Matrix Algebra From a Statistician's Perspective

By D. Harville. 1998

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High-Dimensional Multivariate Forecasting with Low-Rank Gaussian Copula Processes (2019, methodology): “valuated by rst computing an r-by- r matrix C = Ir + V T D 1 V in O (dr 2 ) time, followed by computing its Cholesky decomposition C = L C L T C in O (r3 ) time. Using the matrix determinant lemma in [7] we can write log j j = log jD + VV T j = log jC j+log jD j = 2log jL C j+log jD j which can be c”
Gradient Descent with Identity Initialization Efficiently Learns Positive-Definite Linear Transformations by Deep Residual Networks (2018, methodology): “We rely on the following facts (Horn, 1986; Harville, 1997).”
Weed Abundance, Distribution, Diversity, and Community Analyses (2015, unknown): “Summary details about matrix algebra are presented by Digby and Kempton (1987), and Khattree and Naik (2000), whereas further details with statistical viewpoints can be found in books by Harville (1997), Rao and Rao (1998), and Schott (2005).”
IDENTIFYING LOCALLY OPTIMAL DESIGNS FOR NONLINEAR MODELS: A SIMPLE EXTENSION WITH PROFOUND CONSEQUENCES (2012, unknown): “Following Karlin and Studden (1966) and Dette and Melas (2011), a set of k + 1 real-valued continuous functions u0, .”
About predictions in spatial autoregressive models: optimal and almost optimal strategies (2017, methodology): “…and almost optimal strategies 309 SPATIAL ECONOMIC ANALYSIS Another formulation of the Goldberger formula for SAR models Using algebra results from Harville (1997), the Goldberger formula can be written in terms of the precision matrix Q , as in the prediction formula for Markov Gaussian vector…”
Testing statistical hypotheses based on the density power divergence (2012, background): “3 of Harville (2008). Remark 2 The most common situation with usual density functions is r = rank (Kβ(θ0)) = rank(Aγ (θ0)) = p.”
QML ESTIMATION OF A CLASS OF MULTIVARIATE ASYMMETRIC GARCH MODELS (2011, methodology): “We shall use standard results on the differentiation of matrices that can be found in Magnus and Neudecker (1988) and Harville (1997), in particular the chain rule for differentiation of composed functions: ∂ f (A)/∂x = Tr { (∂ f (A)/∂ A′)(∂ A/∂x) } .”
Reinforcement learning of motor skills with policy gradients (2008, methodology): “When making use of thematrix inversion theorem (using Harville (2000), pages 98–101, or Moon and Stirling (2000), pages 258–259; for details see also Peters (2005)), we can simplify this approach to”
Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees (2013, background): “, [43]) states that for every nonsingular matrix A and every real number t,”
Block Variable Selection in Multivariate Regression and High-dimensional Causal Inference (2010, background): “Howeve r, a closed form solution for (4) can be derived by recalling the following matrix identities [8],”
Quantile regression methods for recursive structural equation models (2004, unknown): “In that case we have, e.g. Harville (1997), A ∂A− ∂x A = −AA−∂A ∂x A−A. Multiplying from the left and right by A−, and noting that A−A = Ip by the rank hypothesis, yields (A.6).”
Multiple Imputation of Multilevel Missing Data—Rigor Versus Simplicity (2015, unknown): “…m2ð Þ;Σ11 Σ12Σ 122 Σ21 : Thus, the conditional distribution for af is given by: af jDobs;s2; bf N â f; ðR0RÞ 111 ðR0RÞ 1 12 ðR0RÞ22ðR0RÞ 1 21 : ð3Þ It also holds (see, e.g., Harville, 1999, p. 100) for any arbitrarily partitioned nonsingular matrix M ¼ T U V W , for which the inverse matrix B ¼…”
Principal component analysis for compositional data with outliers (2009, methodology): “…on the simplex sample space (for details, see Pawlowsky-Glahn et al., 2007) for the ilr transformed data matrix Z of dimension n × (D − 1) follows immediately from (6) using V V = I D−1 (identity matrix of order D − 1) and basic properties of matrix transposition, see e.g. Harville (1997).”
Complex-Valued Matrix Differentiation: Techniques and Key Results (2007, background): “Differential of the Moore-Penrose Inverse: The differential of the real-valued Moore-Penrose inverse can be found in in [1], [3], but the fundamental result of the complex-valued version is derived here.”
Restricted likelihood ratio lack‐of‐fit tests using mixed spline models (2004, background): “space. Theorem 18.3 of Harville (1997, p.433) ensures the existence of a strictly positive value c such that V is symmetric positive denite for > c. This guarantees that it is possible to reparametrise the likelihood and embed the current parameter space in a bigger space, namely ~ = ( c;1) for wh”
Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models (2011, background): “(see Harville (1997)), the expressions for the derivatives are as follows (all right-hand side terms are transformed versions): @log |S| @ρj =λj tr.”