32. Zhang, B., W. Li, and C. Zhang. 2021. Sensitivity analysis of the MCRF model to different transiogram joint modeling methods for simulating categorical spatial variables. arXiv preprint arXiv:2112.09178.
31. Li, W., and C. Zhang. 2019. Markov chain random fields in the perspective of spatial Bayesian networks and optimal neighborhoods for simulation of categorical fields. Computational Geosciences, 23(5): 1087-1106. doi: 10.1007/s10596-019-09874-z.
30. Zhai, R., W. Li, C. Zhang, W. Zhang, and W. Wang. 2019. The transiogram as a graphic metric for characterizing the spatial patterns of landscapes. Landscape Ecology, 34(9): 2103–2121. doi: 10.1007/s10980-018-0760-7.
29. Zhang, W., W. Li, C. Zhang, and T. Zhao. 2019. Parallel computing solutions for Markov chain sequential simulation of categorical fields. International Journal of Digital Earth, 12(5): 566-582. doi: 10.1080/17538947.2018.1464073.
28. Yu, J., W. Li and C. Zhang. 2019. A framework of experimental transiogram modelling for Markov chain geostatistical simulation of landscape categories. Computers, Environment and Urban Systems, 73: 16–26. doi: 10.1016/j.compenvurbsys.2018.07.007.
27. Li, W., and C. Zhang. 2018. Markov chain random fields, spatial Bayesian networks, and optimal neighborhoods for simulation of categorical fields. arXiv preprint arXiv:1807.06111. (This manuscript was initially prepared in Sept. 2014; see Li and Zhang (2019) for a formal published version)
26. Zhang, W., W. Li, C. Zhang, D. Hanink, Y. Liu, and R. Zhai. 2018. Analyzing horizontal and vertical urban expansions in three East Asian megacities with the SS-coMCRF model. Landscape and Urban Planning, 177: 114-127. doi: 10.1016/j.landurbplan.2018.04.010.
25. Wang, W., W. Li, C. Zhang and W. Zhang. 2018. Improving object-based land use/cover classification from medium resolution imagery by Markov chain geostatistical post-classification. Land, 7(1), 31. doi: 10.3390/land7010031.
24. Zhang, W., W. Li, C. Zhang, and W.B. Ouimet. 2017b. Detecting horizontal and vertical urban growth from medium resolution imagery and its relationships with major socioeconomic factors. International Journal of Remote Sensing, 38(12): 3704-3734. doi: 10.1080/01431161.2017.1302113.
23. Li, W. and C. Zhang, 2017. Comments on "Spatial hidden Markov chain models for estimation of petroleum reservoir categorical variables". Journal of Petroleum Exploration and Production Technology, 7(3): 905-909.
22. Zhang, W., W. Li, C. Zhang, and X. Li. 2017a. Incorporating spectral similarity into Markov chain geostatistical cosimulation for reducing smoothing effect in land cover postclassification. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 10(3): 1082-1095.
21. Zhang, W., W. Li and C. Zhang. 2016. Land cover post-classifications by Markov chain geostatistical cosimulation based on pre-classifications by different conventional classifiers. International Journal of Remote Sensing, 37(4): 926-949.
20. Li, W., C. Zhang, M.R. Willig, D.K. Dey, G. Wang, and L. You. 2015. Bayesian Markov chain random field cosimulation for improving land cover classification accuracy. Mathematical Geosciences, 47(2): 123-148. (Note: This article was initially prepared in 2011)
19. Li, W., C. Zhang, D. K. Dey, and M. R. Willig. 2013. Updating categorical soil maps using limited survey data by Bayesian Markov chain cosimulation. The Scientific World Journal (Soil science section), vol. 2013, Article ID 587284, doi:10.1155/2013/587284. (Note: This article was prepared during the winter season of 2011-2012).
18. Li, W. and C. Zhang. 2013. Some further clarification on Markov chain random fields and transiograms. International Journal of Geographical Information Science, 27(3): 423-430.
