A Statistical Investigation to Monitor and Understand Atmospheric CFC Decline with the Spatial-longitudinal Bent-cable Model

Abstract

Concerns about the concentrations of chlorofluorocarbons (CFCs) in the atmosphere are based on their effects on the ozone layer by catalytically destroying ozone. The recent steady decline in atmospheric concentration of CFC could be a direct result of the Montréal Protocol's ban on CFC products, in effect since 1989. However, CFCs have long atmospheric residence times because of their low chemical reactivity, and as a consequence have already been distributed globally. To study the spatial effects and extent of the decline, we apply the proposed spatial-longitudinal bent-cable model to CFC data observed over a global detection network. The bent cable is a parametric regression model to study data that exhibits a trend change. It comprises two linear segments to describe the incoming and the outgoing phases, joined by a quadratic bend to model the transition period. For spatial longitudinal data, measurements taken over time are nested within spatially dependent locations. Here, it is useful to extend the existing longitudinal bent-cable regression to handle spatial effects. We do so in a hierarchical Bayesian framework by allowing the error terms to be correlated across space. The methodology is illustrated with applications to CFC-11 and 12 data. Our analysis reveals that: (a) there is a strong spatial relationship among all the monitoring locations across the globe; (b) both CFC-11 and CFC-12 increased significantly before entering into a transition zone; (c) after completing the transition, CFC-11 has been decreasing significantly from the atmosphere, but a slow (insignificant) decrease for CFC-12 is observed; and (d) it may take almost 5 times longer to diminish CFC-12 from the atmosphere compared to CFC-11.

Atmospheric concentration of fine particulate matter and its impact on daily hospital admissions for respiratory disease in the prairies of Canada

“…The Bayesian analogue of a frequentist confidence interval (CI) is usually referred to as a credible interval (CI). A 100 1 À 2a ð Þ% CI is (r 1 , r 2 ), where r 1 and r 1 are the ath and (1 − a)th quantiles of the posterior density, respectively [24].…”

“…Note that W = I, where I is the identity matrix, implies no spatial effects. Assumptions for v ij and g i added to the bent-cable model constitute the spatiallongitudinal bent-cable model (see [24] for technical details). Statistical inference is carried out via a Bayesian approach: the posterior median can be considered a point estimate of the parameter, whereas the posterior standard deviation measures the uncertainty of the parameter.…”

Modeling a Mixture of Linear and Changepoint Trajectories for Longitudinal Time-Series Data

“…Broken-stick models, however, may not be realistic in many applications where the longitudinal trajectories exhibit gradual changes (e.g., atmospheric concentration of CFCs (Khan et al, 2012)), or simply linear trends. So, it is desirable to relax the assumption of abruptness beforehand, and to formulate a model that is flexible enough to handle either type of changes, gradual or abrupt, as well as linear trends over time.…”

Generalized bent-cable methodology for changepoint data: a Bayesian approach

“…Since the second derivative of the likelihood function does not exist everywhere, the asymptotics were developed under non-standard regularity conditions (Chiu et al 2006). Khan et al (2013;2012) subsequently proposed the Bent-cable function,…”

“…Khan et al (2012) studied the existing longitudinal bent-cable model to handle spatial effects where they allowed the error terms to be correlated across space in a hierarchical Bayesian framework. It might be the another possibility to extend the generalized bent-cable by considering spatial effects into the model; that would be possible for CFC data and others that comprise spatial data.…”