# Confidence bands for time series data

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## Abstract

Simultaneous confidence intervals, or *confidence bands*, provide an intuitive description of the variability of a time series. Given a set of \(N\) time series of length \(M\), we consider the problem of finding a confidence band that contains a \((1-\alpha )\)-fraction of the observations. We construct such confidence bands by finding the set of \(N\!\!-\!\!K\) time series whose envelope is minimized. We refer to this problem as the *minimum width envelope* problem. We show that the minimum width envelope problem is \(\mathbf {NP}\)-hard, and we develop a greedy heuristic algorithm, which we compare to quantile- and distance-based confidence band methods. We also describe a method to find an effective confidence level \(\alpha _{\mathrm {eff}}\) and an effective number of observations to remove \(K_{\mathrm {eff}}\), such that the resulting confidence bands will keep the family-wise error rate below \(\alpha \). We evaluate our methods on synthetic and real datasets. We demonstrate that our method can be used to construct confidence bands with guaranteed family-wise error rate control, also when there is too little data for the quantile-based methods to work.

### Keywords

Simultaneous confidence interval Confidence band Time series Multiplicity correction Family-wise error rate## Notes

### Acknowledgments

The authors would like to thank Andreas Henelius for helpful discussions and suggestions. The work of J. Korpela and K. Puolamäki was supported in part by the Revolution of Knowledge Work Project, funded by Tekes (The Finnish Funding Agency for Innovation).

## Supplementary material

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