# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
# 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
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# 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
# OTHER AUTHORS - MACROSCOPIC CHANGES
r"""Polynomial regression model.
Polynomial regression 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
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::
\begin{pmatrix}
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\\
\end{pmatrix}
= (x_i^d)_{i=1, \dots, n;\ d=1, \dots, D}.
The output variable 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 :math:`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`.
Dependence
----------
The polynomial regression model relies
on the `LinearRegression <https://scikit-learn.org/stable/modules/
linear_model.html>`_ and `PolynomialFeatures <https://scikit-learn.org/stable/
modules/generated/sklearn.preprocessing.PolynomialFeatures.html>`_ classes of
the `scikit-learn library <https://scikit-learn.org/stable/modules/
linear_model.html>`_.
"""
from __future__ import annotations
from typing import TYPE_CHECKING
from typing import ClassVar
from numpy import concatenate
from numpy import newaxis
from numpy import zeros
from sklearn.preprocessing import PolynomialFeatures
from gemseo.mlearning.regression.algos.linreg import LinearRegressor
from gemseo.mlearning.regression.algos.polyreg_settings import (
PolynomialRegressor_Settings,
)
if TYPE_CHECKING:
from gemseo.mlearning.core.algos.ml_algo import DataType
from gemseo.typing import RealArray
[docs]
class PolynomialRegressor(LinearRegressor):
"""Polynomial regression model."""
SHORT_ALGO_NAME: ClassVar[str] = "PolyReg"
Settings: ClassVar[type[PolynomialRegressor_Settings]] = (
PolynomialRegressor_Settings
)
def _post_init(self):
"""
Raises:
ValueError: If the degree is lower than one.
""" # noqa: D205 D212
super()._post_init()
self._poly = PolynomialFeatures(
degree=self._settings.degree, include_bias=False
)
def _fit(
self,
input_data: RealArray,
output_data: RealArray,
) -> None:
super()._fit(self._poly.fit_transform(input_data), output_data)
def _predict(
self,
input_data: RealArray,
) -> RealArray:
return super()._predict(self._poly.transform(input_data))
def _predict_jacobian(
self,
input_data: RealArray,
) -> RealArray:
# 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
powers = self._poly.powers_
n_inputs = self._poly.n_features_in_
n_outputs = self.algo.coef_.shape[0]
coefs = self.get_coefficients()
jac_intercept = zeros((n_outputs, n_inputs))
jac_coefs = zeros((n_outputs, self._poly.n_output_features_, n_inputs))
# Compute partial derivatives with respect to each input separately
for index in range(n_inputs):
# Coefficients of monomial derivatives
dcoefs = powers[newaxis, :, 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 = [
((powers == dpowers[i]).prod(axis=1) == 1).nonzero()[0]
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)
vandermonde = self._poly.transform(input_data)
contributions = jac_coefs[newaxis] * vandermonde[:, newaxis, :, newaxis]
return jac_intercept + contributions.sum(axis=2)
[docs]
def get_coefficients(
self,
as_dict: bool = False,
) -> 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.
"""
if as_dict:
msg = (
"For now the coefficients can only be obtained "
"in the form of a NumPy array"
)
raise NotImplementedError(msg)
return self.coefficients
