Confidence bands for time series data

Abstract

Simultaneous confidence intervals, or confidence bands, provide an intuitive description of the variability of a time series. Given a set of $$N$$ N time series of length $$M$$ M , we consider the problem of finding a confidence band that contains a $$(1-\alpha )$$ ( 1 - ? ) -fraction of the observations. We construct such confidence bands by finding the set of $$N\!\!-\!\!K$$ 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 $$\mathbf {NP}$$ 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 $$\alpha _{\mathrm {eff}}$$ ? eff and an effective number of observations to remove $$K_{\mathrm {eff}}$$ K 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.