17. Li, W. and C. Zhang, 2012b. Comments on ‘Combining spatial transition probabilities for stochastic simulation of categorical fields’ with communications on some issues related to Markov chain geostatistics. International Journal of Geographical Information Science, 26(10): 1725–1739.
16. Li, W. and C. Zhang, 2012a. Comments on ‘An efficient maximum entropy approach for categorical variable prediction’ by D. Allard, D. D’Or & R. Froidevaux. European Journal of Soil Science, 63(1): 120-124.
15. Li, W., C. Zhang and D.K. Dey, 2012. Modeling experimental cross transiograms of neighboring landscape categories with the gamma distribution. International Journal of Geographical Information Science, 26(4): 599-620.
14. Li, W. and C. Zhang, 2011. A Markov chain geostatistical framework for land-cover classification with uncertainty assessment based on expert-interpreted pixels from remotely sensed imagery. IEEE Transactions on Geoscience and Remote Sensing, 49(8): 2983-2992.
13. Li, W., C. Zhang, D.K. Dey, and S. Wang, 2010. Estimating threshold-exceeding probability maps of continuous environmental variables with Markov chain random fields. Stochastic Environmental Research and Risk Assessment, 24(8): 1113-1126.
12. Li, W. and C. Zhang, 2010b. Linear interpolation and joint model fitting of experimental transiograms for Markov chain simulation of categorical spatial variables. International Journal of Geographical Information Science, 24(6): 821-839.
11. Li, W. and C. Zhang, 2010a. Simulating the spatial distribution of clay layer occurrence depth in alluvial soils with a Markov chain geostatistical approach. Environmetrics, 21(1): 21–32.
10. Li, W. and C. Zhang, 2009. Markov chain analysis. International Encyclopedia of Human Geography, 6: 455-460.
9. Zhang, C, and W. Li, 2008b. Regional-scale modeling of the spatial distribution of surface and subsurface textural classes in alluvial soils using Markov chain geostatistics. Soil Use and Management, 24(3): 263-272.
8. Zhang, C. and W. Li, 2008a. A comparative study of nonlinear Markov chain models in conditional simulation of categorical variables from regular samples. Stochastic Environmental Research and Risk Assessment, 22(2): 217-230.
7. Li, W., and C. Zhang, 2008. A single-chain-based multidimensional Markov chain model for subsurface characterization. Environmental and Ecological Statistics, 15(2): 157-174. (Note: This is the first manuscript we wrote for the single-chain-based multi-dimensional Markov chain model, aiming to tackle the small class underestimation problem of the CMC model. Its publication was delayed due to misunderstandings about the defects of the CMC model in hydrogeology. The conditional independence assumption of nearest data within a neighborhood was also presented).
6. Zhang, C. and W. Li, 2007. Comparing a fixed-path Markov chain geostatistical algorithm with sequential indicator simulation in categorical variable simulation from regular samples. GIScience & Remote Sensing, 44(3): 251-266.
5. Li, W., and C. Zhang, 2007. A random-path Markov chain algorithm for simulating categorical soil variables from random point samples. Soil Science Society of America Journal, 71(3): 656-668. (Note: This paper introduced and tested the random-path quadrantal-neighborhood sequential simulation algorithm we developed during winter 2005 to spring 2006).
4. Li, W. 2007c. A fixed-path Markov chain algorithm for conditional simulation of discrete spatial variables. Mathematical Geology, 39(2): 159-176. (Note: The fixed-path algorithm was developed and tested only for regular sample data. We did not develop any fixed-path algorithm for irregular point sample data).
3. Li, W. 2007b. Transiograms for characterizing spatial variability of soil classes. Soil Science Society of America Journal, 71(3): 881-893. (Note: This is the full paper introducing the transiogram methodology with real soil data).
