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  • 11
    Electronic Resource
    Electronic Resource
    Hierarchical Bayesian space-time models (1998)
    Springer
    Environmental and ecological statistics 5 (1998), S. 117-154 
    Details
    ISSN: 1573-3009
    Keywords: atmospheric science ; dynamical systems ; environmental studies ; Gibbs sampling ; Markov random field ; MCMC ; non-stationarity ; temperature
    Source: Springer Online Journal Archives 1860-2000
    Topics: Energy, Environment Protection, Nuclear Power Engineering
    Notes: Abstract Space-time data are ubiquitous in the environmental sciences. Often, as is the case with atmo- spheric and oceanographic processes, these data contain many different scales of spatial and temporal variability. Such data are often non-stationary in space and time and may involve many observation/prediction locations. These factors can limit the effectiveness of traditional space- time statistical models and methods. In this article, we propose the use of hierarchical space-time models to achieve more flexible models and methods for the analysis of environmental data distributed in space and time. The first stage of the hierarchical model specifies a measurement- error process for the observational data in terms of some 'state' process. The second stage allows for site-specific time series models for this state variable. This stage includes large-scale (e.g. seasonal) variability plus a space-time dynamic process for the ’anomalies'. Much of our interest is with this anomaly proc ess. In the third stage, the parameters of these time series models, which are distributed in space, are themselves given a joint distribution with spatial dependence (Markov random fields). The Bayesian formulation is completed in the last two stages by speci- fying priors on parameters. We implement the model in a Markov chain Monte Carlo framework and apply it to an atmospheric data set of monthly maximum temperature.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1009662704779
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  • 12
    Electronic Resource
    Electronic Resource
    A note on the behaviour of the stable distributions for small index α (1975)
    Springer
    Probability theory and related fields 33 (1975), S. 61-64 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00539862
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  • 13
    Electronic Resource
    Electronic Resource
    A central limit theorem for random sets (1979)
    Springer
    Probability theory and related fields 49 (1979), S. 37-47 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00534339
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  • 14
    Electronic Resource
    Electronic Resource
    Spatial prediction and ordinary kriging (1989)
    Springer
    Mathematical geology 21 (1989), S. 493-494 
    Details
    ISSN: 1573-8868
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00897332
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  • 15
    Electronic Resource
    Electronic Resource
    The origins of kriging (1990)
    Springer
    Mathematical geology 22 (1990), S. 239-252 
    Details
    ISSN: 1573-8868
    Keywords: blue ; blup ; covariance function ; geodesy ; homogeneous structure function ; meteorology ; mining ; optimum interpolation ; spatial blup ; statistics ; variogram
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract In this article, kriging is equated with spatial optimal linear prediction, where the unknown random-process mean is estimated with the best linear unbiased estimator. This allows early appearances of (spatial) prediction techniques to be assessed in terms of how close they came to kriging.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00889887
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  • 16
    Electronic Resource
    Electronic Resource
    The Variance-Based Cross-Variogram: You Can Add Apples and Oranges (1998)
    Springer
    Mathematical geology 30 (1998), S. 789-799 
    Details
    ISSN: 1573-8868
    Keywords: cokriging ; equivariance ; pseudo cross-variogram
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The variance-based cross-variogram between two spatial processes, Z1 (·) and Z2 (·), is var (Z1 ( u ) − Z2 ( v )), expressed generally as a bivariate function of spatial locations uandv. It characterizes the cross-spatial dependence between Z1 (·) and Z2 (·) and can be used to obtain optimal multivariable predictors (cokriging). It has also been called the pseudo cross-variogram; here we compare its properties to that of the traditional (covariance-based) cross-variogram, cov (Z1 ( u ) − Z1 ( v ), Z2 ( u ) − Z2 ( v )). One concern with the variance-based cross-variogram has been that Z1 (·) and Z2 (·) might be measured in different units (“apples” and “oranges”). In this note, we show that the cokriging predictor based on variance-based cross-variograms can handle any units used for Z1 (·) and Z2 (·); recommendations are given for an appropriate choice of units. We review the differences between the variance-based cross-variogram and the covariance-based cross-variogram and conclude that the former is more appropriate for cokriging. In practice, one often assumes that variograms and cross-variograms are functions of uandv only through the difference u − v. This restricts the types of models that might be fitted to measures of cross-spatial dependence.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1021770324434
