Estimating the Confidence Interval of Evolutionary Stochastic Process Mean from Wavelet Based Bootstrapping

  • Aline Edlaine de Medeiros
  • Eniuce Menezes de Souza
Part of the ICSA Book Series in Statistics book series (ICSABSS)


A time series is a realization of a stochastic process, where each observation is considered in general as the mean of a Gaussian distribution for each time point t. The classical theory is built based on this supposition. However, this assumption may be frequently broken, mainly for non-stationary or evolutionary stochastic process. Thus, in this work we proposed to estimate the uncertainty for the evolutionary mean, μt, of a stochastic process based on bootstrapping of wavelet coefficients. The wavelet multiscale decomposition provides wavelet coefficients that have less autocorrelation than the observations in time domain, allowing to apply bootstrap methodologies. Several bootstrap methodologies based on discrete wavelet transform (DWT), also called wavestrapping, have been proposed in the literature to estimate the confidence interval of some statistics for a time series, such as the autocorrelation. In this paper we implemented these methods with few modifications and compared them to newly proposed methods based on non-decimated wavelet transform (NDWT), which is a translation invariant transform and more adequate for dealing with time series. Each realization of the bootstrap provides a surrogate time series, that imitates the trajectories of the original stochastic process, allowing to build a confidence interval for its mean for both stationary and non-stationary processes. As an application, the confidence interval of the mean rate of bronchiolitis hospitalizations for Paraná-BR state was estimated as well as its bias and standard errors.



The authors acknowledge and appreciate the anonymous reviewers for the valuable comments and such a positive feedback. The authors also thank the financial support of the Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES).


  1. Angelini, C., Cava, D., Katul, G., & Vidakovic, B. (2005). Resampling hierarchical processes in the wavelet domain: A case study using atmospheric turbulence. Physica D: Nonlinear Phenomena, 207(1), 24–40.MathSciNetCrossRefGoogle Scholar
  2. Breakspear, M., Brammer, M., & Robinson, P. A. (2003). Construction of multivariate surrogate sets from nonlinear data using the wavelet transform. Physica D: Nonlinear Phenomena, 182(1), 1–22.MathSciNetCrossRefGoogle Scholar
  3. Chatfield, C. (2016). The analysis of time series: an introduction. Boca Raton: CRC Press.zbMATHGoogle Scholar
  4. Daubechies, I. (1992). Ten lectures on wavelets —— 1. The what, why, and how of wavelets.
  5. Efron, B., & Gong, G. (1983). A leisurely look at the bootstrap, the jackknife, and cross-validation. The American Statistician, 37(1), 36–48.MathSciNetGoogle Scholar
  6. Golia, S. (2002). Evaluating the GPH estimator via bootstrap technique. In W. Härdle, B. Rönz (Eds.), Compstat (pp. 343–348). Heidelberg: Physica.CrossRefGoogle Scholar
  7. Kang, M., & Vidakovic, B. (2017). MEDL and MEDLA: Methods for assessment of scaling by medians of log-squared nondecimated wavelet coefficients. arXiv preprint arXiv:1703.04180.Google Scholar
  8. Mallat, S. G. (1989). A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7) , 674–693.CrossRefGoogle Scholar
  9. Morettin, P. A., & Toloi, C. (2006). Análise de séries temporais. Spindale: Blucher.Google Scholar
  10. Nason, G. P., & Silverman, B. W. (1995). The stationary wavelet transform and some statistical applications. In A. Antoniadis & G. Oppenheim (Eds.), Wavelets and statistics (pp. 281–299). New York: Springer.CrossRefGoogle Scholar
  11. Percival, D., Sardy, S., & Davison, A. (2000). Wavestrapping time series: Adaptive wavelet-based bootstrapping. In Nonlinear and nonstationary signal processing (pp. 442–471). Cambridge: Cambridge University Press.Google Scholar
  12. Politis, D. N., & Romano, J. P. (1994). The stationary bootstrap. Journal of the American Statistical association, 89(428), 1303–1313.MathSciNetCrossRefGoogle Scholar
  13. R Core Team (2016). R: A language and environment for statistical computing. Vienna: R Foundation for Statistical Computing.Google Scholar
  14. Tang, L., Woodward, W. A., & Schucany, W. R. (2008). Undercoverage of wavelet-based resampling confidence intervals. Communications in Statistics – Simulation and Computation®, 37(7), 1307–1315.MathSciNetCrossRefGoogle Scholar
  15. Wei, W. S. W. (2006). Time series analysis: univariate and multivariate methods. Boston: Pearson Addison Wesley.zbMATHGoogle Scholar
  16. Wornell, G., & Oppenheim, A. V. (1996). Signal processing with fractals: A wavelet-based approach. Upper Saddle River: Prentice Hall Press.Google Scholar
  17. Yi, J.-S., Jung, Y.-Y., Jacko, J., Sainfort, F., & Vidakovic, B. (2007). Parallel wavestrap: Simulating acceleration data for mobile context simulator. Current Development in Theory and Applications of Wavelets, 1, 251–272.MathSciNetGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Aline Edlaine de Medeiros
    • 1
  • Eniuce Menezes de Souza
    • 2
  1. 1.Graduate Program in BiostatisticsState University of MaringaMaringaBrazil
  2. 2.Department of StatisticsState University of MaringaMaringaBrazil

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