# -*- coding: utf-8 -*-
# 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"""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::
\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 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.
Dependence
----------
The polynomial regression model relies on the LinearRegression class
of 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 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
