skfolio.optimization.SchurComplementary

class skfolio.optimization.SchurComplementary(gamma=0.5, keep_monotonic=True, prior_estimator=None, distance_estimator=None, hierarchical_clustering_estimator=None, min_weights=0.0, max_weights=1.0, transaction_costs=0.0, management_fees=0.0, previous_weights=None, portfolio_params=None, fallback=None, raise_on_failure=True)

Schur Complementary Allocation estimator.

Schur Complementary Allocation is a portfolio allocation method developed by Peter Cotton [1].

It uses Schur-complement-inspired augmentation of sub-covariance matrices, revealing a link between Hierarchical Risk Parity (HRP) and minimum-variance portfolios (MVP).

By tuning the regularization factor gamma, which governs how much off-diagonal information is incorporated into the augmented covariance blocks, the method smoothly interpolates from the heuristic divide-and-conquer allocation of HRP (gamma = 0) to the MVP solution (gamma -> 1).

The algorithm begins by computing a distance matrix and performing hierarchical clustering, then applies seriation to reorder assets in the dendrogram so that adjacent leaves have minimal distance.

Next, it uses recursive bisection: starting with the top-level cluster, each cluster is split into two sub-clusters in a top-down traversal.

For each sub-cluster, an augmented covariance matrix is built based on the Schur complement to incorporate off-diagonal block information. From this matrix, the total cluster variance under an inverse-variance allocation is computed, and a weighting factor derived from the variances of the two sub-clusters is used to update their cluster weights.

Parameters:

gammafloat

Regularization factor in [0, 1]. When gamma is zero, no off-diagonal information is used (equivalent to HRP). As gamma approaches one, the allocation moves toward the minimum variance solution. The better the conditioning of the initial covariance matrix, the closer the allocation will get to the MVP solution when gamma is near one.

keep_monotonicbool, default=True

If True, ensures that the portfolio variance decreases monotonically with respect to gamma. This is achieved by capping gamma at its maximum permissible value (effective_gamma_). This constraint guarantees that the solution remains variance-bounded by the HRP portfolio (variance(Schur) <= variance(HRP)), even in the presence of ill-conditioned covariance matrices. If False, no monotonicity enforcement or gamma capping is applied. For more details, see: skfolio/skfolio#3

prior_estimatorBasePrior, optional

Prior estimator. The prior estimator is used to estimate the ReturnDistribution containing the estimation of assets expected returns, covariance matrix and returns. The moments and returns estimations are used for the risk computation and the returns estimation are used by the distance matrix estimator. The default (None) is to use EmpiricalPrior.

distance_estimatorBaseDistance, optional

Distance estimator. The distance estimator is used to estimate the codependence and the distance matrix needed for the computation of the linkage matrix. The default (None) is to use PearsonDistance.

hierarchical_clustering_estimatorHierarchicalClustering, optional

Hierarchical Clustering estimator. The hierarchical clustering estimator is used to compute the linkage matrix and the hierarchical clustering of the assets based on the distance matrix. The default (None) is to use HierarchicalClustering.

min_weightsfloat | dict[str, float] | array-like of shape (n_assets, ), default=0.0

Minimum assets weights (weights lower bounds). The default is 0.0 (no short selling). Negative weights are not allowed. If a float is provided, it is applied to each asset. None is equivalent to the default 0.0. If a dictionary is provided, its (key/value) pair must be the (asset name/asset minimum weight) and the input X of the fit methods must be a DataFrame with the asset names in columns. When using a dictionary, assets values that are not provided are assigned the default minimum weight of 0.0.

Example:

• min_weights = 0.0 –> long only portfolio (default).

• min_weights = {"SX5E": 0.1, "SPX": 0.2}

• min_weights = [0.1, 0.2]

max_weightsfloat | dict[str, float] | array-like of shape (n_assets, ), default=1.0

Maximum assets weights (weights upper bounds). The default is 1.0 (each asset is below 100%). Weights above 1.0 are not allowed. If a float is provided, it is applied to each asset. None is equivalent to the default 1.0. If a dictionary is provided, its (key/value) pair must be the (asset name/asset maximum weight) and the input X of the fit method must be a DataFrame with the asset names in columns. When using a dictionary, assets values that are not provided are assigned the default maximum weight of 1.0.

Example:

• max_weights = 1.0 –> each weight must be below 100% (default).

• max_weights = 0.5 –> each weight must be below 50%.

