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Li and Zhang (2012b, Comments on ‘Combining spatial transition probabilities for stochastic simulation of categorical fields’ with communications on some issues related to Markov chain geostatics) raised a series of comments on our recent paper (Cao, G., Kyriakidis, P.C., and Goodchild, M.F., 2011. Combining spatial transition probabilities for stochastic simulation of categorical fields. International Journal of Geographical Information Science, 25 (11), 1773–1791), which include a notation error in the model equation provided for the Markov chain random field (MCRF) or spatial Markov chain model (SMC), originally proposed by Li (2007b, Markov chain random fields for estimation of categorical variables. Mathematical Geology, 39 (3), 321–335), and followed by Allard et al. (2011, An efficient maximum entropy approach for categorical variable prediction. European Journal of Soil Science, 62, 381–393) about the misinterpretation of MCRF (or SMC) as a simplified form of the Bayesian maximum entropy (BME)-based approach, the so-called Markovian-type categorical prediction (MCP) (Allard, D., D'Or, D., and Froideveaux, R., 2009. Estimating and simulating spatial categorical data using an efficient maximum entropy approach. Avignon: Unite Biostatisque et Processus Spatiaux Institute National de la Recherche Agronomique. Technical Report No. 37; Allard, D., D'Or, D., and Froideveaux, R., 2011. An efficient maximum entropy approach for categorical variable prediction. European Journal of Soil Science, 62, 381–393). Li and Zhang (2012b, Comments on ‘Combining spatial transition probabilities for stochastic simulation of categorial fields’ with communication on some issues related to Markov chain geostatistics. International Journal of Geographical Information Science) also raised concerns regarding several statements Cao et al. (2011, Combining spatial transition probabilities for stochastic simulation of categorical fields. International Journal of Geographical Information Science, 25 (11), 1773–1791) had made, which mainly include connections between permanence of ratios and conditional independence, connections between MCRF and Bayesian networks and transiograms as spatial continuity measures. In this response, all of the comments and concerns will be addressed, while also communicating with Li and other colleagues on general topics in Markov chain geostatistics. [ABSTRACT FROM AUTHOR]

In this article, the prediction problem of categorical spatial data, that is, the estimation of class occurrence probability for (target) locations with unknown class labels given observed class labels at sample (source) locations, is analyzed in the framework of generalized linear mixed models, where intermediate, latent (unobservable) spatially correlated Gaussian variables (random effects) are assumed for the observable non-Gaussian responses to account for spatial dependence information. Within such a framework, a spatial multinomial logistic mixed model is proposed specifically to model categorical spatial data. Analogous to the dual form of kriging family, the proposed model is represented as a multinomial logistic function of spatial covariances between target and source locations. The associated inference problems, such as estimation of parameters and choice of the spatial covariance function for latent variables, and the connection of the proposed model with other methods, such as the indicator variants of the kriging family (indicator kriging and indicator cokriging) and Bayesian maximum entropy, are discussed in detail. The advantages and properties of the proposed method are illustrated via synthetic and real case studies. [ABSTRACT FROM AUTHOR]

Categorical spatial data, such as land use classes and socioeconomic statistics data, are important data sources in geographical information science (GIS). The investigation of spatial patterns implied in these data can benefit many aspects of GIS research, such as classification of spatial data, spatial data mining, and spatial uncertainty modeling. However, the discrete nature of categorical data limits the application of traditional kriging methods widely used in Gaussian random fields. In this article, we present a new probabilistic method for modeling the posterior probability of class occurrence at any target location in space-given known class labels at source data locations within a neighborhood around that prediction location. In the proposed method, transition probabilities rather than indicator covariances or variograms are used as measures of spatial structure and the conditional or posterior (multi-point) probability is approximated by a weighted combination of preposterior (two-point) transition probabilities, while accounting for spatial interdependencies often ignored by existing approaches. In addition, the connections of the proposed method with probabilistic graphical models (Bayesian networks) and weights of evidence method are also discussed. The advantages of this new proposed approach are analyzed and highlighted through a case study involving the generation of spatial patterns via sequential indicator simulation. [ABSTRACT FROM AUTHOR]

Areal interpolation is the procedure of using known attribute values at a set of (source) areal units to predict unknown attribute values at another set of (target) units. Geostatistical areal interpolation employs spatial prediction algorithms, that is, variants of Kriging, which explicitly incorporate spatial autocorrelation and scale differences between source and target units in the interpolation endeavor. When all the available source measurements are used for interpolation, that is, when a global search neighborhood is adopted, geostatistical areal interpolation is extremely computationally intensive. Interpolation in this case requires huge memory space and massive computing power, even with the dramatic improvement introduced by the spectral algorithms developed by Kyriakidis et al. (2005. Improving spatial data interoperability using geostatistical support-to-support interpolation. In: Proceedings of geoComputation. Ann Arbor, MI: University of Michigan) and Liu et al. (2006. Calculation of average covariance using fast Fourier transform (FFT). Menlo Park, CA: Stanford Center for Reservoir Forecasting, Petroleum Engineering Department, Stanford University) based on the fast Fourier transform (FFT). In this study, a parallel FFT-based geostatistical areal interpolation algorithm was developed to tackle the computational challenge of such problems. The algorithm includes three parallel processes: (1) the computation of source-to-source and source-to-target covariance matrices by means of FFT; (2) the QR factorization of the source-to-source covariance matrix; and (3) the computation of source-to-target weights via Kriging, and the subsequent computation of predicted attribute values for the target supports. Experiments with real-world datasets (i.e., predicting population densities of watersheds from population densities of counties in the Eastern Time Zone and in the continental United States) showed that the parallel algorithm drastically reduced the computing time to a practical length that is feasible for actual spatial analysis applications, and achieved fairly high speed-ups and efficiencies. Experiments also showed the algorithm scaled reasonably well as the number of processors increased and as the problem size increased. [ABSTRACT FROM PUBLISHER]

Three forms of linear interpolation are routinely implemented in geographical information science, by interpolating between measurements made at the endpoints of a line, the vertices of a triangle, and the vertices of a rectangle (bilinear interpolation). Assuming the linear form of interpolation to be correct, we study the propagation of error when measurement error variances and covariances are known for the samples at the vertices of these geometric objects. We derive prediction error variances associated with interpolated values at generic points in the above objects, as well as expected (average) prediction error variances over random locations in these objects. We also place all the three variants of linear interpolation mentioned above within a geostatistical framework, and illustrate that they can be seen as particular cases of Universal Kriging (UK). We demonstrate that different definitions of measurement error in UK lead to different UK variants that, for particular expected profiles or surfaces (drift models), yield weights and predictions identical with the interpolation methods considered above, but produce fundamentally different (yet equally plausible from a pure data standpoint) prediction error variances. [ABSTRACT FROM AUTHOR]