Showing papers on "Covariance mapping published in 2011"

A Penalized Spline Approach to Functional Mixed Effects Model Analysis

Abstract: In this article, we propose penalized spline (P-spline)-based methods for functional mixed effects models with varying coefficients. We decompose longitudinal outcomes as a sum of several terms: a population mean function, covariates with time-varying coefficients, functional subject-specific random effects, and residual measurement error processes. Using P-splines, we propose nonparametric estimation of the population mean function, varying coefficient, random subject-specific curves, and the associated covariance function that represents between-subject variation and the variance function of the residual measurement errors which represents within-subject variation. Proposed methods offer flexible estimation of both the population- and subject-level curves. In addition, decomposing variability of the outcomes as a between- and within-subject source is useful in identifying the dominant variance component therefore optimally model a covariance function. We use a likelihood-based method to select multiple smoothing parameters. Furthermore, we study the asymptotics of the baseline P-spline estimator with longitudinal data. We conduct simulation studies to investigate performance of the proposed methods. The benefit of the between- and within-subject covariance decomposition is illustrated through an analysis of Berkeley growth data, where we identified clearly distinct patterns of the between- and within-subject covariance functions of children's heights. We also apply the proposed methods to estimate the effect of antihypertensive treatment from the Framingham Heart Study data.

Some covariance models based on normal scale mixtures

Abstract: Modelling spatio-temporal processes has become an important issue in current research. Since Gaussian processes are essentially determined by their second order structure, broad classes of covariance functions are of interest. Here, a new class is described that merges and generalizes various models presented in the literature, in particular models in Gneiting (J. Amer. Statist. Assoc. 97 (2002) 590--600) and Stein (Nonstationary spatial covariance functions (2005) Univ. Chicago). Furthermore, new models and a multivariate extension are introduced.

Relationships between correlation lengths and integral scales for covariance models with more than two parameters

Abstract: In geostatistical applications, the terms correlation length and range are often used interchangeably and refer to a characteristic covariance length ξ that normalizes the lag distance in the variogram or the covariance model. We present equations that strictly define the correlation length (r c ) and integral range (l c ). We derive analytical expressions for r c and l c of the Whittle–Matern, fluctuation gradient curvature and rational quadratic covariances. For these covariances, we show that the correlation length and integral range for a given model are not fully determined by ξ. We define non-trivial covariance functions, and we formulate an ergodicity index based on l c . We propose using the ergodicity index to compare coarse-grained measures corresponding to non-trivial covariance functions with different parameters. Finally, we discuss potential applications of the proposed covariance models in stochastic subsurface hydrology.

Covariance Matrices for Second-Order Vector Random Fields in Space and Time

Abstract: This paper deals with vector (or multivariate) random fields in space and/or time with second-order moments, for which a framework is needed for specifying not only the properties of each component but also the possible cross relationships among the components. We derive basic properties of the covariance matrix function of the vector random field and propose three approaches to construct covariance matrix functions for Gaussian or non-Gaussian random fields. The first approach is to take derivatives of a univariate covariance function, the second one is to work on the univariate random field whose index domain is in a higher dimension and the third one is based on the scale mixture of separable spatio-temporal covariance matrix functions. To illustrate these methods, many parametric or semiparametric examples are formulated.

Estimation of noise covariance matrices for periodic systems

Abstract: Estimation of the noise covariance matrices for linear time-variant stochastic dynamic periodic systems is treated. The novel offline method for estimation of the covariance matrices of the state and measurement noises is designed. The method is based on analysis of second-order statistics of the state estimate produced by the linear multi-step predictor. The estimates of the noise covariance matrices are unbiased and converge to the true values with increasing number of data. The theoretical results are illustrated in numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.

On limiting spectral distribution of large sample covariance matrices by VARMA(p,q)

Abstract: We studied the limiting spectral distribution of large-dimensional sample covariance matrices of a stationary and invertible VARMA(p,q) model. Relationship of the power spectral density and limiting spectral distribution of large population dimensional covariance matrices of ARMA(p,q) is established. The equation about Stieltjes transform of large-dimensional sample covariance matrices is also derived. As applications, the classical M-P law, VAR(1) and VMA(1) can be regarded as special examples.

