"""Bootstrap Uncertainty Set estimators."""
# Copyright (c) 2023-2026
# Author: Hugo Delatte <hugo.delatte@skfoliolabs.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.
from __future__ import annotations
import numpy as np
import scipy.linalg as sla
import scipy.stats as st
import sklearn.utils.metadata_routing as skm
from skfolio.prior import BasePrior, EmpiricalPrior
from skfolio.typing import ArrayLike
from skfolio.uncertainty_set._base import (
BaseCovarianceUncertaintySet,
BaseMuUncertaintySet,
)
from skfolio.uncertainty_set._model import UncertaintySet
from skfolio.utils.bootstrap import stationary_bootstrap
from skfolio.utils.tools import check_estimator
[docs]
class BootstrapMuUncertaintySet(BaseMuUncertaintySet):
r"""Bootstrap Mu Uncertainty set.
Compute the expected returns ellipsoidal uncertainty set using stationary bootstrap:
.. math::
U_{\mu}
=
\left\{
\mu :
(\mu - \hat{\mu})^\top S^{-1}(\mu - \hat{\mu})
\le \kappa^2
\right\}.
The radius of the ellipsoid :math:`\kappa` (confidence region) is computed using:
.. math:: \kappa^2 = \chi^2_{n_{\text{assets}}}(\beta)
with :math:`\chi^2_{n_{\text{assets}}}(\beta)` the inverse cumulative distribution
function of the chi-squared distribution with :math:`n_{\text{assets}}` degrees of
freedom at the :math:`\beta` confidence level.
The shape matrix :math:`S` of the ellipsoid is computed using stationary bootstrap,
with the option to retain only its diagonal. The estimator stores the square-root
factor as the linear geometry map:math:`L = S^{1/2}`.
Parameters
----------
prior_estimator : BasePrior, optional
The :ref:`prior estimator <prior>` used to estimate the assets return
distribution. 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 `True`, only the diagonal of the ellipsoid shape matrix is retained.
n_bootstrap_samples : int, default=1000
Number of bootstrap samples to generate. The default value is `1000`.
block_size : float, optional
Bootstrap block size. The default (`None`) is to estimate the optimal block size
using Politis & White algorithm for all individual assets.
seed : int, optional
Random seed used to initialize the pseudo-random number generator.
Attributes
----------
uncertainty_set_ : UncertaintySet
Mu Uncertainty set :class:`~skfolio.uncertainty_set.UncertaintySet`.
prior_estimator_ : BasePrior
Fitted `prior_estimator`.
References
----------
.. [1] "Portfolio Optimization: Theory and Application", Chapter 14,
Daniel P. Palomar (2025)
.. [2] "Robustness properties of mean-variance portfolios",
Optimization: A Journal of Mathematical Programming and Operations Research,
Schöttle & Werner (2009).
.. [3] "Automatic Block-Length Selection for the Dependent Bootstrap",
Politis & White (2004).
.. [4] "Correction to Automatic Block-Length Selection for the Dependent
Bootstrap",
Patton, Politis & White (2009).
"""
def __init__(
self,
prior_estimator: BasePrior | None = None,
confidence_level: float = 0.95,
diagonal: bool = True,
n_bootstrap_samples: int = 1000,
block_size: float | None = None,
seed: int | None = None,
):
super().__init__(prior_estimator=prior_estimator)
self.confidence_level = confidence_level
self.diagonal = diagonal
self.n_bootstrap_samples = n_bootstrap_samples
self.block_size = block_size
self.seed = seed
[docs]
def fit(
self, X: ArrayLike, y: ArrayLike | None = None, **fit_params
) -> BootstrapMuUncertaintySet:
"""Fit the Bootstrap 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 : BootstrapMuUncertaintySet
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)
mu = self.prior_estimator_.return_distribution_.mu
returns = self.prior_estimator_.return_distribution_.returns
n_assets = returns.shape[1]
radius = np.sqrt(st.chi2.ppf(q=self.confidence_level, df=n_assets))
samples = stationary_bootstrap(
returns=returns,
block_size=self.block_size,
n_bootstrap_samples=self.n_bootstrap_samples,
seed=self.seed,
)
mus = np.mean(samples, axis=1)
deviations = mus - mu
if self.diagonal:
geometry = np.diag(np.sqrt(np.var(deviations, axis=0, ddof=1)))
else:
geometry = sla.sqrtm(np.cov(deviations, rowvar=False)).real
self.uncertainty_set_ = UncertaintySet(radius=radius, geometry=geometry, norm=2)
return self
[docs]
class BootstrapCovarianceUncertaintySet(BaseCovarianceUncertaintySet):
r"""Bootstrap Covariance Uncertainty set.
