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scikit-learn 1.3.2
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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.

## Dataset based latent variables model

    import numpy as np

    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.    0.45 -0.04  0.  ]
     [ 0.45  1.   -0.1  -0.02]
     [-0.04 -0.1   1.    0.42]
     [ 0.   -0.02  0.42  1.  ]]
    Corr(Y)
    [[ 1.    0.48 -0.12 -0.05]
     [ 0.48  1.    0.07  0.04]
     [-0.12  0.07  1.    0.5 ]
     [-0.05  0.04  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 = 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.   2.   0.  -0.  -0.   0.   0.  -0.   0.   0. ]
     [ 1.   1.9  0.  -0.  -0.   0.1  0.   0.   0.   0. ]
     [ 1.   2.1 -0.   0.  -0.   0.   0.   0.   0.  -0. ]]

    array([[4.11693539, 4.19803308, 4.12190903],
           [8.77322639, 8.77777215, 9.04995982],
           [5.34990341, 5.37257991, 5.27597342],
           ...,
           [5.95433992, 5.9403917 , 6.02818216],
           [5.06880943, 5.08604995, 5.05216586],
           [9.72295655, 9.70432034, 9.79769376]])

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

    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
    [[ 1.1  2.  -0.1  0.  -0.  -0.  -0.   0.   0.  -0.1]]

## 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 0.147 seconds)

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