In this work, we propose a spatio-temporal Markovian-like model for ordinal observations to predict in time the spread of disease in a discrete rectangular grid of plants. This model is constructed from a logistic distribution and some simple assumptions that reflect the conditions present in a series of studies carried out to understand the dissemination of a particular infection in plants. After constructing the model, we establish conditions for the existence and uniqueness of the maximum likelihood estimator (MLE) of the model parameters. In addition, we show that, under further restrictions based on Partially Ordered Markov Models (POMMs), the MLE of the model is consistent and normally asymptotic. We then employ the MLE’s asymptotic normality to propose methods for testing spatio-temporal and spatial dependencies. The model is estimated from the real data on plants that inspired the model, and we used its results to construct prediction maps to better understand the transmission of plant illness in time and space.
Besag J. Spatial interaction and the statistical analysis of lattice systems. J R Stat Soc: Ser B (Method). 1974;36:192–225.
Besag J. Nearest-neighbour systems and the auto-logistic model for binary data. J R Stat Soc: Ser B (Method). 1972;34:75–83.
Besag J. On the statistical analysis of dirty pictures. J R Stat Soc: Ser B (Method). 1986;48:259–79.
Gumpertz ML, Graham JM, Ristaino JB. Autologistic model of spatial pattern of phytophthora epidemic in bell pepper: effects of soil variables on disease presence. J Agric, Biolo, Environ Stat. 1997;2:131–56.
Huffer FW, Wu H. Markov chain monte carlo for autologistic regression models with application to the distribution of plant species. Biometrics. 1998;54:509–24.
Heikkinen J, Högmander H. Fully bayesian approach to image restoration with an application in biogeography. J R Stat Socy: Ser C (Appl Stat). 1994;43:569–82.
Zhu W, Fan Y. A novel approach for markov random field with intractable normalizing constant on large lattices. J Comput Graphical Stat. 2018;27:59–70.
Zhu J, Zheng Y, Carroll AL, Aukema BH. Autologistic regression analysis of spatial-temporal binary data via monte carlo maximum likelihood. J Agric Biol Environ Stat. 2008;13:84–98.10.1198/108571108X273566)| false
Schliep EM, Lany NK, Zarnetske PL, Schaeffer RN, Orians CM, Orwig DA, et al. Joint species distribution modelling for spatio-temporal occurrence and ordinal abundance data. Global Ecol. Biogeogr. 2018;27:142–55.
Schliep EM, Lany NK, Zarnetske PL, Schaeffer RN, Orians CM, Orwig DA, et al. Joint species distribution modelling for spatio-temporal occurrence and ordinal abundance data. Global Ecol. Biogeogr. 2018;27:142–55.10.1111/geb.12666)| false
Huang H-C, Cressie N. Asymptotic properties of maximum (composite) likelihood estimators for partially ordered Markov models. Statistica Sinica. 2000;10:1325–1344, http://www.jstor.org/stable/24306782.
Deveaux V. Partially oriented markov models. Geometrical approach of Probabilistic Cellular Automata, Ph.D. thesis, Université de Rouen, 2008.
Deveaux V, Fernández R. Partially ordered models. J Stat Phys. 2010;141:476–516.
Jiménez-Hidalgo I, Virgen-Calleros G, Martinez-de La Vega O, Vandemark G, Olalde-Portugal V. Identification and characterisation of bacteria causing soft-rot in agave tequilana. Eur J Plant Pathol. 2004;110:317–31.
Ramírez-Ramírez MJ, Mancilla-Margalli NA, Meza-Álvarez L, Turincio-Tadeo R, Guzmán-de Pena D, Avila-Miranda EM. Epidemiology of fusarium agave wilt in agave tequilana weber var. azul. Plant Prot Sci. 2017;53:144–52.
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