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Long-Range Dependence in Directional Data

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Directional Statistics for Innovative Applications

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Abstract

The necessity of extending circular methods to time series data was recognized in early papers by Wehrly and Johnson (Biometrika 66:255–256, 1980 [57]), Breckling (Springer, Berlin, 1989 [8]), and Fisher and Lee (J R Stat Soc Ser B (Methodological) 56(2) 327–339, 1994 [17]), among others. Most methods for circular time series assume short-range dependence. In this paper, we review some recent results on circular time series with long-range dependence. In contrast to weak dependence, circular autocorrelations are nonsummable. This has major implications for statistical inference. Extensions to nonstationary processes are also discussed.

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References

  1. Abry, P., Flandrin, P., Taqqu, M.S., Veitch, D.: Self-similarity and long-range dependence through the wavelet lens. In: Doukhan, P., Oppenheim, G., Taqqu, M.S. (eds.) Long-range Dependence: Theory and Applications, pp. 527–556. Birkhäuser, Boston (2003)

    MATH  Google Scholar 

  2. Bai, Z.D., Rao, C.R., Zhao, L.C.: Kernel estimators of density function of directional data. J. Multivar. Anal. 27(1), 24–39 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beran, J.: Fitting long-memory models by generalized linear regression. Biometrika 80, 817–822 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beran, J.: Statistics for Long-Memory Processes. Monographs on Statistics and Applied Probability, vol. 61. Chapman and Hall/CRC, New York (1994)

    Google Scholar 

  5. Beran, J., Feng, Y., Ghosh, S., Kulik, R.: Long Memory Processes: Probabilistic Properties and Statistical Methods. Springer, Berlin/Heidelberg (2013)

    Book  MATH  Google Scholar 

  6. Beran, J., Ghosh, S.: Estimating the mean direction of strongly dependent circular time series. J. Time Ser. Anal. 41, 210–228 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beran, J., Steffens, B., Ghosh, S.: On nonparametric regression for bivariate circular long-memory time series. Preprint (2020)

    Google Scholar 

  8. Breckling, J.: The Analysis of Directional Time Series: Application to Wind Speed and Direction. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  9. Di Marzio, M., Panzera, A., Taylor, C.C.: Non-parametric smoothing and prediction for nonlinear circular time series. J. Time Ser. Anal. 33(4), 620–630 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dobrushin, R.L.: Gaussian random fields—Gibbsian point of view. In: Dobrushin, R.L., Sinai, Ya.G. (Eds.), Multicomponent Random Systems. Advances in Probability and Related Topics, vol. 6, pp. 119–152. Dekker, New York (1980)

    Google Scholar 

  11. Dobrushin, R.L., Major, P.: Non-central limit theorems for nonlinear functionals of Gaussian fields. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 50(1), 27–52 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fay, G., Moulines, E., Roueff, F., Taqqu, M.S.: Asymptotic normality of wavelet estimators of the memory parameter for linear processes. J. Time Ser. Anal. 30(5), 534–558 (2009)

    Google Scholar 

  13. Fisher, N.I.: Smoothing a sample of circular data. J. Struct. Geol. 11, 775–778 (1989)

    Article  Google Scholar 

  14. Fisher, N.I.: Statistical Analysis of Circular Data. Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  15. Fisher, N.I., Lewis, T.: Estimating the common mean direction of several circular or spherical distributions with different dispersions. Biometrika 70, 333–341 (1983)

    Google Scholar 

  16. Fisher, N.I., Lee, A.: Regression models for an angular response. Biometrics 48(3), 665–677 (1992)

    Article  MathSciNet  Google Scholar 

  17. Fisher, N.I., Lee, A.: Time series analysis of circular data. J. R. Stat. Soc. Ser. B (Methodological) 56(2), 327–339 (1994)

    MathSciNet  MATH  Google Scholar 

  18. Geweke, J., Porter-Hudak, S.: The estimation and application of long memory time series models. J. Time Ser. Anal. 4, 221–238 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ghosh, S.: On local slope estimation in partial linear models under Gaussian subordination. J. Stat. Plann. Inference 155, 42–53 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Giraitis, L., Koul, H.L., Surgailis, D.: Large Sample Inference for Long Memory Processes. Imperial College Press, London (2012)

    Book  MATH  Google Scholar 

  21. Gould, A.L.: A regression technique for angular variates. Biometrics 25, 683–700 (1969)

    Article  Google Scholar 

  22. Hall, P., Watson, G.S., Cabrera, J.: Kernel density estimation with spherical data. Biometrika 74(4), 751–762 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hurvich, C.M.: Model selection for broadband semiparametric estimation of long memory in time series. J. Time Ser. Anal. 22, 679–709 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hurvich, C.M., Brodsky, J.: Broadband semiparametric estimation of the memory parameter of a long-memory time series using fractional exponential models. J. Time Ser. Anal. 22, 221–249 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hurvich, C.M., Moulines, E., Soulier, P.: The FEXP estimator for potentially nonstationary linear time series. Stoch. Proc. Their Appl. 97, 307–340 (2002)

    Article  MATH  Google Scholar 

  26. Jammalamadaka, S.R., Sarma, Y.R.: A correlation coefficient for angular variables. In: Theory, Statistical, Analysis, Data, pp. 349–364. Matusita, North Holland, New York (1988)

    Google Scholar 

  27. Jammalamadaka, S.R., SenGupta, A.: Topics in Circular Statistics. World Scientific, Singapore (2001)

    Book  Google Scholar 

  28. Johnson, R.A., Wehrly, T.E.: Some angular-linear distributions and related regression models. J. Am. Stat. Assoc. 73(363), 602–606 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kato, S.: A Markov process for circular data. J. R. Stat. Soc. Ser. B 72, 655–672 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kato, S., Shimizu, K., Shieh, G.S.: A circular-circular regression model. Stat. Sinica 18(2), 633–645 (2008)

