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Compare Cross Decomposition Methods in Scikit-learn

Simple usage of various cross decomposition algorithms: - PLSCanonical - PLSRegression, with multivariate response, a.k.a. PLS2 - PLSRegression, with univariate response, a.k.a. PLS1 - CCA

Given 2 multivariate covarying two-dimensional datasets, X, and Y, PLS extracts the ‘directions of covariance’, i.e. the components of each datasets that explain the most shared variance between both datasets. This is apparent on the scatterplot matrix display: components 1 in dataset X and dataset Y are maximally correlated (points lie around the first diagonal). This is also true for components 2 in both dataset, however, the correlation across datasets for different components is weak: the point cloud is very spherical.

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Version

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

Imports

This tutorails imports PLSCanonical, PLSRegression and CCA.

In [2]:
print(__doc__)

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.cross_decomposition import PLSCanonical, PLSRegression, CCA
Automatically created module for IPython interactive environment

Calculations

Dataset based latent variables model

In [3]:
n = 500
# 2 latents vars:
l1 = np.random.normal(size=n)
l2 = np.random.normal(size=n)

latents = np.array([l1, l1, l2, l2]).T
X = latents + np.random.normal(size=4 * n).reshape((n, 4))
Y = latents + np.random.normal(size=4 * n).reshape((n, 4))

X_train = X[:n / 2]
Y_train = Y[:n / 2]
X_test = X[n / 2:]
Y_test = Y[n / 2:]

print("Corr(X)")
print(np.round(np.corrcoef(X.T), 2))
print("Corr(Y)")
print(np.round(np.corrcoef(Y.T), 2))
Corr(X)
[[ 1.    0.5  -0.06 -0.1 ]
 [ 0.5   1.    0.02 -0.01]
 [-0.06  0.02  1.    0.43]
 [-0.1  -0.01  0.43  1.  ]]
Corr(Y)
[[ 1.    0.48 -0.11 -0.06]
 [ 0.48  1.   -0.03  0.01]
 [-0.11 -0.03  1.    0.51]
 [-0.06  0.01  0.51  1.  ]]

Canonical (symmetric) PLS

Transform Data

In [4]:
plsca = PLSCanonical(n_components=2)
plsca.fit(X_train, Y_train)
X_train_r, Y_train_r = plsca.transform(X_train, Y_train)
X_test_r, Y_test_r = plsca.transform(X_test, Y_test)

Scatter plot of scores

In [5]:
fig = tools.make_subplots(rows=2, cols=2,
                          print_grid=False,
                          subplot_titles=('Comp. 1: X vs Y (test corr = %.2f)' %
                                          np.corrcoef(X_test_r[:, 0], Y_test_r[:, 0])[0, 1],
                                          'X comp. 1 vs X comp. 2 (test corr = %.2f)'
                                          % np.corrcoef(X_test_r[:, 0], X_test_r[:, 1])[0, 1],
                                          'Y comp. 1 vs Y comp. 2 , (test corr = %.2f)'
                                          % np.corrcoef(Y_test_r[:, 0], Y_test_r[:, 1])[0, 1],
                                          'Comp. 2: X vs Y (test corr = %.2f)' %
                                          np.corrcoef(X_test_r[:, 1], Y_test_r[:, 1])[0, 1]))

# 1) On diagonal plot X vs Y scores on each components

comp1 = go.Scatter(x=X_train_r[:, 0], 
                   y=Y_train_r[:, 0], 
                   name="train",
                   mode='markers',
                   marker=dict(color='red',
                               line=dict(color='black', width=1))
                  )
comp1_ = go.Scatter(x=X_test_r[:, 0],
                    y=Y_test_r[:, 0], 
                    name="test",
                    mode='markers',
                    marker=dict(color='green',
                               line=dict(color='black', width=1))
                   )
fig.append_trace(comp1, 1, 1)
fig.append_trace(comp1_, 1, 1)

fig['layout']['xaxis1'].update(title='x scores', zeroline=False,
                               showgrid=False)
fig['layout']['yaxis1'].update(title='y scores', zeroline=False,
                               showgrid=False)

comp2 = go.Scatter(x=X_train_r[:, 1], 
                   y=Y_train_r[:, 1],
                   name="train",
                   showlegend=False,
                   mode='markers',
                   marker=dict(color='red',
                               line=dict(color='black', width=1))
                  )

comp2_ = go.Scatter(x=X_test_r[:, 1], 
                    y=Y_test_r[:, 1], 
                    name="test",
                    showlegend=False,
                    mode='markers',
                    marker=dict(color='green',
                                line=dict(color='black', width=1))
                   )
fig.append_trace(comp2, 2, 2)
fig.append_trace(comp2_, 2, 2)

fig['layout']['xaxis4'].update(title='x scores', zeroline=False,
                               showgrid=False)
fig['layout']['yaxis4'].update(title='y scores', zeroline=False,
                               showgrid=False)

