4 attributes: sepal length, sepal width, petal length and petal width.





In [8]:



    


from sklearn import datasets
iris = datasets.load_iris()
x = pd.DataFrame(iris.data)
x.columns = ['SepalLength','SepalWidth','PetalLength','PetalWidth'] 
x.head()



    


















    
Out[8]:










  
    

      
      SepalLength
      SepalWidth
      PetalLength
      PetalWidth
    

  
  
    

      0
      5.1
      3.5
      1.4
      0.2
    

    

      1
      4.9
      3.0
      1.4
      0.2
    

    

      2
      4.7
      3.2
      1.3
      0.2
    

    

      3
      4.6
      3.1
      1.5
      0.2
    

    

      4
      5.0
      3.6
      1.4
      0.2

We apply Principal Component Analysis to the Iris dataset with 4-dimensions (all components are keeped).





In [9]:



    


pca = PCA(n_components=4)
pca.fit(iris.data)



    


















    
Out[9]:








PCA(copy=True, iterated_power='auto', n_components=4, random_state=None,
  svd_solver='auto', tol=0.0, whiten=False)

















In [10]:



    


eigen_values =pca.explained_variance_
print(eigen_values)



    


















    






[ 4.19667516  0.24062861  0.07800042  0.02352514]

















In [11]:



    


eigen_vectors = pca.components_
print(eigen_vectors)



    


















    






[[ 0.36158968 -0.08226889  0.85657211  0.35884393]
 [ 0.65653988  0.72971237 -0.1757674  -0.07470647]
 [-0.58099728  0.59641809  0.07252408  0.54906091]
 [ 0.31725455 -0.32409435 -0.47971899  0.75112056]]

We project data in the PCA 4-dimensionnal space.





In [12]:



    


projection = pca.transform(iris.data)
x = pd.DataFrame(projection)
x.columns = ['PC1','PC2','PC3','PC4'] 
x.head()



    


















    
Out[12]:










  
    

      
      PC1
      PC2
      PC3
      PC4
    

  
  
    

      0
      -2.684207
      0.326607
      -0.021512
      0.001006
    

    

      1
      -2.715391
      -0.169557
      -0.203521
      0.099602
    

    

      2
      -2.889820
      -0.137346
      0.024709
      0.019305
    

    

      3
      -2.746437
      -0.311124
      0.037672
      -0.075955
    

    

      4
      -2.728593
      0.333925
      0.096230
      -0.063129

The following figure shows successively the projections on (x=PC1,y=PC2), (x=PC1,y=PC3),(x=PC1,y=PC4),(x=PC2,y=PC3),(x=PC2,y=PC4) and (x=PC3,y=PC4). Data is best separated with the components with largest eigenvalues (highest variance).





In [13]:



    


# Show projections

y = iris.target
target_names = iris.target_names

colors = ['navy', 'turquoise', 'darkorange']
lw = 2

plt.figure(figsize=(25,30))

plt.subplot(231)
for color, i, target_name in zip(colors, [0, 1,2], target_names):
    plt.scatter(projection[y == i, 0], projection[y == i, 1], color=color, alpha=.8, lw=lw,
                label=target_name)
plt.legend(loc='best', shadow=False, scatterpoints=1)
plt.xlabel('PC1')
plt.ylabel('PC2')
plt.title('PCA((x=PC1,y=PC2))')

plt.subplot(232)
for color, i, target_name in zip(colors, [0, 1,2], target_names):
    plt.scatter(projection[y == i, 0], projection[y == i, 2], color=color, alpha=.8, lw=lw,
                label=target_name)
plt.legend(loc='best', shadow=False, scatterpoints=1)
plt.xlabel('PC1')
plt.ylabel('PC3')
plt.title('PCA((x=PC1,y=PC3))')

plt.subplot(233)
for color, i, target_name in zip(colors, [0, 1,2], target_names):
    plt.scatter(projection[y == i, 0], projection[y == i, 3], color=color, alpha=.8, lw=lw,
                label=target_name)
plt.legend(loc='best', shadow=False, scatterpoints=1)
plt.xlabel('PC1')
plt.ylabel('PC4')
plt.title('PCA((x=PC1,y=PC4))')

plt.subplot(234)
for color, i, target_name in zip(colors, [0, 1,2], target_names):
    plt.scatter(projection[y == i, 1], projection[y == i, 2], color=color, alpha=.8, lw=lw,
                label=target_name)
plt.legend(loc='best', shadow=False, scatterpoints=1)
plt.xlabel('PC2')
plt.ylabel('PC3')
plt.title('PCA((x=PC2,y=PC3))')

plt.subplot(235)
for color, i, target_name in zip(colors, [0, 1,2], target_names):
    plt.scatter(projection[y == i, 1], projection[y == i, 3], color=color, alpha=.8, lw=lw,
                label=target_name)
plt.legend(loc='best', shadow=False, scatterpoints=1)
plt.xlabel('PC2')
plt.ylabel('PC4')
plt.title('PCA((x=PC2,y=PC4))')

plt.subplot(236)
for color, i, target_name in zip(colors, [0, 1,2], target_names):
    plt.scatter(projection[y == i, 2], projection[y == i, 3], color=color, alpha=.8, lw=lw,
                label=target_name)
plt.legend(loc='best', shadow=False, scatterpoints=1)
plt.xlabel('PC3')
plt.ylabel('PC4')
plt.title('PCA((x=PC3,y=PC4))')


plt.show()



    


















    
























In [ ]:

Content source: chseifert/tutorials

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