On the Distribution of Matrix Quadratic Forms

Singull, Martin; Koski, Timo

doi:10.1080/03610926.2011.563009

Cited by 6 publications

(2 citation statements)

References 23 publications

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“…The presentation of distribution of the matrix quadratic form, done by [23], can also be implemented in the context of the MVMMN family of distributions. Referring to theorem 2.2 of [23], they defined the distribution of quadratic form Q ¼ XAX > to be Q p ðA; M; ΣÞ where A is a n × n symmetric real matrix of rank r, and X � N p;n ðM; Σ; I n Þ.…”

Section: Plos Onementioning

confidence: 99%

A theoretical framework for Landsat data modeling based on the matrix variate mean-mixture of normal model

et al. 2020

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This paper introduces a new family of matrix variate distributions based on the mean-mixture of normal (MMN) models. The properties of the new matrix variate family, namely stochastic representation, moments and characteristic function, linear and quadratic forms as well as marginal and conditional distributions are investigated. Three special cases including the restricted skew-normal, exponentiated MMN and the mixed-Weibull MMN matrix variate distributions are presented and studied. Based on the specific presentation of the proposed model, an EM-type algorithm can be directly implemented for obtaining maximum likelihood estimate of the parameters. The usefulness and practical utility of the proposed methodology are illustrated through two conducted simulation studies and through the Landsat satellite dataset analysis.

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Section: Plos Onementioning

confidence: 99%

A theoretical framework for Landsat data modeling based on the matrix variate mean-mixture of normal model

et al. 2020

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“…Moreover, the matrix A in our work is stochastic but existing results on the distribution of the quadratic form are limited to the constant matrix A. It is worth to mention that there have been fruitful results on the distribution of the matrix quadratic form in random vector X when X is real [9], [10]. For complex random vectors, [11] considers the quadratic form in a zero-mean complex random vector and the case of a non-zero mean complex random vector is studied in [12], [13].…”

Section: Introductionmentioning

confidence: 99%

Quasi-Degradation Probability of Two-User NOMA over Rician Fading Channels

Liu

2020

Preprint

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Non-orthogonal multiple access (NOMA) has a great potential to offer a higher spectral efficiency of multi-user wireless networks than orthogonal multiples access (OMA). Given two users, previous work has established the condition, referred to quasi-degradation (QD) probability, under which NOMA has no performance loss compared to the capacity-achieving dirty paper coding. Existing results assume Rayleigh fading channels without line-of-sight (LOS). In many practical scenarios, the channel LOS component is critical to the link quality where the channel gain follows a Rician distribution instead of a Rayleigh distribution. In this work, we analyze the QD probability over multi-input and single-output (MISO) channels subject to Rician fading. The QD probability heavily depends on the angle between two user channels, which involves a matrix quadratic form in random vectors and a stochastic matrix.With the deterministic LOS component, the distribution of the matrix quadratic form is non-central that dramatically complicates the derivation of the QD probability. To remedy this difficulty, a series of approximations is proposed that yields a closed-form expression for the QD probability over MISO Rician channels. Numerical results are presented to assess the analysis accuracy and get insights into the optimality of NOMA over Rician fading channels.

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A graphical model for skewed matrix-variate non-randomly missing data

Zhang

Bandyopadhyay

2018

Biostatistics

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Summary Epidemiological studies on periodontal disease (PD) collect relevant bio-markers, such as the clinical attachment level (CAL) and the probed pocket depth (PPD), at pre-specified tooth sites clustered within a subject’s mouth, along with various other demographic and biological risk factors. Routine cross-sectional evaluation are conducted under a linear mixed model (LMM) framework with underlying normality assumptions on the random terms. However, a careful investigation reveals considerable non-normality manifested in those random terms, in the form of skewness and tail behavior. In addition, PD progression is hypothesized to be spatially-referenced, i.e. disease status at proximal tooth-sites may be different from distally located sites, and tooth missingness is non-random (or informative), given that the number and location of missing teeth informs about the periodontal health in that region. To mitigate these complexities, we consider a matrix-variate skew-$t$ formulation of the LMM with a Markov graphical embedding to handle the site-level spatial associations of the bivariate (PPD and CAL) responses. Within the same framework, the non-randomly missing responses are imputed via a latent probit regression of the missingness indicator over the responses. Our hierarchical Bayesian framework powered by relevant Markov chain Monte Carlo steps addresses the aforementioned complexities within an unified paradigm, and estimates model parameters with seamless sharing of information across various stages of the hierarchy. Using both synthetic and real clinical data assessing PD status, we demonstrate a significantly improved fit of our proposition over various other alternative models.

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On the Distribution of Matrix Quadratic Forms

Cited by 6 publications

References 23 publications

A theoretical framework for Landsat data modeling based on the matrix variate mean-mixture of normal model

A theoretical framework for Landsat data modeling based on the matrix variate mean-mixture of normal model

Quasi-Degradation Probability of Two-User NOMA over Rician Fading Channels

A graphical model for skewed matrix-variate non-randomly missing data

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