Response to ‘Comments on “Combining Spatial Transition Probabilities for Stochastic Simulation of Categorical Fields” with Communications on Some Issues Related to Markov Chain Geostatistics

Abstract

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.