.. _`Partial Least Squares cross-decomposition (PLS regression)`:

.. _`org.sysess.sympathy.machinelearning.pls`:

Partial Least Squares cross-decomposition (PLS regression)
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.. image:: PCA.svg
   :width: 48


Finds the fundamental relations between two matrices X and Y, ie. it finds the (multidimensional) direction in X that best explains maximum multidimensional direction in Y. See also PCA-analysis




Documentation
:::::::::::::

Partial Least Squares (PLS) cross-decomposition is a statistical method used to find the
    fundamental relations between two matrices, typically predictors (X) and responses (Y). It
    projects both X and Y into a lower-dimensional subspace such that the covariance between
    transformed(X) and transformed(Y) is maximal.

    PLS draws similarities with Principal Component Regression (PCR), where the samples are first
    projected into a lower-dimensional subspace, and the targets y are predicted using
    transformed(X). One issue with PCR is that the dimensionality reduction is unsupervised,
    and may lose some important variables: PCR would keep the features with the most variance, but
    it's possible that features with a small variances are relevant from predicting the target. In
    a way, PLS allows for the same kind of dimensionality reduction, but by taking into account the
    targets y.


Attributes
==========

    **coef_**
        The coefficients of the linear model such that `Y` is approximated as
        `Y = X @ coef_.T + intercept_`.


    **n_iter_**
        Number of iterations of the power method, for each
        component.


    **x_loadings_**
        The loadings of `X`.


    **x_rotations_**
        The projection matrix used to transform `X`.


    **x_scores_**
        The transformed training samples.


    **x_weights_**
        The left singular vectors of the cross-covariance matrices of each
        iteration.


    **y_loadings_**
        The loadings of `Y`.


    **y_rotations_**
        The projection matrix used to transform `Y`.


    **y_scores_**
        The transformed training targets.


    **y_weights_**
        The right singular vectors of the cross-covariance matrices of each
        iteration.


Definition
::::::::::


Output ports
============

    **model**
        | Type: model
        | Description: Model



Configuration
=============

    **Max iterations** (max_iter)
        The maximum number of iterations of the power method when
        `algorithm='nipals'`. Ignored otherwise.
    **Number of components to keep** (n_components)
        Number of components to keep. Should be in `[1, n_features]`.
    **Scale the data** (scale)
        Whether to scale `X` and `Y`.
    **Tolerance** (tol)
        The tolerance used as convergence criteria in the power method: the
        algorithm stops whenever the squared norm of `u_i - u_{i-1}` is less
        than `tol`, where `u` corresponds to the left singular vector.













Implementation
==============

.. automodule:: node_decomposition
    :noindex:

.. class:: PLSRegressionCrossDecomposition
    :noindex: