[论文解读] Identifiability and Estimation in Continuous Lyapunov Models

Cecilie Olesen Recke, Niels Richard Hansen|arXiv (Cornell University)|Mar 17, 2026

Target Tracking and Data Fusion in Sensor Networks被引用 0

一句话总结

This paper proves generic identifiability up to scaling for drift matrices in continuous Lyapunov models with non-Gaussian Lévy noise and proposes a semiparametric estimator for the drift using higher-order cumulants, with consistency and asymptotic normality results.

ABSTRACT

Cross-sectional observations from a dynamical system can be modeled via steady-state distributions of Markov processes. The major challenge is then to determine whether the process parameters can be identified and estimated from the steady-state distributions. We study this problem for continuous Lyapunov models that arise as steady-state distributions of the solution to a multivariate stochastic differential equation, whose linear drift matrix is parametrized by a directed graph. We derive equations for the cumulant tensors of any order for this distribution, which generalize the well-known covariance Lyapunov equation. Under a non-Gaussianity assumption we prove generic identifiability of the drift matrix for any connected graph using the equations for the higher-order cumulants. Based on the identifiability result, we propose a new semiparametric estimator of the drift matrix, and we derive its asymptotic distribution. A simulation study demonstrates the asymptotic validity of the estimator but shows that it is only accurate for relatively large sample sizes, illustrating the hardness of the unconstrained estimation problem.

研究动机与目标

• Motivate the problem of parameter identifiability from cross-sectional distributions of stationary Markov processes.
• Develop a cumulant-based framework for analysing steady-state distributions of continuous Lyapunov models.
• Prove generic identifiability of the drift matrix up to a scaling factor under non-Gaussian Lévy noise and connected graph structure.
• Propose a semiparametric estimator for the drift matrix and establish its consistency and asymptotic normality.
• Provide computational tools and simulation evidence illustrating the estimator’s finite-sample performance.

提出的方法

• Derive k-th order cumulant equations (k-th order continuous Lyapunov equations) that relate cumulants to the drift M and Lévy cumulants Ck.
• Show that the second- and r-th order cumulants (with appropriate diagonal conditions) determine (Σ, K) via a linear system obtained from the Lyapunov equations.
• Demonstrate that identifiability reduces to rank conditions of a block matrix built from A2(Σ)off and Ar(K).
• Prove that for connected graphs with all self-loops, M and C2, Cr are generically identifiable up to a common scaling factor from (Σ, K).
• Form a practical estimator based on a least singular value approach to recover vec(M) from a linearized system using empirical cumulants, and derive its asymptotic distribution.
• Provide the Julia implementation SteadyStateStatistics.jl for computation and inference.

实验结果

研究问题

• RQ1 Can the drift parameters of a continuous Lyapunov model be identified from cross-sectional steady-state data when the driving Lévy process is non-Gaussian?
• RQ2 Under what graph connectivity and cumulant-diagonal assumptions is the drift matrix identifiable up to scaling from the steady-state cumulants?
• RQ3 What is the estimator that leverages higher-order cumulants to recover the drift matrix, and what are its asymptotic properties?
• RQ4 How do the second- and higher-order Lyapunov equations organize into a tractable linear system for identification and estimation?
• RQ5 What are the practical implications and finite-sample behavior of the proposed estimator?

主要发现

• The k-th order cumulant tensor of the steady-state distribution satisfies a k-th order continuous Lyapunov equation K ×1 M + ... + K ×k M + Ck = 0.
• If the Lévy process coordinates are independent and the graph is connected with non-Gaussianity, M and C2, Cr are generically identifiable up to a common scaling from (Σ, K).
• Identifiability up to scaling can fail in certain non-connected graphs or under Gaussian Lévy noise; for connected graphs with diagonal C2 and Cr, identifiability holds generically.
• A linearized system combining the second- and r-th order Lyapunov equations yields a matrix whose kernel corresponds to vec(M), enabling a least singular value estimator for M.
• The estimator is consistent and asymptotically normal as the sample size grows, with an explicit asymptotic covariance expression for vec(M).
• A Julia package SteadyStateStatistics.jl implements the method and simulations show asymptotic validity but finite-sample bias is notable, requiring fairly large sample sizes.