2. Li, W. 2007a. Markov chain random fields for estimation of categorical variables. Mathematical Geology, 39(3): 321-335. (Note: This paper is my second manuscript for the single-chain-based multi-dimensional Markov chain model. The model was generalized for possibly dealing with irregular point data and random path with any number of nearest neighbors in further studies. The small class underestimation problem of the CMC model was also proved by calculated local probabiliities for estimating the state of the current grid cell (i.e., assuming the CMC moves one step forward along a fixed-path). Considering that the term "spatial Markov chain model" was not sufficiently unique to represent the generalized “single-chain-based multidimensional Markov chain model", "MCRF" was used as a unique name to differentiate the new multi-dimensional Markov chain model from other Markov chain models for spatial data. The spatial conditional independence assumption of nearest data within a neighborhood, extended from the conditional independence property of Pickard random fields for adjacent cells in cardinal directions, was provided in Appendix).
1. Li, W. 2006. Transiogram: A spatial relationship measure for categorical data (technique note). International Journal of Geographical Information Science, 20(6): 693-699. (Note: This technique note replaced a full manuscript, of which the main content was integrated into Li (2007b) published on SSSAJ).
11. Li, W. 2018. Introduction to transiograms – A spatial correlation measure for categorical data (ppt). Prepared material for Proseminar on Advanced Topics in Spatial Analysis. University of Connecticut, 2018.
10. Li, W. and C. Zhang. 2015. Spatiotemporal Markov chain modeling of land use/cover changes: A preliminary study. The 23rd International Conference on Geoinformatics, June 19-21, 2015, Wuhan, China.
9. Zhang, W., W. Li, and C. Zhang. 2015. Comparison of land cover post-classifications by Markov chain random field cosimulation with different conventional classifiers. The 23rd International Conference on Geoinformatics, June 19-21, 2015, Wuhan, China.
8. Li, W. and C. Zhang. 2013. Updating categorical soil map with limited survey data by Bayesian Markov chain co-simulation. GeoComputation 2013 Extended Abstracts, May 23-25, Wuhan University, China. http://www.geocomputation.org/2013/papers/55.pdf.
7. Li, W. and C. Zhang. 2012. A Bayesian Markov chain approach for land use classification based on expert interpretation and auxiliary data. GIScience 2012, Sept. 19-21, Columbus, Ohio. http://www.giscience.org/past/2012/proceedings/abstracts/giscience2012_paper_137.pdf.
6. Li, W. and C. Zhang. 2008. Simulating the vertical two-dimensional structures of alluvial soil textural layers from borehole observations. Proceedings of 3rd Global Workshop on Digital Soil Mapping, Logan, UT, Sept 30 - Oct 3, 2008.
5. Li, W. and C. Zhang, 2008. Mapping the probabilities of soil clay layer thickness exceeding some threshold values with Markov chain geostatistics. In: T.J. Cova, H.J. Miller, K. Beard, A.U. Frank, M.F. Goodchild (eds.) GIScience 2008, Park City, UT, Sept 23-26, 2008, pp. 270-274.
4. Li, W., and C. Zhang, 2007. The nonlinear Markov chain geostatistics. In: IAMG 2007, Geomathematics and GIS Analysis of Resources, Environment, and Hazards, IAMG 2007 Annul Conference. Aug. 26-31, Beijing, China. pp. 573-578.
3. Li, W., and C. Zhang, 2007. A middle-insertion algorithm for Markov chain simulation of soil layering. In: ACMGIS 2007, 15th ACM International Symposium on Advances in Geographic Information Systems, Nov. 7-9, 2007, Seattle, USA. pp. 328-331.
2. Li, W. and C. Zhang. 2006. Visualizing spatial uncertainty of multinomial classes in area-class mapping (online article). AutoCarto 2006, June 26-28, Vancouver, Washington, 13 pages. http://www.cartogis.org/docs/proceedings/2006/li_zhang.pdf.
1. Li, W. and C. Zhang. 2005. Transiograms for characterizing soil spatial variability (extended abstract). GeoComputation 2005. University of Michigan, Aug. 1-3, 2005, Ann Arbor, Michigan. http://www.geocomputation.org/2005/LiW.pdf.
1. Li, W., and C. Zhang, 2006. A generalized Markov chain approach for conditional simulation of categorical variables from grid samples. Transactions in GIS, 10(4): 651-669. (Note: This paper first used the math model method for joint transiogram modeling and applied transiograms to the TMC model, an extension of the CMC model, for dealing with regular point sample data. It was prepared earlier than Li and Zhang (2005) but published later).
