Detecting and modeling changes in a time series of proportions

Abstract

We propose a framework to detect and model shifts in a time series of continuous proportions, that is, a vector of proportions measuring the parts of a whole. By reparameterizing the shape of a Dirichlet distribution, we can model the location and scale separately through generalized linear models. A hidden Markov model allows the coefficients of the generalized linear models to change, thus allowing for the time series to undergo multiple regimes. This framework allows a practitioner to adequately model seasonality, trends, or include covariate information as well as detect change points. The model’s behavior is studied via simulation and through the analysis of lake phytoplankton data from 1992 through 2012. Our analyses demonstrate that the model can be effective in detecting and modeling changes in a time series of proportions. Pertaining to the phytoplankton data, the overall biomass has grown with some changes to the community level dynamics occurring circa 2000. Specifically, the proportion of cyanobacteria appears to have increased to the detriment of diatoms.