Showing papers on "Covariance mapping published in 1999"

Construction of correlation functions in two and three dimensions

Abstract: This article focuses on the construction, directly in physical space, of simply parametrized covariance functions for data-assimilation applications. A self-contained, rigorous mathematical summary of relevant topics from correlation theory is provided as a foundation for this construction. Covariance and correlation functions are defined, and common notions of homogeneity and isotropy are clarified. Classical results are stated, and proven where instructive. Included are smoothness properties relevant to multivariate statistical-analysis algorithms where wind/wind and wind/mass correlation models are obtained by differentiating the correlation model of a mass variable. the Convolution Theorem is introduced as the primary tool used to construct classes of covariance and cross-covariance functions on three-dimensional Euclidean space R3. Among these are classes of compactly supported functions that restrict to covariance and cross-covariance functions on the unit sphere S2, and that vanish identically on subsets of positive measure on S2. It is shown that these covariance and cross-covariance functions on S2, referred to as being space-limited, cannot be obtained using truncated spectral expansions. Compactly supported and space-limited covariance functions determine sparse covariance matrices when evaluated on a grid, thereby easing computational burdens in atmospheric data-analysis algorithms. Convolution integrals leading to practical examples of compactly supported covariance and cross-covariance functions on R3 are reduced and evaluated. More specifically, suppose that gi and gj are radially symmetric functions defined on R3 such that gi(x) = 0 for |x| > di and gj(x) = 0 for |xv > dj, O di + dj and |x - y| > 2di, respectively, Additional covariance functions on R3 are constructed using convolutions over the real numbers R, rather than R3. Families of compactly supported approximants to standard second- and third-order autoregressive functions are constructed as illustrative examples. Compactly supported covariance functions of the form C(x,y) := Co(|x - y|), x,y ∈ R3, where the functions Co(r) for r ∈ R are 5th-order piecewise rational functions, are also constructed. These functions are used to develop space-limited product covariance functions B(x, y) C(x, y), x, y ∈ S2, approximating given covariance functions B(x, y) supported on all of S2 × S2.

Classes of nonseparable, spatio-temporal stationary covariance functions

Abstract: Suppose that a random process Z(s;t), indexed in space and time, has spatio-temporal stationary covariance C(h;u), where h ∈ ℝd (d ≥ 1) is a spatial lag and u ∈ ℝ is a temporal lag. Separable spatio-temporal covariances have the property that they can be written as a product of a purely spatial covariance and a purely temporal covariance. Their ease of definition is counterbalanced by the rather limited class of random processes to which they correspond. In this article we derive a new approach that allows one to obtain many classes of nonseparable, spatio-temporal stationary covariance functions and fit several such classes to spatio-temporal data on wind speed over a region in the tropical western Pacific ocean.

Stochastic significance of peaks in the least-squares spectrum

Abstract: The least-squares spectral analysis method is reviewed, with emphasis on its remarkable property to accept time series with an associated, fully populated covariance matrix. Two distinct cases for the input covariance matrix are examined: (a) it is known absolutely (a-priori variance factor known); and (b) it is known up to a scale factor (a-priori variance factor unknown), thus the estimated covariance matrix is used. For each case, the probability density function that underlines the least-squares spectrum is derived and criteria for the statistical significance of the least-squares spectral peaks are formulated. It is shown that for short series (up to about 150 values) with an estimated covariance matrix (case b), the spectral peaks must be stronger to be statistically significant than in the case of a known covariance matrix (case a): the shorter the series and the lower the significance level, the higher the difference becomes. For long series (more than about 150 values), case (b) converges to case (a) and the least-squares spectrum follows the beta distribution. The results of this investigation are formulated in two new theorems.

Bayesian spatial prediction

Abstract: This paper presents a complete Bayesian methodology for analyzing spatial data, one which employs proper priors and features diagnostic methods in the Bayesian spatial setting. The spatial covariance structure is modeled using a rich class of covariance functions for Gaussian random fields. A general class of priors for trend, scale, and structural covariance parameters is considered. In particular, we obtain analytic results that allow easy computation of the predictive distribution for an arbitrary prior on the parameters of the covariance function using importance sampling. The computations, as well as model diagnostics and sensitivity analysis, are illustrated with a set of precipitation data.

Stereological estimation of covariance using linear dipole probes.

Abstract: Classical stereology is capable of quantifying the total amount or ‘density’ of a geometrical feature from sampled information, but gives no information about the local spatial arrangement of the feature. However, stereological methods also exist for quantifying the ‘local’ spatial architecture of a 3D microstructure from sampled information. These methods are capable of quantifying, in a statistical manner, the spatial interaction in a structure over a range of distances. One of the key quantities used in a second-order analysis of a volumetric feature is the set covariance. Previous applications of covariance analysis have been ‘model-based’ and relied upon computerized image analysis. In this paper we describe a new ‘design-based’ manual method, known as linear dipole probes, that is suitable for estimating covariance from microscopic images. The approach is illustrated in practice on vertically sectioned lung tissue. We find that only relatively sparse sampling per animal is required to obtain estimates of covariance that have low inter-animal variability.

