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A diagram to detect serial dependencies: an application to transport time series

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Abstract

The Ljung–Box test is typically used to test serial independence even if, by construction, it is generally powerful only in presence of pairwise linear dependence between lagged variables. To overcome this problem, Bagnato et al. recently proposed a simple statistic defining a serial independence test which, differently from the Ljung–Box test, is powerful also under a linear/nonlinear dependent process characterized by pairwise independence. The authors also introduced a normalized bar diagram, based on p-values from the proposed test, to investigate serial dependence. This paper proposes a balanced normalization of such a diagram taking advantage of the concept of reproducibility probability. This permits to study the strength and the stability of the evidence about the presence of the dependence under investigation. An illustrative example based on an artificial time series, as well as an application to a transport time series, are considered to appreciate the usefulness of the proposal.

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References

  1. Agresti, A.: Categorical Data Analysis. Wiley, Hoboken, NJ (2002)

  2. Anderson, H.M., Vahid, F.: Nonlinear correlograms and partial autocorrelograms. Oxford Bull. Econ. Stat. 67, 957–982 (2005)

  3. Bagnato, L., Punzo, A.: On the use of χ2-test to check serial independence. Stat. Appl. VIII(1), 57–74 (2010)

  4. Bagnato, L., Punzo, A., Nicolis, O.: The autodependogram: a graphical device to investigate serial dependences. J. Time Ser. Anal. 33(2), 233–254 (2012)

  5. Bagnato, L., De Capitani, L., Punzo, A.: Detecting serial dependencies with the reproducibility probability autodependogram. Adv. Stat. Anal. 98(1), 35–61 (2014a)

  6. Bagnato, L., De Capitani, L., Punzo, A.: Testing serial independence via density-based measures of divergence. Method. Comput. Appl. Probab. 16(3), 627–641 (2014b)

  7. Bagnato, L., De Capitani, L., Mazza, A., Punzo, A.: SDD: an R package for serial dependence diagrams. J. Stat. Softw. 64(Code Snippet 2):1–19 (2015)

  8. Bagnato, L., De Capitani, L., Punzo, A.: The Kullback–Leibler autodependogram. J. Appl. Stat. 43(14), 2574–2594 (2016a)

  9. Bagnato, L., De Capitani, L., Punzo, A.: Testing for serial independence: beyond the Portmanteau approach. Am. Stat. (accepted) (2016b)

  10. Box, G.E.P., Tiao, G.C.: Intervention analysis with applications to economic and environmental problems. J. Am. Stat. Assoc. 70(349), 70–79 (1975)

  11. Chan, K.S.: TSA: time series analysis. R package version 1.01. http://CRAN.R-project.org/package=TSA(2012)

  12. Cochran, W.G.: Some methods for strengthening the common \(\chi ^2\) tests. Biometrics 10(4), 417–451 (1954)

  13. Cryer, J.D., Chan, K.S.: Time Series Analysis: With Applications in R. Springer Texts in Statistics. Springer, New York (2010)

  14. De Capitani, L.: An introduction to RP-testing. Epidemiol. Biostat. Public Health 10(1) (2013)

  15. De Capitani, L., De Martini, D.: On stochastic orderings of the Wilcoxon rank sum test statisticwith applications to reproducibility probability estimation testing. Stat. Probab. Lett. 81(8), 937–946 (2011)

  16. De Capitani, L., De Martini, D.: Reproducibility probability estimation and testing for the Wilcoxon rank-sum test. J. Stat. Comput. Simul. 85(3), 468–493 (2015)

  17. De Capitani, L., De Martini, D.: Reproducibility probability estimation and RP-testing for some nonparametric tests. Entropy 18(4), 142 (2016)

  18. De Martini, D.: Reproducibility probability estimation for testing statistical hypotheses. Stat. Probab. Lett. 78(9), 1056–1061 (2008)

  19. Diks, C.: Nonparametric Tests for Independence. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science, pp. 6252–6271. Springer, New York (2009)

  20. Genest, C., Rémillard, B.: Test of independence and randomness based on the empirical copula process. Test 13(2), 335–369 (2004)

  21. Goodman, S.N.: A comment on replication, \(p\)-values and evidence. Stat. Med. 11(7), 875–879 (1992)

  22. Hall, P., Wolff, R.: On the strength of dependence of a time series generated by a chaotic map. J. Time Ser. Anal. 16(6), 571–583 (1995)

  23. Hallin, M., Mélard, G.: Rank-based tests for randomness against first-order serial dependence. J. Am. Stat. Assoc. 83(404), 1117–1128 (1988)

  24. Johnson, N., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, vol. 2. Wiley, New York (1995)

  25. King, M.: Testing for autocorrelation in linear regression models: a survey. In: King, M.L., Giles, D.E.A. (eds.) Specification Analysis in the Linear Model, pp. 19–73. Routledge Kegan & Paul, London (1987)

  26. Lehmann, E.L.: Testing Statistical Hypotheses. Springer, New York (1997)

  27. Ljung, G.M., Box, G.E.P.: On a measure of lack of fit in time series models. Biometrika 65(2), 297–303 (1978)

  28. Zhou, Z.: Measuring nonlinear dependence in time-series, a distance correlation approach. J. Time Ser. Anal. 33(3), 438–457 (2012)

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Correspondence to Antonio Punzo.

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Bagnato, L., De Capitani, L. & Punzo, A. A diagram to detect serial dependencies: an application to transport time series. Qual Quant 51, 581–594 (2017). https://doi.org/10.1007/s11135-016-0426-y

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Keywords

  • Chi-squared statistic
  • Serial dependence
  • Reproducibility probability
  • Multi-way contingency table
  • Ljung–Box test