# 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]](#r8132b7cb9480-1). 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:** **gamma** *float* : 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_monotonic** *bool, 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: [https://github.com/skfolio/skfolio/discussions/3](https://github.com/skfolio/skfolio/discussions/3) **prior_estimator** *BasePrior, optional* : [Prior estimator](https://skfolio.org/user_guide/prior.html.md#prior). The prior estimator is used to estimate the [`ReturnDistribution`](https://skfolio.org/generated/skfolio.prior.ReturnDistribution.html.md#skfolio.prior.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`](https://skfolio.org/generated/skfolio.prior.EmpiricalPrior.html.md#skfolio.prior.EmpiricalPrior). **distance_estimator** *BaseDistance, optional* : [Distance estimator](https://skfolio.org/user_guide/distance.html.md#distance). 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`](https://skfolio.org/generated/skfolio.distance.PearsonDistance.html.md#skfolio.distance.PearsonDistance). **hierarchical_clustering_estimator** *HierarchicalClustering, optional* : [Hierarchical Clustering estimator](https://skfolio.org/user_guide/cluster.html.md#hierarchical-clustering). 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`](https://skfolio.org/generated/skfolio.cluster.HierarchicalClustering.html.md#skfolio.cluster.HierarchicalClustering). **min_weights** *float | 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_weights** *float | 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_costs** *float | 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_fees** *float | 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_weights** *float | 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_params** *dict, 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`. **fallback** *BaseOptimization | “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_failure** *bool, 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`](#skfolio.optimization.SchurComplementary.fit)(X[, y]) | Fit the Schur Complementary estimator. | |-------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------------| | [`fit_predict`](#skfolio.optimization.SchurComplementary.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`](#skfolio.optimization.SchurComplementary.get_metadata_routing)() | Get metadata routing of this object. | | [`get_params`](#skfolio.optimization.SchurComplementary.get_params)([deep]) | Get parameters for this estimator. | | [`predict`](#skfolio.optimization.SchurComplementary.predict)(X) | Predict the `Portfolio` or a `Population` of portfolios on `X`. | | [`score`](#skfolio.optimization.SchurComplementary.score)(X[, y]) | Prediction score using the Sharpe Ratio. | | [`set_params`](#skfolio.optimization.SchurComplementary.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](https://skfolio.org/auto_examples/clustering/plot_6_schur.html.md#sphx-glr-auto-examples-clustering-plot-6-schur-py). ```pycon >>> 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) 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), ... 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) Fit the Schur Complementary 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. * **Returns:** **self** *SchurComplementary* : 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:** **X** *array-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](https://skfolio.org/user_guide/metadata_routing.html.md#metadata-routing) on how the routing mechanism works. * **Returns:** **routing** *MetadataRequest* : A `MetadataRequest` encapsulating routing information. #### 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. #### *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. #### 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:** **X** *array-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:** **X** *array-like of shape (n_observations, n_assets)* : Price returns of the assets. **y** *Ignored* : Not used, present here for API consistency by convention. * **Returns:** **score** *float* : 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 `__` so that it’s possible to update each component of a nested object. * **Parameters:** **\*\*params** *dict* : Estimator parameters. * **Returns:** **self** *estimator instance* : Estimator instance.