# Source code for gemseo.mlearning.regression.polyreg

# -*- 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
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

# 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)
]
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)