Influence of resonance parameters' correlations on the resonance integral uncertainty; 55Mn case

Abstract: For nuclides with a large number of resonances the covariance matrix of resonance parameters can become very large and expensive to process in terms of the computation time. By converting covariance matrix of resonance parameters into covariance matrices of background cross-section in a more or less coarse group structure a considerable amount of computer time and memory can be saved. The question is how important is the information that is discarded in the process. First, the uncertainty of the 55Mn resonance integral was estimated in narrow resonance approximation for different levels of self-shielding using Bondarenko method by random sampling of resonance parameters according to their covariance matrices from two different 55Mn evaluations: one from Nuclear Research and Consultancy Group NRG (with large uncertainties but no correlations between resonances), the other from Oak Ridge National Laboratory (with smaller uncertainties but full covariance matrix). We have found out that if all (or at least significant part of the) resonance parameters are correlated, the resonance integral uncertainty greatly depends on the level of self-shielding. Second, it was shown that the commonly used 640-group SAND-II representation cannot describe the increase of the resonance integral uncertainty. A much finer energy mesh for the background covariance matrix would have to be used to take the resonance structure into account explicitly, but then the objective of a more compact data representation is lost.

Joint Mean-Covariance Models with Applications to Longitudinal Data in Partially Linear Model

Abstract: Semiparametric regression models and estimating covariance functions are very useful in longitudinal study. Unfortunately, challenges arise in estimating the covariance function of longitudinal data collected at irregular time points. In this article, for mean term, a partially linear model is introduced and for covariance structure, a modified Cholesky decomposition approach is proposed to heed the positive-definiteness constraint. We estimate the regression function by using the local linear technique and propose quasi-likelihood estimating equations for both the mean and covariance structures. Moreover, asymptotic normality of the resulting estimators is established. Finally, simulation study and real data analysis are used to illustrate the proposed approach.

Flexible estimation of covariance function by penalized spline with application to longitudinal family data.

Abstract: Longitudinal data are routinely collected in biomedical research studies. A natural model describing longitudinal data decomposes an individual's outcome as the sum of a population mean function and random subject-specific deviations. When parametric assumptions are too restrictive, methods modeling the population mean function and the random subject-specific functions nonparametrically are in demand. In some applications, it is desirable to estimate a covariance function of random subject-specific deviations. In this work, flexible yet computationally efficient methods are developed for a general class of semiparametric mixed effects models, where the functional forms of the population mean and the subject-specific curves are unspecified. We estimate nonparametric components of the model by penalized spline (P-spline, Biometrics 2001; 57:253–259), and reparameterize the random curve covariance function by a modified Cholesky decomposition (Biometrics 2002; 58:121–128) which allows for unconstrained estimation of a positive-semidefinite matrix. To provide smooth estimates, we penalize roughness of fitted curves and derive closed-form solutions in the maximization step of an EM algorithm. In addition, we present models and methods for longitudinal family data where subjects in a family are correlated and we decompose the covariance function into a subject-level source and observation-level source. We apply these methods to the multi-level Framingham Heart Study data to estimate age-specific heritability of systolic blood pressure nonparametrically. Copyright © 2011 John Wiley & Sons, Ltd.

Estimation of hyperspectral covariance matrices

Abstract: Estimation of covariance matrices is a fundamental step in hyperspectral remote sensing where most detection algorithms make use of the covariance matrix in whitening procedures. We present a simple method to improve the estimation of the eigenvalues of a sample covariance matrix. With the improved eigenvalues we construct an improved covariance matrix. Our method is based on the Marcenko-Pastur law, theory of eigenvalue bounds, and energy conservation. Our objective is to add a new method for estimating the eigenvalues of Wishart covariance matrices in scenarios where the sample size is small. Our method is simple, practical and easy to implement (it consists of a multiplication of 3 matrices). We did our study with extensive simulations and a few examples of hyperspectral remote sensing data that were measured in the long infrared wavelength region (8–12µm). We show examples of the improved eigenvalues over the sampled eigenvalues. We choose the following five figures-of-merit for evaluating our method as ratios of properties between sampled data and our solution: (i) residual (rms) that gives the improvement of the solution over the sampled data with respect to the population eigenvalues, (ii) area under the scree-plot, (iii) condition number that gives the improvement in the stability (regularization), (iv) a distance-measure that gives the average statistical improvement between the improved and the sampled eigenvalues, and (v) Kullback-Leibler distance. We show hyperspectral matched-filter detection performance (ROC curves) for TELOPS data where we use our improved covariance matrix. We compare the improved ROC to the one that are obtained with sampled (data) covariance matrix.

