Uncertainty Set Estimator

The Uncertainty Set estimator builds an ellipsoidal UncertaintySet of the distribution moments.

An ellipsoidal uncertainty set is defined by its size \(\kappa\) and shape \(S\). Ellipsoidal uncertainty set can be used with both expected returns and covariance:

Expected returns ellipsoidal uncertainty set:

\[U_{\mu}=\left\{\mu\,|\left(\mu-\hat{\mu}\right)S^{-1}\left(\mu-\hat{\mu}\right)^{T}\leq\kappa^{2}\right\}\]

Covariance ellipsoidal uncertainty set:

\[U_{\Sigma}=\left\{\Sigma\,|\left(\text{vec}(\Sigma)-\text{vec}(\hat{\Sigma})\right)S^{-1}\left(\text{vec}(\Sigma)-\text{vec}(\hat{\Sigma})\right)^{T}\leq k^{2}\,,\,\Sigma\succeq 0\right\}\]

It follows the same API as scikit-learn’s estimator: the fit method takes X as the assets returns and stores the UncertaintySet in its uncertainty_set_ attribute.

X can be any array-like structure (numpy array, pandas DataFrame, etc.)

Available estimators for the expected returns:

• EmpiricalMuUncertaintySet

• BootstrapMuUncertaintySet

Available estimators for the covariance:

• EmpiricalCovarianceUncertaintySet

• BootstrapCovarianceUncertaintySet

Example:

from skfolio.datasets import load_sp500_dataset
from skfolio.preprocessing import prices_to_returns
from skfolio.uncertainty_set import EmpiricalMuUncertaintySet

prices = load_sp500_dataset()
X = prices_to_returns(prices)

model = EmpiricalMuUncertaintySet()
model.fit(X)
print(model.uncertainty_set_)

It is used to solve worst-case optimization using the MeanRisk estimator. Worst-case optimization is a class of robust optimization. It reduces the instability that arises from the estimation errors of the expected returns and the covariance matrix.

Example:

from skfolio.datasets import load_sp500_dataset
from skfolio.optimization import MeanRisk, ObjectiveFunction
from skfolio.preprocessing import prices_to_returns
from skfolio.uncertainty_set import (
    BootstrapMuUncertaintySet,
    EmpiricalCovarianceUncertaintySet,
)

prices = load_sp500_dataset()
prices = prices["2020":]
X = prices_to_returns(prices)

model = MeanRisk(
    objective_function=ObjectiveFunction.MAXIMIZE_RATIO,
    mu_uncertainty_set_estimator=BootstrapMuUncertaintySet(confidence_level=0.5),
    covariance_uncertainty_set_estimator=EmpiricalCovarianceUncertaintySet(
        confidence_level=0.5
    ),
)
model.fit(X)
print(model.weights_)