# Data-Driven Observability Analysis for Nonlinear Stochastic Systems

## TL;DR

This paper introduces a data-driven method to analyze the observability of nonlinear stochastic systems by quantifying distributional distinguishability from measurement data, addressing challenges posed by noise and lack of analytical models.

## Contribution

It presents a novel approach to assess and quantify distributional distinguishability directly from data, including a statistical test for high-confidence state differentiation, applicable to nonlinear stochastic systems.

## Key findings

- Distributional distinguishability is equivalent to classical for linear systems.
- The method quantifies how much data is needed to differentiate initial states.
- The approach successfully maps distinguishability in simulation and hardware experiments.

## Abstract

Distinguishability and, by extension, observability are key properties of dynamical systems. Establishing these properties is challenging, especially when no analytical model is available and they are to be inferred directly from measurement data. The presence of noise further complicates this analysis, as standard notions of distinguishability are tailored to deterministic systems. We build on distributional distinguishability, which extends the deterministic notion by comparing distributions of outputs of stochastic systems. We first show that both concepts are equivalent for a class of systems that includes linear systems. We then present a method to assess and quantify distributional distinguishability from output data. Specifically, our quantification measures how much data is required to tell apart two initial states, inducing a continuous spectrum of distinguishability. We propose a statistical test to determine a threshold above which two states can be considered distinguishable with high confidence. We illustrate these tools by computing distinguishability maps over the state space in simulation, then leverage the test to compare sensor configurations on hardware.