A comparison of sampling grids,cut-off distance and type of residuals in parametric variogram estimation

In spatial statistics, the correct identification of a variogram model when fitted to an empirical variogram depends on many factors. Here, simulation experiments show fitting based on the variogram cloud is preferable to that based on Matheron's and Cressie–Hawkins empirical variogram estimators. For correct model specification, a number of models should be fitted to the empirical variogram using a grid of cut-off values, and recommendations are given for best choice. A design where roughly half the maximum distance between points equals the practical range works well for correct variogram identification of any model, with varying nugget sizes and sample sizes.     

Least squares variogram fitting by spatial subsampling

Summary. Least squares methods are popular for fitting valid variogram models to spatial data. The paper proposes a new least squares method based on spatial subsampling for variogram model fitting. We show that the method proposed is statistically efficient among a class of least squares methods, including the generalized least squares method. Further, it is computationally much simpler than the generalized least squares method. The method produces valid variogram estimators under very mild regularity conditions on the underlying random field and may be applied with different choices of the generic variogram estimator without analytical calculation. An extension of the method proposed to a class of spatial regression models is illustrated with a real data example. Results from a simulation study on finite sample properties of the method are also reported.     

On strict positive definiteness of product and product-sum covariance models

Although positive definiteness is a sufficient condition for a function to be a covariance, the stronger strict positive definiteness is important for many applications, especially in spatial statistics, since it ensures that the kriging equations have a unique solution. In particular, spatial-temporal prediction has received a lot of attention, hence strictly positive definite spatial-temporal covariance models (or equivalently strictly conditionally negative definite variogram models) are needed.In this paper the necessary and sufficient condition for the product and the product-sum space-time covariance models to be strictly positive definite (or the variogram function to be strictly conditionally negative definite) is given. In addition it is shown that an example appeared in the recent literature which purports to show that product-sum covariance functions may be only semi-definite is itself invalid. Strict positive definiteness of the sum of products model is also discussed.     

On nonparametric variogram estimation

A critical step for geostatistical prediction is estimation of variogram from the data. One of the popular methods estimating variogram is a smoothed version of classical nonparametric variogram estimator. In this paper we investigate its theoretical and empirical properties to provide useful information for using it. The main results are based on asymptotic theories (i.e., risk and central limit theorem) under nearly infill domain sampling. Simulation is also employed to make our points.     

On asymptotic properties of the mark variogram estimator of a marked point process

The mark variogram [Cressie, 1993. Statistics for Spatial Data. Wiley, New York] is a useful tool to analyze data from marked point processes. In this paper, we investigate the asymptotic properties of its estimator. Our main findings are that the sample mark variogram is a consistent estimator for the true mark variogram and is asymptotically normal under some mild conditions. These results hold for both the geostatistical marking case (i.e., the case where the marks and points are independent) and the non-geostatistical marking case (i.e., the case where the marks and points are dependent). As an application we develop a general test for spatial isotropy and study our methodology through a simulation study and an application to a data set on long leaf pine trees.     

Constructing and fitting models for cokriging and multivariable spatial prediction

We consider best linear unbiased prediction for multivariable data. Minimizing mean-squared-prediction errors leads to prediction equations involving either covariances or variograms. We discuss problems with multivariate extensions that include the construction of valid models and the estimation of their parameters. In this paper, we develop new methods to construct valid crossvariograms, fit them to data, and then use them for multivariable spatial prediction, including cokriging. Crossvariograms are constructed by explicitly modeling spatial data as moving averages over white noise random processes. Parameters of the moving average functions may be inferred from the variogram, and with few additional parameters, crossvariogram models are constructed. Weighted least squares is then used to fit the crossvariogram model to the empirical crossvariogram for the data. We demonstrate the method for simulated data, and show a considerable advantage of cokriging over ordinary kriging.     

A ROLE FOR VARIOGRAMS

In time series texts and journals, variograms are mentioned seldom, if at all. The autocovariance function is preferred. However there are situations where the variogram can be estimated with moderate precision but the autocovariance function cannot, because the variance of the process is not well known. If the problem to be solved does not require the process variance for its solution then it is generally more straightforward to use the variogram rather than the autocovariance function in solving this problem.     

