New Frontiers of Biostatistics and Bioinformatics pp 161-174 | Cite as

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

## Abstract

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, *μ*_{t}, 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.

## Notes

### Acknowledgements

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).

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