Identifiability in Continuous Lyapunov Models

IF 1.5 2区数学 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications Pub Date : 2023-12-04 DOI:10.1137/22m1520311

Philipp Dettling, Roser Homs, Carlos Améndola, Mathias Drton, Niels Richard Hansen

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引用次数: 1

Abstract

SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1799-1821, December 2023.
Abstract. The recently introduced graphical continuous Lyapunov models provide a new approach to statistical modeling of correlated multivariate data. The models view each observation as a one-time cross-sectional snapshot of a multivariate dynamic process in equilibrium. The covariance matrix for the data is obtained by solving a continuous Lyapunov equation that is parametrized by the drift matrix of the dynamic process. In this context, different statistical models postulate different sparsity patterns in the drift matrix, and it becomes a crucial problem to clarify whether a given sparsity assumption allows one to uniquely recover the drift matrix parameters from the covariance matrix of the data. We study this identifiability problem by representing sparsity patterns by directed graphs. Our main result proves that the drift matrix is globally identifiable if and only if the graph for the sparsity pattern is simple (i.e., does not contain directed 2-cycles). Moreover, we present a necessary condition for generic identifiability and provide a computational classification of small graphs with up to 5 nodes.

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连续Lyapunov模型的可辨识性

SIAM矩阵分析与应用学报，第44卷，第4期，1799-1821页，2023年12月。摘要。最近引入的图形连续李雅普诺夫模型为相关多元数据的统计建模提供了一种新的方法。该模型将每次观测视为平衡状态下多元动态过程的一次性横截面快照。通过求解由动态过程漂移矩阵参数化的连续Lyapunov方程得到数据的协方差矩阵。在这种情况下，不同的统计模型在漂移矩阵中假设不同的稀疏性模式，并且澄清给定的稀疏性假设是否允许人们从数据的协方差矩阵中唯一地恢复漂移矩阵参数成为一个关键问题。我们通过用有向图表示稀疏模式来研究这个可辨识性问题。我们的主要结果证明了漂移矩阵是全局可识别的当且仅当稀疏模式的图是简单的(即，不包含有向2环)。此外，我们提出了一般可识别性的必要条件，并提供了一个多达5个节点的小图的计算分类。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Matrix Analysis and Applications 数学-应用数学

CiteScore

2.90

自引率

6.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.