Fluctuations of Matrix Entries of Regular Functions of Sample Covariance Random Matrices

Abstract: We extend the results about the fluctuations of the matrix entries of regular functions of Wigner matrices to the case of sample covariance random matrices.

Joint mean-covariance modelling and variable selection for longitudinal data analysis

Abstract: Briefly saying, in this thesis, I endeavor to deliver both the parametric and non-parametric modelling and selection tools for longitudinal data analysis.The first part of my work, is to extend the GEEs with random effects into thejoint modelling of longitudinal data. This is a parametric approach in which theheterogeneity and heteroscedasticity for different individuals are taken into account.With the only assumption about the existence of the first four order moments of theresponses, random effects are treated as a kind of penalty in the extended GEEs. Thisapproach includes both the virtues of GEEs and joint modelling with random effects.The modified Cholesky decomposition is used here for joint modelling because it hasa explicit statistical interpretation. This work could be applied to the longitudinaldata analysis in which the individual performance is of our main interest.The second part of this thesis, dedicates to the selection of random effects in theGeneralized Linear Mixed Model (GLMM). In this work, the penalized functions areimplemented into the selection of random effects covariance components. And the Pe-nalized Quasi-Likelihood (PQL) is recruited to deal with the integration of likelihood.When nonzero random effects covariance components are selected, their correspond-ing random effects are selected and other zero ones are eliminated. A backfittingalgorithm is proposed here for variable estimation and the leave-one-subject-out CV(SCV) method is used to select the optimal value of tuning parameter in penaltyfunction. This work is valuable in the aspect that random effects could also be par-simoniously selected with penalty functions. Besides, extension of this work to theselection of both fixed-effects and random effects are quite straight forward and therefore applicable to a more general area.The last part of this thesis aims to utilize a nonparametric data-driven approach,i.e., polynomial techniques to analyze the longitudinal data. Also based on the mod-ified Cholesky decomposition, the within subject covariance matrix is decomposedinto a unit lower triangle matrix involving generalized autoregressive coefficients anda diagonal matrix involving innovation variances. Local polynomial smoothing esti-mation is proposed to model the nonparametric smoothing functions of mean, gen-eralized autoregressive parameters and log-innovation variance, simultaneously. Theleave-one-subject-out CV (SCV) method is also implemented for the bandwidth se-lection. This work is creative in joint-modelling mean and covariance parameters bylocal polynomial method with the modified Choleskey decomposition. Besides, theproposed approach shows the robustness in computation in application.

Using log-constrast relations to generate Normal variables on theSimplex

Abstract: It is well known that the covariance structure of a D-part composition is completely determined by D(D 1)

Semi-parametric X2 Testing in Mean Structure and Covariance Structure Analysis Using Projections

Abstract: Mean structures form a basis for mean, covariance, and other forms of moment structure analysis including structural equation modeling. It is shown how to analyze mean structures using projections. These are used to derive a simple general goodness of fit test statistic that is asymptotically chi-squared and robust to departures from normality. Projections are also used to derive two goodness of fit test statistics for mean structures that are substructures of a more general mean structure. One of these uses the difference of two goodness of fit test statistics, one for the general structure and one for the substructure. It is shown how to use the mean structure results for covariance structure analysis. Best generalized least squares, or ADF estimates are not required. Any asymptotically normal estimates may be use. The primary methods used for testing mean and covariance structures are orthogonal complement methods. A basic difficulty with using these is identified. Specific examples show how the general results may be applied to generalized nonlinear regression and to autoregression with measurement errors. Simulation studies investigate the type one errors and power of the test statistics involved. An appendix contains a review of the basic asymptotic and projection methods used. It also gives conditions that lead to the commonly made assumption that the asymptotic covariance matrix of a vectorized form of a sample covariance matrix is positive definite and that this is a very mild assumption.