skfolio.uncertainty_set.OrthogonalMuUncertaintySet#

class skfolio.uncertainty_set.OrthogonalMuUncertaintySet(confidence_level=0.95, cs_weighting=INVERSE_IDIO_VARIANCE, uncertainty_shape='identity')[source]#

Expected return uncertainty set estimator for directions outside the factor span.

This estimator builds a norm-ball uncertainty set for expected returns that are in the subspace orthogonal to the factor-model loading matrix, under the selected cross-sectional weighting metric.

It is intended for cases where orthogonal expected returns are considered less reliable than spanned expected returns and is designed to reduce the tendency of optimizers to overallocate in these directions. Rather than shrinking the orthogonal expected returns in the prior, this estimator keeps the point estimate $\hat{\mu}$ unchanged and adds a portfolio-dependent worst-case penalty that grows with exposure to the orthogonal subspace [1] [2] [3].

Under this uncertainty set, the worst-case expected return for a portfolio with weights $w$ is

\[\inf_{\mu \in U_\mu} w^\top \mu \;=\; w^\top \hat{\mu} - \kappa \, \lVert L^\top w \rVert_2,\]

where the low-rank geometry factor is

\[L = G \Lambda^{1/2}.\]

Here, $G$ is a basis for the subspace orthogonal to the factor-model span and $\Lambda$ is a positive semidefinite scaling matrix that controls the uncertainty assigned to each orthogonal direction.

Equivalently, the ellipsoidal shape matrix is [4]:

\[S_\mu = L L^\top = G \Lambda G^\top.\]

If the factor model uses basket-neutral constraints, the loading matrix is first reduced to its effective full-rank basis before the orthogonal subspace is computed.

Parameters:

confidence_levelfloat, default=0.95

Confidence level $\beta$ used to set the uncertainty size

\[\kappa = \sqrt{\chi^2_{\mathrm{rank}}(\beta)}.\]

cs_weightingCSWeighting, default=CSWeighting.INVERSE_IDIO_VARIANCE

Cross-sectional weighting used to define the orthogonality metric.

uncertainty_shape{“identity”, “idio_variance”}, default=”identity”

Shape used inside the orthogonal subspace.

"identity" assigns the same uncertainty to all orthogonal directions.
"idio_variance" scales uncertainty using projected idiosyncratic variance in the orthogonal subspace.

Attributes:

uncertainty_set_UncertaintySet

Fitted solver-ready uncertainty set with:

uncertainty_set_.radius = kappa
uncertainty_set_.geometry = L = G Lambda^{1/2}
uncertainty_set_.norm = 2

Methods

`fit`(X[, y, return_distribution])	Fit the orthogonal mu uncertainty set.
`get_metadata_routing`()	Get metadata routing of this object.
`get_params`([deep])	Get parameters for this estimator.
`partial_fit`(X[, y, return_distribution])	Update the orthogonal mu uncertainty set.
`set_fit_request`(*[, return_distribution])	Configure whether metadata should be requested to be passed to the `fit` method.
`set_params`(**params)	Set the parameters of this estimator.
`set_partial_fit_request`(*[, return_distribution])	Configure whether metadata should be requested to be passed to the `partial_fit` method.

Notes

This estimator requires a factor model in the fitted return distribution. When used inside MeanRisk, the return_distribution metadata is passed automatically by fit and partial_fit.

References

[1]

“Robustness properties of mean-variance portfolios”, Optimization: A Journal of Mathematical Programming and Operations Research, Schöttle & Werner (2009).

[2]

“Portfolio Optimization: Theory and Application”, Chapter 14, Daniel P. Palomar (2025)

[3]

“Robust Portfolio Selection Problems”, Mathematics of Operations Research, Goldfarb and Iyengar (2003).

[4]

“Robust Convex Optimization”, Mathematics of Operations Research, Ben-Tal and Nemirovski (1998).

fit(X, y=None, *, return_distribution=None, **fit_params)[source]#

Fit the orthogonal mu uncertainty set.

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.
return_distributionReturnDistribution, optional: The fitted return distribution from the prior estimator. Passed internally by MeanRisk. Must contain a factor_model with loading_matrix and idio_covariance.
**fit_paramsdict: Additional parameters (unused).

Returns:

selfOrthogonalMuUncertaintySet: 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.

partial_fit(X, y=None, *, return_distribution=None, **fit_params)[source]#

Update the orthogonal mu uncertainty set.

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.
return_distributionReturnDistribution, optional: The fitted return distribution from the prior estimator. Passed internally by MeanRisk. Must contain a factor_model with loading_matrix and idio_covariance.
**fit_paramsdict: Additional parameters (unused).

Returns:

selfOrthogonalMuUncertaintySet: Updated estimator.

set_fit_request(*, return_distribution='$UNCHANGED$')#

Configure whether metadata should be requested to be passed to the fit method.

Note that this method is only relevant when this estimator is used as a sub-estimator within a meta-estimator and metadata routing is enabled with enable_metadata_routing=True (see sklearn.set_config). Please check the User Guide on how the routing mechanism works.

The options for each parameter are:

True: metadata is requested, and passed to fit if provided. The request is ignored if metadata is not provided.
False: metadata is not requested and the meta-estimator will not pass it to fit.
None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

Added in version 1.3.

Parameters:

return_distributionstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED: Metadata routing for return_distribution parameter in fit.

Returns:

selfobject: The updated object.

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.

set_partial_fit_request(*, return_distribution='$UNCHANGED$')#

Configure whether metadata should be requested to be passed to the partial_fit method.

The options for each parameter are:

True: metadata is requested, and passed to partial_fit if provided. The request is ignored if metadata is not provided.
False: metadata is not requested and the meta-estimator will not pass it to partial_fit.
None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

Added in version 1.3.

Parameters:

return_distributionstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED: Metadata routing for return_distribution parameter in partial_fit.

Returns:

selfobject: The updated object.