High-Confidence Data-Driven Ambiguity Sets for Time-Varying Linear Systems

TL;DR

This paper develops Wasserstein ambiguity sets for time-varying linear systems using noisy partial observations, enabling robust stochastic optimization with quantifiable confidence in the true distribution.

Contribution

It introduces a method to construct high-confidence ambiguity sets for dynamic systems with unknown distributions based on partial noisy data.

Findings

Ambiguity sets contain the true distribution with high probability.

The approach provides out-of-sample guarantees for stochastic optimization.

Application to economic dispatch demonstrates practical effectiveness.

Abstract

This paper builds Wasserstein ambiguity sets for the unknown probability distribution of dynamic random variables leveraging noisy partial-state observations. The constructed ambiguity sets contain the true distribution of the data with quantifiable probability and can be exploited to formulate robust stochastic optimization problems with out-of-sample guarantees. We assume the random variable evolves in discrete time under uncertain initial conditions and dynamics, and that noisy partial measurements are available. All random elements have unknown probability distributions and we make inferences about the distribution of the state vector using several output samples from multiple realizations of the process. To this end, we leverage an observer to estimate the state of each independent realization and exploit the outcome to construct the ambiguity sets. We illustrate our results in an economic dispatch problem involving distributed energy resources over which the scheduler has no direct control.