### Browsing by Department "University of Montana"

Now showing 1 - 2 of 2

###### Results Per Page

###### Sort Options

Item Determinantal Generalizations of Instrumental Variables(DE GRUYTER, 2018-03) Weihs, Luca; Robinson, Bill; Dufresne, Emilie; Kenkel, Jennifer; Kubjas, Kaie; McGee, Reginald L., II; Nguyen, Nhan; Robeva, Elina; Drton, Mathias; University of Washington; Denison University; University of Nottingham; University of Utah; Statistics and Mathematical Data Science; Ohio State University; University of Montana; Massachusetts Institute of Technology MIT; Department of Mathematics and Systems AnalysisLinear structural equation models relate the components of a random vector using linear interdependencies and Gaussian noise. Each such model can be naturally associated with a mixed graph whose vertices correspond to the components of the random vector. The graph contains directed edges that represent the linear relationships between components, and bidirected edges that encode unobserved confounding. We study the problem of generic identifiability, that is, whether a generic choice of linear and confounding effects can be uniquely recovered from the joint covariance matrix of the observed random vector. An existing combinatorial criterion for establishing generic identifiability is the half-trek criterion (HTC), which uses the existence of trek systems in the mixed graph to iteratively discover generically invertible linear equation systems in polynomial time. By focusing on edges one at a time, we establish new sufficient and necessary conditions for generic identifiability of edge effects extending those of the HTC. In particular, we show how edge coefficients can be recovered as quotients of subdeterminants of the covariance matrix, which constitutes a determinantal generalization of formulas obtained when using instrumental variables for identification.Item Perturbation Theory of Transfer Function Matrices(Society for Industrial and Applied Mathematics (SIAM), 2023) Noferini, V; Nyman, L; Pérez, J.; Quintana, M. C.; Statistics and Mathematical Data Science; Department of Mathematics and Systems Analysis; University of MontanaZeros of rational transfer function matrices R(λ ) are the eigenvalues of associated polynomial system matrices P(λ ) under minimality conditions. In this paper, we define a structured condition number for a simple eigenvalue λ 0 of a (locally) minimal polynomial system matrix P(λ ), which in turn is a simple zero λ 0 of its transfer function matrix R(λ ). Since any rational matrix can be written as the transfer function of a polynomial system matrix, our analysis yields a structured perturbation theory for simple zeros of rational matrices R(λ ). To capture all the zeros of R(λ ), regardless of whether they are poles, we consider the notion of root vectors. As corollaries of the main results, we pay particular attention to the special case of λ 0 being not a pole of R(λ ) since in this case the results get simpler and can be useful in practice. We also compare our structured condition number with Tisseur's unstructured condition number for eigenvalues of matrix polynomials and show that the latter can be unboundedly larger. Finally, we corroborate our analysis by numerical experiments.