Statistical Methods for Ultrafine Particle Distributions
We develop statistical methods to model ultrafine particle (UFP) counts as a function of size and time using Bayesian functional time-series methods. Because of their small diameter and relatively large surface area, UFPs can penetrate the lung, enter the circulatory system, and deposit in the brain. Health effects of UFPs may be strongly linked to particle size as this characteristic determines which region in the lung the particles deposit. UFP counts can increase by orders of magnitude above natural levels in urban environments from anthropogenic processes such as the operation of combustion engines. Understanding how UFP counts of various sizes change over time near combustion engines and high traffic roadways is crucial in setting idling, vehicle reduction, and vehicle emissions policy.
We first propose Bayesian longitudinal functional time series models to understand the impact of engine idling on UFP counts inside school buses. Our approach is a varying coefficient model and we model UFP counts over size at a given time with a cubic B-spline basis. UFP counts across a range of sizes were collected over time during multiple experiments while the bus engines were off and then turned on, making the data multivariate longitudinal. Steady meteorological and background traffic conditions during individual experiments imply that UFP counts over size do not vary greatly with time before the engine is on, but variation in engine-off UFP counts is observed between experiments. We model this using a spline random effect model across all UFP sizes. We allow counts to increase over time at a size dependent rate once the engine is turned on. We explore alternate models for the engine-on increase: a possible jump in counts at engine-on followed by either a quadratic or bent line time trend. A random jump by experiment is also explored. Residuals are modeled with an autoregressive model over time to accommodate correlation over time. To account for larger count variance for smaller particles relative to larger particles, the log residual variance over size bin is modeled using a quadratic B-spline basis.
We next propose distinct Bayesian functional time series models to model the how UFP counts change over time near the I-405 roadway. As with the bus experiment, we model UFP counts over size at a given time with a cubic B-spline basis. Our methods allow UFP counts to then change over time at a size bin dependent rate by letting B-spline coefficients evolve over time following a random walk model. This model smooths UFP counts both over size and over time. To account for larger count variance for smaller particles relative to larger particle size bins, the log residual variance over size bin is again modeled using a quadratic B-spline basis. We also explore modeling residuals with an autoregressive model to accommodate correlation over time.
In both studies, we provide statistical summaries of how particle counts evolve over time as a function of size bin to estimate the contribution of emissions to UFP counts. We utilize plots to aid in the interpretation of model inferences and make model output interpretable to non-statisticians. Graphs also aid in diagnosis of lack of fit and can help suggest model improvements. Taken jointly, our methods can be used to understand the behavior of UFP counts over size and over time near combustion engines and high traffic roadways.