A Kernel Variogram Estimator for Clustered Data

Abstract.  The variogram provides an important method for measuring the dependence of attribute values between spatial locations. Suppose that the nature of the sampling process leads to the presence of clustered data; it would be advisable to use a variogram estimator that aims to adjust for clustering of samples. In this setting, the use of a non-parametric weighted estimator, obtained by considering an inverse weight to a given neighbourhood density combined with the kernel method, seems to have a satisfactory behaviour in practice. This paper pursues a theoretical study of the cluster robust estimator, by proving that it is asymptotically unbiased as well as consistent and by providing criteria for selection of the bandwidth parameter and the neighbourhood radius. Numerical studies are also included to illustrate the performance of the considered estimator and the suggested approaches.     

Analysis of the Covariance Structure in Manufactured Parts

We discuss the use of variograms for covariance modeling under the Kriging model to assess tolerances on manufactured parts. The variogram is very informative about the spatial dependence and it is favored by researchers in the choice of a correlation function. It may give evidence of anisotropy and of nugget effect too. In this paper, various variogram estimators from literature are used for comparing and evaluating their properties in assessing the surface error of three workpieces manufactured with different machining precisions. Contrary to the common engineering belief, the variograms give evidence of both technological signatures and systematic errors of the measurement machines.     

EXPERIMENTAL DESIGN FOR CONTINUOUS PROCESSES

Measurements taken on a continuous process (e.g. production of chemicals, ore processing, steel production) often exhibit autocorrelation. Such correlation between successive measurements can be avoided only by taking measurements at widely separated times, but this delays the results. We discuss the impact of this correlation on experimental designs which for a continuous process must include the sequence in which treatment combinations are applied and the spacing between samples. The variogram of the process is a convenient Summary of the correlation. Formulae for the variance of the mean of measurements taken on a continuous process and of contrasts calculated from such measurements are simple expressions in terms of the variogram. These formulae allow a simple approach to experimental designs that avoid unnecessary delays.     

Statistical comparisons of methods for interpolating the output of a numerical air quality model

This paper compares Models-3/Community Multiscale Air Quality (CMAQ) outputs at multiple resolutions by interpolating from coarse resolution to fine resolution and analyzing the interpolation difference. Spatial variograms provide a convenient way to investigate the spatial character of interpolation differences and, importantly, to distinguish between naive (nearest neighbor) interpolation and bilinear interpolation, which takes a weighted average of four neighboring cells. For example, when the higher resolution is three times the lower, the variogram of the difference between naive interpolation of the lower resolution output and the higher resolution output shows a depression at every third lag. This phenomenon is related to the blocky nature of naive interpolation and demonstrates the inferiority of naive interpolation to bilinear interpolation in a way that pixelwise comparisons cannot. Theoretical investigations show when one can expect to observe this periodic depression in the variogram of interpolation differences. Naive interpolation is in fact used widely in a number of settings; our results suggest that it should be routinely replaced by bilinear interpolation.     

Estimation of causal continuous‐time autoregressive moving average random fields

We estimate model parameters of Lévy‐driven causal continuous‐time autoregressive moving average random fields by fitting the empirical variogram to the theoretical counterpart using a weighted least squares (WLS) approach. Subsequent to deriving asymptotic results for the variogram estimator, we show strong consistency and asymptotic normality of the parameter estimator. Furthermore, we conduct a simulation study to assess the quality of the WLS estimator for finite samples. For the simulation, we utilize numerical approximation schemes based on truncation and discretization of stochastic integrals and we analyze the associated simulation errors in detail. Finally, we apply our results to real data of the cosmic microwave background.     

New compactly supported spatiotemporal covariance functions from SPDEs

Potential theory and Dirichlet’s priciple constitute the basic elements of the well-known classical theory of Markov processes and Dirichlet forms. This paper presents new classes of fractional spatiotemporal covariance models, based on the theory of non-local Dirichlet forms, characterizing the fundamental solution, Green kernel, of Dirichlet boundary value problems for fractional pseudodifferential operators. The elements of the associated Gaussian random field family have compactly supported non-separable spatiotemporal covariance kernels admitting a parametric representation. Indeed, such covariance kernels are not self-similar but can display local self-similarity, interpolating regular and fractal local behavior in space and time. The associated local fractional exponents are estimated from the empirical log-wavelet variogram. Numerical examples are simulated for illustrating the properties of the space–time covariance model class introduced.     

