Estimation Of Covariance Parameters Research Articles

A COVARIANCE PARAMETER ESTIMATION METHOD FOR POLAR-ORBITING SATELLITE DATA

We consider the problem of estimating an unknown covariance function of a Gaussian random field for data collected by a polar-orbiting satellite. The complex and asynoptic nature of such data requires a parameter estimation method that scales well with the number of observations, can accommodate many covariance functions, and uses information throughout the full range of spatio-temporal lags present in the data. Our solution to this problem is to develop new estimating equations using composite likelihood methods as a base. We modify composite likelihood methods through the inclusion of an approximate likelihood of interpolated points in the estimating equation. The new estimating equation is denoted the I-likelihood. We provide a simulation study of the I-likelihood estimator, and we show this estimator to provide consistent performance for a variety of composite likelihood bases and interpolation methods. We apply the I-likelihood method to 30 days of ozone data occurring in a single degree latitude band collected by a polar orbiting satellite. In this application, we compare I-likelihood methods to competing composite likelihood methods. The I-likelihood is shown capable of producing covariance parameter estimates that are equally or more statistically efficient than competing composite likelihood methods and to be more computationally scalable.

Asymptotic analysis of the role of spatial sampling for covariance parameter estimation of Gaussian processes

Covariance parameter estimation of Gaussian processes is analyzed in an asymptotic framework. The spatial sampling is a randomly perturbed regular grid and its deviation from the perfect regular grid is controlled by a single scalar regularity parameter. Consistency and asymptotic normality are proved for the Maximum Likelihood and Cross Validation estimators of the covariance parameters. The asymptotic covariance matrices of the covariance parameter estimators are deterministic functions of the regularity parameter. By means of an exhaustive study of the asymptotic covariance matrices, it is shown that the estimation is improved when the regular grid is strongly perturbed. Hence, an asymptotic confirmation is given to the commonly admitted fact that using groups of observation points with small spacing is beneficial to covariance function estimation. Finally, the prediction error, using a consistent estimator of the covariance parameters, is analyzed in detail.

A priori noise and regularization in least squares collocation of gravity anomalies

Abstract The paper describes the estimation of covariance parameters in least squares collocation (LSC) by the cross-validation (CV) technique called leave-one-out (LOO). Two parameters of Gauss-Markov third order model (GM3) are estimated together with a priori noise standard deviation, which contributes significantly to the covariance matrix composed of the signal and noise. Numerical tests are performed using large set of Bouguer gravity anomalies located in the central part of the U.S. Around 103 000 gravity stations are available in the selected area. This dataset, together with regular grids generated from EGM2008 geopotential model, give an opportunity to work with various spatial resolutions of the data and heterogeneous variances of the signal and noise. This plays a crucial role in the numerical investigations, because the spatial resolution of the gravity data determines the number of gravity details that we may observe and model. This establishes a relation between the spatial resolution of the data and the resolution of the gravity field model. This relation is inspected in the article and compared to the regularization problem occurring frequently in data modeling.

DiceKriging,DiceOptim: TwoRPackages for the Analysis of Computer Experiments by Kriging-Based Metamodeling and Optimization

We present two recently released R packages, DiceKriging and DiceOptim, for the approximation and the optimization of expensive-to-evaluate deterministic functions. Following a self-contained mini tutorial on Kriging-based approximation and optimization, the functionalities of both packages are detailed and demonstrated in two distinct sections. In particular, the versatility of DiceKriging with respect to trend and noise specifications, covariance parameter estimation, as well as conditional and unconditional simulations are illustrated on the basis of several reproducible numerical experiments. We then put to the fore the implementation of sequential and parallel optimization strategies relying on the expected improvement criterion on the occasion of DiceOptim’s presentation. An appendix is dedicated to complementary mathematical and computational details.

Corrected prediction intervals for change detection in paired watershed studies

Hydrological data may be temporally autocorrelated requiring autoregressive process parameters to be estimated. Current statistical methods for hydrological change detection in paired watershed studies rely on prediction intervals, but the current form of prediction intervals does not include all appropriate sources of variation. Corrected prediction intervals for the analysis of paired watershed study data that include variation associated with covariance and linear model parameter estimation are presented. We provide an example of their application to data from the Hinkle Creek Paired Watershed Study located in the western Cascade foothills of Southern Oregon, USA. Research implications of using the correct prediction limits and incorporating the estimation uncertainty of autoregressive process parameters are discussed. Editor D. Koutsoyiannis Citation Som, N.A., Zégre, N.P., Ganio, L.M. and Skaugset, A.E., 2012. Corrected prediction intervals for change detection in paired watershed studies. Hydrological Sciences Journal, 57 (1), 134–143.