Generalized covariance functions associated with the laplace equation and their use in interpolation and inverse problems

Abstract: This work examines which generalized covariance function when used in the stochastic approach produces the flattest possible estimate of an unknown function that is consistent with the data. Such an estimate is the plainest possible continuous function, thus in a sense eliminating details that are irrelevant or unsupported by data. The answer is found from the solution of the following variational problem: Determine the function that reproduces the data, has the smallest gradient (in the square norm sense), and has a gradient that vanishes at large distances from the observations. The generalized covariance functions are shown to be the Green's functions for the free-space Laplace equation: the linear distance, in one dimension; the logarithmic distance in two dimensions; and the inverse distance in three dimensions. It is demonstrated that they are appropriate covariance functions for intrinsic random fields, a modification is proposed to facilitate numerical implementation, and a couple of examples are presented to illustrate the applicability of the methodology.

The characterization problem for isotropic covariance functions

Abstract: Isotropic covariance functions are successfully used to model spatial continuity in a multitude of scientific disciplines. Nevertheless, a satisfactory characterization of the class of permissible isotropic covariance models has been missing. The intention of this note is to review, complete, and extend the existing literature on the problem. As it turns out, a famous conjecture of Schoenberg (1938) holds true: any measurable, isotropic covariance function on ℝ d (d ≥ 2) admits a decomposition as the sum of a pure nugget effect and a continuous covariance function. Moreover, any measurable, isotropic covariance function defined on a ball in ℝd can be extended to an isotropic covariance function defined on the entire space ℝ d .

Stochastic three‐mode models for mean and covariance structures

Abstract: With three-mode models, the three modes are analysed simultaneously. Examples are the analysis of multitrait-multimethod data where the modes are traits, methods and subjects, and the analysis of multivariate longitudinal data where the modes are variables, occasions and subjects. If we consider the subjects mode as random, and the other modes as fixed, such data can be analysed using stochastic three-mode models. Three-mode factor analysis models and composite direct product models are special cases, but they are models for the covariance structure only. Stochastic three-mode models for mean and covariance structures are presented, and the identification, estimation and interpretation of the model parameters are discussed. Interpretation is facilitated by introducing a new terminology and by considering various special cases. Analyses of real data from the field of economic psychology serve as an illustration.

Angle-resolved covariance mapping of spatially-aligned in an intense picosecond laser field

Abstract: Angle-resolved covariance mapping has been applied to measure the angular distributions of - and - ion pairs resulting from double ionization of using picosecond-duration, linearly-polarized laser pulses of intensity is spatially aligned along the laser polarization vector much more efficiently than . The field-induced dipole moment of the dication is calculated to be 21% more than that of the cation. The following sequence of events is inferred upon immersion of in the intense field: ionization and double ionization spatial alignment dissociation.

An efficient method for determining the significance of covariance estimates

Abstract: A technique is presented for determining if an estimate of covariance (or correlation) of two time series is statistically significantly different from zero. The technique makes no assumptions about the spectrum or the probability distribution of either time series. It is more efficient than the method of Yamazaki and Osborn and Fleury and Lueck by several factors of 10.

Approximation of harmonic covariance functions on the sphere by non-harmonic locally supported functions

Abstract: Three methods to construct positive definite functions with compact support for the approximation of general geophysical harmonic covariance functions are presented. The theoretical background is given and simulations carried out, for three types of covariance functions associated with the determination of the anomalous gravity potential from gravity anomalies. The results are compared with those of the finite covariance function of Sanso and Schuh (1987).

The interaction of titanium-carbon cluster fragments

Abstract: Covariance and correlation coefficient mapping techniques are utilized to investigate the formation mechanism of titanium carbon clusters in a laser vaporization source, and also to elucidate the details of fragmentation of these clusters under excitation with an intense femtosecond laser pulse (∼1015 W/cm2 at 120 fs). This study, and a previous investigation of the Coulomb explosion of ammonia clusters, demonstrates the potential which covariance mapping has in discerning the details of cluster formation and fragmentation processes. Particular attention is paid to the titanium metallocarbohedrene (Met-Car), Ti8C12.

Multichannel signal processing using spatial rank covariance matrices.

Abstract: This paper addresses the problem of estimating the covariance matrix reliably when the assumptions, such as Gaussianity, on the probabilistic nature of multichannel data do not necessarily hold. Multivariate spatial sign and rank functions, which are generalizations of univariate sign and centered rank, are introduced. Furthermore, spatial rank covariance matrix and spatial Kendall’s tau covariance matrix based new robust covariance matrix estimators are proposed. Efficiency of the estimators is discussed and their qualitative robustness is demonstrated using a empirical influence function concept. The use and reliable performance of the proposed methods is demonstrated in color image filtering, image analysis, principal component analysis and blind source separation tasks.

A Simulation Study To Compare Various Covariance Adjustment Techniques

Abstract: A common procedure when combining two multivariate unbiased estimates (or forecasts) is the covariance adjustment technique (CAT). Here the optimal combination weights depend on the covariance structure of the estimators. In practical applications, however, this covariance structure is hardly ever known and, thus, has to be estimated. An effect of this drawback may be that the theoretically best method is no longer the best. In a simulation study (using normally distributed data) three different variants of CAT are compared with respect to their accuracy. These variants are different in the portion of the covariance structure that is estimated. We characterize which variant is appropriate in different situations and quantify the gains and losses that occur.