skfolio.uncertainty_set.BootstrapMuUncertaintySet

class skfolio.uncertainty_set.BootstrapMuUncertaintySet(prior_estimator=None, confidence_level=0.95, diagonal=True, n_bootstrap_samples=1000, block_size=None, seed=None)

Bootstrap Mu Uncertainty set.

Compute the expected returns ellipsoidal uncertainty set using stationary bootstrap:

\[U_{\mu} = \left\{ \mu : (\mu - \hat{\mu})^\top S^{-1}(\mu - \hat{\mu}) \le \kappa^2 \right\}.\]

The radius of the ellipsoid \(\kappa\) (confidence region) is computed using:

\[\kappa^2 = \chi^2_{n_{\text{assets}}}(\beta)\]

with \(\chi^2_{n_{\text{assets}}}(\beta)\) the inverse cumulative distribution function of the chi-squared distribution with \(n_{\text{assets}}\) degrees of freedom at the \(\beta\) confidence level.

The shape matrix \(S\) 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_estimatorBasePrior, optional

The prior estimator used to estimate the assets return distribution. The default (None) is to use EmpiricalPrior.

confidence_levelfloat , default=0.95

Confidence level \(\beta\) of the inverse cumulative distribution function of the chi-squared distribution. The default value is 0.95.

diagonalbool, default=True

If True, only the diagonal of the ellipsoid shape matrix is retained.

n_bootstrap_samplesint, default=1000

Number of bootstrap samples to generate. The default value is 1000.

block_sizefloat, optional

Bootstrap block size. The default (None) is to estimate the optimal block size using Politis & White algorithm for all individual assets.

seedint, optional

Random seed used to initialize the pseudo-random number generator.

Attributes:

uncertainty_set_UncertaintySet

Mu Uncertainty set UncertaintySet.

prior_estimator_BasePrior

Fitted prior_estimator.

Methods

fit(X[, y])	Fit the Bootstrap Mu Uncertainty set estimator.
get_metadata_routing()	Get metadata routing of this object.
get_params([deep])	Get parameters for this estimator.
set_params(**params)	Set the parameters of this 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).

fit(X, y=None, **fit_params)

Fit the Bootstrap Mu Uncertainty set estimator.

Parameters:

Xarray-like of shape (n_observations, n_assets)

Price returns of the assets.

yarray-like of shape (n_observations, n_factors), optional

Price returns of factors. The default is None.

**fit_paramsdict

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 Metadata Routing User Guide for more details.

Returns:

selfBootstrapMuUncertaintySet

Fitted estimator.

get_metadata_routing()

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Returns:

routingMetadataRequest

A MetadataRequest encapsulating routing information.

get_params(deep=True)

Get parameters for this estimator.

Parameters:

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:

paramsdict

Parameter names mapped to their values.

set_params(**params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters:

**paramsdict

Estimator parameters.

Returns:

selfestimator instance

Estimator instance.