<a id="skfolio-uncertainty-set-empiricalcovarianceuncertaintyset"></a>

# skfolio.uncertainty_set.EmpiricalCovarianceUncertaintySet

<a id="skfolio.uncertainty_set.EmpiricalCovarianceUncertaintySet"></a>

### *class* skfolio.uncertainty_set.EmpiricalCovarianceUncertaintySet(prior_estimator=None, confidence_level=0.95, diagonal=True, n_eff=None)

Empirical Covariance Uncertainty set.

Compute the covariance ellipsoidal uncertainty set [[1]](#r6b35a4f504b5-1):

$$
U_{\Sigma}
=
\left\{
    \Sigma :
    d^\top S^{-1} d \le \kappa^2,
    \Sigma \succeq 0
\right\},
\quad
d =
\operatorname{vec}(\Sigma) - \operatorname{vec}(\hat{\Sigma}).
$$

We consider the Wishart distribution for the covariance matrix:

$$
\hat{\Sigma}\sim W(\frac{1}{T-1}\Sigma, T-1)

$$

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

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

$$

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

The shape matrix $S$ of the ellipsoid is based on the covariance matrix of the
Wishart distributed random variable using the vector notation
$\operatorname{vec}(x)$:

$$
\operatorname{Cov}[\operatorname{vec}(\hat{\Sigma})]
=
\frac{1}{n_{\text{eff}}}
(I_{n^2} + K_{nn})(\Sigma \otimes \Sigma).
$$

where $K_{nn}$ denotes a commutation matrix and $\otimes$ represents
the Kronecker product. If `diagonal` is `True`, the asset covariance estimate is
diagonalized and the linear geometry map $L$ is built directly from the
diagonal of $S$. Otherwise, the estimator stores a full square-root factor
$L = S^{1/2}$.

* **Parameters:**
  **prior_estimator** *BasePrior, optional*
  : The [prior estimator](https://skfolio.org/user_guide/prior.html.md#prior) used to estimate the assets covariance
    matrix. The default (`None`) is to use [`EmpiricalPrior`](https://skfolio.org/generated/skfolio.prior.EmpiricalPrior.html.md#skfolio.prior.EmpiricalPrior).

  **confidence_level** *float , default=0.95*
  : Confidence level $\beta$ of the inverse cumulative distribution function
    of the chi-squared distribution. The default value is `0.95`.

  **diagonal** *bool, default=True*
  : If `True`, the non-diagonal elements of the asset covariance matrix are set to
    zero before building the ellipsoid shape matrix.

  **n_eff** *float, optional*
  : Effective number of observations used for the covariance estimator. If `None`,
    the number of observations in `X` is used. This is useful when the covariance
    matrix is estimated using a different window length or a weighted estimator
    (e.g. EWMA), in which case `n_eff` should be set to the corresponding effective
    sample size.
* **Attributes:**
  **uncertainty_set_** *UncertaintySet*
  : Covariance Uncertainty set [`UncertaintySet`](https://skfolio.org/generated/skfolio.uncertainty_set.UncertaintySet.html.md#skfolio.uncertainty_set.UncertaintySet).

  **prior_estimator_** *BasePrior*
  : Fitted `prior_estimator`.

  **n_eff_** *float*
  : Effective number of observations actually used to build the uncertainty set.

### Methods

| [`fit`](#skfolio.uncertainty_set.EmpiricalCovarianceUncertaintySet.fit)(X[, y])            | Fit the Empirical Covariance Uncertainty set estimator.   |
|-------------------------------------------------------------------------|-----------------------------------------------------------|
| [`get_metadata_routing`](#skfolio.uncertainty_set.EmpiricalCovarianceUncertaintySet.get_metadata_routing)() | Get metadata routing of this object.                      |
| [`get_params`](#skfolio.uncertainty_set.EmpiricalCovarianceUncertaintySet.get_params)([deep])     | Get parameters for this estimator.                        |
| [`set_params`](#skfolio.uncertainty_set.EmpiricalCovarianceUncertaintySet.set_params)(\*\*params) | Set the parameters of this estimator.                     |

### References

* <a id='r6b35a4f504b5-1'>**[1]**</a> “Robustness properties of mean-variance portfolios”, Optimization: A Journal of Mathematical Programming and Operations Research, Schöttle & Werner (2009).
* <a id='r6b35a4f504b5-2'>**[2]**</a> “Portfolio Optimization: Theory and Application”, Chapter 14, Daniel P. Palomar (2025)

<a id="skfolio.uncertainty_set.EmpiricalCovarianceUncertaintySet.fit"></a>

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

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 [Metadata Routing User Guide](https://skfolio.org/user_guide/metadata_routing.html.md#metadata-routing) for
    more details.
* **Returns:**
  **self** *EmpiricalCovarianceUncertaintySet*
  : Fitted estimator.

<a id="skfolio.uncertainty_set.EmpiricalCovarianceUncertaintySet.get_metadata_routing"></a>

#### 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.

<a id="skfolio.uncertainty_set.EmpiricalCovarianceUncertaintySet.get_params"></a>

#### 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.

<a id="skfolio.uncertainty_set.EmpiricalCovarianceUncertaintySet.set_params"></a>

#### 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:**
  **\*\*params** *dict*
  : Estimator parameters.
* **Returns:**
  **self** *estimator instance*
  : Estimator instance.