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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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.

  • [1]

    Besag J. Spatial interaction and the statistical analysis of lattice systems. J R Stat Soc: Ser B (Method). 1974;36:192–225.

  • [2]

    Besag J. Nearest-neighbour systems and the auto-logistic model for binary data. J R Stat Soc: Ser B (Method). 1972;34:75–83.

  • [3]

    Besag J. On the statistical analysis of dirty pictures. J R Stat Soc: Ser B (Method). 1986;48:259–79.

  • [4]

    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.

  • [5]

    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.

  • [6]

    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.

  • [7]

    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.

    • Crossref
    • Export Citation
  • [8]

    Strauss D. Clustering on coloured lattices. J Appl Probab. 1977;14:135–43.

    • Crossref
    • Export Citation
  • [9]

    Kutsyy V, Nair VN. Modeling and inference for spatial processes with ordinal data. Ph.D. thesis, University of Michigan, 2001.

  • [10]

    Linero AR, Bradley JR, Desai A. Multi-rubric models for ordinal spatial data with application to online ratings data. Ann Appl Stat. 2018;12:2054–74.

    • Crossref
    • Export Citation
  • [11]

    Besag J. Some methods of statistical analysis for spatial data. Bull Int Statist Inst. 1978;47:77–92.

  • [12]

    Zhu J, Huang H-C, Wu J.Modeling spatial-temporal binary data using markov random fields. J. Agric Biol Environ Stat. 2005;10:212.

    • Crossref
    • Export Citation
  • [13]

    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.

    • Crossref
    • Export Citation
  • [14]

    Wang Z, Zheng Y. Analysis of binary data via a centered spatial-temporal autologistic regression model. Environ Ecol Stat. 2013;20:37–57.

    • Crossref
    • Export Citation
  • [15]

    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.

    • Crossref
    • Export Citation
  • [16]

    Silvapulle MJ. On the existence of maximum likelihood estimators for the binomial response models. J R Stat Soc. Ser B (Method). 1981;43:310–3.

  • [17]

    Cressie N, Davidson JL. Image analysis with partially ordered markov models. Comput Stat Data Anal. 1998;29:1–26.

    • Crossref
    • Export Citation
  • [18]

    Davidson JL, Cressie N, Hua X. Texture synthesis and pattern recognition for partially ordered markov models. Pattern Recogn. 1999;32:1475–505.

    • Crossref
    • Export Citation
  • [19]

    Huang H-C, Cressie N. Asymptotic properties of maximum (composite) likelihood estimators for partially ordered Markov models. Statistica Sinica. 2000;10:1325–1344,

  • [20]

    Deveaux V. Partially oriented markov models. Geometrical approach of Probabilistic Cellular Automata, Ph.D. thesis, Université de Rouen, 2008.

  • [21]

    Deveaux V, Fernández R. Partially ordered models. J Stat Phys. 2010;141:476–516.

    • Crossref
    • Export Citation
  • [22]

    Hammersley JM, Clifford P. Markov fields on finite graphs and lattices. Unpublished, 1971.

  • [23]

    Strauss D. The many faces of logistic regression. Am Stat. 1992;46:321–7.

  • [24]

    Severini TA. Likelihood methods in statistics. New York: Oxford University Press, 2000.

  • [25]

    Gil-Vega K, González-Chavira M, Martínez-de la Vega O, Simpson J, Vandemark G. Analysis of genetic diversity in agave tequilana var. azul using rapd markers. Euphytica. 2001;119:335–41.

  • [26]

    Alegria M, González V. Mientras el tequila repunta en el mundo, el agave padece enfermedades en su tierra. Gaceta Universitaria, Universidad de Guadalajara. 1998:10,

  • [27]

    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.

  • [28]

    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.

    • Crossref
    • Export Citation
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IJB publishes biostatistical models and methods, statistical theory, as well as original applications of statistical methods, for important practical problems arising from various sciences. It covers the entire range of biostatistics, from theoretical advances to relevant and sensible translations of a practical problem into a statistical framework, including advances in biostatistical computing.