# skfolio.uncertainty_set.OrthogonalCovarianceUncertaintySet ### *class* skfolio.uncertainty_set.OrthogonalCovarianceUncertaintySet(radius=1.0, cs_weighting=INVERSE_IDIO_VARIANCE) Covariance uncertainty set estimator for directions outside the factor span. This estimator builds a compact covariance uncertainty set for robust portfolio optimization [[1]](#read94a27047e-1) [[2]](#read94a27047e-2) [[3]](#read94a27047e-3). The robust penalty assigns additional covariance uncertainty to portfolio directions that are in the subspace orthogonal to the factor-model loading matrix, under the selected cross-sectional weighting metric. The base covariance is assumed to have the factor structure $$ \Sigma = B F B^\top + D $$ where $B$ is the loading matrix, $F$ is the factor covariance matrix and $D$ is the idiosyncratic covariance matrix. Under this uncertainty set, the worst-case variance for a portfolio with weights $w$ is $$ \sup_{\Sigma \in U} w^\top \Sigma\, w = w^\top \hat{\Sigma} w + \kappa \min_z \lVert C w - Q z \rVert_2^2. $$ Here, $Q$ is an orthonormal basis of the weighted factor span $\operatorname{col}(W^{1/2} B)$ and $C = W^{-1/2}$. The expression is the compact form of the quadratic penalty $$ \kappa C^\top (I - Q Q^\top) C. $$ This structured form avoids the lifted SDP formulation used for fully generic covariance uncertainty sets and avoids materializing the dense matrix [[4]](#read94a27047e-4) $C^\top (I - Q Q^\top) C$. 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:** **radius** *float, default=1.0* : Penalty radius $\kappa$. Controls the magnitude of the orthogonal covariance penalty. Must be non-negative. **cs_weighting** *CSWeighting, default=CSWeighting.INVERSE_IDIO_VARIANCE* : Cross-sectional weighting used to define the orthogonality metric. * **Attributes:** **uncertainty_set_** *CompactCovarianceUncertaintySet* : Fitted solver-ready uncertainty set containing the radius, diagonal metric square root and basis. ### Methods | [`fit`](#skfolio.uncertainty_set.OrthogonalCovarianceUncertaintySet.fit)(X[, y, return_distribution]) | Fit the orthogonal covariance uncertainty set. | |-----------------------------------------------------------------------------------------------------|------------------------------------------------------------------------------------------| | [`get_metadata_routing`](#skfolio.uncertainty_set.OrthogonalCovarianceUncertaintySet.get_metadata_routing)() | Get metadata routing of this object. | | [`get_params`](#skfolio.uncertainty_set.OrthogonalCovarianceUncertaintySet.get_params)([deep]) | Get parameters for this estimator. | | [`partial_fit`](#skfolio.uncertainty_set.OrthogonalCovarianceUncertaintySet.partial_fit)(X[, y, return_distribution]) | Update the orthogonal covariance uncertainty set. | | [`set_fit_request`](#skfolio.uncertainty_set.OrthogonalCovarianceUncertaintySet.set_fit_request)(\*[, return_distribution]) | Configure whether metadata should be requested to be passed to the `fit` method. | | [`set_params`](#skfolio.uncertainty_set.OrthogonalCovarianceUncertaintySet.set_params)(\*\*params) | Set the parameters of this estimator. | | [`set_partial_fit_request`](#skfolio.uncertainty_set.OrthogonalCovarianceUncertaintySet.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 return distribution. When used inside [`MeanRisk`](https://skfolio.org/generated/skfolio.optimization.MeanRisk.html.md#skfolio.optimization.MeanRisk), the `return_distribution` metadata is passed automatically by `fit` and `partial_fit`. Covariance uncertainty is applied when `risk_measure=RiskMeasure.VARIANCE` or when `max_variance` is set. ### 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) Fit the orthogonal covariance uncertainty set. * **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`. **return_distribution** *ReturnDistribution, optional* : The fitted return distribution from the prior estimator. Passed internally by [`MeanRisk`](https://skfolio.org/generated/skfolio.optimization.MeanRisk.html.md#skfolio.optimization.MeanRisk). Must contain a `factor_model` with `loading_matrix` and `idio_covariance`. **\*\*fit_params** *dict* : Additional parameters (unused). * **Returns:** **self** *OrthogonalCovarianceUncertaintySet* : Fitted estimator. #### get_metadata_routing() Get metadata routing of this object. Please check [User Guide](https://skfolio.org/user_guide/metadata_routing.html.md#metadata-routing) on how the routing mechanism works. * **Returns:** **routing** *MetadataRequest* : A `MetadataRequest` encapsulating routing information. #### get_params(deep=True) Get parameters for this estimator. * **Parameters:** **deep** *bool, default=True* : If True, will return the parameters for this estimator and contained subobjects that are estimators. * **Returns:** **params** *dict* : Parameter names mapped to their values. #### partial_fit(X, y=None, \*, return_distribution=None, \*\*fit_params) Update the orthogonal covariance uncertainty set. * **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`. **return_distribution** *ReturnDistribution, optional* : The fitted return distribution from the prior estimator. Passed internally by [`MeanRisk`](https://skfolio.org/generated/skfolio.optimization.MeanRisk.html.md#skfolio.optimization.MeanRisk). Must contain a `factor_model` with `loading_matrix` and `idio_covariance`. **\*\*fit_params** *dict* : Additional parameters (unused). * **Returns:** **self** *OrthogonalCovarianceUncertaintySet* : 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](https://skfolio.org/user_guide/metadata_routing.html.md#metadata-routing) 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. #### Versionadded Added in version 1.3. * **Parameters:** **return_distribution** *str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED* : Metadata routing for `return_distribution` parameter in `fit`. * **Returns:** **self** *object* : 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 `__` so that it’s possible to update each component of a nested object. * **Parameters:** **\*\*params** *dict* : Estimator parameters. * **Returns:** **self** *estimator instance* : Estimator instance. #### set_partial_fit_request(\*, return_distribution='$UNCHANGED$') Configure whether metadata should be requested to be passed to the `partial_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](https://skfolio.org/user_guide/metadata_routing.html.md#metadata-routing) on how the routing mechanism works. 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. #### Versionadded Added in version 1.3. * **Parameters:** **return_distribution** *str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED* : Metadata routing for `return_distribution` parameter in `partial_fit`. * **Returns:** **self** *object* : The updated object.