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Compare cross decomposition methods
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.
# Authors: The scikit-learn developers # SPDX-License-Identifier: BSD-3-Clause
Dataset based latent variables model
import numpy as np
from sklearn.model_selection import train_test_split
rng = np.random.default_rng(42)
n = 500
# 2 latents vars:
l1 = rng.normal(size=n)
l2 = rng.normal(size=n)
latents = np.array([l1, l1, l2, l2]).T
X = latents + rng.normal(size=(n, 4))
Y = latents + rng.normal(size=(n, 4))
X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size=0.5, shuffle=False)
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. 0.48 -0.01 0. ] [ 0.48 1. -0.01 0. ] [-0.01 -0.01 1. 0.49] [ 0. 0. 0.49 1. ]] Corr(Y) [[ 1. 0.47 0.03 0.05] [ 0.47 1. 0.03 -0.01] [ 0.03 0.03 1. 0.5 ] [ 0.05 -0.01 0.5 1. ]]
Canonical (symmetric) PLS
Transform data
from sklearn.cross_decomposition import PLSCanonical 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
import matplotlib.pyplot as plt
# On diagonal plot X vs Y scores on each components
plt.figure(figsize=(12, 8))
plt.subplot(221)
plt.scatter(X_train_r[:, 0], Y_train_r[:, 0], label="train", marker="o", s=25)
plt.scatter(X_test_r[:, 0], Y_test_r[:, 0], label="test", marker="o", s=25)
plt.xlabel("x scores")
plt.ylabel("y scores")
plt.title(
"Comp. 1: X vs Y (test corr = %.2f)"
% np.corrcoef(X_test_r[:, 0], Y_test_r[:, 0])[0, 1]
)
plt.xticks(())
plt.yticks(())
plt.legend(loc="best")
plt.subplot(224)
plt.scatter(X_train_r[:, 1], Y_train_r[:, 1], label="train", marker="o", s=25)
plt.scatter(X_test_r[:, 1], Y_test_r[:, 1], label="test", marker="o", s=25)
plt.xlabel("x scores")
plt.ylabel("y scores")
plt.title(
"Comp. 2: X vs Y (test corr = %.2f)"
% np.corrcoef(X_test_r[:, 1], Y_test_r[:, 1])[0, 1]
)
plt.xticks(())
plt.yticks(())
plt.legend(loc="best")
# Off diagonal plot components 1 vs 2 for X and Y
plt.subplot(222)
plt.scatter(X_train_r[:, 0], X_train_r[:, 1], label="train", marker="*", s=50)
plt.scatter(X_test_r[:, 0], X_test_r[:, 1], label="test", marker="*", s=50)
plt.xlabel("X comp. 1")
plt.ylabel("X comp. 2")
plt.title(
"X comp. 1 vs X comp. 2 (test corr = %.2f)"
% np.corrcoef(X_test_r[:, 0], X_test_r[:, 1])[0, 1]
)
plt.legend(loc="best")
plt.xticks(())
plt.yticks(())
plt.subplot(223)
plt.scatter(Y_train_r[:, 0], Y_train_r[:, 1], label="train", marker="*", s=50)
plt.scatter(Y_test_r[:, 0], Y_test_r[:, 1], label="test", marker="*", s=50)
plt.xlabel("Y comp. 1")
plt.ylabel("Y comp. 2")
plt.title(
"Y comp. 1 vs Y comp. 2 , (test corr = %.2f)"
% np.corrcoef(Y_test_r[:, 0], Y_test_r[:, 1])[0, 1]
)
plt.legend(loc="best")
plt.xticks(())
plt.yticks(())
plt.show()
PLS regression, with multivariate response, a.k.a. PLS2
from sklearn.cross_decomposition import PLSRegression
n = 1000
q = 3
p = 10
X = rng.normal(size=(n, p))
B = np.array([[1, 2] + [0] * (p - 2)] * q).T
# each Yj = 1*X1 + 2*X2 + noize
Y = np.dot(X, B) + rng.normal(size=(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. 2. 0. -0. 0. 0. 0. 0. 0. -0. ]
[ 1. 2. 0. -0. -0. -0. -0. -0.1 -0. 0. ]
[ 1. 2. 0. -0. 0. 0. -0. -0. 0. 0. ]]
array([[4.09928294, 4.27252412, 4.116446 ],
[3.22383315, 3.36186659, 3.2829478 ],
[6.40665836, 6.45699286, 6.28414926],
...,
[1.50716084, 1.50460976, 1.5177967 ],
[6.67188307, 6.51139993, 6.47838503],
[5.93803911, 5.99272896, 5.91191611]], shape=(1000, 3))
PLS regression, with univariate response, a.k.a. PLS1
n = 1000
p = 10
X = rng.normal(size=(n, p))
y = X[:, 0] + 2 * X[:, 1] + rng.normal(size=n) + 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. -0.1 0. -0. -0. ]]
CCA (PLS mode B with symmetric deflation)
from sklearn.cross_decomposition import CCA cca = CCA(n_components=2) cca.fit(X_train, Y_train) X_train_r, Y_train_r = cca.transform(X_train, Y_train) X_test_r, Y_test_r = cca.transform(X_test, Y_test)
Total running time of the script: (0 minutes 0.174 seconds)
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https://scikit-learn.org/1.9/auto_examples/cross_decomposition/plot_compare_cross_decomposition.html
