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A latent discrete Markov random field approach to identifying and classifying historical forest communities based on spatial multivariate tree species counts. (English) Zbl 1435.62411

Summary: The Wisconsin Public Land Survey database describes historical forest composition at high spatial resolution and is of interest in ecological studies of forest composition in Wisconsin just prior to significant Euro-American settlement. For such studies it is useful to identify recurring subpopulations of tree species known as communities, but standard clustering approaches for subpopulation identification do not account for dependence between spatially nearby observations. Here, we develop and fit a latent discrete Markov random field model for the purpose of identifying and classifying historical forest communities based on spatially referenced multivariate tree species counts across Wisconsin. We show empirically for the actual dataset and through simulation that our latent Markov random field modeling approach improves prediction and parameter estimation performance. For model fitting we introduce a new stochastic approximation algorithm which enables computationally efficient estimation and classification of large amounts of spatial multivariate count data.

MSC:

62P12 Applications of statistics to environmental and related topics
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62H11 Directional data; spatial statistics
62M05 Markov processes: estimation; hidden Markov models
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[1] Barnes, B., Zak, D., Denton, S. and Spurr, S. (2010). Forest Ecology, 4th ed. Wiley, New York.
[2] Benveniste, A., Priouret, P. and Métivier, M. (1990). Adaptive Algorithms and Stochastic Approximations. Applications of Mathematics (New York) 22. Springer, Berlin. · Zbl 0752.93073
[3] Berg, S., Zhu, J., Clayton, M. K, Shea, M. E and Mladenoff, D. J (2019). Supplement to “A latent discrete Markov random field approach to identifying and classifying historical forest communities based on spatial multivariate tree species counts.” DOI:10.1214/19-AOAS1259SUPP.
[4] Burns, R. M. and Honkala, B. H. (1990). Silvics of North America 2. U.S. Department of Agriculture, Washington, DC.
[5] Chen, J. (2017). Consistency of the MLE under mixture models. Statist. Sci. 32 47-63. · Zbl 1442.62064
[6] Comets, F. and Gidas, B. (1992). Parameter estimation for Gibbs distributions from partially observed data. Ann. Appl. Probab. 2 142-170. · Zbl 0742.60026
[7] Curtis, J. T. (1959). The Vegetation of Wisconsin: An Ordination of Wisconsin Plant Communities. Univ. Wisconsin Press, Madison, WI.
[8] Davis, M. B., Schwartz, M. W. and Woods, K. (1991). Detecting a species limit from pollen in sediments. J. Biogeogr. 18 653-668.
[9] Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39 1-38. · Zbl 0364.62022
[10] Egan, D. (2005). The Historical Ecology Handbook: A Restorationist’s Guide to Reference Ecosystems. Island Press.
[11] Fahey, R. T., Lorimer, C. G. and Mladenoff, D. J. (2012). Habitat heterogeneity and life-history traits influence presettlement distributions of early-successional tree species in a late-successional, hemlock-hardwood landscape. Landsc. Ecol. 27 999-1013.
[12] Finley, R. W. (1976). The original vegetation cover of Wisconsin. Wisconsin Department of Natural Resources.
[13] Forbes, F. and Fort, G. (2007). Combining Monte Carlo and mean-field-like methods for inference in hidden Markov random fields. IEEE Trans. Image Process. 16 824-837.
[14] Forbes, F., Charras-Garrido, M., Azizi, L., Doyle, S. and Abrial, D. (2013). Spatial risk mapping for rare disease with hidden Markov fields and variational EM. Ann. Appl. Stat. 7 1192-1216. · Zbl 1288.62158
[15] Fort, G. and Moulines, E. (2003). Convergence of the Monte Carlo expectation maximization for curved exponential families. Ann. Statist. 31 1220-1259. · Zbl 1043.62015
[16] Gaetan, C. and Guyon, X. (2010). Spatial Statistics and Modeling. Springer Series in Statistics. Springer, New York. · Zbl 1271.62214
