<a id="skfolio-moments-geodesicshrinkagecovariance"></a>

# skfolio.moments.GeodesicShrinkageCovariance

<a id="skfolio.moments.GeodesicShrinkageCovariance"></a>

### *class* skfolio.moments.GeodesicShrinkageCovariance(covariance_estimator=None, shrinkage=0.1, target=SCALED_IDENTITY, nearest=True, higham=False, higham_max_iteration=100)

Covariance estimator with geodesic shrinkage.

Classical shrinkage estimators (e.g. [`ShrunkCovariance`](https://skfolio.org/generated/skfolio.moments.ShrunkCovariance.html.md#skfolio.moments.ShrunkCovariance),
[`LedoitWolf`](https://skfolio.org/generated/skfolio.moments.LedoitWolf.html.md#skfolio.moments.LedoitWolf)) regularize the sample covariance matrix `S`
towards a well-conditioned target `T` using a linear (Euclidean) combination:

$$
(1 - \alpha) \cdot S + \alpha \cdot T

$$

`GeodesicShrinkageCovariance` instead moves `S` towards `T` along the
geodesic of the manifold of Symmetric Positive Definite (SPD) matrices under
the affine-invariant Riemannian metric (AIRM) [[1]](#r68ff7faa0105-1) [[2]](#r68ff7faa0105-2):

$$
\Sigma(\alpha) = S^{1/2} \left(S^{-1/2} \, T \, S^{-1/2}\right)^{\alpha} S^{1/2}

$$

At `shrinkage` $\alpha = 0$, the estimate is the starting covariance
from `covariance_estimator`, after any repair requested by `nearest`. At
$\alpha = 1$, it is `target`, subject to the final `nearest` repair.

The estimate remains SPD for every `shrinkage` in [0, 1] whenever `S` and
`target` are themselves positive definite.

When `target` commutes with `S`, their shared eigenvectors are preserved and
their eigenvalues are interpolated geometrically rather than arithmetically.
For the default `SCALED_IDENTITY` target $\mu I$, with
$\mu = \mathrm{trace}(S) / n$, the interpolated eigenvalues are
$\lambda_i^{1-\alpha} \mu^{\alpha}$, whereas linear shrinkage gives
$(1 - \alpha) \lambda_i + \alpha \mu$. The condition number of the
geodesic estimate is exactly $\kappa(S)^{1-\alpha}$. Unlike linear
shrinkage, the trace is generally not preserved at intermediate values.

The general interpolation is computed with a generalized eigendecomposition
relative to the target, avoiding an explicit inverse square root of `S`.

* **Parameters:**
  **covariance_estimator** *BaseCovariance, optional*
  : [Covariance estimator](https://skfolio.org/user_guide/covariance.html.md#covariance-estimator) used to compute the
    starting covariance matrix `S` (the estimate returned when `shrinkage=0`).
    The default (`None`) is to use [`EmpiricalCovariance`](https://skfolio.org/generated/skfolio.moments.EmpiricalCovariance.html.md#skfolio.moments.EmpiricalCovariance).

  **shrinkage** *float, default=0.1*
  : Shrinkage intensity `alpha` in the geodesic interpolation. Must be between 0
    (no shrinkage) and 1 (fully shrunk to `target`) inclusive. The default value
    is `0.1`.

  **target** *GeodesicShrinkageTarget or array-like of shape (n_assets, n_assets), default=GeodesicShrinkageTarget.SCALED_IDENTITY*
  : The shrinkage target `T`. The string values `"scaled_identity"` and
    `"diagonal"` are also accepted:
    > - `SCALED_IDENTITY`: $\mu \cdot I$, with $\mu = \mathrm{trace}(S) / n$.
    >   This is the same default target used by scikit-learn’s
    >   `ShrunkCovariance`. Its condition number is exactly 1, so `shrinkage`
    >   controls how far the estimate moves from `S` towards a perfectly
    >   well-conditioned matrix.
    > - `DIAGONAL`: $\mathrm{diag}(S)$. At `shrinkage=1` the
    >   variances equal those of `S` and all correlations are zero. At
    >   intermediate values the geodesic also changes the variances.
    > - array-like: a user-provided SPD target matrix of shape
    >   `(n_assets, n_assets)`. A well-conditioned target is recommended
    >   for numerical accuracy.