2. Li, W. and C. Zhang, 2005. Application of transiograms to Markov chain modeling and spatial uncertainty assessment of land cover classes. GIScience & Remote Sensing, 42(4): 297-319. (Note: This paper first used the linear interpolation method for joint transiogram modeling. Due to the difficulty in publishing the single-chain-based multi-dimensional Markov chain model (later generalized and called the MCRF model) with the transiogram methodology at that time, the linear interpolation method for joint transiogram modeling was first used to the TMC model - an extension of the CMC model).
3. Zhang, C. and W. Li, 2005. Markov chain modeling of multinomial land-cover classes. GIScience & Remote Sensing, 42(1): 1-18.
4. Li, W., C. Zhang, J.E. Burt and A. Zhu, 2005. A Markov chain-based probability vector approach for modeling spatial uncertainties of soil classes. Soil Science Society of America Journal, 69(6): 1931-1942.
5. Li, W., C. Zhang, J.E. Burt, A. Zhu and J. Feyen, 2004. Two-dimensional Markov chain simulation of soil type spatial distribution. Soil Science Society of America Journal, 68(5): 1479-1490. (Note: This paper modified and extended the CMC model for simulating the horizontal spatial distribution of soil types with survey line data. It used an alternate advancing method with three Markov chains (forming two CMCs) to tackle the parcel inclination tendency (actually directional correlation due to asymmetric neighborhood and simulation path). It also discussed the small class underestimation problem with data analysis. Although the small class underestimation problem was reduced by model extension and the use of survey line data in four directions, it had not been effectively solved by the extended model).
6. Zhang, C., and W. Li. 2004. Predictive area class mapping of multinomial land-cover categories using Markov chains. pp. 239-242. In: GIScience 2004 - The Third International Conference on Geographic Information Science, Maryland.
Li, W., Zhang, C., et al. 2004. 2D Markov-chain simulation and prediction of shallow subsurface alluvial soil textural layers. Unpublished manuscript. (Note: This manuscript was not published althought it was suggested for a resubmission after a review by WRR, because we found that one drawback raised by a reviewer could not be solved: The inserted simulated line data generated by one-dimensional Markov chain methods were not spatially correlated, thus messing the pattern. Our further testing found that if we used the CMC model to generate the simulated lines to make them correlated with each other, the small class underestimation probelm would occur in the simulated virtual line data, then the effect of reducing small class underestimation would be gone. Later, we developed the MCRF idea which theoretically and also effectively solved the small class underestimation problem of the CMC model. Apparently, there was no need to use or modify the CMC model anymore (unless somebody wanted to mislead readers, as occurred later in geology/geological engineering). Therefore, this mauscript was thrown away.)
Li, W., Zhang, C. 2003. Application of the coupled Markov chain model to categorical soil data: Feasibility and constraints. Unpublished manuscript. (Note: This manuscript were given up because a reviewer from the Netherlands was unhappy with its contents in 2003. So although we revised it, we still decided not trying to publish it anymore. The defects of the CMC model was not difficult to be seen, especially after we explained them, proved and demonstrated the small class underestimation problem in the article for the MCRF model (Li 2007). However, some people refused to adimit the CMC model has any defects using various tricks in published journal articles. The things that occurred later and have last for nearly two decades are really out of our imagination)
Li, W. 1999. Two-Dimensional Stochastic Simulation of Spatial Distributions of Soil Layers and Types Using the Coupled Markov Chain Method. Postdoctoral report, Institute for Land and Water Management, Catholic University of Leuven, Leuven, Belgium. (Note: This postdoc research was done with Professor Jan Feyen. The content of this report was presented in Leuven in 1999, with main audiences from the Institute for Land and Water Management. In addition, D. Jacques attended this research by providing assistance, suggesting research method, discussing results and checking the report; C. Zhang attended this research by helping in GIS skill and image visualization. The computer program for implementing the CMC model was developed by W. Li based on the decription of the CMC model provided in Elfeki's PhD dissertation)