Annals of Applied Statistics

Functional time series models for ultrafine particle distributions

Heidi J. Fischer, Qunfang Zhang, Yifang Zhu, and Robert E. Weiss

Full-text: Open access

Abstract

We propose Bayesian functional mixed effect time series models to explain the impact of engine idling on ultrafine particle (UFP) counts inside school buses. UFPs are toxic to humans and school engines emit particles primarily in the UFP size range. As school buses idle at bus stops, UFPs penetrate into cabins through cracks, doors, and windows. Counts increase over time at a size dependent rate once the engine turns on. How UFP counts inside buses vary by particle size over time and under different idling conditions is not yet well understood. We model UFP counts at a given time using a mixed effect model with a cubic B-spline basis as a function of size. The log residual variance over size is modeled using a quadratic B-spline basis to account for heterogeneity in error across size bin, and errors are autoregressive over time. Model predictions are communicated graphically. These methods provide information needed to quantify UFP counts by size and possibly minimize UFP exposure in the future.

Article information

Source
Ann. Appl. Stat., Volume 11, Number 1 (2017), 297-319.

Dates
Received: December 2014
Revised: November 2016
First available in Project Euclid: 8 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1491616882

Digital Object Identifier
doi:10.1214/16-AOAS1004

Mathematical Reviews number (MathSciNet)
MR3634325

Keywords
Bayesian statistics hierarchical models varying coefficient models heteroskedasticity

Citation

Fischer, Heidi J.; Zhang, Qunfang; Zhu, Yifang; Weiss, Robert E. Functional time series models for ultrafine particle distributions. Ann. Appl. Stat. 11 (2017), no. 1, 297--319. doi:10.1214/16-AOAS1004. https://projecteuclid.org/euclid.aoas/1491616882


