Abstract: Inferring large-scale covariance matrices from sparse genomic data is an ubiquitous problem in bioinformatics. Clearly, the widely used standard covariance and correlation estimators are ill-suited for this purpose. As statistically efficient and computationally fast alternative we propose a novel shrinkage covariance estimator that exploits the Ledoit-Wolf (2003) lemma for analytic calculation of the optimal shrinkage intensity. Subsequently, we apply this improved covariance estimator (which has guaranteed minimum mean squared error, is well-conditioned, and is always positive definite even for small sample sizes) to the problem of inferring large-scale gene association networks. We show that it performs very favorably compared to competing approaches both in simulations as well as in application to real expression data.

Abstract: We propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data. The method aims at irregularly spaced longitudinal data, where the number of repeated measurements available per subject is small. In contrast, classical functional data analysis requires a large number of regularly spaced measurements per subject. We assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random (subject-specific) trajectory plus measurement errors. Basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories, and the estimation of the variance of the measurement errors. The eigenfunction basis is estimated from the data, and functional principal components score estimates are obtained by a conditioning step. This conditional estimation method is conceptually simple and straightforward to implement...

Abstract: This work considers a number of properties of space–time covariance functions and how these relate to the spatial-temporal interactions of the process. First, it examines how the smoothness away from the origin of a space–time covariance function affects, for example, temporal correlations of spatial differences. Models that are not smoother away from the origin than they are at the origin, such as separable models, have a kind of discontinuity to certain correlations that one might wish to avoid in some circumstances. Smoothness away from the origin of a covariance function is shown to follow from the corresponding spectral density having derivatives with finite moments. These results are used to obtain a parametric class of spectral densities whose corresponding space–time covariance functions are infinitely differentiable away from the origin and that allows for essentially arbitrary and possibly different degrees of smoothness for the process in space and time. Second, this work considers models that ...

Abstract: We consider the problem of estimating the covariance of two diffusion processes when they are observed only at discrete times in a non-synchronous manner. The modern, popular approach in the literature, the realized covariance estimator, which is based on (regularly spaced) synchronous data, is problematic because the choice of regular interval size and data interpolation scheme may lead to unreliable estimation. We propose a new estimator which is free of any `synchronization' processing of the original data, hence free of bias or other problems caused by it.

Abstract: SUMMARY Two asymptotic frameworks, increasing domain asymptotics and infill asymptotics, have been advanced for obtaining limiting distributions of maximum likelihood estimators of covariance parameters in Gaussian spatial models with or without a nugget effect. These limiting distributions are known to be different in some cases. It is therefore of interest to know, for a given finite sample, which framework is more appropriate. We consider the possibility of making this choice on the basis of how well the limiting distri butions obtained under each framework approximate their finite-sample counterparts. We investigate the quality of these approximations both theoretically and empirically, showing that, for certain consistently estimable parameters of exponential covariograms, approximations corresponding to the two frameworks perform about equally well. For those parameters that cannot be estimated consistently, however, the infill asymptotic approximation is preferable.

Abstract: Summary. Meteorological and environmental data that are collected at regular time intervals on a fixed monitoring network can be usefully studied combining ideas from multiple time series and spatial statistics, particularly when there are little or no missing data. This work investigates methods for modelling such data and ways of approximating the associated likelihood functions. Models for processes on the sphere crossed with time are emphasized, especially models that are not fully symmetric in space–time. Two approaches to obtaining such models are described. The first is to consider a rotated version of fully symmetric models for which we have explicit expressions for the covariance function. The second is based on a representation of space–time covariance functions that is spectral in just the time domain and is shown to lead to natural partially nonparametric asymmetric models on the sphere crossed with time. Various models are applied to a data set of daily winds at 11 sites in Ireland over 18 years. Spectral and space–time domain diagnostic procedures are used to assess the quality of the fits. The spectral-in-time modelling approach is shown to yield a good fit to many properties of the data and can be applied in a routine fashion relative to finding elaborate parametric models that describe the space–time dependences of the data about as well.

