"""Empirical Uncertainty Set estimators."""
# Copyright (c) 2023
# Author: Hugo Delatte <delatte.hugo@gmail.com>
# SPDX-License-Identifier: BSD-3-Clause
# Implementation derived from:
# Riskfolio-Lib, Copyright (c) 2020-2023, Dany Cajas, Licensed under BSD 3 clause.
# scikit-learn, Copyright (c) 2007-2010 David Cournapeau, Fabian Pedregosa, Olivier
# Grisel Licensed under BSD 3 clause.
import numpy as np
import numpy.typing as npt
import scipy.stats as st
import sklearn.utils.metadata_routing as skm
from skfolio.prior import BasePrior, EmpiricalPrior
from skfolio.uncertainty_set._base import (
BaseCovarianceUncertaintySet,
BaseMuUncertaintySet,
UncertaintySet,
)
from skfolio.utils.stats import commutation_matrix
from skfolio.utils.tools import check_estimator
[docs]
class EmpiricalMuUncertaintySet(BaseMuUncertaintySet):
r"""Empirical Mu Uncertainty Set.
Compute the expected returns ellipsoidal uncertainty set [1]_:
.. math:: U_{\mu}=\left\{\mu\,|\left(\mu-\hat{\mu}\right)S^{-1}\left(\mu-\hat{\mu}\right)^{T}\leq\kappa^{2}\right\}
Under the assumption that :math:`\Sigma` is given, the distribution of the sample
estimator :math:`\hat{\mu}` based on an i.i.d. sample
:math:`R_{t}\sim N(\mu, \Sigma), t=1,...,T` is given by
:math:`\hat{\mu}\sim N(\mu, \frac{1}{T}\Sigma)`.
The size of the ellipsoid :math:`\kappa` (confidence region), is computed using:
.. math:: \kappa^2 = \chi^2_{n\_assets} (\beta)
with :math:`\chi^2_{n\_assets}(\beta)` the inverse cumulative distribution function
of the chi-squared distribution with `n_assets` degrees of freedom at the
:math:`\beta` confidence level.
The Shape of the ellipsoid :math:`S` is computed using:
.. math:: S = \frac{1}{T}\Sigma
with the option to force the non-diagonal elements of the covariance matrix to zero.
Parameters
----------
prior_estimator : BasePrior, optional
The :ref:`prior estimator <prior>` used to estimate the assets covariance
matrix. The default (`None`) is to use :class:`~skfolio.prior.EmpiricalPrior`.
confidence_level : float , default=0.95
Confidence level :math:`\beta` of the inverse cumulative distribution function
of the chi-squared distribution. The default value is `0.95`.
diagonal : bool, default=True
If this is set to True, the non-diagonal elements of the covariance matrix are
set to zero.
Attributes
----------
uncertainty_set_ : UncertaintySet
Mu Uncertainty set :class:`~skfolio.uncertainty_set.UncertaintySet`.
prior_estimator_ : BasePrior
Fitted `prior_estimator`.
References
----------
.. [1] "Robustness properties of mean-variance portfolios",
Optimization: A Journal of Mathematical Programming and Operations Research,
Schöttle & Werner (2009).
"""
def __init__(
self,
prior_estimator: BasePrior | None = None,
confidence_level: float = 0.95,
diagonal: bool = True,
):
super().__init__(prior_estimator=prior_estimator)
self.confidence_level = confidence_level
self.diagonal = diagonal
[docs]
def fit(
self, X: npt.ArrayLike, y: npt.ArrayLike | None = None, **fit_params
) -> "EmpiricalMuUncertaintySet":
"""Fit the Empirical Mu Uncertainty set estimator.
Parameters
----------
X : array-like of shape (n_observations, n_assets)
Price returns of the assets.
y : array-like of shape (n_observations, n_factors), optional
Price returns of factors.
The default is `None`.
**fit_params : dict
Parameters to pass to the underlying estimators.
Only available if `enable_metadata_routing=True`, which can be
set by using ``sklearn.set_config(enable_metadata_routing=True)``.
See :ref:`Metadata Routing User Guide <metadata_routing>` for
more details.
Returns
-------
self : EmpiricalMuUncertaintySet
Fitted estimator.
"""
routed_params = skm.process_routing(self, "fit", **fit_params)
self.prior_estimator_ = check_estimator(
self.prior_estimator,
default=EmpiricalPrior(),
check_type=BasePrior,
)
# fitting estimators
self.prior_estimator_.fit(X, y, **routed_params.prior_estimator.fit)
prior_model = self.prior_estimator_.prior_model_
n_observations, n_assets = prior_model.returns.shape
k = np.sqrt(st.chi2.ppf(q=self.confidence_level, df=n_assets))
sigma = prior_model.covariance / n_observations
if self.diagonal:
sigma = np.diag(np.diag(sigma))
self.uncertainty_set_ = UncertaintySet(k=k, sigma=sigma)
return self
[docs]
class EmpiricalCovarianceUncertaintySet(BaseCovarianceUncertaintySet):
r"""Empirical Covariance Uncertainty set.