[17] Gangnon, R. E. and Clayton, M. K. (2003). A hierarchical model for spatially clustered disease rates. Stat. Med. 22 3213-3228.
[18] Gelman, A. and Meng, X.-L. (1998). Simulating normalizing constants: From importance sampling to bridge sampling to path sampling. Statist. Sci. 13 163-185. · Zbl 0966.65004
[19] Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6 721-741. · Zbl 0573.62030
[20] Hong, C., Ning, Y., Wang, S., Wu, H., Carroll, R. J. and Chen, Y. (2017). PLEMT: A novel pseudolikelihood-based EM test for homogeneity in generalized exponential tilt mixture models. J. Amer. Statist. Assoc. 112 1393-1404.
[21] Knorr-Held, L. and Raßer, G. (2004). Bayesian detection of clusters and discontinuities in disease maps. Biometrics 56 13-21. · Zbl 1060.62629
[22] Kushner, H. J. and Yin, G. G. (1997). Stochastic Approximation Algorithms and Applications. Applications of Mathematics (New York) 35. Springer, New York. · Zbl 0914.60006
[23] Lawson, A. B. (2010). Hotspot detection and clustering: Ways and means. Environ. Ecol. Stat. 17 231-245.
[24] Liu, F., Mladenoff, D. J., Keuler, N. S. and Moore, L. S. (2011). Broadscale variability in tree data of the historical Public Land Survey and its consequences for ecological studies. Ecol. Monogr. 81 259-275.
[25] Mladenoff, D. J., Sickley, T. A., Schulte, L. A., Rhemtulla, J. M. and Bolliger, J. (2009). Wisconsin’s Land Cover in the 1800s. Wisconsin Department of Natural Resources.
[26] Mladenoff, D. J., White, M. A., Pastor, J. and Crow, T. R. (1993). Comparing spatial pattern in unaltered old-growth and disturbed forest landscapes. Ecol. Appl. 3 294-306.
[27] Neal, R. M. (1993). Probabilistic Inference Using Markov Chain Monte Carlo Methods. Technical Report.
[28] Paciorek, C. J., Goring, S. J., Thurman, A. L., Cogbill, C. V., Williams, J. W., Mladenoff, D. J., Peters, J. A., Zhu, J. and McLachlan, J. S. (2016). Statistically-estimated tree composition for the northeastern United States at Euro-American settlement. PLoS ONE 11 1-20.
[29] Peterson, D. W. and Reich, P. B. (2001). Prescribed fire in oak savanna: Fire frequency effects on stand structure and dynamics. Ecol. Appl. 11 914-927.
[30] Radeloff, V. C., Mladenoff, D. J., He, H. S. and Boyce, M. S. (1999). Forest landscape change in the northwestern Wisconsin Pine Barrens from pre-European settlement to the present. Can. J. For. Res. 29 1649-1659.
[31] Robbins, H. and Monro, S. (1951). A stochastic approximation method. Ann. Math. Stat. 22 400-407. · Zbl 0054.05901
[32] Schulte, L. and Mladenoff, D. (2001). The original US public land survey records: Their use and limitations in reconstructing pre-European settlement vegetation. J. For. 99 5-10.
[33] Schulte, L. A., Mladenoff, D. J. and Nordheim, E. V. (2002). Quantitative classification of a historic northern Wisconsin (USA) landscape: Mapping forests at regional scales. Can. J. For. Res. 32 1616-1638.
[34] Shao, J. (2003). Mathematical Statistics, 2nd ed. Springer Texts in Statistics. Springer, New York. · Zbl 1018.62001
[35] Shea, M. E., Schulte, L. A. and Palik, B. J. (2014). Reconstructing vegetation past: Pre-Euro-American vegetation for the midwest Driftless area, USA. Ecol. Restor. 32 417-433.
[36] Städler, N., Bühlmann, P. and van de Geer, S. (2010). \( \ell_1\)-penalization for mixture regression models. TEST 19 209-256. · Zbl 1203.62128
[37] Stambaugh, M. C. and Guyette, R. P. (2008). Predicting spatio-temporal variability in fire return intervals using a topographic roughness index. For. Ecol. Manag. 254 463-473.
[38] Waller, L. A. (2009). Detection of clustering in spatial data. In The SAGE Handbook of Spatial Analysis (J. Fagerberg, D. C. Mowery and R. R. Nelson, eds.) 299-321 10. Sage, London.
[39] Wu, F. Y. (1982). The Potts model. Rev. Modern Phys. 54 235-268.
[40] Wu, C.-F. J. (1983). On the convergence properties of the EM algorithm. Ann. Statist. 11 95-103. · Zbl 0517.62035
[41] Younes, L. (1989). Parametric inference for imperfectly observed Gibbsian fields. Probab. Theory Related Fields 82 625-645. · Zbl 0659.62115
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