Product Covariance Structure Research Articles

Testing Kronecker product covariance matrices for high-dimensional matrix-variate data

Summary The Kronecker product covariance structure provides an efficient way to model the inter-correlations of matrix-variate data. In this paper, we propose test statistics for the Kronecker product covariance matrix based on linear spectral statistics of renormalized sample covariance matrices. A central limit theorem is proved for the linear spectral statistics, with explicit formulas for the mean and covariance functions, thereby filling a gap in the literature. We then show theoretically that the proposed test statistics have well-controlled size and high power. We further propose a bootstrap resampling algorithm to approximate the limiting distributions of the associated linear spectral statistics. Consistency of the bootstrap procedure is guaranteed under mild conditions. The proposed test procedure is also applicable to the Kronecker product covariance model with additional random noise. In our simulations, the empirical sizes of the proposed test procedure and its bootstrapped version are close to the corresponding theoretical values, while the power converges to $1$ quickly as the dimension and sample size increase.

A Novel Distributed Data-Driven Strategy for Fault Detection of Multi-Source Dynamic Systems

In this brief, a distributed data-driven fault detection method based on sparse integrated dynamic principal component analysis is designed to enhance the fault detection performance of multi-source dynamic systems. First, by dividing the complex system into multiple sub-sources according to the structural principle, the fusion and complementation of process mechanism knowledge and data-driven are realized. Second, by introducing the Kronecker product covariance structure, the individual modes within each sub-source as well as joint modes shared between sub-sources are captured. Notably, by developing a truncated contribution power method, key variables are selected and the interpretability of principal components is improved. Finally, experimental results demonstrate that the proposed strategy can significantly improve the detection performance compared with the existing PCA-based methods.

Low-power laser and potassium oxalate gel in the treatment of cervical dentin hypersensitivity-a randomized clinical trial.

To evaluate the effectiveness of different protocols for the treatment of cervical dentin hypersensitivity (CDH) in non-carious cervical lesions (NCCLs). The CONSORT checklist was used to design this study. The sample with n = 74 participants (389 NCCLs) was randomly allocated into three groups: G1, potassium oxalate (Oxa-Gel BF); G2, GaAlAs (gallium-aluminum-arsenate) low-power laser (100mW, 808nn, 60J/cm2); and G3, potassium oxalate (Oxa-Gel BF) associated with the GaAlAs low-power laser. The CDH was triggered by the evaporative stimulus test (EST) and by the tactile stimulus test (TST). The visual analog scale (VAS) was used to quantify the degree of CDH. Changes in sensitivity were assessed from baseline over 3weeks. Data were analyzed for NCCLs using mixed-effects models with unstructured direct product covariance structure (α = 0.05). After the first application, participants from G1 and G3 had a reduction in CDH (p < 0.05) compared with group G2 for TST. After the second application, G3 participants had a reduction in CDH (p < 0.05) in relation to G2 for both stimuli. Reduction in CDH (p < 0.05) occurred over 3weeks for EST and TST for all groups; however, there was no difference between groups at the end of the therapies. Potassium oxalate was more effective in reducing immediate CDH. After four applications, all groups showed similar results for the reduction of CDH. GaAlAS laser irradiation and oxalate potassium gel could reduce the symptoms of CDH; thus, they are viable alternatives for the treatment of this condition. Chemical occlusion of dental tubules showed effective results after a shorter time interval. Brazilian Clinical Trials Registration Platform under protocol number RBR-4ybjmt. http://www.braziliantrials.com/?keywords=RBR-4ybjmt&order=%7Eensaios.patrocinador_primario.

Testing variance parameters in models with a Kronecker product covariance structure

Under a model having a Kronecker product covariance structure with compound symmetry, hypothesis testing for a correlation is investigated. Several tests are suggested and practical recommendations are made based on their type I error probabilities and powers.

Existence and uniqueness of the maximum likelihood estimator for models with a Kronecker product covariance structure

This paper deals with multivariate Gaussian models for which the covariance matrix is a Kronecker product of two matrices. We consider maximum likelihood estimation of the model parameters, in particular of the covariance matrix. There is no explicit expression for the maximum likelihood estimator of a Kronecker product covariance matrix. We investigate whether the maximum likelihood estimator of the covariance matrix exists and whether it is unique. We consider models with general, with double diagonal, and with one diagonal Kronecker product covariance matrices, and find different results.

Approximate tail probabilities of the maximum of a chi-square field on multi-dimensional lattice points and their applications to detection of loci interactions

In this study, we define a chi-square random field on a multi-dimensional lattice points index set with a direct product covariance structure and consider the distribution of the maximum of this random field. We provide two approximate formulas for the upper tail probability of the distribution based on nonlinear renewal theory and an integral-geometric approach called the volume-of-tube method. This study is motivated by the detection problem of the interactive loci pairs which play an important role in forming biological species. The joint distribution of scan statistics for detecting the pairs is regarded as the chi-square random field above, and hence the multiplicity-adjusted $$p$$ -value can be calculated using the proposed approximate formulas. By using these formulas, we examine the data of Mizuta, Harushima and Kurata (Proc Nat Acad Sci USA 107(47):20417–20422, 2010) who reported a new interactive loci pair of rice inter-subspecies.