# 2) Off diagonal plot components 1 vs 2 for X and Y

xcomp = go.Scatter(x=X_train_r[:, 0], 
                   y=X_train_r[:, 1],
                   name="train",
                   showlegend=False,
                   mode='markers',
                   marker=dict(color='red',
                              line=dict(color='black', width=1))
                  )
xcomp_ = go.Scatter(x=X_test_r[:, 0], 
                    y=X_test_r[:, 1], 
                    name="test",
                    showlegend=False,
                    mode='markers',
                    marker=dict(color='green',
                                line=dict(color='black', width=1))
                   )

fig.append_trace(xcomp, 1, 2)
fig.append_trace(xcomp_, 1, 2)

fig['layout']['xaxis2'].update(title='X comp. 1', zeroline=False,
                               showgrid=False)
fig['layout']['yaxis2'].update(title='X comp. 2', zeroline=False,
                               showgrid=False)

ycomp1 = go.Scatter(x=Y_train_r[:, 0], 
                    y=Y_train_r[:, 1], 
                    name="train",
                    showlegend=False,
                    mode='markers',
                    marker=dict(color='red',
                              line=dict(color='black', width=1))
                   )

ycomp1_ = go.Scatter(x=Y_test_r[:, 0], 
                     y=Y_test_r[:, 1],
                     name="test",
                     showlegend=False,
                     mode='markers',
                     marker=dict(color='green',
                                 line=dict(color='black', width=1))
                    )
fig.append_trace(ycomp1, 2, 1)
fig.append_trace(ycomp1_, 2, 1)

fig['layout']['xaxis3'].update(title='Y comp. 1', zeroline=False,
                               showgrid=False)
fig['layout']['yaxis3'].update(title='Y comp. 2', zeroline=False,
                               showgrid=False)

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

PLS Regression

PLS regression, with multivariate response, a.k.a. PLS2

In [7]:
n = 1000
q = 3
p = 10
X = np.random.normal(size=n * p).reshape((n, p))
B = np.array([[1, 2] + [0] * (p - 2)] * q).T
# each Yj = 1*X1 + 2*X2 + noize
Y = np.dot(X, B) + np.random.normal(size=n * q).reshape((n, q)) + 5

pls2 = PLSRegression(n_components=3)
pls2.fit(X, Y)
print("True B (such that: Y = XB + Err)")
print(B)
# compare pls2.coef_ with B
print("Estimated B")
print(np.round(pls2.coef_, 1))
pls2.predict(X)
True B (such that: Y = XB + Err)
[[1 1 1]
 [2 2 2]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]
 [0 0 0]]
Estimated B
[[ 1.   1.   1. ]
 [ 2.   2.   2. ]
 [ 0.   0.   0. ]
 [ 0.   0.  -0. ]
 [-0.  -0.1 -0. ]
 [ 0.   0.   0. ]
 [-0.   0.  -0. ]
 [-0.  -0.  -0. ]
 [-0.  -0.  -0. ]
 [ 0.1  0.1 -0. ]]
Out[7]:
array([[ 2.2696561 ,  2.32455797,  2.40508248],
       [ 9.78256978,  9.95096021,  9.88771175],
       [ 6.86635142,  6.99968069,  6.7605165 ],
       ..., 
       [ 6.75327401,  6.87061734,  6.72441019],
       [ 7.57312605,  7.71579299,  7.56704522],
       [ 3.11627201,  3.19632039,  3.08518641]])

PLS regression, with univariate response, a.k.a. PLS1

In [8]:
n = 1000
p = 10
X = np.random.normal(size=n * p).reshape((n, p))
y = X[:, 0] + 2 * X[:, 1] + np.random.normal(size=n * 1) + 5
pls1 = PLSRegression(n_components=3)
pls1.fit(X, y)
# note that the number of components exceeds 1 (the dimension of y)
print("Estimated betas")
print(np.round(pls1.coef_, 1))
Estimated betas
[[ 1. ]
 [ 2. ]
 [-0. ]
 [ 0. ]
 [-0. ]
 [-0.1]
 [-0. ]
 [-0. ]
 [-0. ]
 [ 0. ]]

CCA (PLS mode B with symmetric deflation)

In [9]:
cca = CCA(n_components=2)
cca.fit(X_train, Y_train)
X_train_r, Y_train_r = plsca.transform(X_train, Y_train)
X_test_r, Y_test_r = plsca.transform(X_test, Y_test)
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