Source code for gemseo.mlearning.regression.polyreg

# -*- coding: utf-8 -*-
# Copyright 2021 IRT Saint Exupéry,
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public
# License version 3 as published by the Free Software Foundation.
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# Lesser General Public License for more details.
# You should have received a copy of the GNU Lesser General Public License
# along with this program; if not, write to the Free Software Foundation,
# Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.

# Contributors:
#    INITIAL AUTHORS - initial API and implementation and/or initial
#                         documentation
#        :author: Syver Doving Agdestein
r"""The polynomial model for regression.

Polynomial regression class is a particular case of the linear regression,
where the input data is transformed before the regression is applied.
This transform consists of creating a matrix of monomials (Vandermonde)
by raising the input data to different powers up to a certain degree :math:`D`.
In the case where there is only one input variable,
the input data :math:`(x_i)_{i=1, \dots, n}\in\mathbb{R}^n` is transformed
into the Vandermonde matrix

.. math::

        x_1^1  & x_1^2  & \cdots & x_1^D\\
        x_2^1  & x_2^2  & \cdots & x_2^D\\
        \vdots & \vdots & \ddots & \vdots\\
        x_n^1  & x_n^2  & \cdots & x_n^D\\
    = (x_i^d)_{i=1, \dots, n;\ d=1, \dots, D}.

The output is expressed as a weighted sum of monomials:

.. math::

     y = w_0 + w_1 x^1 + w_2 x^2 + ... + w_D x^D,

where the coefficients :math:`(w_1, w_2, ..., w_d)` and the intercept :math:`w_0`
are estimated by least square regression.

In the case of a multidimensional input,
i.e. :math:`X = (x_{ij})_{i=1,\dots,n; j=1,\dots,m}`,
where :math:`n` is the number of samples and :math:`m` is the number of input variables,
the Vandermonde matrix is expressed
through different combinations of monomials of degree :math:`d, (1 \leq d \leq D)`;
e.g. for three variables :math:`(x, y, z)` and degree :math:`D=3`,
the different terms are
:math:`x`, :math:`y`, :math:`z`, :math:`x^2`, :math:`xy`, :math:`xz`,
:math:`y^2`, :math:`yz`, :math:`z^2`, :math:`x^3`, :math:`x^2y` etc.
More generally,
for m input variables,
the total number of monomials of degree :math:`1 \leq d \leq D` is given
by :math:`P = \binom{m+D}{m} = \frac{(m+D)!}{m!D!}`.
In the case of 3 input variables given above,
the total number of monomial combinations of degree lesser than or equal to three
is thus :math:`P = \binom{6}{3} = 20`.
The linear regression has to identify the coefficients :math:`(w_1, \dots, w_P)`,
in addition to the intercept :math:`w_0`.

This concept is implemented through the :class:`.PolynomialRegression` class
which inherits from the :class:`.MLRegressionAlgo` class.

The polynomial regression model relies on the LinearRegression class
of the `LinearRegression <
linear_model.html>`_ and  `PolynomialFeatures <
modules/generated/sklearn.preprocessing.PolynomialFeatures.html>`_ classes of
the `scikit-learn library <
from __future__ import division, unicode_literals

import logging
import pickle
from typing import Iterable, Optional, Union

from numpy import concatenate, ndarray, where, zeros
from sklearn.preprocessing import PolynomialFeatures

from gemseo.core.dataset import Dataset
from gemseo.mlearning.core.ml_algo import DataType, TransformerType
from gemseo.mlearning.regression.linreg import LinearRegression
from gemseo.utils.py23_compat import Path

LOGGER = logging.getLogger(__name__)

