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

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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.

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Notes

  1. Data are available for download at http://aqsdr1.epa.gov/aqsweb/aqstmp/airdata/download_files.html.

  2. https://www.epa.gov/criteria-air-pollutants/naaqs-table#3.

  3. Following Berman (1981) and Lasko (2014), we could adopt a more general Ga(a, b) model for \(\varLambda (x)\). We do not explore this here or in the sequel due to the computational demands imposed under Gaussian process modulation for each of the alternating renewal processes.

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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.

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Authors and Affiliations

  1. University of Missouri, 146 Middlebush Hall, Columbia, MO, 65211-6100, USA

    Erin M. Schliep

  2. Duke University, Durham, NC, 27708-0251, USA

    Alan E. Gelfand

  3. National Exposure Research Laboratory, U.S. Environmental Protection Agency, Research Triangle Park, NC, 27711, USA

    David M. Holland

Authors
  1. Erin M. Schliep
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  2. Alan E. Gelfand
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  3. David M. Holland
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Correspondence to Erin M. Schliep.

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Disclaimer: The U.S. Environmental Protection Agency through its Office of Research and Development partially collaborated in this research. Although it has been reviewed by the Agency and approved for publication, it does not necessarily reflect the Agency’s policies or views.

Appendix

Appendix

See Fig. 9.

Fig. 9
figure 9

Observed hourly ozone for the week July 1–7, 2014 for the four locations, Los Angeles, Phoenix, Colorado Springs, and Oklahoma City

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Schliep, E.M., Gelfand, A.E. & Holland, D.M. Alternating Gaussian process modulated renewal processes for modeling threshold exceedances and durations. Stoch Environ Res Risk Assess 32, 401–417 (2018). https://doi.org/10.1007/s00477-017-1417-9

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