Multidimensional prewhitening for enhanced signal reconstruction and parameter estimation in colored noise with Kronecker correlation structure

Parameter estimation of multidimensional data in the presence of colored noise or interference with a Kronecker product covariance structure, which appears in electroencephalogram/magnetoencephalogram and multiple-input multiple-output applications, is addressed. In order to improve the accuracy of the multidimensional subspace-based estimation techniques designed for white noise, prewhitening algorithms are devised by exploiting the Kronecker structure of the noise covariance matrix. We first contribute to the development of the multidimensional prewhitening (MD-PWT) scheme which assumes that noise-only measurements are available. By applying prewhitening sequentially along various dimensions using the corresponding correlation factors estimated from the noise-only measurements, the MD-PWT significantly improves the performance of the closed-form parallel factor decomposition based parameter estimator (CFP-PE) with a small number of noise-only snapshots. When noise-only measurements are unavailable, an iterative joint estimation of noise and signal parameters and prewhitening algorithm is proposed by iteratively applying the MD-PWT and CFP-PE. Adaptive convergence thresholds are designed as the stopping conditions such that the optimal number of iterations is automatically determined. Simulation results show that the iterative scheme performs nearly the same as the MD-PWT with noise statistics, in all scenarios except for a special one of intermediate signal-to-noise ratios and high noise correlation levels.

Flexible marginalized models for bivariate longitudinal ordinal data

Random effects models are commonly used to analyze longitudinal categorical data. Marginalized random effects models are a class of models that permit direct estimation of marginal mean parameters and characterize serial correlation for longitudinal categorical data via random effects (Heagerty, 1999). Marginally specified logistic-normal models for longitudinal binary data. Biometrics 55, 688-698; Lee and Daniels, 2008. Marginalized models for longitudinal ordinal data with application to quality of life studies. Statistics in Medicine 27, 4359-4380). In this paper, we propose a Kronecker product (KP) covariance structure to capture the correlation between processes at a given time and the correlation within a process over time (serial correlation) for bivariate longitudinal ordinal data. For the latter, we consider a more general class of models than standard (first-order) autoregressive correlation models, by re-parameterizing the correlation matrix using partial autocorrelations (Daniels and Pourahmadi, 2009). Modeling covariance matrices via partial autocorrelations. Journal of Multivariate Analysis 100, 2352-2363). We assess the reasonableness of the KP structure with a score test. A maximum marginal likelihood estimation method is proposed utilizing a quasi-Newton algorithm with quasi-Monte Carlo integration of the random effects. We examine the effects of demographic factors on metabolic syndrome and C-reactive protein using the proposed models.

Planning Planning of experiments for a nonautonomous Ornstein-Uhlenbeck process

We study exact optimal designs for processes governed by mean-reversion stochastic differential equations with a time dependent volatility and known mean-reversion speed. It turns out that any mean-reversion Itō process has a product covariance structure. We prove the existence of a nondegenerate optimal sampling design for the parameter estimation and derive the information matrix corresponding to the observation of the full path. The results are demonstrated on a process with exponential volatility.

Multivariate Model with Correlated Observation Units

In this paper we consider $p$ characteristics at $n$ time-points. These observations can be represented as $n$ random $p$-vectors $X_1, \ldots, X_n$. We assume that a $pn$-vector ${\rm vec}({\bf X}) = \left(X_1^T, \ldots, X_n^T\right)^T$ is distributed normally and follows a Kronecker product covariance structure. New methods to test linear hypotheses in the considered model are presented.

Estimating and Testing a Structured Covariance Matrix for Three-Level Multivariate Data

This article considers an approach to estimating and testing a new Kronecker product covariance structure for three-level (multiple time points (p), multiple sites (u), and multiple response variables (q)) multivariate data. Testing of such covariance structure is potentially important for high dimensional multi-level multivariate data. The hypothesis testing procedure developed in this article can not only test the hypothesis for three-level multivariate data, but also can test many different hypotheses, such as blocked compound symmetry, for two-level multivariate data as special cases. The tests are implemented with two real data sets.

Multivariate model with a Kronecker product covariance structure: S. N. Roy method of estimation

In this paper we consider a (p × q)-matrix X = (X1, ..., Xq), where a pq-vector vec (X) = (X1T, ...,XqT)T is assumed to be distributed normally with mean vector vec (M) = (M1T, ...,MqT)T and a positive definite covariance matrix Λ. Suppose that Λ follows a Kronecker product covariance structure, that is Λ = Φ⊗Σ, where Φ = (ϕij) is a (q × q)-matrix and Σ = (σij) is a (p × p)-matrix and the matrices Φ, Σ are positive definite. Such a model is considered in [4], where the maximum likelihood estimates of the parameters M, Φ, Σ are obtained. Using S. N. Roy’s technique (see, e.g., [3]) of the multivariate statistical analysis, we obtain consistent and unbiased estimates of M, Φ, Σ as in [4], but with less calculations.

