scikit learn机器学习包中包含了偏最小二乘回归，所以可以调用对应的函数来实现

class sklearn.cross_decomposition.PLSRegression(n_components=2, scale=True, max_iter=500, tol=1e-06, copy=True)

参数信息：

Parameters:

n_components : int, (default 2)

Number of components to keep.(要保留的主成分数量,默认为2)

scale : boolean, (default True)

whether to scale the data (是否归一化数据，默认为是)

max_iter : an integer, (default 500)

the maximum number of iterations of the NIPALS inner loop (used only if algorithm=”nipals”) （使用NIPALS时的最大迭代次数）

tol : non-negative real

Tolerance used in the iterative algorithm default 1e-06. （迭代截止条件）

copy : boolean, default True

Whether the deflation should be done on a copy. Let the default value to True unless you don’t care about side effect
Attributes:

x_weights_ : array, [p, n_components]

X block weights vectors.

y_weights_ : array, [q, n_components]

Y block weights vectors.

x_loadings_ : array, [p, n_components]

X block loadings vectors.

y_loadings_ : array, [q, n_components]

Y block loadings vectors.

x_scores_ : array, [n_samples, n_components]

X scores.

y_scores_ : array, [n_samples, n_components]

Y scores.

x_rotations_ : array, [p, n_components]

X block to latents rotations.

y_rotations_ : array, [q, n_components]

Y block to latents rotations.

coef_: array, [p, q] :

The coefficients of the linear model: Y = X coef_ + Err

n_iter_ : array-like

Number of iterations of the NIPALS inner loop for each component.

Notes

Matrices:

T: x_scores_
U: y_scores_
W: x_weights_
C: y_weights_
P: x_loadings_
Q: y_loadings__

Are computed such that:

X = T P.T + Err and Y = U Q.T + Err
T[:, k] = Xk W[:, k] for k in range(n_components)
U[:, k] = Yk C[:, k] for k in range(n_components)
x_rotations_ = W (P.T W)^(-1)
y_rotations_ = C (Q.T C)^(-1)

where Xk and Yk are residual matrices at iteration k.

Slides explaining PLS <http://www.eigenvector.com/Docs/Wise_pls_properties.pdf>

For each component k, find weights u, v that optimizes: max corr(Xk u, Yk v) * std(Xk u) std(Yk u), such that |u| = 1

Note that it maximizes both the correlations between the scores and the intra-block variances.

The residual matrix of X (Xk+1) block is obtained by the deflation on the current X score: x_score.

The residual matrix of Y (Yk+1) block is obtained by deflation on the current X score. This performs the PLS regression known as PLS2. This mode is prediction oriented.

This implementation provides the same results that 3 PLS packages provided in the R language (R-project):

  • “mixOmics” with function pls(X, Y, mode = “regression”)
  • “plspm ” with function plsreg2(X, Y)
  • “pls” with function oscorespls.fit(X, Y)

References

Jacob A. Wegelin. A survey of Partial Least Squares (PLS) methods, with emphasis on the two-block case. Technical Report 371, Department of Statistics, University of Washington, Seattle, 2000.

In french but still a reference: Tenenhaus, M. (1998). La regression PLS: theorie et pratique. Paris: Editions Technic.

Examples

>>>

>>> from sklearn.cross_decomposition import PLSRegression
>>> X = [[0., 0., 1.], [1.,0.,0.], [2.,2.,2.], [2.,5.,4.]]
>>> Y = [[0.1, -0.2], [0.9, 1.1], [6.2, 5.9], [11.9, 12.3]]
>>> pls2 = PLSRegression(n_components=2)
>>> pls2.fit(X, Y)
...
PLSRegression(copy=True, max_iter=500, n_components=2, scale=True,
        tol=1e-06)
>>> Y_pred = pls2.predict(X)

Methods

fit(X, Y) Fit model to data.
fit_transform(X[, y]) Learn and apply the dimension reduction on the train data.
get_params([deep]) Get parameters for this estimator.
predict(X[, copy]) Apply the dimension reduction learned on the train data.
score(X, y[, sample_weight]) Returns the coefficient of determination R^2 of the prediction.
set_params(**params) Set the parameters of this estimator.
transform(X[, Y, copy]) Apply the dimension reduction learned on the train data.
__init__(n_components=2, scale=True, max_iter=500, tol=1e-06, copy=True)[source]
fit(X, Y)[source]

Fit model to data.

Parameters:

X : array-like, shape = [n_samples, n_features]

Training vectors, where n_samples in the number of samples and n_features is the number of predictors.

Y : array-like of response, shape = [n_samples, n_targets]

Target vectors, where n_samples in the number of samples and n_targets is the number of response variables.
fit_transform(X, y=None, **fit_params)[source]

Learn and apply the dimension reduction on the train data.

Parameters:

X : array-like of predictors, shape = [n_samples, p]

Training vectors, where n_samples in the number of samples and p is the number of predictors.

Y : array-like of response, shape = [n_samples, q], optional

Training vectors, where n_samples in the number of samples and q is the number of response variables.

copy : boolean, default True

Whether to copy X and Y, or perform in-place normalization.
Returns:

x_scores if Y is not given, (x_scores, y_scores) otherwise. :
get_params(deep=True)[source]

Get parameters for this estimator.

Parameters:

deep: boolean, optional :

If True, will return the parameters for this estimator and contained subobjects that are estimators.
Returns:

params : mapping of string to any

Parameter names mapped to their values.
predict(X, copy=True)[source]

Apply the dimension reduction learned on the train data.

Parameters:

X : array-like of predictors, shape = [n_samples, p]

Training vectors, where n_samples in the number of samples and p is the number of predictors.

copy : boolean, default True

Whether to copy X and Y, or perform in-place normalization.

Notes

This call requires the estimation of a p x q matrix, which may be an issue in high dimensional space.

score(X, y, sample_weight=None)[source]

Returns the coefficient of determination R^2 of the prediction.

The coefficient R^2 is defined as (1 - u/v), where u is the regression sum of squares ((y_true - y_pred) ** 2).sum() and v is the residual sum of squares ((y_true - y_true.mean()) ** 2).sum(). Best possible score is 1.0 and it can be negative (because the model can be arbitrarily worse). A constant model that always predicts the expected value of y, disregarding the input features, would get a R^2 score of 0.0.

Parameters:

X : array-like, shape = (n_samples, n_features)

Test samples.

y : array-like, shape = (n_samples) or (n_samples, n_outputs)

True values for X.

sample_weight : array-like, shape = [n_samples], optional

Sample weights.
Returns:

score : float

R^2 of self.predict(X) wrt. y.
set_params(**params)[source]

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). The former have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Returns: self :
transform(X, Y=None, copy=True)[source]

Apply the dimension reduction learned on the train data.

Parameters:

X : array-like of predictors, shape = [n_samples, p]

Training vectors, where n_samples in the number of samples and p is the number of predictors.

Y : array-like of response, shape = [n_samples, q], optional

Training vectors, where n_samples in the number of samples and q is the number of response variables.

copy : boolean, default True

Whether to copy X and Y, or perform in-place normalization.
Returns:

x_scores if Y is not given, (x_scores, y_scores) otherwise. :

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