Abstract: The effects of positional uncertainty on the Gaussian probability computation for orbit conjunction are examined and an upper bound determined. Relative motion between two objects is assumed linear for a given encounter with time-invariant position covariance. A method is developed to assess the maximum probability for various satellite sizes, encounter geometries, and covariance sizes and shapes. The associated standard deviation then defines the boundary of probability dilution. The assertion is made that orbit positions should be sufficiently accurate to avoid this dilution region. This work shows how to calculate the upper bounds of probability by assuming worst-case covariance orientation and size. Power series approximations are developed for aspect ratios ranging from 1 to 50 to capture 99% of all conjunction possibilities. An analytical approximation is also given for an infinite aspect ratio to capture all possibilities. These expressions can be used as a simple pre-filter or to determine worst-case scenarios. Although desired, the actual covariances are not needed. What is needed is the ratio of major-to-minor axes of the projected combined covariance ellipse, the object sizes, and the relative distance at the point of closest approach.

Abstract: A method is described for producing explicit nonstationary spatial covariance functions that, for example, allows both the local geometric anisotropy and the degree of differentiability to vary spatially.

Abstract: [1] We present a novel approach to modeling stochastic multiphase flow problems, for example, nonaqueous phase liquid flow, in a heterogeneous subsurface medium with random soil properties, in particular, with randomly heterogeneous intrinsic permeability and soil pore size distribution. A stochastic numerical model for steady state water-oil flow in a random soil property field is developed using the Karhunen-Loeve moment equation (KLME) approach and is numerically implemented. An exponential model is adopted to define the constitutive relationship between phase relative permeability and capillary pressure. The log-transformed intrinsic permeability Y(x) and soil pore size distribution b(x) are assumed to be Gaussian random functions with a separable exponential covariance function. The perturbation part of these two log-transformed soil properties is then decomposed into an infinite series based on a set of orthogonal normal random variables {xn}. The phase pressure, capillary pressure, and phase mobility are decomposed by polynomial expansions and the perturbation method. Combining these expansions of Y(x), b(x) and dependent pressures, the steady state water-oil flow equations and corresponding boundary conditions are reformulated as a series of differential equations up to second order. These differential equations are solved numerically, and the solutions are directly used to construct moments of phase pressure and capillary pressure. We demonstrate the validity of the proposed KLME model by favorably comparing firstand second-order approximations to Monte Carlo simulations. The significant computational efficiency of the KLME approach over Monte Carlo simulation is also illustrated.

Abstract: Smoothed nonparametric estimates of the spectral density matrix at zero frequency have been widely used in econometric inference, because they can consistently estimate the covariance matrix of a partial sum of a possibly dependent vector process. When elements of the vector process exhibit long memory or antipersistence such estimates are inconsistent. We propose estimates which are still consistent in such circumstances, adapting automatically to memory parameters that can vary across the vector and be unknown.

Abstract: A bias-corrected estimator of noncentrality parameters of covariance structure models is discussed. The approach represents an application of the bootstrap methodology for purposes of bias correction, and utilizes the relation between average of resample conventional noncentrality parameter estimates and their sample counterpart. The bias-corrected bootstrap estimator can be viewed as a possible alternative to the traditionally used one that is presently implemented in popular covariance structure modeling programs, and is illustrated by means of a numerical example.

Abstract: Compact analytic expressions are derived for the mean, mutual intensity, and spatial covariance of the acoustic field forward propagated though a stratified ocean waveguide containing three-dimensional random surface and volume inhomogeneities. The inhomogeneities need not obey a stationary random process in space, can be of arbitrary composition and size relative to the wavelength, or can have large surface roughness and slope. The form of the mean forward field after multiple scattering through the random waveguide is similar to that of the incident field, except for a complex change in the horizontal wave number of each mode. This change describes attenuation and dispersion induced by the medium’s inhomogeneities, including potential mode coupling along the propagation path. The spatial covariance of the forward field between two receivers includes the accumulated effects of both coherent and incoherent multiple forward scattering through the random waveguide. It is expressed as a sum of modal covariance terms. Each term depends on the medium’s expected modal extinction densities as well as the covariance of its scattering properties, which potentially couple each mode to every other mode. Three-dimensional scattering effects can become important at ranges where the Fresnel width exceeds the cross-range coherence scale of the medium’s inhomogeneities.

