Selecting a binary Markov model for a precipitation process

Reza Hosseini, Nhu Le and Jim Zidek have a new paper in Environmental and Ecological Statistics. This considers modelling a 0-1 precipitation process (e.g. daily data with 1 if amount of precipitation is above some threshold) as a binary discrete-time Markov process of some fixed order. The main focus of the paper is model selection where the model with the smallest BIC is chosen. Unsurprisingly simulations find that BIC is better than AIC at picking the "true" model where the "true" model is some model with a relatively small number of parameters.

The data example is rainfall in Calgary over 5 year time periods. Model building is detailed somewhat laboriously. Some of this seems unnecessary. For instance, the exploratory analysis shows clear seasonality in marginal probabilities of rain. However, models without seasonality are first considered leading to choosing a first order Markov model with the number of rainy days that month (i.e. a proxy for season) as a covariate. An obvious difficulty with fitting high order Markov models is that the number of parameters in the general case increases exponentially with the order of the process. Once seasonality is included only a first order Markov model is needed.

There is a disappointing lack of any reference to either hidden Markov models or (hidden) semi-Markov models. In particular the work of Hughes et al (1999, Applied Statistics) and Sansom and Thomson (2001, Journal of Applied Probability) are highly relevant. A hidden process perhaps makes more sense from the perspective of rain occurring due to e.g. the passing of a low-pressure system but the presence of such a system not necessarily manifesting itself with rain.