Python偏最小二乘回归(PLSR)测试

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

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 __ 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. :