Abstract: This article provides a general framework for Bayesian variable selection and covariance selection in a multivariate regression model with Gaussian errors. By variable selection we mean allowing certain regression coefficients to be zero. By covariance selection we mean allowing certain elements of the inverse covariance matrix to be zero. We estimate all the model parameters by model averaging using a Markov chain Monte Carlo simulation method. The methodology is illustrated by applying it to four real data sets. The effectiveness of variable selection and covariance selection in estimating the multivariate regression model is assessed by using four loss functions and four simulated data sets. Each of the simulated data sets is based on parameter estimates obtained from a corresponding real data set.

Abstract: We investigate the fluctuations around the average density profile in the weakly asymmetric exclusion process with open boundaries in the steady state. We show that these fluctuations are given, in the macroscopic limit, by a centered Gaussian field and we compute explicitly its covariance function. We use two approaches. The first method is dynamical and based on fluctuations around the hydrodynamic limit. We prove that the density fluctuations evolve macroscopically according to an autonomous stochastic equation, and we search for the stationary distribution of this evolution. The second approach, which is based on a representation of the steady state as a sum over paths, allows one to write the density fluctuations in the steady state as a sum over two independent processes, one of which is the derivative of a Brownian motion, the other one being related to a random path in a potential.

Abstract: In the cost function of three- or four-dimensional variational data assimilation, each term is weighted by the inverse of its associated error covariance matrix and the background error covariance matrix is usually much larger than the other covariance matrices. Although the background error covariances are traditionally normalized and parameterized by simple smooth homogeneous correlation functions, the covariance matrices constructed from these correlation functions are often too large to be inverted or even manipulated. It is thus desirable to find direct representations of the inverses of background error correlations. This problem is studied in this paper. In particular, it is shown that the background term can be written into ∫ dx|Dv(x)|2, that is, a squaredL2 norm of a vector differential operatorD, called the D-operator, applied to the field of analysis increment v(x). For autoregressive correlation functions, the D-operators are of finite orders. For Gaussian correlation functions, the D-operators are of infinite order. For practical applications, the Gaussian D-operators must be truncated to finite orders. The truncation errors are found to be small even when the Gaussian D-operators are truncated to low orders. With a truncated D-operator, the background term can be easily constructed with neither inversion nor direct calculation of the covariance matrix. D-operators are also derived for non-Gaussian correlations and transformed into non-isotropic forms.

Abstract: We consider goodness-of fit tests of symmetric stable distributions based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard symmetric stable distribution with the characteristic exponentƒ? estimated from the data. We treat ƒ? as an unknown parameter, but for theoretical simplicity we also consider the case that ƒ? is fixed. For estimation of parameters and the standardization of data we use maximum likelihood estimator (MLE) and an equivariant integrated squared error estimator (EISE) which minimizes the weighted integral. We derive the asymptotic covariance function of the characteristic function process with parameters estimated by MLE and EISE. For the case of MLE, the eigenvalues of the covariance function are numerically evaluated and asymptotic distribution of the test statistic is obtained using complex integration. Simulation studies show that the asymptotic distribution of the test statistics is very accurate. We also present a formula of the asymptotic covariance function of the characteristic function process with parameters estimated by an efficient estimator for general distributions.

