Showing papers on "Covariance mapping published in 2007"

Separable approximations of space‐time covariance matrices

Abstract: SUMMARY Statistical modeling of space-time data has often been based on separable covariance functions, that is, covariances that can be written as a product of a purely spatial covariance and a purely temporal covariance. The main reason is that the structure of separable covariances dramatically reduces the number of parameters in the covariance matrix and thus facilitates computational procedures for large space-time data sets. In this paper, we discuss separable approximations of nonseparable space-time covariance matrices. Specifically, we describe the nearest Kronecker product approximation, in the Frobenius norm, of a space-time covariance matrix. The algorithm is simple to implement and the solution preserves properties of the space-time covariance matrix, such as symmetry, positive definiteness, and other structures. The separable approximation allows for fast kriging of large space-time data sets. We present several illustrative examples based on an application to data of Irish wind speeds, showing that only small differences in prediction error arise while computational savings for large data sets can be obtained. Copyright © 2007 John Wiley & Sons, Ltd.

Research Article Comparing covariance matrices: random skewers method compared to the common principal components model

Abstract: Comparisons of covariance patterns are becoming more common as interest in the evolution of relationships between traits and in the evolutionary phenotypic diversification of clades have grown. We present parallel analyses of covariance matrix similarity for cranial traits in 14 New World Monkey genera using the Random Skewers (RS), T-statistics, and Common Principal Components (CPC) approaches. We find that the CPC approach is very powerful in that with adequate sample sizes, it can be used to detect significant differences in matrix structure, even between matrices that are virtually identical in their evolutionary properties, as indicated by the RS results. We suggest that in many instances the assumption that population covariance matrices are identical be rejected out of hand. The more interesting and relevant question is, How similar are two covariance matrices with respect to their predicted evolutionary responses? This issue is addressed by the random skewers method described here.

Multivariate Spatial Modeling for Geostatistical Data Using Convolved Covariance Functions

Abstract: Soil pollution data collection typically studies multivariate measurements at sampling locations, e.g., lead, zinc, copper or cadmium levels. With increased collection of such multivariate geostatistical spatial data, there arises the need for flexible explanatory stochastic models. Here, we propose a general constructive approach for building suitable models based upon convolution of covariance functions. We begin with a general theorem which asserts that, under weak conditions, cross convolution of covariance functions provides a valid cross covariance function. We also obtain a result on dependence induced by such convolution. Since, in general, convolution does not provide closed-form integration, we discuss efficient computation. We then suggest introducing such specification through a Gaussian process to model multivariate spatial random effects within a hierarchical model. We note that modeling spatial random effects in this way is parsimonious relative to say, the linear model of coregionalization. Through a limited simulation, we informally demonstrate that performance for these two specifications appears to be indistinguishable, encouraging the parsimonious choice. Finally, we use the convolved covariance model to analyze a trivariate pollution dataset from California.

Knowledge-Aided Bayesian Detection in Heterogeneous Environments

Abstract: We address the problem of detecting a signal of interest in the presence of noise with unknown covariance matrix, using a set of training samples. We consider a situation where the environment is not homogeneous, i.e., when the covariance matrices of the primary and the secondary data are different. A knowledge-aided Bayesian framework is proposed, where these covariance matrices are considered as random, and some information about the covariance matrix of the training samples is available. Within this framework, the maximum a priori (MAP) estimate of the primary data covariance matrix is derived. It is shown that it amounts to colored loading of the sample covariance matrix of the secondary data. The MAP estimate is in turn used to yield a Bayesian version of the adaptive matched filter. Numerical simulations illustrate the performance of this detector, and compare it with the conventional adaptive matched filter

Analytic Properties and Covariance Functions for a New Class of Generalized Gibbs Random Fields

Abstract: Spartan spatial random fields (SSRFs) are generalized Gibbs random fields, equipped with a coarse-graining kernel that acts as a low-pass filter for the fluctuations. SSRFs are defined by means of physically motivated spatial interactions and a small set of free parameters (interaction couplings). This paper focuses on the fluctuation-gradient-curvature (FGC) SSRF model, henceforth referred to as FGC-SSRF. This model is defined on the Euclidean space R by means of interactions proportional to the squares of the field realizations, as well as their gradient and curvature. The permissibility criteria of FGC-SSRFs are extended by considering the impact of a finite-bandwidth kernel. It is proved that the FGC-SSRFs are almost surely differentiable in the case of finite bandwidth. Asymptotic explicit expressions for the Spartan covariance function are derived for d = 1 and d= 3; both known and new covariance functions are obtained depending on the value of the FGC-SSRF shape parameter. Nonlinear dependence of the covariance integral scale on the FGC-SSRF characteristic length is established, and it is shown that the relation becomes linear asymptotically. The results presented in this paper are useful in random field parameter inference, and in spatial interpolation of irregularly spaced samples.