[docs]class PolynomialRegression(LinearRegression): """Polynomial regression.""" LIBRARY = "scikit-learn" ABBR = "PolyReg" def __init__( self, data, # type: Dataset degree, # type: int transformer=None, # type: Optional[TransformerType] input_names=None, # type: Optional[Iterable[str]] output_names=None, # type: Optional[Iterable[str]] fit_intercept=True, # type: bool penalty_level=0.0, # type: float l2_penalty_ratio=1.0, # type: float **parameters # type: Optional[Union[float,int,str,bool]] ): # type: (...) -> None """ Args: degree: The polynomial degree. fit_intercept: Whether to fit the intercept. penalty_level: The penalty level greater or equal to 0. If 0, there is no penalty. l2_penalty_ratio: The penalty ratio related to the l2 regularization. If 1, the penalty is the Ridge penalty. If 0, this is the Lasso penalty. Between 0 and 1, the penalty is the ElasticNet penalty. Raises: ValueError: If the degree is lower than one. """ super(PolynomialRegression, self).__init__( data, degree=degree, transformer=transformer, input_names=input_names, output_names=output_names, fit_intercept=fit_intercept, penalty_level=penalty_level, l2_penalty_ratio=l2_penalty_ratio, **parameters ) self._poly = PolynomialFeatures(degree=degree, include_bias=False) self.parameters["degree"] = degree if degree < 1: raise ValueError("Degree must be >= 1.") def _fit( self, input_data, # type: ndarray output_data, # type: ndarray ): # type: (...) -> None input_data = self._poly.fit_transform(input_data) super(PolynomialRegression, self)._fit(input_data, output_data) def _predict( self, input_data, # type: ndarray ): # type: (...) -> ndarray input_data = self._poly.transform(input_data) return super(PolynomialRegression, self)._predict(input_data) def _predict_jacobian( self, input_data, # type: ndarray ): # type: (...) -> ndarray # Dimensions: # powers: ( , , n_powers , n_inputs ) # coefs: ( , n_outputs , n_powers , ) # jac_coefs: ( , n_outputs , n_powers , n_inputs ) # vandermonde: ( n_samples , , n_powers , ) # contributions: ( n_samples , n_outputs , n_powers , n_inputs ) # jacobians: ( n_samples , n_outputs , , n_inputs ) # # n_powers is given by the formula # n_powers = binom(n_inputs+degree, n_inputs)+1 vandermonde = self._poly.transform(input_data) powers = self._poly.powers_ n_inputs = self._poly.n_input_features_ n_powers = self._poly.n_output_features_ n_outputs = self.algo.coef_.shape[0] coefs = self.get_coefficients(False) jac_intercept = zeros((n_outputs, n_inputs)) jac_coefs = zeros((n_outputs, n_powers, n_inputs)) # Compute partial derivatives with respect to each input separately for index in range(n_inputs): # Coefficients of monomial derivatives dcoefs = powers[None, :, index] * coefs # Powers of monomial derivatives dpowers = powers.copy() dpowers[:, index] -= 1 # Keep indices of remaining monomials only mask_zero = (dpowers == 0).prod(axis=1) == 1 mask_keep = dpowers[:, index] >= 0 mask_keep[mask_zero == 1] = False # Extract intercept for Jacobian (0th order term) dintercept = dcoefs[:, mask_zero].flatten() # Filter kept terms dcoefs = dcoefs[:, mask_keep] # Coefficients of kept terms dpowers = dpowers[mask_keep] # Power keys of kept terms # Find indices for the given powers inds_keep = [ where((powers == dpowers[i]).prod(axis=1) == 1) for i in range(dpowers.shape[0]) ] if len(inds_keep) > 0: inds_keep = concatenate(inds_keep).flatten() # Coefficients of partial derivatives in terms of original powers jac_intercept[:, index] = dintercept jac_coefs[:, inds_keep, index] = dcoefs # Assemble polynomial (sum of weighted monomials) contributions = jac_coefs[None] * vandermonde[:, None, :, None] jacobians = jac_intercept + contributions.sum(axis=2) return jacobians
[docs] def get_coefficients( self, as_dict=False, # type:bool ): # type: (...) -> DataType """Return the regression coefficients of the linear model. Args: as_dict: If True, return the coefficients as a dictionary of Numpy arrays indexed by the names of the coefficients. Otherwise, return the coefficients as a Numpy array. For now the only valid value is False. Returns: The regression coefficients of the linear model. Raises: NotImplementedError: If the coefficients are required as a dictionary. """ coefficients = self.coefficients if as_dict: raise NotImplementedError( "For now the coefficients can only be obtained " "in the form of a NumPy array" ) return coefficients
def _save_algo( self, directory, # type: Path ): # type: (...) -> None super(PolynomialRegression, self)._save_algo(directory) with (directory / "poly.pkl").open("wb") as handle: pickle.dump(self._poly, handle)
[docs] def load_algo( self, directory, # type: Union[str,Path] ): # type: (...) -> None directory = Path(directory) super(PolynomialRegression, self).load_algo(directory) with (directory / "poly.pkl").open("rb") as handle: poly = pickle.load(handle) self._poly = poly