• max_weights = {"SX5E": 0.8, "SPX": 0.9}

• max_weights = [0.8, 0.9]

transaction_costsfloat | dict[str, float] | array-like of shape (n_assets, ), default=0.0

Transaction costs of the assets. If a float is provided, it is applied to each asset. If a dictionary is provided, its (key/value) pair must be the (asset name/asset cost) and the input X of the fit method must be a DataFrame with the asset names in columns. The default value is 0.0.

management_feesfloat | dict[str, float] | array-like of shape (n_assets, ), default=0.0

Management fees of the assets. If a float is provided, it is applied to each asset. If a dictionary is provided, its (key/value) pair must be the (asset name/asset fee) and the input X of the fit method must be a DataFrame with the asset names in columns. The default value is 0.0.

previous_weightsfloat | dict[str, float] | array-like of shape (n_assets, ), optional

Previous weights of the assets. Previous weights are used to compute the portfolio total cost. If a float is provided, it is applied to each asset. If a dictionary is provided, its (key/value) pair must be the (asset name/asset previous weight) and the input X of the fit method must be a DataFrame with the asset names in columns. The default (None) means no previous weights. Additionally, when fallback="previous_weights", failures will fall back to these weights if provided.

portfolio_paramsdict, optional

Portfolio parameters forwarded to the resulting Portfolio in predict. If not provided and if available on the estimator, the following attributes are propagated to the portfolio by default: name, transaction_costs, management_fees, previous_weights and risk_free_rate.

fallbackBaseOptimization | “previous_weights” | list[BaseOptimization | “previous_weights”], optional

Fallback estimator or a list of estimators to try, in order, when the primary optimization raises during fit. Alternatively, use "previous_weights" (alone or in a list) to fall back to the estimator’s previous_weights. When a fallback succeeds, its fitted weights_ are copied back to the primary estimator so that fit still returns the original instance. For traceability, fallback_ stores the successful estimator (or the string "previous_weights")

and fallback_chain_ stores each attempt with the associated outcome.

raise_on_failurebool, default=True

Controls error handling when fitting fails. If True, any failure during fit is raised immediately, no weights_ are set and subsequent calls to predict will raise a NotFittedError. If False, errors are not raised; instead, a warning is emitted, weights_ is set to None and subsequent calls to predict will return a FailedPortfolio. When fallbacks are specified, this behavior applies only after all fallbacks have been exhausted.

Attributes:

weights_ndarray of shape (n_assets,)

Weights of the assets.

effective_gamma_float

If keep_monotonic is True, the highest permissible gamma that preserves monotonic variance decrease; otherwise, equal to the input gamma.

distance_estimator_BaseDistance

Fitted distance_estimator.

hierarchical_clustering_estimator_HierarchicalClustering

Fitted hierarchical_clustering_estimator.

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 asset names that are all strings.

fallback_BaseOptimization | “previous_weights” | None

The fallback estimator instance, or the string "previous_weights", that produced the final result. None if no fallback was used.

fallback_chain_list[tuple[str, str]] | None

Sequence describing the optimization fallback attempts. Each element is a pair (estimator_repr, outcome) where estimator_repr is the string representation of the primary estimator or a fallback (e.g. "EqualWeighted()", "previous_weights"), and outcome is "success" if that step produced a valid solution, otherwise the stringified error message. For successful fits without any fallback, this is None.

error_str | list[str] | None

Captured error message(s) when fit fails. For multi-portfolio outputs (weights_ is 2D), this is a list aligned with portfolios.

Methods

fit(X[, y])	Fit the Schur Complementary estimator.
fit_predict(X)	Perform fit on X and returns the predicted Portfolio or Population of Portfolio on X based on the fitted weights.
get_metadata_routing()	Get metadata routing of this object.
get_params([deep])	Get parameters for this estimator.
predict(X)	Predict the Portfolio or a Population of portfolios on X.
score(X[, y])	Prediction score using the Sharpe Ratio.
set_params(**params)	Set the parameters of this estimator.

Notes

A poorly conditioned covariance matrix can prevent convergence to the MVP solution as gamma approaches one. Setting keep_monotonic=True (the default) ensures that the portfolio variance decreases monotonically with respect to gamma and remains bounded by the variance of the HRP portfolio (variance(Schur) <= variance(HRP)), even in the presence of ill-conditioned covariance matrices. Additionally, you can apply shrinkage or other conditioning techniques via the prior_estimator parameter to improve numerical stability and estimation accuracy.