  **nearest** *bool, default=True*
  : If this is set to True, the starting covariance is repaired before
    interpolation and the resulting covariance is repaired afterwards. Custom
    targets must be positive definite and are never repaired.
    The covariance is replaced by the nearest covariance
    matrix that is positive definite and with a Cholesky decomposition that can be
    computed. The variance is left unchanged.
    A covariance matrix that is not positive definite often occurs in high
    dimensional problems. It can be due to multicollinearity, floating-point
    inaccuracies, or when the number of observations is smaller than the number of
    assets. For more details, see [`cov_nearest`](https://skfolio.org/generated/skfolio.utils.stats.cov_nearest.html.md#skfolio.utils.stats.cov_nearest).
    The default is `True`.

  **higham** *bool, default=False*
  : If this is set to True, the Higham (2002) algorithm is used to find the
    nearest PD covariance, otherwise the eigenvalues are clipped to a threshold
    above zeros (1e-13). The default is `False` and uses the clipping method as
    the Higham algorithm can be slow for large datasets.

  **higham_max_iteration** *int, default=100*
  : Maximum number of iterations of the Higham (2002) algorithm.
    The default value is `100`.
* **Attributes:**
  **covariance_** *ndarray of shape (n_assets, n_assets)*
  : Estimated covariance.

  **covariance_estimator_** *BaseCovariance*
  : Fitted `covariance_estimator`.

  **location_** *ndarray of shape (n_assets,)*
  : Estimated mean, available when the fitted covariance estimator exposes it.

  **n_features_in_** *int*
  : Number of assets seen during `fit`.

  **feature_names_in_** *ndarray of shape (`n_features_in_`,)*
  : Names of assets seen during `fit`. Defined only when `X`
    has assets names that are all strings.

### Methods

| [`fit`](#skfolio.moments.GeodesicShrinkageCovariance.fit)(X[, y])                     | Fit the Geodesic Shrinkage Covariance estimator.                                   |
|----------------------------------------------------------------------------------|------------------------------------------------------------------------------------|
| [`get_metadata_routing`](#skfolio.moments.GeodesicShrinkageCovariance.get_metadata_routing)()          | Get metadata routing for this estimator.                                           |
| [`get_params`](#skfolio.moments.GeodesicShrinkageCovariance.get_params)([deep])              | Get parameters for this estimator.                                                 |
| [`mahalanobis`](#skfolio.moments.GeodesicShrinkageCovariance.mahalanobis)([X, X_test])        | Compute the squared Mahalanobis distance of observations.                          |
| [`score`](#skfolio.moments.GeodesicShrinkageCovariance.score)(X_test[, y])              | Compute the mean log-likelihood of observations under the estimated model.         |
| [`set_params`](#skfolio.moments.GeodesicShrinkageCovariance.set_params)(\*\*params)          | Set the parameters of this estimator.                                              |
| [`set_score_request`](#skfolio.moments.GeodesicShrinkageCovariance.set_score_request)(\*[, X_test]) | Configure whether metadata should be requested to be passed to the `score` method. |

### References

* <a id='r68ff7faa0105-1'>**[1]**</a> “Positive Definite Matrices”. Bhatia, R. (2007). Princeton University Press.
* <a id='r68ff7faa0105-2'>**[2]**</a> “Geodesically parameterized covariance estimation”. Musolas, A., Smith, S.T. & Marzouk, Y. (2021). SIAM Journal on Matrix Analysis and Applications.

<a id="skfolio.moments.GeodesicShrinkageCovariance.fit"></a>

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

Fit the Geodesic Shrinkage Covariance estimator.

* **Parameters:**
  **X** *array-like of shape (n_observations, n_assets)*
  : Price returns of the assets.

  **y** *Ignored*
  : Not used, present for API consistency by convention.

  **\*\*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** *GeodesicShrinkageCovariance*
  : Fitted estimator.

<a id="skfolio.moments.GeodesicShrinkageCovariance.get_metadata_routing"></a>

#### get_metadata_routing()

Get metadata routing for this estimator.

Routes metadata passed to `fit` to the `fit` method of `covariance_estimator`.

* **Returns:**
  **routing** *MetadataRouter*
  : Metadata routing configuration.

<a id="skfolio.moments.GeodesicShrinkageCovariance.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.moments.GeodesicShrinkageCovariance.mahalanobis"></a>

#### mahalanobis(X=None, \*, X_test=None)

Compute the squared Mahalanobis distance of observations.

