Modelling biological rhythms using order restricted inference

Abstract

Many biological processes, such as cell cycle, circadian clock or menstrual cycles, are governed by oscillatory systems consisting of numerous components that exhibit periodic patterns over time. Modelling these rhythms is a challenge in literature since usually the sampling density is low, the number of periods is generally two and the underlying signals adopt a wide range of temporal patterns, see Larriba et al. (2016). Several authors proposed parametric functions of time, such as the sinusoidal function, to model these signals. However these parametric functions might be too rigid for data derived from cell-cycle or circadian clock. Among these, a common shape of interest to a biologist is the circular up-down-up signal with a unique peak (U) and a unique trough (L) within each period. The shape of these signals is entirely described by mathematical inequalities among their components which allow to establish a relationship between the euclidean and the circular space using circular isotonic regression. In this work we state this connection between the euclidean and the circular space based on circular isotonic regression, formulate the ML isotonic regression estimator under circular up-down-up constraints and assess its computational advantages to calculate the circular isotonic regression estimator (CIRE), see Rueda et al. (2009). Results are shown both on simulations and on real data.