skfolio.moments.GraphicalLassoCV#

class skfolio.moments.GraphicalLassoCV(alphas=4, n_refinements=4, cv=None, tol=0.0001, enet_tol=0.0001, max_iter=100, mode='cd', n_jobs=None, verbose=False, assume_centered=False, nearest=True, higham=False, higham_max_iteration=100)[source]#

Sparse inverse covariance with cross-validated choice of the l1 penalty.

Read more in scikit-learn.

Parameters:

alphasint or array-like of shape (n_alphas,), dtype=float, default=4

If an integer is given, it fixes the number of points on the grids of alpha to be used. If a list is given, it gives the grid to be used. See the notes in the class docstring for more details. Range is [1, inf) for an integer. Range is (0, inf] for an array-like of floats.

n_refinementsint, default=4

The number of times the grid is refined. Not used if explicit values of alphas are passed. Range is [1, inf).

cvint, cross-validation generator or iterable, default=None

Determines the cross-validation splitting strategy. Possible inputs for cv are:

None, to use the default 5-fold cross-validation,
integer, to specify the number of folds.
CV splitter,
An iterable yielding (train, test) splits as arrays of indices.

For integer/None inputs KFold is used.

tolfloat, default=1e-4

The tolerance to declare convergence: if the dual gap goes below this value, iterations are stopped. Range is (0, inf].

enet_tolfloat, default=1e-4

The tolerance for the elastic net solver used to calculate the descent direction. This parameter controls the accuracy of the search direction for a given column update, not of the overall parameter estimate. Only used for mode=’cd’. Range is (0, inf].

max_iterint, default=100

Maximum number of iterations.

mode{‘cd’, ‘lars’}, default=’cd’

The Lasso solver to use: coordinate descent or LARS. Use LARS for very sparse underlying graphs, where number of features is greater than number of samples. Elsewhere prefer cd which is more numerically stable.

n_jobsint, default=None

Number of jobs to run in parallel. None means 1 unless in a joblib.parallel_backend context. -1 means using all processors.

verbosebool, default=False

If verbose is True, the objective function and duality gap are printed at each iteration.

assume_centeredbool, default=False

If True, data are not centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If False, data are centered before computation.

Attributes:

covariance_ndarray of shape (n_assets, n_assets)

Estimated covariance.

location_ndarray of shape (n_assets,)

Estimated location, i.e. the estimated mean.

precision_ndarray of shape (n_assets, n_assets)

Estimated pseudo inverse matrix. (stored only if store_precision is True)

alpha_float

Penalization parameter selected.

cv_results_dict of ndarrays

A dict with keys:

alphasndarray of shape (n_alphas,): All penalization parameters explored.
split(k)_test_scorendarray of shape (n_alphas,): Log-likelihood score on left-out data across (k)th fold.

Added in version 1.0.
mean_test_scorendarray of shape (n_alphas,): Mean of scores over the folds.

Added in version 1.0.
std_test_scorendarray of shape (n_alphas,): Standard deviation of scores over the folds.

Added in version 1.0.

n_iter_int

Number of iterations run for the optimal alpha.

n_features_in_int

Number of assets seen during fit.

feature_names_in_ndarray of shape (n_features_in_,)

Names of features seen during fit. Defined only when X has feature names that are all strings.

Methods

`error_norm`(comp_cov[, norm, scaling, squared])	Compute the Mean Squared Error between two covariance estimators.
`fit`(X[, y])	Fit the GraphicalLasso covariance model to X.
`get_metadata_routing`()	Get metadata routing of this object.
`get_params`([deep])	Get parameters for this estimator.
`get_precision`()	Getter for the precision matrix.
`mahalanobis`(X_test)	Compute the squared Mahalanobis distance of observations.
`score`(X_test[, y])	Compute the mean log-likelihood of observations under the estimated model.
`set_params`(**params)	Set the parameters of this estimator.
`set_score_request`(*[, X_test])	Configure whether metadata should be requested to be passed to the `score` method.

Notes

The search for the optimal penalization parameter (alpha) is done on an iteratively refined grid: first the cross-validated scores on a grid are computed, then a new refined grid is centered around the maximum, and so on.

One of the challenges which is faced here is that the solvers can fail to converge to a well-conditioned estimate. The corresponding values of alpha then come out as missing values, but the optimum may be close to these missing values.

In fit, once the best parameter alpha is found through cross-validation, the model is fit again using the entire training set.

error_norm(comp_cov, norm='frobenius', scaling=True, squared=True)#

Compute the Mean Squared Error between two covariance estimators.

Parameters:

comp_covarray-like of shape (n_features, n_features): The covariance to compare with.
norm{“frobenius”, “spectral”}, default=”frobenius”: The type of norm used to compute the error. Available error types: - ‘frobenius’ (default): sqrt(tr(A^t.A)) - ‘spectral’: sqrt(max(eigenvalues(A^t.A)) where A is the error (comp_cov - self.covariance_).
scalingbool, default=True: If True (default), the squared error norm is divided by n_features. If False, the squared error norm is not rescaled.
squaredbool, default=True: Whether to compute the squared error norm or the error norm. If True (default), the squared error norm is returned. If False, the error norm is returned.

Returns:

resultfloat: The Mean Squared Error (in the sense of the Frobenius norm) between self and comp_cov covariance estimators.

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

Fit the GraphicalLasso covariance model to X.

Parameters:

Xarray-like of shape (n_observations, n_assets): Price returns of the assets.
yIgnored: Not used, present for API consistency by convention.

Returns:

selfGraphicalLassoCV: Fitted estimator.

get_metadata_routing()[source]#

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Added in version 1.5.

Returns:

routingMetadataRouter: A MetadataRouter 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.

get_precision()#

Getter for the precision matrix.

Returns:

precision_array-like of shape (n_features, n_features): The precision matrix associated to the current covariance object.

mahalanobis(X_test)#

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_testarray-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.

Returns:

distancesndarray of shape (n_observations,) or float: Squared Mahalanobis distance for each observation. Returns a scalar if input is 1D.

Examples

>>> import numpy as np
>>> from skfolio.moments import EmpiricalCovariance
>>> X = np.random.randn(100, 3)
>>> model = EmpiricalCovariance()
>>> model.fit(X)
>>> distances = model.mahalanobis(X)
>>> # Distances follow approximately chi-squared distribution with n_assets DoF
>>> print(f"Mean distance: {distances.mean():.2f}, Expected: {3:.2f}")

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_testarray-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.
yIgnored: Not used, present for scikit-learn API consistency.

Returns:

scorefloat: Mean log-likelihood of the observations. Higher values indicate better fit. The score is averaged over all observations.

Examples

>>> import numpy as np
>>> from skfolio.moments import EmpiricalCovariance, LedoitWolf
>>> X_train = np.random.randn(100, 5)
>>> X_test = np.random.randn(50, 5)
>>> emp = EmpiricalCovariance().fit(X_train)
>>> lw = LedoitWolf().fit(X_train)
>>> # Compare models on held-out data
>>> print(f"Empirical: {emp.score(X_test):.2f}")
>>> print(f"LedoitWolf: {lw.score(X_test):.2f}")

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_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 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.

Added in version 1.3.

Parameters:

X_teststr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED: Metadata routing for X_test parameter in score.

Returns:

selfobject: The updated object.