A test of spatial independence

Mardia (1980) proposed a bi-directional test of spatial independence based upon the sample variogram which has a X2-distribution under the null hypothesis. We give here a modified test whichleads to improvement to the X2-distribution approximation to the size. We introduce some other tests including a four-directional extension, and analogous tests based on the sample correlation coefficient. Their power is compared with the modified bi-directional test for a doubly geometric alternative, and this limited power study indicates that the modified bi-directional test is more powerful.     

Modelling of connected processes

The problem of comparing, contrasting and combining information from different sets of data is an enduring one in many practical applications of statistics. A specific problem of combining information from different sources arose in integrating information from three different sets of data generated by three different sampling campaigns at the input stage as well as at the output stage of a grey-water treatment process. For each stage, a common process trend function needs to be estimated to describe the input and output material process behaviours. Once the common input and output process models are established, it is required to estimate the efficiency of the grey-water treatment method. A synthesized tool for modelling different sets of process data is created by assembling and organizing a number of existing techniques: (i) a mixed model of fixed and random effects, extended to allow for a nonlinear fixed effect, (ii) variogram modelling, a geostatistical technique, (iii) a weighted least squares regression embedded in an iterative maximum-likelihood technique to handle linear/nonlinear fixed and random effects and (iv) a formulation of a transfer-function model for the input and output processes together with a corresponding nonlinear maximum-likelihood method for estimation of a transfer function. The synthesized tool is demonstrated, in a new case study, to contrast and combine information from connected process models and to determine the change in one quality characteristic, namely pH, of the input and output materials of a grey-water filtering process.     

A regression approach for estimating the parameters of the covariance function of a stationary spatial random process

We consider the problem of estimating the parameters of the covariance function of a stationary spatial random process. In spatial statistics, there are widely used parametric forms for the covariance functions, and various methods for estimating the parameters have been proposed in the literature. We develop a method for estimating the parameters of the covariance function that is based on a regression approach. Our method utilizes pairs of observations whose distances are closest to a value h>0$h>0$ which is chosen in a way that the estimated correlation at distance h is a predetermined value. We demonstrate the effectiveness of our procedure by simulation studies and an application to a water pH data set. Simulation studies show that our method outperforms all well-known least squares-based approaches to the variogram estimation and is comparable to the maximum likelihood estimation of the parameters of the covariance function. We also show that under a mixing condition on the random field, the proposed estimator is consistent for standard one parameter models for stationary correlation functions.     

LOCATION OF THE OPTIMAL SAMPLING POINT FOR THE QUALITY ASSESSMENT OF CONTINUOUS STREAMS

The paper makes an appraisal of the most appropriate sampling point for situations where a single sample must be used to estimate the mean flow of a continuous stream during a set time interval. Taking ‘optimal’ to mean the point at which the estimation error variance is minimised, optimal sampling locations are obtained for constant, linear and exponential flow rates when the process variogram is assumed linear or exponential. Numerical results illustrate the significance of failing to sample at the optimal point.     

The practice of mining gegstatistics in 1980

The scope of the term “Mining Geostatistics” is defined in the context of a brief historical review of the topic. Assumptions andtools adapted from probability theory are applied first to linear estimation problems leading to the iging” estimator and then shown capable of extension to non- ationary and non-linear situa­tions, in which conditional simulation, conditional distributions and “disjunctive Kriging” become applicable. Directions for further research work are indicated regarding variogram identification and modelling of random functions where stationarity or linearity may not hold. The brief review suggests that the mutual benefits of practice and theory will continue to develop.     

Variance function estimation of a one-dimensional nonstationary process

We propose a flexible nonparametric estimation of a variance function from a one-dimensional process where the process errors are nonstationary and correlated. Due to nonstationarity a local variogram is defined, and its asymptotic properties are derived. We include a bandwidth selection method for smoothing taking into account the correlations in the errors. We compare the proposed difference-based nonparametric approach with Anderes and Stein(2011)’s local-likelihood approach. Our method has a smaller integrated MSE, easily fixes the boundary bias, and requires far less computing time than the likelihood-based method.     

Construction of non-Gaussian random fields with any given correlation structure

A positive definite function can be thought of as the covariance function of a Gaussian random field, according to the celebrated Kolmogorov existence theorem. A question of great theoretical and practical interest is: how could one construct a non-Gaussian random field with the given positive definite function as its covariance function? In this paper we demonstrate a novel and simple method for constructing many such non-Gaussian random fields, with the corresponding finite-dimensional distributions identified. Also, we show how to construct a non-Gaussian random field with a given negative definite function as its variogram.