Objectively mapping HF radar-derived surface current data using measured and idealized data covariance matrices

Abstract: [1] Surface currents measured by high-frequency radars are objectively mapped using covariance matrices computed from hourly surface current vectors spanning two years. Since retrievals of surface radial velocities are inherently gappy in space and time, the irregular density of surface current data leads to negative eigenvalues in the sample covariance matrix. The number and the magnitude of the negative eigenvalues depend on the degree of data continuity used in the matrix computation. In a region of 90% data coverage, the negative eigenvalues of the sample covariance matrix are small enough to be removed by adding a noise term to the diagonal of the matrix. The mapping is extended to regions of poorer data coverage by applying a smoothed covariance matrix obtained by spatially averaging the sample covariance matrix. This approach estimates a stable covariance matrix of surface currents for regions with the intermittent radar coverage. An additional benefit is the removal of baseline errors that often exist between two radar sites. The covariance matrices and the correlation functions of the surface currents are exponential in space rather than Gaussian, as is often assumed in the objective mapping of oceanographic data sets. Patterns in the decorrelation length scale provide the variabilities of surface currents and the insights on the influence of topographic features (bathymetry and headlands). The objective mapping approach presented herein lends itself to various applications, including the Lagrangian transport estimates, dynamic analysis through divergence and vorticity of current vectors, and statistical models of surface currents.

Individualization of gasoline samples by covariance mapping and gas chromatography/mass spectrometry.

Abstract: A set of 10 fresh (unevaporated) gasoline samples from a single metropolitan area were differentiated based on a covariance mapping method combined with a t-test statistic. The covariance matrix for each sample was calculated from the retention time−ion abundance data set obtained by gas chromatography/mass spectrometry analysis. Distance metrics were calculated between the covariance matrices from replicate analyses of the same sample and between the replicate analyses of different samples. The distance metric for the same-sample comparisons were shown to constitute a population significantly different from the distance metric for the different-sample comparisons. A power analysis was performed to estimate the number of analyses needed to discriminate between two samples while maintaining a probability of type II error, β, below 1%, e.g., a test power greater than 99%. Triplicate analyses of two gasoline samples was shown to be sufficient to discriminate between the two using a t-test, while keeping β < ...

First- and Second-Order Moments of the Normalized Sample Covariance Matrix of Spherically Invariant Random Vectors

Abstract: Under Gaussian assumptions, the sample covariance matrix (SCM) is encountered in many covariance based processing algorithms. In case of impulsive noise, this estimate is no more appropriate. This is the reason why when the noise is modeled by spherically invariant random vectors (SIRV), a natural extension of the SCM is extensively used in the literature: the well-known normalized sample covariance matrix (NSCM), which estimates the covariance of SIRV. Indeed, this estimate gets rid of a fluctuating noise power and is widely used in radar applications. The aim of this paper is to derive closed-form expressions of the first- and second-order moments of the NSCM

Testing for the Mean of Random Curves: A Penalization Approach

Abstract: Let X1,...,Xn be an i.i.d. sample of random curves, viewed as Hilbert space valued random elements, with mean curve m. An asymptotic test of m = m0 vs m ≠ m0 is proposed, when m0 is a fixed known function. The test statistics converges under very mild assumptions and relies on the pseudo-inversion of the covariance operator (leading to a non standard inverse problem). The power against local alternatives is investigated.

