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Kernel PCA in Scikit-learn

This example shows that Kernel PCA is able to find a projection of the data that makes data linearly separable.

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Version

In [1]:
import sklearn
sklearn.__version__
Out[1]:
'0.18'

Imports

This tutorial imports PCA, KernelPCA and make_circles.

In [2]:
import plotly.plotly as py
import plotly.graph_objs as go
from plotly import tools

import numpy as np
import matplotlib.pyplot as plt

from sklearn.decomposition import PCA, KernelPCA
from sklearn.datasets import make_circles

Calculations

In [3]:
np.random.seed(0)

X, y = make_circles(n_samples=400, factor=.3, noise=.05)

kpca = KernelPCA(kernel="rbf", fit_inverse_transform=True, gamma=10)
X_kpca = kpca.fit_transform(X)
X_back = kpca.inverse_transform(X_kpca)
pca = PCA()
X_pca = pca.fit_transform(X)

Plot Results

In [4]:
fig = tools.make_subplots(rows=2, cols=2,
                          subplot_titles=("Original space",
                                          "Projection by PCA",
                                          "Projection by KPCA",
                                          "Original space after inverse transform"))
This is the format of your plot grid:
[ (1,1) x1,y1 ]  [ (1,2) x2,y2 ]
[ (2,1) x3,y3 ]  [ (2,2) x4,y4 ]

In [5]:
reds = y == 0
blues = y == 1

original_space1 = go.Scatter(x=X[reds, 0], 
                             y=X[reds, 1],
                             mode='markers',
                             showlegend=False,
                             marker=dict(color='red',
                                         line=dict(color='black', width=1))
                            )
original_space2 = go.Scatter(x=X[blues, 0],
                             y=X[blues, 1],
                             mode='markers',
                             showlegend=False,
                             marker=dict(color='blue',
                                         line=dict(color='black', width=1))
                             
                            )

X1, X2 = np.linspace(-1.5, 1.5, 50), np.linspace(-1.5, 1.5, 50)
X_grid = np.array([np.ravel(X1), np.ravel(X2)]).T

# projection on the first principal component (in the phi space)
Z_grid = kpca.transform(X_grid)[:, 0].reshape(X1.shape)
lines = go.Contour(x=X1, 
                   y=X2, 
                   z=Z_grid, 
                   showscale=False,
                   colorscale=[[0,'white'],[1, 'black']],
                   contours=dict(coloring='lines')
                  )

fig.append_trace(lines, 1, 1)
fig.append_trace(original_space1, 1, 1)
fig.append_trace(original_space2, 1, 1)

fig['layout']['xaxis1'].update(title='x<sub>1</sub>',
                               zeroline=False, showgrid=False)
fig['layout']['yaxis1'].update(title='x<sub>2</sub>',
                               zeroline=False, showgrid=False)


projection_pca1 = go.Scatter(x=X_pca[reds, 0], 
                             y=X_pca[reds, 1], 
                             mode='markers',
                             showlegend=False,
                             marker=dict(color='red',
                                         line=dict(color='black', width=1)) 
                            )
projection_pca2 = go.Scatter(x=X_pca[blues, 0], 
                             y=X_pca[blues, 1], 
                             mode='markers',
                             showlegend=False,
                             marker=dict(color='blue',
                                         line=dict(color='black', width=1))
                            )

fig.append_trace(projection_pca1, 1, 2)
fig.append_trace(projection_pca2, 1, 2)

fig['layout']['xaxis2'].update(title='1st principal component',
                               zeroline=False, showgrid=False)
fig['layout']['yaxis2'].update(title='2nd component',
                               zeroline=False, showgrid=False)

projection_kpca1 = go.Scatter(x=X_kpca[reds, 0], 
                              y=X_kpca[reds, 1],
                              mode='markers',
                              showlegend=False,
                              marker=dict(color='red',
                                          line=dict(color='black', width=1)) 
                             )
projection_kpca2 = go.Scatter(x=X_kpca[blues, 0], 
                              y=X_kpca[blues, 1], 
                              mode='markers',
                              showlegend=False,
                              marker=dict(color='blue',
                                          line=dict(color='black', width=1))  
                             )

fig.append_trace(projection_kpca1, 2, 1)
fig.append_trace(projection_kpca2, 2, 1)

fig['layout']['xaxis3'].update(title="1st principal component in space induced by Phi",
                               zeroline=False, showgrid=False)
fig['layout']['yaxis3'].update(title='2nd component',
                               zeroline=False, showgrid=False)

after_inverse1 = go.Scatter(x=X_back[reds, 0], 
                            y=X_back[reds, 1], 
                            mode='markers',
                            showlegend=False,
                            marker=dict(color='red',
                                        line=dict(color='black', width=1))  
                           )
after_inverse2 = go.Scatter(x=X_back[blues, 0],
                            y=X_back[blues, 1], 
                            mode='markers',
                            showlegend=False,
                            marker=dict(color='blue',
                                        line=dict(color='black', width=1))  
                           )

fig.append_trace(after_inverse1, 2, 2)
fig.append_trace(after_inverse2, 2, 2)

fig['layout']['xaxis4'].update(title='x<sub>1</sub>',
                               zeroline=False, showgrid=False)
fig['layout']['yaxis4'].update(title='x<sub>2</sub>',
                               zeroline=False, showgrid=False)

fig['layout'].update(height=800)
In [6]:
py.iplot(fig)
Out[6]:

License

Authors:

      Mathieu Blondel

      Andreas Mueller

License:

      BSD 3 clause
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