Michael L. Stein

Large sample properties of simulations using latin hypercube sampling

Latin hypercube sampling (McKay, Conover, and Beckman 1979) is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables. The asymptotic variance of such an estimate is obtained. The estimate is also shown to be asymptotically normal. Asymptotically, the variance is less than that obtained using simple random sampling, with the degree of variance reduction depending on the degree of additivity in the function being integrated. A method for producing Latin hypercube samples when the components of the input variables are statistically dependent is also described. These techniques are applied to a simulation of the performance of a printer actuator.

A Bayesian Analysis of Kriging

This article is concerned with predicting for Gaussian random fields in a way that appropriately deals with uncertainty in the covariance function. To this end, we analyze the best linear unbiased prediction procedure within a Bayesian framework. Particular attention is paid to the treatment of parameters in the covariance structure and their effect on the quality, both real and perceived, of the prediction. These ideas are implemented using topographical data from Davis.

Space–Time Covariance Functions

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 are asymmetric in space–time; the covariance between site x at time t and site y at time s is different than the covariance between site x at time s and site y at time t. A general approach is described for generating asymmetric models from symmetric models by taking derivatives. Finally, the implications of a Markov assumption in time on space–time covariance functions for Gaussian processes are examined, and an explicit characterization of all such continuous covariance functions is given. Several of the new models described in this work are applied to wind data from Ireland.

Spatial sampling design for prediction with estimated parameters

We study spatial sampling design for prediction of stationary isotropic Gaussian processes with estimated parameters of the covariance function. The key issue is how to incorporate the parameter uncertainty into design criteria to correctly represent the uncertainty in prediction. Several possible design criteria are discussed that incorporate the parameter uncertainty. A simulated annealing algorithm is employed to search for the optimal design of small sample size and a two-step algorithm is proposed for moderately large sample sizes. Simulation results are presented for the Matérn class of covariance functions. An example of redesigning the air monitoring network in EPA Region 5 for monitoring sulfur dioxide is given to illustrate the possible differences our proposed design criterion can make in practice.

NONSTATIONARY COVARIANCE MODELS FOR GLOBAL DATA

With the widespread availability of satellite-based instruments, many geophysical processes are measured on a global scale and they often show strong nonstationarity in the covariance structure. In this paper we present a flexible class of parametric covariance models that can capture the nonstationarity in global data, especially strong dependency of covariance structure on latitudes. We apply the Discrete Fourier Transform to data on regular grids, which enables us to calculate the exact likelihood for large data sets. Our covariance model is applied to global total column ozone level data on a given day. We discuss how our covariance model compares with some existing models.

An approach to producing space-time covariance functions on spheres

For space–time processes on global or large scales, it is critical to use models that respect the Earths spherical shape. The covariance functions of such processes should be not only positive definite on sphere × time, but also capable of capturing the dynamics of the processes well. We develop space–time covariance functions on sphere × time that are flexible in producing space–time interactions, especially space–time asymmetries. Our idea is to consider a sum of independent processes in which each process is obtained by applying a first-order differential operator to a fully symmetric process on sphere × time. The resulting covariance functions can produce various types of space–time interactions and give different covariance structures along different latitudes. Our approach yields explicit expressions for the covariance functions, which has great advantages in computation. Moreover, it applies equally well to generating asymmetric space–time covariance functions on flat or other spatial domains. We study various characteristics of our new covariance functions, focusing on their space–time interactions. We apply our model to a dataset of total column ozone levels in the Northern hemisphere.

Fast and Exact Simulation of Fractional Brownian Surfaces

Fractional Brownian surfaces are commonly used as models for landscapes and other physical processes in space. This work shows how to simulate fractional Brownian surfaces on a grid efficiently and exactly by embedding them in a periodic Gaussian random field and using the fast Fourier transform. Periodic embeddings are given that are proven to yield positive definite covariance functions and hence yield exact simulations for all possible densities of the simulation grid. Numerical results show these embeddings can sometimes be made more efficient in practice. Further numerical results show how the ideas developed for simulating fractional Brownian surfaces can be used for simulating other Gaussian random fields. The simulation methodology is used to study the behavior of a simple estimator of the parameters of a fractional Brownian surface.

Trends in column ozone based on TOMS data: Dependence on month, latitude, and longitude

On the basis of the TOMS satellite column ozone data in latitudes 70°S–70°N from November 1978 to May 1990, we use a statistical model to estimate the trends in ozone as a function of latitude, longitude, and month. The trends in the TOMS ozone data are highly seasonal and dependent on location. Near the equator, the estimated monthly trends are not significantly different from zero. For high latitudes, most of the estimated monthly trends are negative. In January, February, and March, there are some positive trend estimates in the western hemisphere around latitude 60°N. The most negative trends for these 3 months also appear in the high latitudes of the northern hemisphere. Starting in June, more negative trends appear in the latitudes 50°S–70°S than the trends in the rest of the world considered. A large depletion develops during the spring time (September to November) in the southern high-latitude region, and the area of peak ozone decline is moving eastward during the period. The largest negative trends (about −29% per decade) for the area considered in this study appear in October around the latitude 70°S and longitudes 20°W–100°W region. Since the magnitudes of the estimated trends in the southern hemisphere increase toward the pole, more negative trends occur beyond the latitude 70°S. For the northern hemisphere, the year-round trend estimates for latitudes 30°N–70°N range from −0.96% to −7.43% per decade. In the latitudes 30°N–50°N, the winter trend estimates are more negative than those for the summer and the fall. However, this pattern did not hold for latitudes 50°N–70°N.

Spatial variation of total column ozone on a global scale

The spatial dependence of total column ozone varies strongly with latitude, so that homogeneous models (invariant to all rotations) are clearly unsuitable. However, an assumption of axial symmetry, which means that the process model is invariant to rotations about the Earths axis, is much more plausible and considerably simplifies the modeling. Using TOMS (Total Ozone Mapping Spectrometer) measurements of total column ozone over a six-day period, this work investigates the modeling of axially symmetric processes on the sphere using expansions in spherical harmonics. It turns out that one can capture many of the large scale features of the spatial covariance structure using a relatively small number of terms in such an expansion, but the resulting fitted model provides a horrible fit to the data when evaluated via its likelihood because of its inability to describe accurately the processs local behavior. Thus, there remains the challenge of developing computationally tractable models that capture both the large and small scale structure of these data.

Fixed-Domain Asymptotics for Spatial Periodograms

Abstract The periodogram for a spatial process observed on a lattice is often used to estimate the spectral density. The bases for such estimators are two asymptotic properties that periodograms commonly possess: (1) the periodogram at a particular frequency is approximately unbiased for the spectral density, and (2) the correlation of the periodogram at distinct frequencies is approximately zero. For spatial data, it is often appropriate to use fixed-domain asymptotics in which the observations get increasingly dense in some fixed region as their number increases. Using fixed-domain asymptotics, this article shows that standard asymptotic results for periodograms do not apply and that using the periodogram of the raw data can yield highly misleading results. But by appropriately filtering the data before computing the periodogram, it is possible to obtain results similar to the standard asymptotic results for spatial periodograms.