The squared Mahalanobis distance of an observation $r$ is defined as:

$$
d^2 = (r - \mu)^T \Sigma^{-1} (r - \mu)

$$

where $\Sigma$ is the estimated covariance matrix (`self.covariance_`)
and $\mu$ is the estimated mean (`self.location_` if available, otherwise
zero).

This distance measure accounts for correlations between assets and is useful
for:

* Outlier detection in portfolio returns
* Risk-adjusted distance calculations
* Identifying unusual market regimes

* **Parameters:**
  **X** *array-like of shape (n_observations, n_assets) or (n_assets,)*
  : Observations for which to compute the squared Mahalanobis distance.
    Each row represents one observation. If 1D, treated as a single
    observation. Assets with non-finite fitted variance are excluded from
    inference. After this asset-level filtering, each row is evaluated
    using the remaining available values only, covering row-level missing
    values such as market holidays or pre/post-listing. When rows have
    different observation patterns, the returned distances follow
    $\chi^2$ distributions with different degrees of freedom.
    Rows with no finite retained observation return NaN.

  **X_test** *array-like of shape (n_observations, n_assets) or (n_assets,), optional*
  : Deprecated alias for `X`. It will be removed in version 2.0.
    Use `X` instead.
* **Returns:**
  **distances** *ndarray of shape (n_observations,) or float*
  : Squared Mahalanobis distance for each observation. Returns a scalar
    if input is 1D.

### Examples

```pycon
>>> import numpy as np
>>> from skfolio.moments import EmpiricalCovariance
>>> rng = np.random.default_rng(0)
>>> X = rng.standard_normal((100, 3))
>>> model = EmpiricalCovariance()
>>> model.fit(X)
EmpiricalCovariance()
>>> distances = model.mahalanobis(X)
>>> # The mean squared distance should be close to the number of assets (3).
>>> print(distances.mean())
2.9...
```

<a id="skfolio.moments.GeodesicShrinkageCovariance.score"></a>

#### score(X_test, y=None)

Compute the mean log-likelihood of observations under the estimated model.

Evaluates how well the fitted covariance matrix explains new observations,
assuming a multivariate Gaussian distribution. This is useful for:

* Model selection (comparing different covariance estimators)
* Cross-validation of covariance estimation methods
* Assessing goodness-of-fit

The log-likelihood for a single observation $r$ is:

$$
\log p(r | \mu, \Sigma) = -\frac{1}{2} \left[
    n \log(2\pi) + \log|\Sigma| + (r - \mu)^T \Sigma^{-1} (r - \mu)
\right]

$$

where $n$ is the number of assets, $\Sigma$ is the estimated
covariance matrix (`self.covariance_`), and $\mu$ is the estimated
mean (`self.location_` if available, otherwise zero).

* **Parameters:**
  **X_test** *array-like of shape (n_observations, n_assets)*
  : Observations for which to compute the log-likelihood.
    Typically held-out test data not used during fitting.
    Assets with non-finite fitted variance are excluded from inference. This
    typically happens when the fitted covariance cannot be estimated for an
    asset, for example before listing, after delisting, or during a warmup
    period. After this asset-level filtering, each row of `X_test` is scored
    using the remaining available values only. This covers row-level missing
    values in `X_test`, such as market holidays or pre/post-listing.

  **y** *Ignored*
  : Not used, present for scikit-learn API consistency.
* **Returns:**
  **score** *float*
  : Mean log-likelihood of the observations. Higher values indicate better fit.
    The score is averaged over all observations.

### Examples

```pycon
>>> import numpy as np
>>> from skfolio.moments import EmpiricalCovariance, LedoitWolf
>>> rng = np.random.default_rng(0)
>>> X_train = rng.standard_normal((100, 5))
>>> X_test = rng.standard_normal((50, 5))
>>> emp = EmpiricalCovariance().fit(X_train)
>>> lw = LedoitWolf().fit(X_train)
>>> # Compare models on held-out data
>>> print("Empirical:", emp.score(X_test))
Empirical: -6.97...
>>> print("LedoitWolf:", lw.score(X_test))
LedoitWolf: -6.88...
```

<a id="skfolio.moments.GeodesicShrinkageCovariance.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.

<a id="skfolio.moments.GeodesicShrinkageCovariance.set_score_request"></a>

#### set_score_request(\*, X_test='$UNCHANGED$')

Configure whether metadata should be requested to be passed to the `score` 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 `score` 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 `score`.
- `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:**
  **X_test** *str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED*
  : Metadata routing for `X_test` parameter in `score`.
* **Returns:**
  **self** *object*
  : The updated object.