Confidence bands for time series data

摘要

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.