    MathSciNet  MATH  Google Scholar 

  31. Kim., S., SenGupta, A.: Multivariate-multiple circular regression. J. Stat. Comput. Simul. 87, 1–15 (2016)

    Google Scholar 

  32. Künsch, H.R.: Statistical aspects of self-similar processes. In: Prohorov, Yu., Sazonov, V.V. (eds.) Proceeding 1st World Congress of the Bernoulli Society, vol. 1, pp. 67–74. VNU Science Press, Utrecht (1987)

    Google Scholar 

  33. Ley, C., Verdebout, T.: Modern Directional Statistics. Chapman & Hall/CRC, Boca Raton (2017)

    Book  MATH  Google Scholar 

  34. Mardia, K.V.: Statistics of Directional Data. Academic Press, London (1972)

    MATH  Google Scholar 

  35. Mardia, K.V., Jupp, P.E.: Directional Statistics. Wiley, New York (2000)

    Google Scholar 

  36. Mielniczuk, J., Wu, W.B.: On random-design model with dependent errors. Stat. Sin. 14, 1105–1126 (2004)

    MathSciNet  MATH  Google Scholar 

  37. Modlin, D., Fuentes, M., Reich, B.: Circular conditional autoregressive modeling of vector fields. Environmetrics 23, 46–53 (2012)

    Article  MathSciNet  Google Scholar 

  38. Moulines, E., Soulier, P.: Broadband log-periodogram regression of time series with long-range dependence. Ann. Stat. 27, 1415–1439 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Moulines, E., Soulier, P.: Data driven order selection for projection estimation of the spectral density of time series with long range dependence. J. Time Ser. Anal. 21, 193–218 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  40. Narukawa, M., Matsuda, Y.: Broadband semi-parametric estimation of long-memory time series by fractional exponential models. J. Time Ser. Anal. 32, 175–193 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Pearson, K.: The problem of the random walk. Nature 72, 294–342 (1905)

    Article  MATH  Google Scholar 

  42. Pewsey, A., Neuhäuser, M., Ruxton, G.D.: Circular Statistics in R. Oxford University Press, Oxford (2013)

    MATH  Google Scholar 

  43. Polsen, O., Taylor, C.: Parametric circular-circular regression and diagnostic analysis. In: Dryden, I.L., Kent, J.T. (eds.) Geometry driven statistics, pp. 115–128 (2015)

    Google Scholar 

  44. Rayleigh, L.: On the resultant of a large number of vibrations of the same pitch and of arbitrary phase. Philos. Magaz. 10, 73–78 (1880)

    Google Scholar 

  45. Robinson, P.M.: Time series with strong dependence. In: Sims, C.A. (ed.) Advances in Econometrics: Sixth World Congress, vol. 1, pp. 47–95. Cambridge University Press, Cambridge (1994)

    Chapter  Google Scholar 

  46. Robinson, P.M.: Log-periodogram regression of time series with long range dependence. Ann. Stat. 23(3), 1048–1072 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  47. Robinson, P.M.: Gaussian semiparametric estimation of long range dependence. Ann. Stat. 23(5), 1630–1661 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  48. Rosenblatt, M.: Independence and dependence. In: Proceedings of the 4th erkeley Symposium on Mathematical Statistics and Probability, vol. II, pp. 431–443. University of California Press, Berkeley (1961)

    Google Scholar 

  49. Rosenblatt, M.: Some limit theorems for partial sums of quadratic forms in stationary Gaussian variables. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 49(2), 125–132 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  50. Taniguchi, M., Kato, S., Ogata, H., Pewsey, A.: Models for circular data from time series spectra. J. Time Ser. Anal. 41(6), 808–829 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  51. Taqqu, M.S.: Weak convergence to fractional Brownian motion and to the Rosenblatt process. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 31, 287–302 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  52. Taqqu, M.S.: Convergence of integrated processes of arbitrary Hermite rank. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 50, 53–83 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  53. Taylor, C.C.: Automatic bandwidth selection for circular density estimation. Comput. Stat. Data Anal. 52(7), 3493–3500 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  54. Tsurata, Y., Sagae, M.: Higher order kernel density estimation on the circle. Stat. Prob. Lett. 131, 46–50 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  55. Wang, F., Gelfand, A.E.: Space and space-time directional data using projected Gaussian processes. J. Am. Stat. Assoc. 109(508), 1565–1580 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  56. Watson, G.S.: Statistics on Spheres. Wiley, New York (1983)

    MATH  Google Scholar 

  57. Wehrly, T.E., Johnson, R.A.: Bivariate models for dependence of angular observations and a related Markov process. Biometrika 66, 255–256 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  58. Yajima, Y.: On estimation of a regression model with long term errors. Ann. Stat. 16(2), 791–807 (1988)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Konstanz, Konstanz, Germany

    Jan Beran & Britta Steffens

  2. Statistics Lab, Swiss Federal Research Institute WSL, Birmensdorf, Switzerland

    Sucharita Ghosh

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  1. Jan Beran
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  2. Britta Steffens
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  3. Sucharita Ghosh
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Correspondence to Jan Beran .

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Editors and Affiliations

  1. Applied Statistics Unit, Indian Statistical Institute, Kolkata, West Bengal, India

    Ashis SenGupta

  2. Department of Statistics, University of California, Riverside, CA, USA

    Barry C. Arnold

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Beran, J., Steffens, B., Ghosh, S. (2022). Long-Range Dependence in Directional Data. In: SenGupta, A., Arnold, B.C. (eds) Directional Statistics for Innovative Applications. Forum for Interdisciplinary Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-19-1044-9_21

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