Robust Estimation of a Spatiotemporal Model with Structural Change

A spatiotemporal model is postulated and estimated using a procedure that infuses the forward search algorithm and maximum likelihood estimation into the backfitting framework. The forward search algorithm filters the effect of temporary structural change in the estimation of covariate and spatial parameters. Simulation studies illustrate capability of the method in producing robust estimates of the parameters even in the presence of structural change. The method provides good model fit even for small sample sizes in short time series data and good predictions for a wide range of lengths of contamination periods and levels of severity of contamination.

D-Optimal Designs for Covariate Parameters in Block Design Set Up

The problem considered is that of finding D-optimal design for the estimation of covariate parameters and the treatment and block contrasts in a block design set up in the presence of non stochastic controllable covariates, when N = 2(mod 4), N being the total number of observations. It is clear that when N ≠ 0 (mod 4), it is not possible to find designs attaining minimum variance for the estimated covariate parameters. Conditions for D-optimum designs for the estimation of covariate parameters were established when each of the covariates belongs to the interval [−1, 1]. Some constructions of D-optimal design have been provided for symmetric balanced incomplete block design (SBIBD) with parameters b = v, r = k = v − 1, λ =v − 2 when k = 2 (mod 4) and b is an odd integer.

Maximum likelihood and restricted maximum likelihood estimation for a class of Gaussian Markov random fields

This work describes a Gaussian Markov random field model that includes several previously proposed models, and studies properties of its maximum likelihood (ML) and restricted maximum likelihood (REML) estimators in a special case. Specifically, for models where a particular relation holds between the regression and precision matrices of the model, we provide sufficient conditions for existence and uniqueness of ML and REML estimators of the covariance parameters, and provide a straightforward way to compute them. It is found that the ML estimator always exists while the REML estimator may not exist with positive probability. A numerical comparison suggests that for this model ML estimators of covariance parameters have, overall, better frequentist properties than REML estimators.

Optimum covariate designs in a binary proper equi-replicate block design set-up

The choice of covariates values for a given block design attaining minimum variance for estimation of each of the regression parameters of the model has attracted attention in recent times. In this article, we consider the problem of finding the optimum covariate design (OCD) for the estimation of covariate parameters in a binary proper equi-replicate block (BPEB) design model with covariates, which cover a large class of designs in common use. The construction of optimum designs is based mainly on Hadamard matrices.

Optimal combination of estimating equations in the analysis of multilevel nested correlated data

Multilevel nested, correlated data often arise in biomedical research. Examples include teeth nested within quadrants in a mouth or students nested within classrooms in schools. In some settings, cluster sizes may be large relative to the number of independent clusters and the degree of correlation may vary across clusters. When cluster sizes are large, fitting marginal regression models using Generalized Estimating Equations with flexible correlation structures that reflect the nested structure may fail to converge and result in unstable covariance estimates. Also, the use of patterned, nested working correlation structures may not be efficient when correlation varies across clusters. This paper describes a flexible marginal regression modeling approach based on an optimal combination of estimating equations. Particular within-cluster and between-cluster data contrasts are used without specification of the working covariance structure and without estimation of covariance parameters. The method involves estimation of the covariance matrix only for the vector of component estimating equations (which is typically of small dimension) rather than the covariance matrix of the observations within a cluster (which may be of large dimension). In settings where the number of clusters is large relative to the cluster size, the method is stable and is highly efficient, while maintaining appropriate coverage levels. Performance of the method is investigated with simulation studies and an application to a periodontal study.

Predicting Mesoscale Variability of the North Atlantic Using a Physically Motivated Scheme for Assimilating Altimeter and Argo Observations

A computationally efficient scheme is described for assimilating sea level measured by altimeters and vertical profiles of temperature and salinity measured by Argo floats. The scheme is based on a transformation of temperature, salinity, and sea level into a set of physically meaningful variables for which it is easier to specify spatial covariance functions. The scheme also allows for sequential correction of temperature and salinity biases and online estimation of background error covariance parameters. Two North Atlantic applications, both focused on predicting mesoscale variability, are used to assess the effectiveness of the scheme. In the first application the background is a monthly temperature and salinity climatology and skill is assessed by how well the scheme recovers Argo profiles that were not assimilated. In the second application the backgrounds are short-term forecasts made by an eddy-permitting model of the North Atlantic. Skill is assessed by the quality of forecasts with lead times of 1–60 days. Both applications show that the scheme has useful skill.

