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.

中文翻译：

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使用时空模型分析序数数据的推理


在这项工作中，我们提出了一种用于顺序观察的时空马尔可夫模型，以及时预测疾病在离散矩形植物网格中的传播。该模型是根据逻辑分布和一些简单的假设构建的，这些假设反映了为了解植物中特定感染的传播而进行的一系列研究中存在的条件。构建模型后，我们建立模型参数的最大似然估计（MLE）的存在性和唯一性条件。此外，我们还表明，在基于偏序马尔可夫模型 (POMM) 的进一步限制下，模型的 MLE 是一致的并且通常是渐近的。然后，我们利用 MLE 的渐近正态性提出测试时空和空间依赖性的方法。该模型是根据启发该模型的植物的真实数据进行估计的，我们利用其结果构建预测图，以更好地了解植物疾病在时间和空间上的传播。