Modeling basal body temperature data using horseshoe process regression

Biomedical data often exhibit jumps or abrupt changes. For example, women's basal body temperature may jump at ovulation, menstruation, implantation, and miscarriage. These sudden changes make these data challenging to model: many methods will oversmooth the sharp changes or overfit in response to measurement error. We develop horseshoe process regression (HPR) to address this problem. We define a horseshoe process as a stochastic process in which each increment is horseshoe-distributed. We use the horseshoe process as a nonparametric Bayesian prior for modeling a potentially nonlinear association between an outcome and its continuous predictor, which we implement via Stan and in the R package HPR. We provide guidance and extensions to advance HPR's use in applied practice: we introduce a Bayesian imputation scheme to allow for interpolation at unobserved values of the predictor within the HPR; include additional covariates via a partial linear model framework; and allow for monotonicity constraints. We find that HPR performs well when fitting functions that have sharp changes. We apply HPR to model women's basal body temperatures over the course of the menstrual cycle.