Skip to main content
SpringerLink
Log in

Wavelet-Based Detection of Outliers in Poisson INAR(1) Time Series

  • Chapter
  • First Online:
Recent Studies on Risk Analysis and Statistical Modeling

Part of the book series: Contributions to Statistics ((CONTRIB.STAT.))

  • 911 Accesses

Abstract

The presence of outliers or discrepant observations has a negative impact in time series modelling. This paper considers the problem of detecting outliers, additive or innovational, single, multiple or in patches, in count time series modelled by first-order Poisson integer-valued autoregressive, PoINAR(1), models. To address this problem, two wavelet-based approaches that allow the identification of the time points of outlier occurrence are proposed. The effectiveness of the proposed methods is illustrated with synthetic as well as with an observed dataset.

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

Access this chapter

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 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
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

Institutional subscriptions

Notes

  1. 1.

    \(Z_t=\frac {Y_t-\mathrm {E}[Y_t | Y_{t-1}]}{\sqrt {\mathrm {Var}(Y_t | Y_{t-1})}}\).

  2. 2.

    Masking occurs when one outlier prevents others from being detected.

  3. 3.

    Available from the authors.

  4. 4.

    In the performed simulation study, the detail coefficients present a large variability. Therefore, as a compromise between correct and false detection of outliers, it is found that a reasonable acceptance envelope is constructed from the 0.01th and 99.99th extreme percentiles.

  5. 5.

    Available from the authors.

  6. 6.

    Since 241 is not a power of two, by default Matlab extends the signal by using symmetric-padding (symmetric boundary value replication).

References

  1. Al-Osh, M.A., Alzaid, A.A.: First-order integer-valued autoregressive (INAR(1)) process. J. Time Ser. Anal. 8, 261–275 (1987)

    Article  MathSciNet  Google Scholar 

  2. Barczy, M., Ispány, M., Pap, G., Scotto, M., Silva, M.E.: Innovational outliers in INAR(1) models. Commun. Stat. Theory Methods 39, 3343–3362 (2010)

    Article  MathSciNet  Google Scholar 

  3. Barczy, M., Ispány, M., Pap, G., Scotto, M., Silva, M.E.: Additive outliers in INAR(1) models. Stat. Pap. 53, 935–949 (2011)

    Article  MathSciNet  Google Scholar 

  4. Bilen, C., Huzurbazar, S.: Wavelet-based detection of outliers in time series. J. Comput. Graph. Stat. 11, 311–327 (2002)

    Article  MathSciNet  Google Scholar 

  5. Chang, I., Tiao, G.C.,Chen, C.: Estimation of time series parameters in the presence of outliers. Technometrics 30, 193–204 (1988)

    Article  MathSciNet  Google Scholar 

  6. Chen, C., Liu, L.M.: Joint estimation of model parameters and outlier effects in time series. J. Am. Stat. Assoc. 88, 284–297 (1993)

    MATH  Google Scholar 

  7. Fox, A.J.: Outliers in time series. J. R. Stat. Soc. Ser. B 34, 350–363 (1972)

    MathSciNet  MATH  Google Scholar 

  8. Grané, A., Veiga, H.: Wavelet-based detection of outliers in financial time series. Comput. Stat. Data Anal. 54, 2580–2593 (2010)

    Article  MathSciNet  Google Scholar 

  9. Mallat, S.G.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989)

    Article  Google Scholar 

  10. MATLAB Release 2012a. The MathWorks, Inc., Natick, MA, USA

    Google Scholar 

  11. McKenzie, E.: Some simple models for discrete variate time series. Water Resour. Bull. 21, 645–650 (1985)

    Article  Google Scholar 

  12. Percival, D., Walden, A.: Wavelet Methods for Time Series Analysis. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, New York (2006)

    MATH  Google Scholar 

  13. Scotto, M.G., Weiß, C.H., Gouveia, S.: Thinning-based models in the analysis of integer-valued time series: a review. Stat. Model. 15, 590–618 (2015)

    Article  MathSciNet  Google Scholar 

  14. Silva, M.E., Oliveira, V.L.: Difference equations for the higher-order moments and cumulants of the INAR(1) model. J. Time Ser. Anal. 25, 317–333 (2004)

    Article  MathSciNet  Google Scholar 

  15. Silva, M.E., Pereira, I.: Detection of additive outliers in Poisson INAR(1) time series. In: Bourguignon, J.P. et al. (eds.) Mathematics of Energy and Climate Change. CIM Series in Mathematical Sciences, pp. 377–388. Springer, Berlin (2015)

    Chapter  Google Scholar 

  16. Steutel, F.W., Van Harn, K.: Discrete analogues of self-decomposability and stability. Ann. Probab. 7, 893–899 (1979)

    Article  MathSciNet  Google Scholar 

  17. Tsay, R.S.: Time series model specification in the presence of outliers. J. Am. Stat. Assoc. 81, 132–141 (1986)

    Article  Google Scholar 

  18. Tsay, R.S.: Model checking via parametric bootstraps in time series analysis. J. R. Stat. Soc. Ser. C Appl. Stat. 41, 1–15 (1992)

    MATH  Google Scholar 

  19. Weiß, C.H.: Controlling correlated processes of Poisson counts. Qual. Reliab. Eng. Int. 23, 741–754 (2007)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their comments which helped to improve the paper and to Aurea Grané for supplying the programs of the paper [8]. This work is partially supported by Portuguese funds through the CIDMA and the Portuguese Foundation for Science and Technology (“FCT—Fundação para a Ciência e a Tecnologia”), within project UID/MAT/04106/2013.

Author information

Authors and Affiliations

  1. Faculdade de Engenharia, Universidade do Porto and CIDMA, Porto, Portugal

    Isabel Silva

  2. Faculdade de Economia, Universidade do Porto and CIDMA, Porto, Portugal

    Maria Eduarda Silva

Authors
  1. Isabel Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maria Eduarda Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Isabel Silva .

Editor information

Editors and Affiliations

  1. Universidade Aberta, Lisboa, Portugal

    Teresa A. Oliveira

  2. Department of Informatics, Technological Educational Institute of Athens, Egaleo, Greece

    Christos P. Kitsos

  3. Universidade Aberta, Lisboa, Portugal

    Amílcar Oliveira

  4. Unidade Dept de Matematica e Fisica, Instituto Politécnico de Tomar, Tomar, Portugal

    Luís Grilo

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Silva, I., Silva, M.E. (2018). Wavelet-Based Detection of Outliers in Poisson INAR(1) Time Series. In: Oliveira, T., Kitsos, C., Oliveira, A., Grilo, L. (eds) Recent Studies on Risk Analysis and Statistical Modeling. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-76605-8_13

Download citation

Publish with us

Policies and ethics

Search

Navigation