skfolio.moments.EWCovariance#

class skfolio.moments.EWCovariance(half_life=None, assume_centered=True, min_observations=None, window_size=None, alpha=None, nearest=True, higham=False, higham_max_iteration=100)[source]#

Exponentially Weighted Covariance estimator with NaN-aware pairwise updates.

This estimator uses the recursive EWMA formula:

\[\Sigma_t = \lambda \Sigma_{t-1} + (1-\lambda) r_t r_t^\top\]

where $\lambda$ is the decay factor, which determines how much weight is given to past observations. It is computed from the half-life parameter:

\[\lambda = 2^{-1/\text{half-life}}\]

The half-life is the number of observations for the weight to decay to 50%.

This estimator supports both batch fitting via fit and incremental updates via partial_fit, making it suitable for online learning.

NaN handling:

The estimator handles missing data (NaN returns) caused by late listings, delistings, and holidays using EWMA updates together with active_mask. An asset with active_mask=True is treated as active at time $t$. If its return is finite, the EWMA is updated normally. If its return is NaN, the observation is treated as a holiday and covariance entries involving this asset are kept unchanged. An asset with active_mask=False is treated as inactive, for example during pre-listing or post-delisting periods, and covariance entries involving this asset are set to NaN.

Active with valid return: Normal EWMA update.
Active with NaN return (holiday): Freeze; covariance entries involving this asset are kept unchanged.
Inactive (active_mask=False): Covariance entries involving this asset are set to NaN.

When active_mask is not provided, trailing NaN returns are ambiguous: they could correspond either to holidays, in which case covariance is frozen, or to inactive periods, in which case covariance is set to NaN.

Late-listing bias correction:

When an asset becomes active (late listing), the EWMA recursion for its covariance entries is initialized at zero rather than at the outer product of the first return. This initialization guarantees that the internal covariance state remains positive semi-definite at every step, but it introduces a transient downward scale bias: after $n_i$ observations, the raw EWMA for asset $i$ is damped by a factor $(1 - \lambda^{n_i})$. At output time, a per-asset correction removes this bias:

\[\hat{\Sigma}_{ij} = \frac{S_{ij}}{\sqrt{(1 - \lambda^{n_i})(1 - \lambda^{n_j})}}\]

where $S$ is the raw internal EWMA. This is a congruence transform $D S D$ with $D = \text{diag}(1 / \sqrt{1 - \lambda^{n_i}})$, which preserves positive semi-definiteness while restoring the correct variance scale. Correlations are unaffected by the correction. For assets with a long history, the correction is negligible ($\lambda^{n_i} \to 0$).

The min_observations parameter controls a warm-up period: an asset’s covariance entries remain NaN in the output until it has accumulated enough valid observations for a reliable estimate.

Parameters:

half_lifefloat, default=40

Half-life of the exponential weights in number of observations.

The half-life controls how quickly older observations lose their influence:

Larger half-life: More stable estimates, slower to adapt (robust to noise)
Smaller half-life: More responsive estimates, faster to adapt (sensitive to noise)

The decay factor $\lambda$ is computed as: $\lambda = 2^{-1/\text{half-life}}$

For example:

half-life = 40: $\lambda \approx 0.983$
half-life = 23: $\lambda \approx 0.970$
half-life = 11: $\lambda \approx 0.939$
half-life = 6: $\lambda \approx 0.891$

Note

For portfolio optimization, larger half-lives (>= 20) are generally preferred to avoid excessive turnover from estimation noise.

assume_centeredbool, default=True

If True (default), the EWMA update uses raw returns without demeaning. This is the standard convention for EWMA covariance estimation in finance. If False, returns are demeaned using an EWMA mean estimate before computing the covariance update, and location_ tracks the EWMA mean.

min_observationsint, optional

Minimum number of valid observations per asset before its covariance entries are considered reliable and exposed in the output covariance_. Until this threshold is reached, the asset’s covariance entries remain NaN.

The default (None) uses int(half_life) as the threshold, ensuring the late-listing initialization bias has decayed to at most 50%. Set to 1 to disable warm-up entirely.

window_sizeint, optional

Window size to truncate data to the last window_size observations before fitting. Only applies to the initial fit call (or equivalently, the first partial_fit call); subsequent partial_fit calls use all provided data.

This is a computational optimization for very long time series. Due to exponential decay, observations far in the past contribute negligibly to the current estimate. For example, with half-life = 23 ($\lambda = 0.97$), observations beyond ~150 periods contribute less than 1% to the estimate. Truncating to a reasonable window (e.g., 252 trading days) speeds up computation without materially affecting results.

