# Data-Driven Observability Analysis for Nonlinear Stochastic Systems

arXiv: https://arxiv.org/abs/2302.11979 · HTML: https://mlsanity.com/paper/2302.11979/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 28 references that are in the map.
- First posted on arXiv 23 Feb 2023.

## Builds on (most cited first, 5 of 5)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Towards gain tuning for numerical KKL observers](https://mlsanity.com/paper/2204.00318/) (2022)
- [A Kernel Two-sample Test for Dynamical Systems](https://mlsanity.com/paper/2004.11098/) (2020)
- [emgr - The Empirical Gramian Framework](https://mlsanity.com/paper/1611.00675/) (2016)
- [Empirical Observability Gramian for Stochastic Observability of Nonlinear Systems](https://mlsanity.com/paper/2006.07451/) (2020)

## Built on by (most cited first, 1 of 1)

- [What Can a Recurrent State Safely Forget?](https://mlsanity.com/paper/2609.23366/) (2026)

## Abstract (arXiv)

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.