Peter Guttorp

Nonparametric Estimation of Nonstationary Spatial Covariance Structure

Abstract Estimation of the covariance structure of spatial processes is a fundamental prerequisite for problems of spatial interpolation and the design of monitoring networks. We introduce a nonparametric approach to global estimation of the spatial covariance structure of a random function Z(x, t) observed repeatedly at times ti (i = 1, …, T) at a finite number of sampling stations xi (i = 1, 2, …, N) in the plane. Our analyses assume temporal stationarity but do not assume spatial stationarity (or isotropy). We analyze the spatial dispersions var(Z(xi, t) − Z(xj, t)) as a natural metric for the spatial covariance structure and model these as a general smooth function of the geographic coordinates of station pairs (xi, xj ). The model is constructed in two steps. First, using nonmetric multidimensional scaling (MDS) we compute a two-dimensional representation of the sampling stations for which a monotone function of interpoint distances δij approximates the spatial dispersions. MDS transforms the problem...

Handbook of Spatial Statistics

Introduction Historical Introduction, Peter J. Diggle Continuous Spatial Variation Continuous Parameter Stochastic Process Theory, Tilmann Gneiting and Peter Guttorp Classical Geostatistical Methods, Dale L. Zimmerman and Michael Stein Likelihood-Based Methods, Dale L. Zimmerman Spectral Domain, Montserrat Fuentes and Brian Reich Asymptotics for Spatial Processes, Michael Stein Hierarchical Modeling with Spatial Data, Christopher K. Wikle Low Rank Representations for Spatial Processes, Christopher K. Wikle Constructions for Nonstationary Spatial Processes, Paul D. Sampson Monitoring Network Design, James V. Zidek and Dale L. Zimmerman Non-Gaussian and Nonparametric Models for Continuous Spatial Data, Mark F.J. Steel and Montserrat Fuentes Discrete Spatial Variation Discrete Spatial Variation, Havard Rue and Leonard Held Conditional and Intrinsic Autoregressions, Leonhard Held and Havard Rue Disease Mapping, Lance Waller and Brad Carlin Spatial Econometrics, R. Kelley Pace and James LeSage Spatial Point Patterns Spatial Point Process Theory, Marie-Colette van Lieshout Spatial Point Process Models, Valerie Isham Nonparametric Methods, Peter J. Diggle Parametric Methods, Jesper Moller Modeling Strategies, Adrian Baddeley Multivariate and Marked Point Processes, Adrian Baddeley Point Process Models and Methods in Spatial Epidemiology, Lance Waller Spatio-Temporal Processes Continuous Parameter Spatio-Temporal Processes, Tilmann Gneiting and Peter Guttorp Dynamic Spatial Models Including Spatial Time Series, Dani Gamerman Spatio-Temporal Point Processes, Peter J. Diggle and Edith Gabriel Modeling Spatial Trajectories, David R. Brillinger Data Assimilation, Douglas W. Nychka and Jeffrey L. Anderson Additional Topics Multivariate Spatial Process Models, Alan E. Gelfand and Sudipto Banerjee Misaligned Spatial Data: The Change of Support Problem, Alan E. Gelfand Spatial Aggregation and the Ecological Fallacy, Jonathan Wakefield and Hilary Lyons Spatial Gradients and Wombling, Sudipto Banerjee Index

A non‐homogeneous hidden Markov model for precipitation occurrence

A non‐homogeneous hidden Markov model is proposed for relating precipitation occurrences at multiple rain‐gauge stations to broad scale atmospheric circulation patterns (the so‐called ‘downscaling problem’). We model a 15‐year sequence of winter data from 30 rain stations in south‐western Australia. The first 10 years of data are used for model development and the remaining 5 years are used for model evaluation. The fitted model accurately reproduces the observed rainfall statistics in the reserved data despite a shift in atmospheric circulation (and, consequently, rainfall) between the two periods. The fitted model also provides some useful insights into the processes driving rainfall in this region.

A class of stochastic models for relating synoptic atmospheric patterns to regional hydrologic phenomena

A model for multistation precipitation, conditional on synoptic atmospheric patterns, is presented. The model, which we call the nonhomogeneous hidden Markov model (NHMM), postulates the existence of an unobserved weather state, which serves as a link between the large-scale atmospheric measures and the small-scale spatially discontinuous precipitation field. The weather state effectively acts as an automatic classifier of atmospheric patterns. The weather state process is assumed to be conditionally Markov, given the atmospheric data. The rainfall process is then assumed to be conditionally independent given the weather state. Various parameterizations for the weather state process and the rainfall process are discussed, and a likelihood-based estimation procedure is described. Model-based estimates of the storm duration distribution and first and second moments of the rainfall process are derived. As an example the model is fit to a four-station network of rain gauge stations in Washington state. The observed first and second moments are reproduced very closely. The fitted duration distributions are somewhat lighter tailed than the observed distribution at two of the four stations but provide a good fit at the other two. We conclude that the NHMM has promise as a method of relating synoptic atmospheric data to rainfall and other regional or local hydrologic processes.

Evidence that hematopoiesis may be a stochastic process in vivo.