Optimum covariate designs in split-plot and strip-plot design set-ups

The problem considered is that of finding optimum covariate designs for estimation of covariate parameters in standard split-plot and strip-plot design set-ups with the levels of the whole-plot factor in r randomised blocks. Also an extended version of a mixed orthogonal array has been introduced, which is used to construct such optimum covariate designs. Hadamard matrices, as usual, play the key role for such construction.

Hierarchical Bayesian statistics: merging experimental and modeling approaches in ecology

covariance estimation. Journal of Agricultural, Biological and Environmental Statistics 12(4)450-469. Latimer, A. M., S. Wu, A. E. Gelfand, and J. A. Silander, Jr. 2006. Building statistical models to analyze species distribu tions. Ecological Applications 16:33-50. Ritter, K., and M. Leecaster. 2007. Multi-lag cluster designs for estimating the semivariogram for sediments affected by effluent discharges offshore in San Diego. Environmental and Ecological Statistics 14:41-53. Schabenberger, O., and C. A. Gotway. 2005. Statistical methods for spatial data analysis. Chapman and Hall/CRC, Boca Raton, Florida, USA. Waller, L. A., B. J. Goodwin, M. L. Wilson, R. S. Ostfeld, S. Marshall, and E. B. Hayes. 2007. Spatio-temporal patterns in county-level incidence and reporting of Lyme disease in the northeastern United States, 1990-2000. Environmental and Ecological Statistics 14:83-100. Waller, L, and C. Gotway. 2004. Applied spatial statistics for public health data. Wiley, New York, New York, USA. Wikle, C. K. 2003. Hierarchical Bayesian models for predicting the spread of ecological processes. Ecology 84:1382-1394. Wikle, C. K., L. M. Berliner, and N. Cressie. 1998. Hierarchical Bayesian space-time models. Environmental and Ecological Statistics 5:117-154. Xia, G, M. L. Miranda, and A. E. Gelfand. 2006. Approx imately optimal spatial design approaches for environmental health data. Environmetrics 17:363-385. Zimmerman, D. L. 2006. Optimal network design for spatial prediction, covariance parameter estimation and empirical prediction. Environmetrics 17:635-652.

MLMATERN: A computer program for maximum likelihood inference with the spatial Matérn covariance model

The Matérn covariance scheme is of great importance in many geostatistical applications where the smoothness or differentiability of the random field that models a natural phenomenon is of interest. In addition to the range and nugget parameters, the flexibility of the Matérn model is provided by the so-called smoothness parameter which controls the degree of smoothness of the random field. It has been the usual practice in geostatistics to fit theoretical semivariograms like the spherical or exponential, thus implicitly assuming the smoothness parameter to be known, without questioning if there is any theoretical or empirical basis to justify such assumption. On the other hand, if only a small number of sparse experimental data are available, it is more critical to ask if the smoothness parameter can be identified with statistical reliability. Maximum likelihood estimation of spatial covariance parameters of the Matérn model has been used to address the previous questions. We have developed a general algorithm for estimating the parameters of a Matérn covariance (or semivariogram) scheme, where the model may be isotropic or anisotropic, the nugget variance can be included in the model if desired, and the uncertainty of the estimates is provided in terms of variance–covariance matrix (or standard error-coefficient of correlation matrix) as well as likelihood profiles for each parameter in the covariance model. It is assumed that the empirical data are a realization of a Gaussian process. Our program allows the presence of a polynomial trend of order zero (constant global mean), one (linear trend) or two (quadratic trend). The restricted maximum likelihood method has also been implemented in the program as an alternative to the standard maximum likelihood. Simulation results are given in order to investigate the sampling distribution of the parameters for small samples. Furthermore, a case study is provided to show a real practical example where the smoothness parameter needs to be estimated.

To the optimal identification of multivariate systems under perturbations of unknown covariances

We consider the problem of parameter and covariance estimation for multivariate stochastic systems described by regression models with special structure perturbations of unknown covariance. Sufficient conditions of uniformly optimal estimations are obtained for system parameters and covariances. The observation vector distribution family is factorized, and the full sufficient statistics is found under those conditions. Equations for uniformely optimal unbiased estimates of covariance parameters are obtained.