References

[1]

“Schur Complementary Allocation: A Unification of Hierarchical Risk Parity and Minimum Variance Portfolios”. Peter Cotton (2024).

[2]

“Portfolio Optimization. Theory and Application”. Chapter 12.3.4 “From Portfolio Risk Minimization to Hierarchical Portfolios” Daniel P. Palomar (2025).

[3]

“Building diversified portfolios that outperform out of sample”, The Journal of Portfolio Management, Marcos López de Prado (2016).

[4]

“A robust estimator of the efficient frontier”, SSRN Electronic Journal, Marcos López de Prado (2019).

[5]

“Machine Learning for Asset Managers”, Elements in Quantitative Finance. Cambridge University Press, Marcos López de Prado (2020).

[6]

“A review of two decades of correlations, hierarchies, networks and clustering in financial markets”, Gautier Marti, Frank Nielsen, Mikołaj Bińkowski, Philippe Donnat (2020).

Examples

For a full tutorial on Schur Complementary Allocation, see Schur Complementary Allocation.

>>> from skfolio import RiskMeasure
>>> from skfolio.cluster import HierarchicalClustering, LinkageMethod
>>> from skfolio.datasets import load_sp500_dataset
>>> from skfolio.distance import KendallDistance
>>> from skfolio.moments import LedoitWolf
>>> from skfolio.optimization import SchurComplementary
>>> from skfolio.preprocessing import prices_to_returns
>>> from skfolio.prior import EmpiricalPrior
>>>
>>> prices = load_sp500_dataset()
>>> X = prices_to_returns(prices)
>>>
>>> # Default Schur Complementary allocation
>>> model = SchurComplementary(gamma=0.5)
>>> model.fit(X)
>>> print(model.weights_)
>>>
>>> # Advanced model:
>>> #    * Ledoit-Wolf covariance shrinkage
>>> #    * Kendall's tau distance (absolute) for asset co-dependence
>>> #    * Hierarchical clustering with Ward's linkage
>>> model = SchurComplementary(
...     gamma=0.5,
...     prior_estimator=EmpiricalPrior(covariance_estimator=LedoitWolf()),
...     distance_estimator=KendallDistance(absolute=True),
...     hierarchical_clustering_estimator=HierarchicalClustering(
...         linkage_method=LinkageMethod.WARD,
... )
>>> model.fit(X)
>>> print(model.weights_)

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

Fit the Schur Complementary estimator.

Parameters:

Xarray-like of shape (n_observations, n_assets)

Price returns of the assets.

yIgnored

Not used, present for API consistency by convention.

Returns:

selfSchurComplementary

Fitted estimator.

fit_predict(X)

Perform fit on X and returns the predicted Portfolio or Population of Portfolio on X based on the fitted weights. For factor models, use fit(X, y) then predict(X) separately.

If fitting fails and raise_on_failure=False, this returns a FailedPortfolio.

Parameters:

Xarray-like of shape (n_observations, n_assets)

Price returns of the assets.

Returns:

Portfolio | Population

The predicted Portfolio or Population based on the fitted weights.

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.

property needs_previous_weights

Whether previous_weights must be propagated between folds/rebalances.

Used by cross_val_predict to decide whether to run sequentially and pass the weights from the previous rebalancing to the next. This is True when transaction costs, a maximum turnover, or a fallback depending on previous_weights are present.

predict(X)

Predict the Portfolio or a Population of portfolios on X.

Optimization estimators can return a 1D or a 2D array of weights. For a 1D array, the prediction is a single Portfolio. For a 2D array, the prediction is a Population of Portfolio.

If name is not provided in the portfolio parameters, the estimator class name is used.

Parameters:

Xarray-like of shape (n_observations, n_assets) | ReturnDistribution

Asset returns or a ReturnDistribution carrying returns and optional sample weights.

Returns:

Portfolio | Population

The predicted Portfolio or Population based on the fitted weights.

score(X, y=None)

Prediction score using the Sharpe Ratio. If the prediction is a single Portfolio, the score is its Sharpe Ratio. If the prediction is a Population, the score is the mean Sharpe Ratio across portfolios.

Parameters:

Xarray-like of shape (n_observations, n_assets)

Price returns of the assets.

yIgnored

Not used, present here for API consistency by convention.

Returns:

scorefloat

The Sharpe Ratio of the portfolio if the prediction is a single Portfolio or the mean of all the portfolios Sharpe Ratios if the prediction is a Population of Portfolio.

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.