回归图与稀疏诱导重参数化
Regression graphs and sparsity-inducing reparametrizations
AI总结：
本文探讨参数化与稀疏性的内在联系，研究在协方差模型中通过重参数化诱导稀疏性的结构，利用Iwasawa分解和链图模型提出新方法，并解释近似零的含义，为建模提供新视角。
摘要：
参数化与稀疏性内在关联，这提出了一个可能性：那些在其自然表述中并不明显稀疏的相关模型，在重参数化后可能表现出总体层面的稀疏性。在协方差模型中，正定性对稀疏性如何合法地表现施加了额外约束。因此，自然要考虑那些稀疏性尊重正定性的重参数化映射。本文的主要目的是深入理解在物理自然尺度上的结构，这些结构在重参数化后诱导稀疏性，并且也被稀疏性所诱导。最初发现的四种结构中最丰富的一种，可以在因果排序下，由Wermuth & Cox (2004)研究的联合响应图生成，而最受限制的一种是由Battey (2017)研究的矩阵对数尺度上的稀疏性所诱导的。一般线性群的Iwasawa分解，结合图模型解释，指向了一类针对链图模型(Andersson et al. 2001)的重参数化，其中无向图和有向无环图作为特例。一个重要的见解是对近似零的解释，阐明了在重参数化后强制稀疏性的建模含义：实际上，如果相对直接的回归效应可忽略，而其他效应通过长路径显现，那么两个变量之间的关系将被宣布为无效。这些见解对方法论有所影响，其中一些方面已呈现。一项详细的模拟利用理论见解进一步探索了重参数化有益的制度。
英文摘要：
That parametrization and sparsity are inherently linked raises the possibility that relevant models, not obviously sparse in their natural formulation, exhibit a population-level sparsity after reparametrization. In covariance models, positive-definiteness enforces additional constraints on how sparsity can legitimately manifest. It is therefore natural to consider reparametrization maps in which sparsity respects positive definiteness. The main purpose of this paper is to provide insight into structures on the physically-natural scale that induce and are induced by sparsity after reparametrization. The richest of the four structures initially uncovered can be generated, under a causal ordering, by the joint-response graphs studied by Wermuth & Cox (2004), while the most restrictive is that induced by sparsity on the scale of the matrix logarithm, studied by Battey (2017). The Iwasawa decomposition of the general linear group, combined with the graphical-models interpretation, points to a class of reparametrizations for the chain-graph models (Andersson et al. 2001), with undirected and directed acyclic graphs as special cases. An important insight is the interpretation of approximate zeros, explaining the modelling implications of enforcing sparsity after reparameterization: in effect, the relation between two variables would be declared null if relatively direct regression effects were negligible and others manifested through long paths. The insights have a bearing on methodology, some aspects of which are presented. A detailed simulation uses the theoretical insights to further explore regimes under which reparametrization is beneficial.
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