Bounds for the covariance of functions of infinite variance stable random variables with applications to central limit theorems and wavelet-based estimation

Abstract: We establish bounds for the covariance of a large class of functions of infinite variance stable random variables, including unbounded functions such as the power function and the logarithm. These bounds involve measures of dependence between the stable variables, some of which are new. The bounds are also used to deduce the central limit theorem for unbounded functions of stable moving average time series. This result extends the earlier results of Tailen Hsing and the authors on central limit theorems for bounded functions of stable moving averages. It can be used to show asymptotic normality of wavelet-based estimators of the self-similarity parameter in fractional stable motions.

On the a priori estimation of collocation error covariance functions: a feasibility study

Abstract: SUMMARY Error covariance estimates are necessary information for the combination of solutions resulting from different kinds of data or methods, or for the assimilation of new results in already existing solutions. Such a combination or assimilation process demands proper weighting of the data, in order for the combination to be optimal and the error estimates of the results realistic. One flexible method for the gravity field approximation is least-squares collocation leading to optimal solutions for the predicted quantities and their error covariance estimates. The drawback of this method is related to the current ability of computers in handling very large systems of linear equations produced by an equally large amount of available input data. This problem becomes more serious when error covariance estimates have to be simultaneously computed. Using numerical experiments aiming at revealing dependencies between error covariance estimates and given features of the input data we investigate the possibility of a straightforward estimation of error covariance functions exploiting known characteristics of the observations. The experiments using gravity anomalies for the computation of geoid heights and the associated error covariance functions were conducted in the Arctic region north of 64° latitude. The correlation between the known features of the data and the parameters variance and correlation length of the computed error covariance functions was estimated using multiple regression analysis. The results showed that a satisfactory a priori estimation of these parameters was not possible, at least in the area considered.

Bayesian Covariance Matrix Estimation with Non-Homogeneous Snapshots

Abstract: We address the problem of estimating the covariance matrix Mp of an observation vector, using K groups of training samples {Zk}Kk=1, of respective size Lk, whose covariance matrices Mk may differ from Mp. A Bayesian model is formulated where we assume that Mp and the matrices Mk are random, with some prior distribution. Within this framework, we derive the minimum mean-square error (MMSE) estimator of Mp which is implemented using a Gibbs-sampling strategy. Moreover, we consider simpler estimators based on a weighted sum of the sample covariance matrices of Zk. We derive an expression for the weights that result in minimum mean square error (MSE), within this class of estimators. Numerical simulations are presented to illustrate the performances of the different estimation schemes.

Self averaging of normalized spectral functions of some product of independent random matrices of growing dimension

Abstract: The problem of the spectral analysis of random matrizant (the product of random matrices), which is the solution of a recurrent system of equations with random coefficients, or the system of stochastic linear differential equations of growing dimension is considered. The growing dimension means that the dimension of matrices and the number of matrices have the same order and both (dimension and number of matrices) tend to infinity. In this paper we give new method of deriving self averaging property for the V.I.C.T.O.R.I.A.-transform of normalized spectral functions (n.s.f.) of random matrizant or the product of independent random matrices. We apply the REFORM method for normalized spectral functions of this matrizant, where random matrices belong to the domain of attraction of the Strong Circular Law.

Cramér-Rao Bound and Maximum Likelihood Estimation of Covariance Matrices With Non-Homogeneous Snapshots

Abstract: We consider the problem of estimating the covariance matrix RT of an observation vector, using K groups of snapshots Zk = [zk.1 ... zk.Lk], of respective size Lk, whose covariance matrices Rk are randomly distributed around Rt, and hence are different from Rt. The Cramer-Rao bound (CRB) for estimation of Rt is derived as well as its maximum likelihood estimator (MLE). We illustrate the behavior of the CRB in the two opposite cases, namely K = 1 where all snapshots share a common covariance matrix, and Lk = 1 where each snapshot has a different covariance matrix. We also discuss the influence of the degree of heterogeneity on the estimation performance.

Covariance transformations for flame-propagation modeling

Abstract: The covariance transformation used in earlier theoretical studies of turbulent-flame and isothermal-front propagation is defined and mathematically derived. This transformation provides the covariance associated with fluctuations of a turbulent scalar or vector-component variable in terms of the statistical properties of another fluctuating scalar or vector-component variable when the Fourier transforms of the two variables in transverse-wave number and frequency space are linearly dependent. Key words: Covariance, turbulence, turbulent combustion, front propagation, flame propagation