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 different 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 coefficients. 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 sufficient 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 ii 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 simulate...

Shifts in vegetation activity of terrestrial ecosystems attributable to climate trends

“…As a second method, we estimate the trend by fitting a bent-cable regression to the remainder. Bent-cable regression is a type of piecewise linear regression for estimating the point of transition between two linear phases in a time series 59,60 . The model takes the form r = b 0 + b 1 x + b 2 q(x, τ, γ) 60 .…”

Refined methods for estimating spirals and spiral deviations

“…Having log-quadratic at either the earlier or the latter part of a simple logarithmic spiral leads, respectively, to the log-quadratic-linear (QL), and the log-linear-quadratic (LQ) spirals (Figure 2, third row). Allowing a transition between two simple logarithmic spirals describes a bent cable (Chiu et al, 2006) as a spiral, and a generalized bent cable (Khan and Kar, 2018) as a spiral. (LQL, Figure 2, bottom left).…”

Random Power Function of Bent-cable Growth Models for Longitudinal Data: Application to AIDS Studies

“…In cases where there are abrupt changes in trajectories, piecewise or ``broken-stick'' models are usually effective (Kiuchi et al, 2001;Slate et al, 2001;Hall et al, 2003;Zhao et al, 2015). In other cases where there are smooth and gradual transitions between incoming and outgoing phases, a power function of bent-cable model is preferred (Khan and Kar, 2018) instead of a piecewise model. For instance, in the case of HIV/AIDS studies, growth curves of longitudinal data may show gradual transition periods between an initial decline in viral load after treatment and rebound at later time in the study.…”

“…The power function, where a quadratic function is a special case (Chiua et al, 2006;Khan et al, 2009), represents the gradual transition period between two unknown time points 1 and 2 . Following Khan and Kar (2018), suppose 1 = 1 − 2 ( 3 − 1) and 2 = 1 + 2 , such that 1 < 1 < 2 and 1 , 2 > 0, and 3 > 1.These parameters can also be considered as random coefficients in the mean structure of PBTM (Tishler et al, 1981;Chiua et al, 2006) as follows.…”

“…The random effects ( 1 , … , 6 )′have a multivariate normal distribution (0, Σ ) where Σ is a variance-covariance matrix with dimension of 6. The critical time point, where the mean trajectory shifts from a decreasing trend to an upward trend, is located at = 1 3 − 1 2 + 2 3 2 3 2 −1 − 3 −1 (Khan and Kar, 2018). In the estimation process of these parameters of the power function of bent-cable Tobit model in (3), measurement errors in covariates (e.g., CD4) (Raboud et al, 1996) are also addressed.…”