Maximum Likelihood based comparison of the specific growth rates for P. aeruginosa and four mutator strains

The specific growth rate for P. aeruginosa and four mutator strains mutT, mutY, mutM and mutY–mutM is estimated by a suggested Maximum Likelihood, ML, method which takes the autocorrelation of the observation into account. For each bacteria strain, six wells of optical density, OD, measurements are used for parameter estimation. The data is log-transformed such that a linear model can be applied. The transformation changes the variance structure, and hence an OD-dependent variance is implemented in the model. The autocorrelation in the data is demonstrated, and a correlation model with an exponentially decaying function of the time between observations is suggested. A model with a full covariance structure containing OD-dependent variance and an autocorrelation structure is compared to a model with variance only and with no variance or correlation implemented. It is shown that the model that best describes data is a model taking into account the full covariance structure. An inference study is made in order to determine whether the growth rate of the five bacteria strains is the same. After applying a likelihood-ratio test to models with a full covariance structure, it is concluded that the specific growth rate is the same for all bacteria strains. This study highlights the importance of carrying out an explorative examination of residuals in order to make a correct parametrization of a model including the covariance structure. The ML method is shown to be a strong tool as it enables estimation of covariance parameters along with the other model parameters and it makes way for strong statistical tools for inference studies.

Robust estimation of covariance parameters in partial linear model for longitudinal data

For longitudinal data, the within-subject dependence structure and covariance parameters may be of practical and theoretical interests. The estimation of covariance parameters has received much attention and been studied mainly in the framework of generalized estimating equations (GEEs). The GEEs method, however, is sensitive to outliers. In this paper, an alternative set of robust generalized estimating equations for both the mean and covariance parameters are proposed in the partial linear model for longitudinal data. The asymptotic properties of the proposed estimators of regression parameters, non-parametric function and covariance parameters are obtained. Simulation studies are conducted to evaluate the performance of the proposed estimators under different contaminations. The proposed method is illustrated with a real data analysis.

Estimation of the Length Distribution of Marine Populations in the Gaussian-multinomial Setting using the Method of Moments

In this paper, the problem of estimation of the length distribution of marine populations in the Gaussian-multinomial model is considered. For the purpose of the mean and covariance parameter estimation, the method of moments estimators are developed. That is, minimum variance linear unbiased estimator for the mean frequency vector is derived and a consistent estimator for the covariance matrix of the length observations is presented. The usefulness of the proposed estimators is illustrated with an analysis of real cod length measurement data.

The Genetic Structure of a Maize Population: The Role of Dominance

ABSTRACTThe combined effects of dominance and inbreeding on covariances between relatives are still poorly understood in maize (Zea mays L.) populations. Our objectives were to address the following questions: (i) What is the importance of dominance in a maize synthetic? (ii) How does inbreeding affect the genetic variance among individuals in a maize synthetic. (iii) How do the covariance parameters compare between populations? (iv) How does breeding design impact estimators? We estimated covariance components for inbred relatives in the maize synthetic BSCB1(R)C13. Previous estimates of covariance parameters have been used to explain the ineffectiveness of inbred progeny selection in the stiff‐stalk population BS13. We found that the dominance variance was larger than the additive variance for grain yield, whereas the additive variance was larger than the dominance variance for all other traits. Negative estimates of the covariance between additive and homozygous dominance deviations were found for all traits with the exception of traits associated with reproductive maturity, suggesting a negative relationship between inbred and outbred performance. The correlation between genotypic values and breeding values was lower for grain yield than for any other trait. Our results were similar to previous results found in the stiff‐stalk maize population BS13, suggesting similarity in structure among populations.

Spatial designs and properties of spatial correlation: Effects on covariance estimation

In a spatial regression context, scientists are often interested in a physical interpretation of components of the parametric covariance function. For example, spatial covariance parameter estimates in ecological settings have been interpreted to describe spatial heterogeneity or “patchiness” in a landscape that cannot be explained by measured covariates. In this article, we investigate the influence of the strength of spatial dependence on maximum likelihood (ML) and restricted maximum likelihood (REML) estimates of covariance parameters in an exponential-with-nugget model, and we also examine these influences under different sampling designs—specifically, lattice designs and more realistic random and cluster designs—at differing intensities of sampling (n=144 and 361). We find that neither ML nor REML estimates perform well when the range parameter and/or the nugget-to-sill ratio is large—ML tends to underestimate the autocorrelation function and REML produces highly variable estimates of the autocorrelation function. The best estimates of both the covariance parameters and the autocorrelation function come under the cluster sampling design and large sample sizes. As a motivating example, we consider a spatial model for stream sulfate concentration.