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  • 17
    Electronic Resource
    Electronic Resource
    Robust kriging—A proposal (1984)
    Springer
    Mathematical geology 16 (1984), S. 3-18 
    Details
    ISSN: 1573-8868
    Keywords: Geostatistics ; kriging ; robust estimation ; time series
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Geological data frequently have a heavy-tailed normal-in-the-middle distribution, which gives rise to grade distributions that appear to be normal except for the occurrence of a few outliers. This same situation also applies to log-transformed data to which lognormal kriging is to be applied. For such data, linear kriging is nonrobust in that (1)kriged estimates tend to infinity as the outliers do, and (2)it is also not minimum mean squared error. The more general nonlinear method of disjunctive kriging is even more nonrobust, computationally more laborious, and in the end need not produce better practical answers. We propose a robust kriging method for such nearly normal data based on linear kriging of an editing of the data. It is little more laborious than conventional linear kriging and, used in conjunction with a robust estimator of the variogram, provides good protection against the effects of data outliers. The method is also applicable to time series analysis.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01036237
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  • 18
    Electronic Resource
    Electronic Resource
    When are relative variograms useful in geostatistics? (1985)
    Springer
    Mathematical geology 17 (1985), S. 693-702 
    Details
    ISSN: 1573-8868
    Keywords: δ method ; location-width plot ; lognormal data ; scaled variogram ; universal transformation principle
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The relative variogram has been employed as a tool for correcting a simple kind of nonstationarity, namely that in which local variance is proportional to local mean squared. In the past, this has been linked in a vague way to the lognormal distribution, although if {Zt; t ∈ D}is strongly stationary and normal over a domain D,then clearly {exp (Zt); t ∈ D}will stillbe stationary, but lognormal. The appropriate link is made in this article through a universal transformation principle. More general situations are considered, leading to the use of a “scaled variogram.”
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01031611
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  • 19
    Electronic Resource
    Electronic Resource
    Spatial prediction and ordinary kriging (1988)
    Springer
    Mathematical geology 20 (1988), S. 405-421 
    Details
    ISSN: 1573-8868
    Keywords: kriging ; nugget effect ; range ; sill ; variogram
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Suppose data {Z(s i ):i=1, ..., n} are observed at spatial locations {s i :i=1, ..., n}. From these data, an unknownZ(s 0) is to be predicted at a known locations 0c, or, ifZ(s0) has a component of measurement error, then a smooth versionS(s 0) should be predicted. This article considers the assumptions needed to carry out the spatial prediction using ordinary kriging, and looks at how nugget effect, range, and sill of the variogram affect the predictor. It is concluded that certain commonly held interpretations of these variogram parameters should be modified.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00892986
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  • 20
    Electronic Resource
    Electronic Resource
    Optimal closed boundary identification in gray-scale imagery (1995)
    Springer
    Journal of mathematical imaging and vision 5 (1995), S. 179-205 
    Details
    ISSN: 1573-7683
    Keywords: closed boundary identification ; Bayesian methods ; Markov chain Monte Carlo algorithms ; image algebra
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Identification of closed boundary contours is an important problem in image analysis because boundaries delineate the structural components, or objects, present in a scene. Most filter-based edge-detection methods do not have a mechanism to identify a group of edge sites that defines a complete closed object boundary. In this paper, we construct a suitable parameter space of one-pixel-wide closed boundaries for gray-scale images that reduces the complexity of the boundary identification problem. An algorithm based on stochastic processes and Bayesian methods is presented to identify an optimal boundary from this space. By defining a prior probability model and appropriately specifying transition probability functions on the space, a Markov chain Monte Carlo algorithm is constructed that theoretically converges to a statistically optimal closed boundary estimate. Moreover, this approach ensures that implementation via computer will result in a final boundary estimate that has the necessary property of closure which previous stochastic approaches have been unable to achieve.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01248371
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