Skip to main content
SpringerLink
Log in
Book cover

Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science pp 113–119Cite as

Discrete Prolate Spheroidal Sequences as Filters in Generalized Additive Models

  • Conference paper
  • First Online:

  • 1752 Accesses

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 117))

Abstract

Air Health Indicator (AHI) is a joint Health Canada/Environment Canada initiative that seeks to model the Canadian national population health risk due to short-term (acute) effects of air pollution. The commonly accepted model in the field uses cubic spline-based temporal smoothers embedded in generalized additive models (GAMs) to account for seasonal and long-term variations in the response. From a spectral point of view, it is natural to think of these smooth, long-term variations as low-frequency components, and the temporal smoother as a linear filter.

Examining the frequency response of the filters typically used, we show that the performance leaves much to be desired. Adapting the discrete prolate spheroidal sequences as filters, taking inspiration from their similar use in the multitaper method, we are able to significantly improve the frequency response of the smoother. We conclude with a discussion of the implications for controlling bias from the long timescale structure of parametric covariates, and suggest a prefiltering stage to such models.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Burr, W.S.: Air Pollution and Health: Time Series Tools and Analysis. PhD thesis, Queen’s University, Kingston, Ontario, Canada, October (2012)

    Google Scholar 

  2. Dominici, F., Samet, J.M., Zeger, S.L.: Combining evidence on air pollution and daily mortality from the 20 largest US cities: a hierarchical modelling strategy. J. R. Stat. Soc. Ser. A (Statistics in Society) 163(3):263–302 (2000)

    Article  Google Scholar 

  3. Dominici, F., McDermott, A., Zeger, S.L., Samet, J.M.: On the use of generalized additive models in time-series studies of air pollution and health. Am. J. Epidemiol. 156(3):193–203 (2002)

    Article  Google Scholar 

  4. Peng, R.D., Welty, L.J.: The NMMAPSdata package. R News 4(2):10–14 (2004)

    Google Scholar 

  5. Peng, R.D., Dominici, F., Louis, T.A.: Model choice in time series studies of air pollution and mortality. J. R. Stat. Soc. Ser. A (Statistics in Society) 169(2):179–203 (2006)

    Article  MathSciNet  Google Scholar 

  6. Papoulis, A., Bertran, M.S.: Digital filtering and prolate functions. IEEE Trans. Circuit Theory 19, 674–681 (1972)

    Article  MathSciNet  Google Scholar 

  7. Ramsay, T.O., Burnett, R.T., Krewski, D.: The effect of concurvity in generalized additive models linking mortality to ambient particulate matter. Epidemiology 14(1):18–23 (2003)

    Article  Google Scholar 

  8. R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2012. ISBN 3-900051-07-0. (2012)

    Google Scholar 

  9. Samet, J., Dominici, F., Zeger, S., Schwartz, J., Dockery, D.W.: National Morbidity, Mortality, and Air Pollution Study. Part I: Methods and Methodologic Issues. Technical Report 94-I, Health Effects Institute, 2000. Part I of IV

    Google Scholar 

  10. Slepian, D.: Prolate spheroidal wave functions, Fourier analysis and Uncertainty: IV. Bell Syst. Tech. J. 43, 3009–3057 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  11. Slepian, D.: Prolate spheroidal wave functions, Fourier analysis, and Uncertainty V: the Discrete Case. Bell Syst. Tech. J. 57, 1371–1429 (1978)

    Article  MATH  Google Scholar 

Download references

Acknowledgment

The authors gratefully acknowledge discussion with Professor Glen Takahara, Queen’s University. Some material contained in this chapter has been previously published as part of the PhD thesis of Wesley S. Burr [1].

Author information

Authors and Affiliations

  1. Queen’s University, Kingston, ON, Canada

    Wesley S. Burr & Hwashin H. Shin

Authors
  1. Wesley S. Burr
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hwashin H. Shin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wesley S. Burr .

Editor information

Editors and Affiliations

  1. Department of Mathematics & Statistics, University of Guelph, Guelph, Ontario, Canada

    Monica G. Cojocaru

  2. Dept of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  3. Department of Mathematics and MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roman N. Makarov

  4. Department of Mathematics & MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roderick V. N. Melnik

  5. MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Hasan Shodiev

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Burr, W., Shin, H. (2015). Discrete Prolate Spheroidal Sequences as Filters in Generalized Additive Models. In: Cojocaru, M., Kotsireas, I., Makarov, R., Melnik, R., Shodiev, H. (eds) Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science. Springer Proceedings in Mathematics & Statistics, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-12307-3_16

Download citation

Publish with us

Policies and ethics

Search

Navigation