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, *, seriation_estimator=None, distance_from_prior=True)[source]#

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 estimates of expected asset 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 used by the seriation estimator. The default (None) is to use PearsonDistance.

hierarchical_clustering_estimatorHierarchicalClustering, optional

Deprecated. Use seriation_estimator=HierarchicalSeriation( hierarchical_clustering_estimator=...). Supplying both parameters is an error. Will be removed in version 2.0.

seriation_estimatorBaseSeriation, optional

Asset ordering estimator. The default is HierarchicalSeriation with Ward linkage and optimal leaf ordering. Use SpectralSeriation for spectral ordering with orientation carried across online updates.

distance_from_priorbool, default=True

If True, fit the distance estimator on the return scenarios produced by the prior. With CovarianceDistance(covariance_estimator="precomputed"), use the prior’s covariance instead.

If False, use the X argument supplied to this optimizer’s fit(X) or partial_fit(X), before the prior processes it.

During online learning, True refits the distance estimator on the prior’s current scenarios or covariance at each update. False updates it from the new observations in X and requires support for partial_fit. Portfolio allocation always uses the prior’s moments and return scenarios. See Seriation for HRP and Schur.

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. With partial_fit, only None or "previous_weights" is supported because fallback estimators have not accumulated the primary model’s online history. See Fallbacks.

raise_on_failurebool, default=True

Controls error handling when fitting fails and no fallback succeeds. If True, the estimator raises the final error. If False, the estimator emits a warning and sets weights_ to None, so subsequent calls to predict return a FailedPortfolio. During fit, raise_on_failure applies to any fitting error, including errors raised by the prior estimator. During partial_fit, raise_on_failure applies only to optimization failures after learning completes. Input validation failures and errors from the prior or other learning estimators are always raised. See Failure Handling for batch recovery and Updates and Failure Handling for online continuation and restart rules.

Attributes:

weights_ndarray of shape (n_assets,): Weights of the assets.
effective_gamma_float or None: If keep_monotonic is True, the highest permissible gamma that preserves monotonic variance decrease. Otherwise, it equals the input gamma. It is None after an optimization failure or when a fallback supplies the allocation.
distance_estimator_BaseDistance or None: Fitted distance_estimator. None when distance_from_prior=True and fewer than two assets are investable.
seriation_estimator_BaseSeriation: Fitted ordering estimator. Its ordering_ contains positions in the full input schema. Hierarchical linkage diagnostics are available through seriation_estimator_.hierarchical_clustering_estimator_.
hierarchical_clustering_estimator_HierarchicalClustering or None: Deprecated alias for seriation_estimator_.hierarchical_clustering_estimator_.
investable_mask_ndarray of shape (n_assets,) or None: Mask of investable assets from the fitted prior. None when all assets are investable. May be absent after batch fallback if fitting failed before the prior’s investable mask was determined.
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] | None: For a single portfolio, this is the recorded error message, or None after a successful allocation or fallback. For multiple portfolios, it is a list with one entry per row of weights_, containing an error message for each failed portfolio and None for each successful portfolio.

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 for the prior, distance and seriation estimators.
`get_params`([deep])	Get parameters for this estimator.
`partial_fit`(X[, y])	Update the prior and compute a Schur allocation.
`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.

For online updates with spectral seriation, late listings, delistings, holidays and asset warm-up, see Online Schur Allocation with a Changing Universe.

>>> 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
>>> from skfolio.seriation import HierarchicalSeriation
>>>
>>> prices = load_sp500_dataset()
>>> X = prices_to_returns(prices)
>>>
>>> # Default Schur Complementary allocation
>>> model = SchurComplementary(gamma=0.5)
>>> model.fit(X)
SchurComplementary()
>>> print(model.weights_)
[0.0358 0.0061 0.0262 ... 0.0426 0.1129 0.048 ]
>>>
>>> # 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),
...     seriation_estimator=HierarchicalSeriation(
...         hierarchical_clustering_estimator=HierarchicalClustering(
...             linkage_method=LinkageMethod.WARD,
...         ),
...     ),
... )
>>> model.fit(X)
SchurComplementary(...)
>>> print(model.weights_)
[0.0323 0.0095 0.0234 ... 0.0402 0.0515 0.0605]

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

Fit the Schur Complementary estimator.

Parameters:

Xarray-like of shape (n_observations, n_assets): Price returns of the assets.
yarray-like, optional: Targets passed to the prior and, when distance_from_prior=False, to the distance estimator.
**fit_paramsdict: Metadata routed to the underlying estimators. Metadata supplied to a distance fitted on prior scenarios must align with those scenarios.

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, factors=...) 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 for the prior, distance and seriation estimators.

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 hierarchical_clustering_estimator_#: Deprecated alias for seriation_estimator_.hierarchical_clustering_estimator_.

property needs_previous_weights#

Whether previous_weights must be propagated between folds/rebalances.

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

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

Update the prior and compute a Schur allocation.

The prior must support incremental learning. With distance_from_prior=False, the distance must also support it. Supply only new observations on each call. See Updates and Failure Handling for failure handling and restarts.

Parameters:

Xarray-like of shape (n_observations, n_assets): New returns with the same full asset schema as previous calls.
yarray-like, optional: Targets passed to the prior and, when distance_from_prior=False, to the distance estimator.
**fit_paramsdict: Metadata routed to the underlying estimators. Requires metadata routing to be enabled. Observation metadata is not forwarded to distances fitted on the prior’s return scenarios or covariance.

Returns:

selfSchurComplementary: Updated estimator.

Raises:

OptimizationError: If optimization fails with raise_on_failure=True and fallback=None.

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 portfolio 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.