Response to comment by C. H. Lee, P. Dutilleul and A. Roy on “Models with a Kronecker Product Covariance Structure: Estimation and Testing”

Response to comment by C. H. Lee, P. Dutilleul and A. Roy on “Models with a Kronecker Product Covariance Structure: Estimation and Testing”

An Application of Linear Mixed Effects Model to Assess the Agreement Between Two Methods With Replicated Observations

We study the problem of assessing the agreement between two methods with any number of replicated observations using linear mixed effects (LME) model with Kronecker product covariance structure in a doubly multivariate set-up. This method can also be used in the case of unbalanced designs when number of replications on each patient is unequal, as well as when the number of replications on each patient by respective methods is unequal. The model is implemented using the MIXED procedure of SAS. We demonstrate our proposed method with three real datasets.

Models with a Kronecker product covariance structure: Estimation and testing

In this article we consider a pq-dimensional random vector x distributed normally with mean vector θ and covariance matrix Λ assumed to be positive definite. On the basis of N independent observations on the random vector x, we want to estimate parameters and test the hypothesis H: Λ = Ψ ⊗ Σ, where Ψ = (ψij): q × q, ψqq = 1, and Σ = (σij): p × p, and Λ = (ψijΣ), the Kronecker product of Ψ and Σ. That is instead of 1/2pq(pq + 1) parameters, it has only 1/2p(p + 1) + 1/2q(q + 1) − 1 parameters. A test based on the likelihood ratio is given to check if this model holds. And, when this model holds, we test the hypothesis that Ψ is a matrix with intraclass correlation structure. The maximum likelihood estimators (MLE) are obtained under the hypothesis as well as under the alternatives. Using these estimators the likelihood ratio tests (LRT) are obtained. One of the main objects of the paper is to show that the likelihood equations provide unique estimators.

A Comparison of Procedures for the Analysis of Multivariate Repeated Measurements

Three procedures for analyzing within-subjects effects in multivariate repeated measures designs are compared when group covariances are heterogeneous: the multiple regression model (MRM) with a structured covariance, Johansen’s (1980) procedure, and the multivariate Brown and Forsythe (1974) procedure. A preliminary likelihood ratio test of a Kronecker product covariance structure is sensitive to sample size and derivational assumption violations. Error rates of the procedures are generally well-controlled except when the distribution is skewed. The MRM procedure displayed few power advantages over the other procedures.

On implementation of a test for Kronecker product covariance structure for multivariate repeated measures data

Under the assumption of multivariate normality the likelihood ratio test is derived to test a hypothesis for Kronecker product structure on a covariance matrix in the context of multivariate repeated measures data. Although the proposed hypothesis testing can be computationally performed by indirect use of Proc Mixed of SAS, the Proc Mixed algorithm often fails to converge. We provide an alternative algorithm. The algorithm is illustrated with two real data sets. A simulation study is also conducted for the purpose of sample size consideration.

Exactly optimal sampling designs for processes with a product covariance structure

AbstractThe author considers the problem of finding exactly optimal sampling designs for estimating a second‐order, centered random process on the basis of finitely many observations. The value of the process at an unsampled point is estimated by the best linear unbiased estimator. A weighted integrated mean squared error or the maximum mean squared error is used to measure the performance of the estimator. The author presents a set of necessary and sufficient conditions for a design to be exactly optimal for processes with a product covariance structure. Expansions of these conditions lead to conditions for asymptotic optimality.

On Sampling Designs for Integral Estimation of a Random Process

In this paper the problem of finding exactly optimal sampling designs for estimating the weighted integral of a stochastic process with a product covariance structure (R(s,t)=u(s)v(t), s<t) is discussed. The sampling designs for certain standard processes belonging to the product class are calculated. An asymptotic solution to the design problem also follows as a consequence.

Bayesian discrimination with longitudinal data.

The motivation for the methodological development is a double-blind clinical trial designed to estimate the effect of regular injection of growth hormone, with the purpose of identifying growth hormone abusers in sport. The data formed part of a multicentre investigation jointly sponsored by the European Union and the International Olympic Committee. The data are such that for each individual there is a matrix of marker variables by time point (nominally 8 markers at each of 7 time points). Data arise out of a double-blind trial in which individuals are given growth hormone at one of two dose levels or placebo daily for 28 days. Monitoring by means of blood samples is at 0, 21, 28, 30, 33, 42 and 84 days. We give a new method of Bayesian discrimination for multivariate longitudinal data. This involves a Kronecker product covariance structure for the time by measurements (markers) data on each individual. This structure is estimated by an empirical Bayes approach, using an ECM algorithm, within a Bayesian Gaussian discrimination model. In future one may have markers for an individual at one or more time points. The method gives probabilities that an individual is on placebo or on one of the two dose regimes.