Accounting for roughness of circular processes: Using Gaussian random processes to model the anisotropic spread of airborne plant disease

Variables with values in the circle or indexed by the circle have been studied in order to investigate questions in ecology, epidemiology, climatology and oceanography for example. To model circular variables with rough behaviors, the use of Gaussian random processes (GRPs) can be particularly convenient as will be seen in this paper. The roughness of a GRP being mainly determined by its correlation function, a circular correlation function convenient for rough processes is proposed. These mathematical tools are applied to describe the anisotropic spread of an airborne plant disease from a point source: a hierarchical model including two circular GRPs is built and used to analyze data coming from a field experiment. This random-effect model is fitted to data using a Monte-Carlo expectation–maximization (MCEM) algorithm.