Compute the covariance ellipsoidal uncertainty set using stationary bootstrap:
.. math::
U_{\Sigma}
=
\left\{
\Sigma :
d^\top S^{-1} d \le \kappa^2,
\Sigma \succeq 0
\right\},
\quad
d =
\operatorname{vec}(\Sigma) - \operatorname{vec}(\hat{\Sigma}).
The radius of the ellipsoid :math:`\kappa` (confidence region) is computed using:
.. math:: \kappa^2 = \chi^2_{n_{\text{assets}}^2}(\beta)
with :math:`\chi^2_{n_{\text{assets}}^2}(\beta)` the inverse cumulative
distribution function of the chi-squared distribution with :math:`n_{\text{assets}}^2`
degrees of freedom at the :math:`\beta` confidence level.
The shape matrix :math:`S` of the ellipsoid is the covariance matrix of the
bootstrapped vectorized covariance estimator. If `diagonal` is `True`, only the
diagonal of :math:`S` is retained and the linear geometry map :math:`L` is built
directly from it. Otherwise, the estimator stores a full square-root factor
:math:`L = S^{1/2}`.
Parameters
----------
prior_estimator : BasePrior, optional
The :ref:`prior estimator <prior>` used to estimate the assets return
distribution. 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 `True`, only the diagonal of the ellipsoid shape matrix in vectorized
covariance space is retained.
n_bootstrap_samples : int, default=1000
Number of bootstrap samples to generate. The default value is `1000`.
block_size : float, optional
Bootstrap block size. The default (`None`) is to estimate the optimal block size
using Politis & White algorithm for all individual assets.
seed : int, optional
Random seed used to initialize the pseudo-random number generator.
Attributes
----------
uncertainty_set_ : UncertaintySet
Covariance Uncertainty set :class:`~skfolio.uncertainty_set.UncertaintySet`.
prior_estimator_ : BasePrior
Fitted `prior_estimator`.
References
----------
.. [1] "Portfolio Optimization: Theory and Application", Chapter 14,
Daniel P. Palomar (2025)
.. [2] "Robustness properties of mean-variance portfolios",
Optimization: A Journal of Mathematical Programming and Operations Research,
Schöttle & Werner (2009).
.. [3] "Automatic Block-Length Selection for the Dependent Bootstrap",
Politis & White (2004).
.. [4] "Correction to Automatic Block-Length Selection for the Dependent
Bootstrap",
Patton, Politis & White (2009).
"""
def __init__(
self,
prior_estimator: BasePrior | None = None,
confidence_level: float = 0.95,
diagonal: bool = True,
n_bootstrap_samples: int = 1000,
block_size: float | None = None,
seed: int | None = None,
):
super().__init__(prior_estimator=prior_estimator)
self.confidence_level = confidence_level
self.diagonal = diagonal
self.n_bootstrap_samples = n_bootstrap_samples
self.block_size = block_size
self.seed = seed
[docs]
def fit(
self, X: ArrayLike, y=None, **fit_params
) -> BootstrapCovarianceUncertaintySet:
"""Fit the Bootstrap 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 : BootstrapCovarianceUncertaintySet
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)
returns = self.prior_estimator_.return_distribution_.returns
n_assets = returns.shape[1]
radius = np.sqrt(st.chi2.ppf(q=self.confidence_level, df=n_assets**2))
samples = stationary_bootstrap(
returns=returns,
block_size=self.block_size,
n_bootstrap_samples=self.n_bootstrap_samples,
seed=self.seed,
)
deviations = np.empty((self.n_bootstrap_samples, n_assets**2))
for i, sample in enumerate(samples):
deviations[i] = np.cov(sample.T).ravel(order="F")
if self.diagonal:
geometry = np.diag(np.sqrt(np.var(deviations, axis=0, ddof=1)))
else:
geometry = sla.sqrtm(np.cov(deviations, rowvar=False)).real
self.uncertainty_set_ = UncertaintySet(radius=radius, geometry=geometry, norm=2)
return self
