Signal definition

Empirical measures

Let P_k denote the multivariate empirical return measure constructed at frequency k, and let H_k be its effective horizon in base periods.

Under power scaling with exponent h, each observation is transformed as:

x_scaled = x / H_k**h

The reference setting uses h = 0.5. Realized-volatility standardization is available as a separate robustness specification; it standardizes every asset within each frequency by its sample volatility.

H_k is declared by the caller. It is not inferred from the pandas resampling rule, because an effective financial horizon is a scientific convention rather than a property of a timestamp label.

Barycenter and dispersion weights

MFDRO distinguishes two weight vectors:

• beta_k: weights used to construct the central barycenter;
• lambda_k: weights used to aggregate dispersion around that center.

Both are uniform by default. Keeping them separate permits a change in the dispersion experiment without silently changing the center.

For a multivariate free-support barycenter Q, the target is conceptually:

Q = argmin_Q sum_k beta_k * W2(P_k, Q)**2

The reported signal is:

rho = sum_k lambda_k * d(P_k, Q)**2

where d is either the configured sliced approximation or the exact discrete transport distance. Consequently, rho is a squared dispersion and sqrt_rho is its square root.

Barycenter choices

free_support

A multivariate free-support Wasserstein barycenter computed by POT. The support is initialized with measure-balanced weighted k-means++.

projected_quantile

For each projection direction, empirical quantile functions are averaged with the barycenter weights. This avoids a multivariate free-support solve. It is a projected construction, not one stored multivariate barycenter.

Distance choices

sliced

Average projected squared W_2 discrepancies over reproducible random unit directions. More projections reduce Monte Carlo error but increase runtime.

exact

Compute the exact discrete squared transport cost between every empirical frequency measure and a free-support barycenter. This mode is generally more expensive and is incompatible with projected_quantile.

Interpretation boundary

A large rho indicates that the scaled empirical measures disagree more strongly around their configured center. It does not identify which asset will rise, estimate expected return, or specify how much robustness a portfolio optimizer should use.

A downstream research design may map rho or sqrt_rho into an ambiguity radius, a regime indicator, or a model-combination weight. That mapping must be estimated point in time and audited separately.

Random approximation

The sliced estimator uses finitely many random directions. Every result records its effective seed, and all frequencies inside one estimate use the same direction stream.

For controlled comparisons, competing configurations should use common random numbers when that is part of the experimental design.