To study the behavior of hematopoietic stem cells in vivo, hematopoiesis was simulated by assuming that all stem cell decisions (that is, replication, apoptosis, initiation of a differentiation/maturation program) were determined by chance. Predicted outcomes from simulated experiments were compared with data obtained in autologous marrow transplantation studies of glucose 6–phosphate dehydrogenase (G6PD) heterozygous female Safari cats. With this approach, we prove that stochastic differentiation can result in the wide spectrum of discrete outcomes observed in vivo, and that clonal dominance can occur by chance. As the analyses also suggest that the frequency of feline hematopoietic stem cells is only 6 per 107 nucleated marrow cells, and that stem cells do not replicate on average more frequently than once every three weeks, these large–animal data challenge clinical strategies for marrow transplantation and gene therapy.

Wavelet analysis of covariance with application to atmospheric time series

Multiscale analysis of univariate time series has appeared in the literature at an ever increasing rate. Here we introduce the multiscale analysis of covariance between two time series using the discrete wavelet transform. The wavelet covariance and wavelet correlation are defined and applied to this problem as an alternative to traditional cross-spectrum analysis. The wavelet covariance is shown to decompose the covariance between two stationary processes on a scale by scale basis. Asymptotic normality is established for estimators of the wavelet covariance and correlation. Both quantities are generalized into the wavelet cross covariance and cross correlation in order to investigate possible lead/lag relationships. A thorough analysis of interannual variability for the Madden-Julian oscillation is performed using a 35+ year record of daily station pressure series. The time localization of the discrete wavelet transform allows the subseries, which are associated with specific physical time scales, to be partitioned into both seasonal periods (such as summer and winter) and also according to El Nino-Southern Oscillation (ENSO) activity. Differences in variance and correlation between these periods may then be firmly established through statistical hypothesis testing. The daily station pressure series used here show clear evidence of increased variance and correlation in winter across Fourier periods of 16-128 days. During warm episodes of ENSO activity, a reduced variance is observed across Fourier periods of 8-512 days for the station pressure series from Truk Island and little or no correlation between station pressure series for the same periods.

Statistical Interpretation of Species Composition

The relative abundance of different species characterizes the structure of a biological community. We analyze an experiment addressing the relationship between omnivorous feeding linkages and community stability. Our goal is to determine whether communities with different predator compositions respond similarly to environmental disturbance. To evaluate these data, we develop a hierarchical statistical model that combines Aitchisons logistic normal distribution with a conditional multinomial observation distribution. In addition, we present an algebra for compositions that includes addition, scalar multiplication, and a metric for differences in compositions. The algebra aids interpretation of treatment effects, treatment interactions, and covariates. Markov chain Monte Carlo (MCMC) is used for inference in a Bayesian framework. Our experimental results indicate that a high degree of omnivory can help to stabilize community dynamics and prevent radical shifts in community composition. This result is at odds with classical food-web predictions, but agrees with recent theoretical formulations.

A review of statistical methods for the meteorological adjustment of tropospheric ozone

Abstract A variety of statistical methods for meteorological adjustment of ozone have been proposed in the literature over the last decade for purposes of forecasting, estimating ozone time trends, or investigating underlying mechanisms from an empirical perspective. The methods can be broadly classified into regression, extreme value, and space–time methods. We present a critical review of these methods, beginning with a summary of what meteorological and ozone monitoring data have been considered and how they have been used for statistical analysis. We give particular attention to the question of trend estimation, and compare selected methods in an application to ozone time series from the Chicago area. We conclude that a number of approaches make useful contributions to the field, but that no one method is most appropriate for all purposes and all meteorological scenarios. Methodological issues such as the need for regional-scale analysis, the nonlinear dependence of ozone on meteorology, and extreme value analysis for trends are addressed. A comprehensive and reliable methodology for space–time extreme value analysis is attractive but lacking.

A Hidden Markov Model for Space-Time Precipitation

A family of multivariate models for the occurrence/nonoccurrence of precipitation at N sites is constructed by assuming a different joint probability of events at the sites for each of a number of unobservable climate states. The climate process is assumed to follow a Markov chain. Simple formulae for first- and second-order parameter functions are derived, and used to find starting values for a numerical maximization of the likelihood. The method is illustrated by applying it to data for one site in Washington and to data for a network in the Great Plains.

The Martian annual atmospheric pressure cycle - Years without great dust storms

A model of the annual cycle of pressure on Mars has been developed for a 2-year period chosen to include 1 year at Lander 2 and to minimize the effect of great dust storms at the 22°N Lander 1 site. The model was developed by weighted least squares fitting of the Viking Lander pressure measurements to an annual mean, and fundamental and the first four harmonics of the annual cycle. The very close agreement between the two years suggests that an accurate representation of the annual CO2 condensation-sublimation cycle can be established for such years. The two annual mean pressures are identical to 0.006 mbar out of 7.9 mbar, and the differences in amplitudes for the first five periodic components between the two years range from 0.017 to 0.001 mbar. The phase angles, primarily dependent on solar insolation determined orbital dynamics, differ by −3.0° Ls for the second harmonic (year 1 minus year 2), and drop to ≤ 0.7° for the fundamental and fourth harmonic. Although the slight year-to-year differences appear to be real, this model is proposed as a “nominal” Martian annual pressure cycle and applications are suggested. By analogy, the corresponding first years representation at Lander 2 is also proposed as the “nominal” cycle, although it has not been verified by data from a subsequent year. These models provide a method of removing low frequencies from the annual pressure cycle for spectral analyses of baroclinic, tidal, and normal mode oscillations, and for comparisons of the interannual variability.