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Alternating Gaussian process modulated renewal processes for modeling threshold exceedances and durations

  • Erin M. Schliep
  • Alan E. Gelfand
  • David M. Holland
Original Paper

Abstract

It is often of interest to model the incidence and duration of threshold exceedance events for an environmental variable over a set of monitoring locations. Such data arrive over continuous time and can be considered as observations of a two-state process yielding, sequentially, a length of time in the below threshold state followed by a length of time in the above threshold state, then returning to the below threshold state, etc. We have a two-state continuous time Markov process, often referred to as an alternating renewal process. The process is observed over a truncated time window and, within this window, duration in each state is modeled using a distinct cumulative intensity specification. Initially, we model each intensity over the window using a parametric regression specification. We extend the regression specification adding temporal random effects to enrich the model using a realization of a log Gaussian process over time. With only one type of renewal, this specification is referred to as a Gaussian process modulated renewal process. Here, we introduce Gaussian process modulation to the intensity for each state. Model fitting is done within a Bayesian framework. We clarify that fitting with a customary log Gaussian process specification over a lengthy time window is computationally infeasible. The nearest neighbor Gaussian process, which supplies sparse covariance structure, is adopted to enable tractable computation. We propose methods for both generating data under our models and for conducting model comparison. The model is applied to hourly ozone data for four monitoring sites at different locations across the United States for the ozone season of 2014. For each site, we obtain estimated profiles of up-crossing and down-crossing intensity functions through time. In addition, we obtain inference regarding the number of exceedances, the distribution of the duration of exceedance events, and the proportion of time in the above and below threshold state for any time interval.

Keywords

Cumulative risk Hazard Log Gaussian process Markov chain Monte Carlo Nearest neighbor Gaussian process Representative points Stochastic integration 

Notes

Acknowledgements

The work of the first author was supported in part by University of Missouri Research Board and the US EPA’s Office of Research and Development under EPA contract EP-13-D-000257.