Compute the covariance ellipsoidal uncertainty set [1]_:
.. math:: U_{\Sigma}=\left\{\Sigma\,|\left(\text{vec}(\Sigma)-\text{vec}(\hat{\Sigma})\right)S^{-1}\left(\text{vec}(\Sigma)-\text{vec}(\hat{\Sigma})\right)^{T}\leq k^{2}\,,\,\Sigma\succeq 0\right\}
We consider the Wishart distribution for the covariance matrix:
.. math:: \hat{\Sigma}\sim W(\frac{1}{T-1}\Sigma, T-1)
The size of the ellipsoid :math:`\kappa` (confidence region), is computed using:
.. math:: \kappa^2 = \chi^2_{n\_assets^2} (\beta)
with :math:`\chi^2_{n\_assets^2}(\beta)` the inverse cumulative distribution
function of the chi-squared distribution with `n_assets` degrees of freedom at the
:math:`\beta` confidence level.
The Shape of the ellipsoid :math:`S` is based on a closed form solution of the
covariance matrix of the Wishart distributed random variable by using the vector
notation :math:`vec(x)`:
.. math:: Cov[vec(\hat{\Sigma})]=\frac{1}{T-1}(I_{n^2} + K_{nn})(\Sigma \otimes \Sigma)
with :math:`K_{nn}` denoting a commutation matrix and :math:`\otimes` representing
the Kronecker product.
Parameters
----------
prior_estimator : BasePrior, optional
The :ref:`prior estimator <prior>` used to estimate the assets covariance
matrix. The default (`None`) is to use :class:`~skfolio.prior.EmpiricalPrior`.
confidence_level : float , default=0.95
Confidence level :math:`\beta` of the inverse cumulative distribution function
of the chi-squared distribution. The default value is `0.95`.
diagonal : bool, default=True
If this is set to True, the non-diagonal elements of the covariance matrix are
set to zero.
Attributes
----------
uncertainty_set_ : UncertaintySet
Covariance Uncertainty set :class:`~skfolio.uncertainty_set.UncertaintySet`.
prior_estimator_ : BasePrior
Fitted `prior_estimator`.
References
----------
.. [1] "Robustness properties of mean-variance portfolios",
Optimization: A Journal of Mathematical Programming and Operations Research,
Schöttle & Werner (2009).
"""
def __init__(
self,
prior_estimator: BasePrior | None = None,
confidence_level: float = 0.95,
diagonal: bool = True,
):
super().__init__(prior_estimator=prior_estimator)
self.confidence_level = confidence_level
self.diagonal = diagonal
[docs]
def fit(
self, X: npt.ArrayLike, y: npt.ArrayLike | None = None, **fit_params
) -> "EmpiricalCovarianceUncertaintySet":
"""Fit the Empirical Covariance Uncertainty set estimator.
Parameters
----------
X : array-like of shape (n_observations, n_assets)
Price returns of the assets.
y : array-like of shape (n_observations, n_factors), optional
Price returns of factors.
The default is `None`.
**fit_params : dict
Parameters to pass to the underlying estimators.
Only available if `enable_metadata_routing=True`, which can be
set by using ``sklearn.set_config(enable_metadata_routing=True)``.
See :ref:`Metadata Routing User Guide <metadata_routing>` for
more details.
Returns
-------
self : EmpiricalCovarianceUncertaintySet
Fitted estimator.
"""
routed_params = skm.process_routing(self, "fit", **fit_params)
self.prior_estimator_ = check_estimator(
self.prior_estimator,
default=EmpiricalPrior(),
check_type=BasePrior,
)
# fitting estimators
self.prior_estimator_.fit(X, y, **routed_params.prior_estimator.fit)
prior_model = self.prior_estimator_.prior_model_
n_observations, n_assets = prior_model.returns.shape
k = np.sqrt(st.chi2.ppf(q=self.confidence_level, df=n_assets**2))
sigma = prior_model.covariance / n_observations
if self.diagonal:
sigma = np.diag(np.diag(sigma))
sigma = np.diag(
np.diag(
n_observations
* (np.identity(n_assets**2) + commutation_matrix(sigma))
@ np.kron(sigma, sigma)
)
)
self.uncertainty_set_ = UncertaintySet(k=k, sigma=sigma)
return self