The default (None) uses all available data.

alphafloat, optional

Deprecated since version 0.17.0: alpha is deprecated and will be removed in a future version. Use half_life instead. Note: alpha = 1 - decay_factor and half_life = -ln(2) / ln(1 - alpha).

nearestbool, default=True

If this is set to True, the covariance is replaced by the nearest covariance matrix that is positive definite and with a Cholesky decomposition than 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. The default is True.

highambool, 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_iterationint, 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. Contains NaN for assets that are inactive or have not yet accumulated min_observations valid observations.
location_ndarray of shape (n_assets,): Estimated location (mean). If assume_centered=True, this is zeros. Otherwise, it tracks the EWMA mean of returns. Contains NaN for inactive assets.
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

`fit`(X[, y, active_mask])	Fit the Exponentially Weighted Covariance estimator.
`get_metadata_routing`()	Get metadata routing of this object.
`get_params`([deep])	Get parameters for this estimator.
`mahalanobis`(X_test)	Compute the squared Mahalanobis distance of observations.
`partial_fit`(X[, y, active_mask])	Incrementally fit the Exponentially Weighted Covariance estimator.
`score`(X_test[, y])	Compute the mean log-likelihood of observations under the estimated model.
`set_fit_request`(*[, active_mask])	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`(*[, active_mask])	Configure whether metadata should be requested to be passed to the `partial_fit` method.
`set_score_request`(*[, X_test])	Configure whether metadata should be requested to be passed to the `score` method.

Examples

>>> import numpy as np
>>> from skfolio.datasets import load_sp500_dataset
>>> from skfolio.moments import EWCovariance
>>> from skfolio.preprocessing import prices_to_returns
>>>
>>> prices = load_sp500_dataset()
>>> X = prices_to_returns(prices)
>>>
>>> # Batch fitting
>>> model = EWCovariance(half_life=40)
>>> model.fit(X)
>>> print(model.covariance_.shape)
>>>
>>> # Streaming updates with partial_fit
>>> model2 = EWCovariance(half_life=20)
>>> model2.partial_fit(X[:100])  # Initial fit
>>> model2.partial_fit(X[100:200])  # Update with new data
>>> model2.partial_fit(X[200:])  # Continue updating
>>>
>>> # NaN-aware fitting with active_mask
>>> # Asset 2 is listed starting from observation 50
>>> active_mask = np.ones(X.shape, dtype=bool)
>>> active_mask[:50, 2] = False
>>> X_nan = X.copy()
>>> X_nan[:50, 2] = np.nan
>>> model3 = EWCovariance(half_life=40)
>>> model3.fit(X_nan, active_mask=active_mask)

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

Fit the Exponentially Weighted Covariance estimator.

Parameters:

Xarray-like of shape (n_observations, n_assets): Price returns of the assets. May contain NaN for missing data (holidays, late listings, delistings).
yIgnored: Not used, present for API consistency by convention.
active_maskarray-like of shape (n_observations, n_assets), optional: Boolean mask indicating whether each asset is structurally active at each observation. Use this to distinguish between holidays (active_mask=True and NaN return: covariance is frozen) and inactive periods such as pre-listing or post-delisting (active_mask=False: covariance is set to NaN). If None (default), all assets are assumed active.

Returns:

selfEWCovariance: 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.

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. Inside the retained inference subspace, the observations must be finite.

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}")

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

Incrementally fit the Exponentially Weighted Covariance estimator.

This method allows for streaming/online updates to the covariance estimate. Each call updates the internal state with new observations.

Parameters:

Xarray-like of shape (n_observations, n_assets): Price returns of the assets. May contain NaN for missing data (holidays, late listings, delistings).
yIgnored: Not used, present for API consistency by convention.
active_maskarray-like of shape (n_observations, n_assets), optional: Boolean mask indicating whether each asset is structurally active at each observation. Use this to distinguish between holidays (active_mask=True and NaN return: covariance is frozen) and inactive periods such as pre-listing or post-delisting (active_mask=False: covariance is set to NaN). If None (default), all assets are assumed active.

Returns:

selfEWCovariance: Fitted estimator.

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_fit_request(*, active_mask='$UNCHANGED$')#

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:

active_maskstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED: Metadata routing for active_mask 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(*, active_mask='$UNCHANGED$')#

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:

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

Returns:

selfobject: The updated object.

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

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

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.