# -*- coding: utf-8 -*-
# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
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# modify it under the terms of the GNU Lesser General Public
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# Contributors:
# INITIAL AUTHORS - initial API and implementation and/or initial
# documentation
# :author: Syver Doving Agdestein
# OTHER AUTHORS - MACROSCOPIC CHANGES
r"""
Polynomial 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 absolute_import, division, unicode_literals
import pickle
from os.path import join
from future import standard_library
from numpy import concatenate, where, zeros
from sklearn.preprocessing import PolynomialFeatures
from gemseo.mlearning.regression.linreg import LinearRegression
standard_library.install_aliases()
from gemseo import LOGGER
[docs]class PolynomialRegression(LinearRegression):
""" Polynomial regression. """
LIBRARY = "scikit-learn"
ABBR = "PolyReg"
def __init__(
self,
data,
degree,
transformer=None,
input_names=None,
output_names=None,
fit_intercept=True,
penalty_level=0.0,
l2_penalty_ratio=1.0,
**parameters
):
"""Constructor.
:param data: learning dataset.
:type data: Dataset
:param degree: Degree of polynomial. Default: 2.
:type degree: int
:param transformer: transformation strategy for data groups.
If None, do not transform data. Default: None.
:type transformer: dict(str)
:param input_names: names of the input variables.
:type input_names: list(str)
:param output_names: names of the output variables.
:type output_names: list(str)
:param fit_intercept: if True, fit intercept. Default: True.
:type fit_intercept: bool
:param penalty_level: penalty level greater or equal to 0.
If 0, there is no penalty. Default: 0.
:type penalty_leve: float
:param l2_penalty_ratio: 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. Default: None.
:type l2_penalty_ratio: float
"""
super(PolynomialRegression, self).__init__(
data,
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, output_data):
"""Fit the regression model.
:param ndarray input_data: input data (2D).
:param ndarray output_data: output data (2D).
"""
input_data = self.poly.fit_transform(input_data)
super(PolynomialRegression, self)._fit(input_data, output_data)
def _predict(self, input_data):
"""Predict output for given input data.
:param ndarray input_data: input data (2D).
:return: output prediction (2D).
:rtype: ndarray.
"""
input_data = self.poly.transform(input_data)
return super(PolynomialRegression, self)._predict(input_data)
def _predict_jacobian(self, input_data):
"""Predict Jacobian of the regression model for the given input data.
:param ndarray input_data: input_data (2D).
:return: Jacobian matrices (3D, one for each sample).
:rtype: 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=True):
"""Return the regression coefficients of the linear fit
as a numpy array or as a dict.
:param bool as_dict: if True, returns coefficients as a dictionary.
Default: True.
"""
coefficients = self.coefficients
if as_dict:
raise NotImplementedError
return coefficients
def _save_algo(self, directory):
"""Save external machine learning algorithm.
:param str directory: algorithm directory.
"""
super(PolynomialRegression, self)._save_algo(directory)
filename = join(directory, "poly.pkl")
with open(filename, "wb") as handle:
pickle.dump(self.poly, handle)
[docs] def load_algo(self, directory):
"""Load external machine learning algorithm.
:param str directory: algorithm directory.
"""
super(PolynomialRegression, self).load_algo(directory)
filename = join(directory, "poly.pkl")
with open(filename, "rb") as handle:
poly = pickle.load(handle)
self.poly = poly
