Inference for the Analysis of Ordinal Data with Spatio-Temporal Models

F. Peraza-Garay 1 , J. U. Márquez-Urbina 2 , 3 ,  and G. González-Farías 4
  • 1 Universidad Autónoma de Sinaloa, Sinaloa, Culiacán, Mexico
  • 2 Probability and Statistics, Center for Research in Mathematics, NL, Monterrey, Mexico
  • 3 Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico
  • 4 Probability and Statistics, Center for Research in Mathematics, Jalisco s/n, Valenciana, Guanajuato, Mexico
F. Peraza-Garay, J. U. Márquez-Urbina
  • Probability and Statistics, Center for Research in Mathematics, Monterrey, NL, Mexico
  • Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico
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and G. González-Farías
  • Corresponding author
  • Probability and Statistics, Center for Research in Mathematics, Jalisco s/n, Valenciana, Guanajuato, GTO 36023, Mexico
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Abstract

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.

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