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Identifiability and Estimation in Continuous Lyapunov Models

Cecilie Olesen Recke, Niels Richard Hansen

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.

Identifiability and Estimation in Continuous Lyapunov Models

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.
Paper Structure (20 sections, 12 theorems, 105 equations, 8 figures)

This paper contains 20 sections, 12 theorems, 105 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main contributions
  4. Relations to existing literature
  5. The Continuous Lyapunov Model
  6. Identifiability
  7. Overview
  8. Limitations to identifiability
  9. Main results
  10. Outline of the proof of Theorem \ref{['thm::GenericIdResult']} via reorganization of the Lyapunov equations
  11. Identification when $G$ is not connected
  12. Estimation
  13. Numerical Experiments
  14. Discussion
  15. Appendix -- proofs of main results
  16. ...and 5 more sections

Key Result

proposition 1

Let $M$ be a $d \times d$ stable matrix and let $Z = (Z_t)_{t \geq 0}$ denote a $d$-dimensional Lévy process with finite $k$-th-order moment. Then the corresponding $M$-selfdecomposable distribution has finite $k$-th-order moment, and the $k$-th-order cumulant tensor $\mathcal{K}$ solves the equatio where $\mathcal{C}_k = \mathrm{cum}_k(Z_1)$ is the $k$-th-order cumulant tensor of $Z_1$. Equation

Figures (8)

  • Figure 1: Example of a directed graph $G = ([4], E)$ and corresponding zero-pattern of $M$ in $\mathbb{R}^{E}$
  • Figure 2: Directed graph, $G$, such that the drift matrix, $M$, is not identifiable up to scaling from the Gaussian model $\psi(M, \mathcal{N}(0, \mathcal{C}_2))$ for any diagonal $\mathcal{C}_2$Dettling:2023.
  • Figure 3: Two directed graphs on three nodes with one and two self-loops missing, respectively, for which $M$ is still identifiable in the sense of Theorem \ref{['thm::GenericIdResult']} for $r = 3$.
  • Figure 4: Relations between model spaces and parametrizations as a commutative diagram.
  • Figure 5: The complete graph on two nodes and four edges corresponding to the parameters in $M$ (left). Example \ref{['ex:twonodes']} shows identifiability of $M$ up to scaling by computing the rank of $A_{\mathrm{off}}$ for a particular choice of parameters (right).
  • ...and 3 more figures

Theorems & Definitions (29)

  • definition 1
  • proposition 1
  • definition 2
  • definition 3
  • theorem 1
  • corollary 1
  • remark 1
  • corollary 2
  • definition 4
  • theorem 2
  • ...and 19 more