Generalized Bent-Cable Methodology for Changepoint Data: A Bayesian Approach

Abstract

The choice of the model framework in a regression setting depends on the nature of the data. The focus of this study is on changepoint data, exhibiting three phases: incoming and outgoing, both of which are linear, joined by a curved transition. These types of data can arise in many applications, including medical, health and environmental sciences. Piecewise linear models have been extensively utilized to characterize such changepoint trajectories in di erent scientific fields. However, although appealing due to its simple structure, a piecewise linear model is not realistic in many applications where data exhibit a gradual change over time. The most important aspect of characterizing a changepoint trajectory involves identifying the transition zone accurately. It is not only because the location of the transition zone is of particular interest in many areas of study, but also because it plays an important role in adequately describing the incoming and the outgoing phases of a changepoint trajectory. Note that once the transition is detected, the incoming and the outgoing phases can be modeled using linear functions. Overall, it is desirable to formulate a model in such a way that it can capture all the three phases satisfactorily, while being parsimonious with greatly interpretable regression coe cients. Since data may exhibit an either gradual or abrupt transition, it is also important for the transition model to be flexible. Bent-cable regression is an appealing statistical tool to characterize such trajectories, quantifying the nature of the transition between the two linear phases by modeling the transition as a quadratic phase with unknown width. We demonstrate that a quadratic function may not be appropriate to adequately describe many changepoint data. In practice, the quadratic function of the bent-cable model may lead to a wider or narrower interval than what could possibly be necessary to adequately describe a transition phase. We propose a generalization of the bent-cable model by relaxing the assumption of the quadratic bend. Specifically, an additional transition parameter is included in the bent-cable model to provide su cient flexibility so that inference about the transition zone (i.e., shape and width of the bend) can be data driven, rather than pre-assumed as a specific type. We discuss the properties of the generalized model, and then propose a Bayesian approach for statistical inference. The generalized model is then demonstrated with applications to three data sets taken from environmental science and economics. We also consider a comparison among the quadratic bent-cable, generalized bent-cable and piecewise linear models in terms of goodness of fit in analyzing both real-world and simulated data. Moreover, we supplement the motivation for our generalized bent-cable methodology via extensive simulations – we simulate changepoint data under some realistic assumptions, and then fit the quadratic bent-cable, generalized bent-cable and piecewise linear models to each of the simulated data sets to compare the performance of these models with respect to the overall quality of fit. A sensitivity analysis is also performed to investigate the sensitivity of Bayesian inference to prior specifications. This study suggests that the proposed generalization of the bent-cable model can be valuable in adequately describing changepoint data that exhibit either an abrupt or gradual transition over time.