Abstract: In continuation to an earlier work, we further consider the problem of robust estimation of a random vector (or signal), with an uncertain covariance matrix, that is observed through a known linear transformation and corrupted by additive noise with a known covariance matrix While, in the earlier work, we developed and proposed a competitive minimax approach of minimizing the worst-case mean-squared error (MSE) difference regret criterion, here, we study, in the same spirit, the minimum worst-case MSE ratio regret criterion, namely, the worst-case ratio (rather than difference) between the MSE attainable using a linear estimator, ignorant of the exact signal covariance, and the minimum MSE (MMSE) attainable by optimum linear estimation with a known signal covariance We present the optimal linear estimator, under this criterion, in two ways: The first is as a solution to a certain semidefinite programming (SDP) problem, and the second is as an expression that is of closed form up to a single parameter whose value can be found by a simple line search procedure We then show that the linear minimax ratio regret estimator can also be interpreted as the MMSE estimator that minimizes the MSE for a certain choice of signal covariance that depends on the uncertainty region We demonstrate that in applications, the proposed minimax MSE ratio regret approach may outperform the well-known minimax MSE approach, the minimax MSE difference regret approach, and the "plug-in" approach, where in the latter, one uses the MMSE estimator with an estimated covariance matrix replacing the true unknown covariance

Abstract: Purpose – Proposes a new covariance matrix robust estimator able to capture the correct orientation of the data and the large unconditional variance caused by occasional high volatility periods.Design/methodology/approach – Derives easy‐to‐compute estimates for the center and covariance matrix of a data set. The method finds the correct orientation of the data through a robust estimator and the variances through the classical sample covariance matrix.Findings – Simulation experiments confirm the good performance of the proposed estimator under e‐contaminated normal models and multivariate t‐distributions.Practical implications – Provides illustrations of the usefulness of this practical tool for the finance industry, in particular when constructing efficient frontiers. Shows that robust portfolios yield higher cumulative returns and possess more stable weights compositions.Originality/value – It provides an alternative estimator for the covariance matrix able to find a good fit for the bulk of the data as...

Abstract: This article considers a class of multiresolution tree-structured models that are spatially shifted versions of each other and proposes a new spatial-prediction method that averages over the optimal spatial predictors produced from members of this class of models. As a consequence, the resulting predicted surface is smooth, even when the predictors generated separately from individual multiresolution tree-structured models are not. We call the new predictor the multiresolution spatial (MURS) predictor and develop a computationally efficient algorithm for it. The algorithm can handle massive datasets even when some observations are missing. Moreover, the MURS predictor can be shown to be the minimum mean squared error predictor for a large class of covariance functions. A simulation example for massive datasets shows that the MURS method consistently outperforms two commonly used filtering methods. Total column ozone data remotely sensed from a satellite are analyzed using the new methodology.

Abstract: The objective of this study was to evaluate the efficiency of spatial statistical analysis in the selection of genotypes in a plant breeding program and, particularly, to demonstrate the benefits of the approach when experimental observations are not spatially independent. The basic material of this study was a yield trial of soybean lines, with five check varieties (of fixed effect) and 110 test lines (of random effects), in an augmented block design. The spatial analysis used a random field linear model (RFML), with a covariance function estimated from the residuals of the analysis considering independent errors. Results showed a residual autocorrelation of significant magnitude and extension (range), which allowed a better discrimination among genotypes (increase of the power of statistical tests, reduction in the standard errors of estimates and predictors, and a greater amplitude of predictor values) when the spatial analysis was applied. Furthermore, the spatial analysis led to a different ranking of the genetic materials, in comparison with the non-spatial analysis, and a selection less influenced by local variation effects was obtained.

Abstract: The principled statistical application of Gaussian random field models used in geostatistics has historically been limited to data sets of a small size. This limitation is imposed by the requirement to store and invert the covariance matrix of all the samples to obtain a predictive distribution at unsampled locations, or to use likelihood-based covariance estimation. Various ad hoc approaches to solve this problem have been adopted, such as selecting a neighborhood region and/or a small number of observations to use in the kriging process, but these have no sound theoretical basis and it is unclear what information is being lost. In this article, we present a Bayesian method for estimating the posterior mean and covariance structures of a Gaussian random field using a sequential estimation algorithm. By imposing sparsity in a well-defined framework, the algorithm retains a subset of “basis vectors” that best represent the “true” posterior Gaussian random field model in the relative entropy sense. This allows a principled treatment of Gaussian random field models on very large data sets. The method is particularly appropriate when the Gaussian random field model is regarded as a latent variable model, which may be nonlinearly related to the observations. We show the application of the sequential, sparse Bayesian estimation in Gaussian random field models and discuss its merits and drawbacks.

