Generalized Bent-Cable Methodology for Changepoint Data: A Bayesian Approach
Date
2017-12-22
Authors
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Type
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Degree Level
Masters
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.
Description
Keywords
Bayesian, bent-cable, changepoint, Markov Chain Monte Carlo, transition
Citation
Degree
Master of Science (M.Sc.)
Department
Mathematics and Statistics
Program
Mathematics