Many problems in ecology and the environmental sciences, such as monitoring the presence/absence of a species, involve the

observation of spatial binary random fields. Markov random field (MRF) models are commonly used to analyze data of this kind. It is becoming more common for these studies to include a time component as well. Markov random field models can be modified to incorporate temporal dependence whether the dependence is on a local level or through a global effect. However, it is important when working with MRF models to ensure the spatial dependence is properly specified. Over the course of three papers, this dissertation explores binary MRF models with an emphasis on arriving at an appropriate model for binary fields observed over time. In the first paper, we explore options to incorporate temporal dependence in MRF models. Our attempts to apply these to data resulted in unexpected results, namely negative spatial dependence estimates. In the second paper, we examine what negative spatial dependence means in the context of binary MRF models. We develop the run length distribution as a tool that can be used to diagnose the strength and direction of the dependence by examining data patterns in realized fields. In the third paper, we shift our focus to exploring the effect of negative dependence on the conditional expectations of MRF models. We develop the empirical conditional expectation as a method of examining the conditional expectation of binary fields in which we do not have knowledge of the true parameters used to generate the field (i.e. real data). The scale and heterogeneity of the empirical conditional expectation fields are statistics we introduce to assess the effect negative dependence has on the conditional expectation field.