Abstract: This paper presents a detailed theoretical analysis of a recently introduced covariance matrix estimator, called the fixed point estimate (FPE). It plays a significant role in radar detection applications. This estimate is provided by the maximum likelihood estimation (MLE) theory when the non-Gaussian noise is modelled as a spherically invariant random process (SIRP). We study in details its properties: existence, uniqueness, unbiasedness, consistency and asymptotic distribution. We propose also an algorithm for its computation and prove the convergence of this numerical procedure. These results allow us to study the performance analysis of the adaptive CFAR radar detectors (GLRT-LQ, BORD, ...).

Abstract: It is frequently claimed in the literature that long-range dependence has equivalent formulations in the time domain and the frequency domain. Although many researchers understand that this is only "operationally true," i.e., it holds in cases of interest, many state this equivalence as a mathematical theorem. In particular, it is claimed as a theorem in the literature that if a covariance function decays like one over a fractional power of n, then the corresponding power spectral density tends to infinity at the origin. It is shown here that the power spectral density need not exist. Conversely, if the power spectral density exists and tends to infinity at the origin, it is shown here that the covariance may not have the claimed decay. To conclude, a new theorem is proved that gives sufficient conditions on the power spectral density to guarantee that a process is asymptotically second-order self-similar (ASOSS). This result is used to provide a counterexample to the claim in the literature that asymptotic second-order self-similarity implies long-range dependence

Abstract: Significant wave height, \(H_s\), is a measure of the variability of the ocean surface and is defined to be four times the standard deviation of the height of the ocean surface. In this paper, we present a methodology for modelling estimates of \(H_s\) over space and time, using data obtained from satellite measurements. These estimates can be thought of as a random surface in space which develops over time. For each fixed time and over some limited region in space, the field consisting of the \(H_s\) estimates may be considered stationary. Furthermore, it is reasonable to assume that the (natural) logarithms of the \(H_s\) estimates are normally distributed. Under these assumptions and for each fixed time, the marginal distribution over space of the random field of the logarithms of the \(H_s\) estimates is fitted by estimating its mean and covariance function, where the form of the covariance function is chosen to allow for correlation patterns at different spatial scales in the data. Both the mean and the covariance function of this model are allowed to be time dependent. A new methodology is developed for estimating the parameters of the chosen covariance structure. The proposed model is validated along the TOPEX-Poseidon satellite tracks by computing distributions of different quantities for the fitted model and comparing these to empirical estimates. Finally, the fitted model is used to compute the distribution of the global maximum over a certain region in the North Atlantic and to reconstruct the \(H_s\) field.

Abstract: . We consider classification of the realization of a multivariate spatial–temporal Gaussian random field into one of two populations with different regression mean models and factorized covariance matrices. Unknown means and common feature vector covariance matrix are estimated from training samples with observations correlated in space and time, assuming spatial–temporal correlations to be known. We present the first-order asymptotic expansion of the expected error rate associated with a linear plug-in discriminant function. Our results are applied to ecological data collected from the Lithuanian Economic Zone in the Baltic Sea.

Abstract: We investigate the fluctuations around the average density profile in the weakly asymmetric exclusion process with open boundaries in the steady state. We show that these fluctuations are given, in the macroscopic limit, by a centered Gaussian field and we compute explicitly its covariance function. We use two approaches. The first method is dynamical and based on fluctuations around the hydrodynamic limit. We prove that the density fluctuations evolve macroscopically according to an autonomous stochastic equation, and we search for the stationary distribution of this evolution. The second approach, which is based on a representation of the steady state as a sum over paths, allows one to write the density fluctuations in the steady state as a sum over two independent processes, one of which is the derivative of a Brownian motion, the other one being related to a random path in a potential.