References

  1. Achcar JA, Fernández-Bremauntz AA, Rodrigues ER, Tzintzun G (2008) Estimating the number of ozone peaks in Mexico City using a non-homogeneous Poisson model. Environmetrics 19(5):469–485CrossRefGoogle Scholar
  2. Achcar JA, Rodrigues ER, Tzintzun G (2011) Using non-homogeneous poisson models with multiple change-points to estimate the number of ozone exceedances in mexico city. Environmetrics 22(1):1–12CrossRefGoogle Scholar
  3. Achcar JA, Coelho-Barros EA, de Souza RM (2016) Use of non-homogeneous Poisson process (NHPP) in presence of change-points to analyze drought periods: a case study in Brazil. Environ Ecol Stat 23(3):1–15CrossRefGoogle Scholar
  4. Banerjee S, Carlin BP, Gelfand AE (2014) Hierarchical modeling and analysis for spatial data. CRC Press, Boca RatonGoogle Scholar
  5. Beichelt F (2006) Stochastic processes in science, engineering and finance. CRC Press, Boca RatonCrossRefGoogle Scholar
  6. Bell ML, McDermott A, Zeger SL, Samet JM, Dominici F (2004) Ozone and short-term mortality in 95 US urban communities, 1987–2000. J Am Med Assoc 292(19):2372–2378CrossRefGoogle Scholar
  7. Bell ML, Dominici F (2008) Effect modification by community characteristics on the short-term effects of ozone exposure and mortality in 98 US communities. Am J Epidemiol 167(8):986–997CrossRefGoogle Scholar
  8. Berman M (1981) Inhomogeneous and modulated gamma processes. Biometrika 68(1):143–152CrossRefGoogle Scholar
  9. Birolini A (2012) On the use of stochastic processes in modeling reliability problems, vol 252. Springer, New YorkGoogle Scholar
  10. Bloomer BJ, Stehr JW, Piety CA, Salawitch RJ, Dickerson RR (2009) Observed relationships of ozone air pollution with temperature and emissions. Geophys Res Lett 36(9), L09803. doi: 10.1029/2009GL037308
  11. Chelani AB (2014) Irregularity analysis of CO, NO\(_2\) and O\(_3\) concentrations at traffic, commercial and low activity sites in Delhi. Stoch Environ Res Risk Assess 28(4):921–925CrossRefGoogle Scholar
  12. Coles S, Bawa J, Trenner L, Dorazio P (2001) An introduction to statistical modeling of extreme values, vol 208. Springer, New YorkCrossRefGoogle Scholar
  13. Cox DR (1972) The statistical analysis of dependencies in point processes. In: Lewis PAW (ed) Stochastic point processes. Wiley, New York, pp 55–66Google Scholar
  14. Cressie N (1993) Statistics for spatial data. WileyGoogle Scholar
  15. Datta A, Banerjee S, Finley AO, Gelfand AE (2016a) Hierarchical nearest-neighbor Gaussian process models for large geostatistical datasets. J Am Stat Assoc Ser B (Stat Methodol) 111(514):800–812Google Scholar
  16. Datta A, Banerjee S, Finley AO, Hamm NA, Schaap M (2016b) Nonseparable dynamic nearest neighbor Gaussian process models for large spatio-temporal data with an application to particulate matter analysis. Ann Appl Stat 10(3):1286–1316CrossRefGoogle Scholar
  17. Diggle PJ (2013) Statistical analysis of spatial and spatio-temporal point patterns. CRC Press, Boca RatonGoogle Scholar
  18. García-Díaz JC (2011) Monitoring and forecasting nitrate concentration in the groundwater using statistical process control and time series analysis: a case study. Stoch Environ Res Risk Assess 25(3):331–339CrossRefGoogle Scholar
  19. Gryparis A, Coull BA, Schwartz J, Suh HH (2007) Semiparametric latent variable regression models for spatiotemporal modelling of mobile source particles in the greater Boston area. J R Stat Soc Ser C (Appl Stat) 56(2):183–209CrossRefGoogle Scholar
  20. Guarnaccia C, Quartieri J, Tepedino C, Rodrigues ER (2015) An analysis of airport noise data using a non-homogeneous poisson model with a change-point. Appl Acoust 91:33–39CrossRefGoogle Scholar
  21. Huerta G, Sansó B, Stroud JR (2004) A spatiotemporal model for Mexico City ozone levels. J R Stat Soc Ser C (Appl Stat) 53(2):231–248CrossRefGoogle Scholar
  22. Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical analysis and modelling of spatial point patterns, vol 70. Wiley, HobokenGoogle Scholar
  23. Jacob DJ, Winner DA (2009) Effect of climate change on air quality. Atmos Environ 43(1):51–63CrossRefGoogle Scholar
  24. Lasko TA (2014) Efficient inference of Gaussian-process-modulated renewal processes with application to medical event data. In: Proceedings of the uncertainty in artificial intelligence (UAI), vol 2014. NIH Public Access, pp 469–476Google Scholar
  25. Leadbetter MR (1991) On a basis for ‘Peaks over Threshold’ modeling. Stat Probab Lett 12(4):357–362CrossRefGoogle Scholar
  26. Lervolino I, Giorgio M, Polidoro B (2014) Sequence-based probabilistic seismic hazard analysis. Bull Seismol Soc Am 104(2):1006–1012CrossRefGoogle Scholar
  27. Murray I, Adams RP, MacKay DJ (2010) Elliptical slice sampling. J Mach Learn Res Workshop Conf Proc (AISTATS) 9:541–548Google Scholar
  28. Norris JR (1998) Markov chains. Number 2008. Cambridge University Press, CambridgeGoogle Scholar
  29. O’Connor JR, Roelle PA, Aneja VP (2005) An ozone climatology: relationship between meteorology and ozone in the Southeast USA. Int J Environ Pollut 23(2):123–139CrossRefGoogle Scholar
  30. Rodrigues ER, Gamerman D, Tarumoto MH, Tzintzun G (2014) A non-homogeneous poisson model with spatial anisotropy applied to ozone data from Mexico City. Environ Ecol Stat 22(2):393–422CrossRefGoogle Scholar
  31. Sahu SK, Gelfand AE, Holland DM (2007) High-resolution space-time ozone modeling for assessing trends. J Am Stat Assoc 102(480):1221–1234CrossRefGoogle Scholar
  32. Teh YW, Rao V (2011) Gaussian process modulated renewal processes. In: Shawe-Taylor J, Zemel RS, Bartlett PL, Pereira F, Weinberger KQ (eds), Advances in neural information processing systems 24. Curran Associates, Inc, pp 2474–2482Google Scholar
  33. Thompson ML, Reynolds J, Cox LH, Guttorp P, Sampson PD (2001) A review of statistical methods for the meteorological adjustment of tropospheric ozone. Atmos Environ 35(3):617–630CrossRefGoogle Scholar
  34. Trivedi KS (2001) Probability and statistics with reliability, queuing and computer science applications. Wiley, HobokenGoogle Scholar
  35. U.S. Environmental Protection Agency (2011) Ozone health risk assessment for selected urban areas. EPA Office of Air Quality Planning and Standards, Research Triangle Park, NC, EPA/452/R-07-009Google Scholar
  36. U.S. Environmental Protection Agency (2013) 2013 Final report: integrated science assessment of ozone and related photochemical oxidants. U.S. Environmental Protection Agency, Washington, DC, EPA/600/R-10/076FGoogle Scholar
  37. Waller LA, Gotway CA (2004) Applied spatial statistics for public health data, vol 368. Wiley, HobokenCrossRefGoogle Scholar
  38. World Health Organization (2006) Air quality guidelines: global update 2005: particulate matter, ozone, nitrogen dioxide, and sulfur dioxide. World Health Organization, GenevaGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.University of MissouriColumbiaUSA
  2. 2.Duke UniversityDurhamUSA
  3. 3.National Exposure Research LaboratoryU.S. Environmental Protection AgencyResearch Triangle ParkUSA

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