Export citation

References

  • Alessandrini, F., Schulz, H., Takenaka, S., Lentner, B., Karg, E., Behrendt, H. and Jakob, T. (2006). Effects of ultrafine carbon particle inhalation on allergic inflammation of the lung. Journal of Allergy and Clinical Immunology 117 824–830.
  • Baladandayuthapani, V., Mallick, B. K. and Carroll, R. J. (2005). Spatially adaptive Bayesian penalized regression splines (P-splines). J. Comput. Graph. Statist. 14 378–394.
  • Bennett, W. D. and Zeman, K. L. (1998). Deposition of fine particles in children spontaneously breathing at rest. Inhalation Toxicology 10 831–842.
  • Berhane, K. and Molitor, N. T. (2008). A Bayesian approach to functional-based multilevel modeling of longitudinal data: Applications to environmental epidemiology. Biostatistics 4 686–699.
  • Chaloner, K. (1994). Residual analysis and outliers in Bayesian hierarchical models. In Aspects of Uncertainty. 149–157. Wiley, Chichester.
  • Chaloner, K. and Brant, R. (1988). A Bayesian approach to outlier detection and residual analysis. Biometrika 75 651–659.
  • Crainiceanu, C. M., Ruppert, D., Carroll, R. J., Joshi, A. and Goodner, B. (2007). Spatially adaptive Bayesian penalized splines with heteroscedastic errors. J. Comput. Graph. Statist. 16 265–288.
  • de Boor, C. (1978). A Practical Guide to Splines. Applied Mathematical Sciences 27. Springer, New York.
  • Delfino, R. J., Sioutas, C. and Malik, S. (2005). Potential role of ultrafine particles in associations between airborne particle mass and cardiovascular health. Environmental Health Perspectives 113 934–946.
  • EPA (2002). Health assessment document for diesel engine exhaust. The National Technical Information Service, Springfield, VA.
  • EPA (2014). Clean school bus USA. Available at http://www.epa.gov/cleanschoolbus/csb-overview.htm.
  • Ferin, J., Oberdorster, G., Penney, D. P., Soderholm, S. C., Gelein, R. and Piper, H. C. (1990). Increased pulmonary toxicity of ultrafine particles? Journal of Aerosol Science 21 384–387.
  • Fischer, H. J, Zhang, Q., Zhu, Y. and Weiss, R. E (2017). Supplement to “Functional time series models for ultrafine particle distributions.” DOI:10.1214/16-AOAS1004SUPPA, DOI:10.1214/16-AOAS1004SUPPB.
  • Frampton, M. W., Stewart, J. C., Oberdorster, G., Morrow, P. E., Chalupa, D., Pietropaoli, A. P., Frasier, L. M., Speers, D. M., Cox, C., Huang, L. S. and Utell, M. J. (2006). Inhalation of ultrafine particles alters blood leukocyte expression of adhesion molecules in humans. Environmental Health Perspectives 114 51–58.
  • Gelfand, A. E. and Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85 398–409.
  • Goldsmith, J. and Kitago, K. (2013). Assessing systematic effects of stroke on motor control using hierarchical scalar-on-function regression. Techical report, Columbia Univ., New York, NY.
  • Hadfield, J. D. (2010). MCMC methods for multi-response generalized linear mixed models: The MCMCglmm R package. J. Stat. Softw. 33 1–22.
  • Hastie, T. and Tibshirani, R. (1993). Varying-coefficient models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 55 757–796.
  • Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97–109.
  • Hussein, T., Maso, M. D., Petäjä, T., Koponen, I. K., Paatero, P., Aalto, P., Hämeri, K. and Kulmala, M. (2005). Evaluation of an automatic algorithm for fitting the particle number size distributions. Boreal Environment Research 10 337–355.
  • Lang, S. and Brezger, A. (2004). Bayesian P-splines. J. Comput. Graph. Statist. 13 183–212.
  • Morawska, L., Ristovski, Z., Jayaratne, E. R., Keogh, D. U. and Ling, X. (2008). Ambient nano and ultrafine particles from motor vehicle emissions: Characteristics, ambient processing and implications on human exposure. Atmospheric Environment 42 8113–8138.
  • Morris, J. (2015). Functional regression. Annual Review of Statistics and Its Application 2 321–359.
  • Morris, J. S., Vannucci, M., Brown, P. J. and Carroll, R. J. (2003). Wavelet-based nonparametric modeling of hierarchical functions in colon carcinogenesis. J. Amer. Statist. Assoc. 98 573–597.
  • Oberdorster, G., Sharp, Z., Atudorei, V., Elder, A., Gelein, R., Kreyling, W. and Cox, C. (2004). Translocation of inhaled ultrafine particles to the brain. Inhalation Toxicology 16 437–445.
  • Palmer, J. and Pettit, L. (1996). Risks of using improper priors with Gibbs sampling and autocorrelated errors. J. Comput. Graph. Statist. 5 245–249.
  • Plummer, M. (2003). JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling In Proceedings of the 3rd International Workshop on Distributed Statistical Computing, Vienna.
  • Prado, R. and West, M. (2010). Time Series: Modeling, Computation, and Inference. CRC Press, Boca Raton, FL.
  • Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
  • Samet, J. M., Rappold, A., Graff, D., Cascio, W. E., Berntsen, J. H., Huang, Y. C. T., Herbst, M., Bassett, M., Montilla, T., Hazucha, M. J., Bromberg, P. A. and Devlin, R. B. (2009). Concentrated ambient ultrafine particle exposure induces cardiac changes in young healthy volunteers. American Journal of Respiratory and Critical Care Medicine 179 1034–1042.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B. Stat. Methodol. 64 583–639.
  • Weiss, R. E. and Lazaro, C. G. (1992). Residual plots for repeated measures. Stat. Med. 11 115–124.
  • Whitby, P. H. (1978). Modal aerosol dynamics modelling. Technical report, U.S. Environment Protection Agency, Atmospheric Research and Exposure Assessment Laboratory, Washington, DC.
  • Whitby, P. H., McMurry, P. H., Shanker, U. and Binkowski, F. S. (1991). Modal aerosol dynamics modeling. Technical report, U.S. Environment Protection Agency, Atmospheric Research and Exposure Assessment Laboratory, Washington, DC.
  • Wraith, D., Alston, C., Mengersen, K. and Hussein, T. (2009). Bayesian mixture model estimation of aerosol particle size distributions. Environmetrics 22 23–34.
  • Wraith, D., Mengersen, K., Alston, C., Rousseu, J. and Hussein, T. (2014). Using informative priors in the estimation of mixtures over time with application to aerosol particle size distributions. Ann. Appl. Stat. 8 232–258.
  • Zhang, Q., Fischer, H. J., Weiss, R. E. and Zhu, Y. (2012). Ultrafine particle concentrations in and around idling school buses. Atmospheric Environment 69 65–75.

Supplemental materials