More on the Kronecker Structured Covariance Matrix

Singull, Martin; Ahmad, M. Rauf; Rosen, Dietrich von

doi:10.1080/03610926.2011.615971

Cited by 17 publications

(11 citation statements)

References 18 publications

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“…Previous work has developed Kronecker‐structured covariance matrices of 2 or 3 components. Repeated measures are the most common application of these methods, as well as applications to spatial and imaging data. Additional structure may be imposed on the subblocks, such as circulant or Toeplitz .…”

Section: Introductionmentioning

confidence: 99%

Separable covariance models for health care quality measures across years and topics

Hatfield

Zaslavsky

2018

Statistics in Medicine

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Public quality reports for Medicare Advantage health plans include 11 measures of patient experiences reported in the annual Consumer Assessment of Healthcare Providers and Systems surveys. Computing summaries at the health plan level (of multiple measures in multiple years) yields an array-structured random variable. To summarize associations among measures and years, we model the variance-covariance matrix governing the plan-level vectors of yearly quality measures as a Kronecker product of an across-measure matrix and an across-year matrix, or a sum of such Kronecker products. This approach extends separable covariance structure to Fay-Herriot models. In addition, we develop linear combinations of Kronecker products similar to principal components for array random variables. To each Kronecker-product term, we apply post hoc analyses suited to the corresponding dimension of the cross-classification: 1-way factor analysis for the across-measure factor and time-series analysis to the across-year factor. These methods draw out key patterns of variation in the quality measures over time and suggest new strategies for reporting quality information to consumers.

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Section: Introductionmentioning

confidence: 99%

Separable covariance models for health care quality measures across years and topics

Hatfield

Zaslavsky

2018

Statistics in Medicine

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“…It was shown how to apply the Tucker product operator in order to substantially simplify the writing of equations and formulas in higher dimensions, including the algorithm derived from solving the system of likelihood equations (Hoff, 2011). Following Srivastava et al (2008), Singull, Ahmad, and von Rosen (2012) proposed two identifiability constraints (i.e., one for each of two of the three factor variance-covariance matrices of the tensor normal distribution of order 3) for the application of the estimation procedure. Of course, some of the remarks made in other sections of this article may apply to the case of the tensor normal distribution of order 3, for example, the nonuniqueness of the MLE when n = 2, p = q = r or n = 3, p = q = r = 2.…”

Section: Discussionmentioning

confidence: 99%

Estimation and testing for separable variance–covariance structures

Dutilleul

2018

WIREs Computational Stats

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The statistical analysis of data for a p‐variate response observed repeatedly on q occasions or of spatiotemporal data recorded at p locations by q times for n individuals may require that constraints be imposed on the modeling of the variance–covariance structure of the underlying process, not because of the repeated‐measures or spatiotemporal nature of the data but because there is not enough data otherwise to estimate the model parameters. Besides stationarity and isotropy, separability is an interesting option for that purpose because it reduces the number of variance‐covariance parameters to estimate, from pq(pq + 1)/2 to the Kronecker product of two matrices with p(p + 1)/2 and q(q + 1)/2 parameters. Originally, in the late 1980s, separability of the variance–covariance structure was assumed. Under this model, combined with the normality assumption on the underlying distribution, novel theoretical developments were thus made. The question of estimation of the parameters of a separable variance–covariance structure, more particularly by maximum likelihood, was raised from the early 1990s on, the question of testing for this structure being effectively addressed several years later. The existence and uniqueness of maximum likelihood estimators for the matrix normal distribution (i.e., the doubly multivariate normal distribution characterized by a simply separable variance–covariance structure) have been and remain questions of interest, as shown by recent results. Below, the reader is guided throughout the field of study of the separable variance–covariance structures as the author provides a fair treatment of the topic, its components, extensions (e.g., double separability), and future perspectives. This article is categorized under Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms

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“…The computations required are typically not too onerous, since for example the Hessian matrix is (v + 1) × (v + 1) (i.e., of order log n by log n), but there is quite complicated non-linearity involved in the definition of the QMLE and so it is not so easy to analyse from a theoretical point of view. See Singull et al (2012) and Ohlson et al (2013) for discussion of estimation algorithms in the case where the data are multi-array and v is of low dimension.…”

Section: The Quasi-maximum Likelihood Estimatormentioning

confidence: 99%

“…There is also a growing Bayesian and Frequentist literature on multiway array or tensor datasets, where this structure is commonly employed. See for example Akdemir and Gupta (2011), Allen (2012), Browne, MacCallum, Kim, Andersen, and Glaser (2002), Cohen, Usevich, and Comon (2016), Constantinou, Kokoszka, and Reimherr (2015), Dobra (2014), Fosdick and Hoff (2014), Gerard and Hoff (2015), Hoff (2011), Hoff (2015, Hoff (2016), Krijnen (2004), Leiva and Roy (2014), Leng and Tang (2012), Li and Zhang (2016), Manceura and Dutilleul (2013), Ning and Liu (2013), Ohlson, Ahmada, and von Rosen (2013), Singull, Ahmad, and von Rosen (2012), Volfovsky and Hoff (2014), Volfovsky and Hoff (2015), and Yin and Li (2012). In both these (apparently separate) literatures the dimension n is fixed and typically there are a small number of products each of whose dimension is of fixed but perhaps moderate size.…”

Section: Introductionmentioning

confidence: 99%

Estimation of a multiplicative covariance structure in the large dimensional case

Hafner

Linton

Tang

2016

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We propose a Kronecker product structure for large covariance or correlation matrices. One feature of this model is that it scales logarithmically with dimension in the sense that the number of free parameters increases logarithmically with the dimension of the matrix. We propose an estimation method of the parameters based on a log-linear property of the structure, and also a quasi-maximum likelihood estimation (QMLE) method. We establish the rate of convergence of the estimated parameters when the size of the matrix diverges. We also establish a central limit theorem (CLT) for our method. We derive the asymptotic distributions of the estimators of the parameters of the spectral distribution of the Kronecker product correlation matrix, of the extreme logarithmic eigenvalues of this matrix, and of the variance of the minimum variance portfolio formed using this matrix. We also develop tools of inference including a test for over-identification. We apply our methods to portfolio choice for S&P500 daily returns and compare with sample covariance-based methods and with the recent Fan, Liao, and Mincheva (2013) method.

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More on the Kronecker Structured Covariance Matrix

Cited by 17 publications

References 18 publications

Separable covariance models for health care quality measures across years and topics

Separable covariance models for health care quality measures across years and topics

Estimation and testing for separable variance–covariance structures

Estimation of a multiplicative covariance structure in the large dimensional case

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