  Confidence intervals are a standard technique for analyzing data. When
applied to time series, confidence intervals are computed for each time point
separately. Alternatively, we can compute confidence bands, where we are
required to find the smallest area enveloping $k$ time series, where $k$ is a
user parameter. Confidence bands can be then used to detect abnormal time
series, not just individual observations within the time series. We will show
that despite being an NP-hard problem it is possible to find optimal confidence
band for some $k$. We do this by considering a different problem: discovering
regularized bands, where we minimize the envelope area minus the number of
included time series weighted by a parameter $\alpha$. Unlike normal confidence
bands we can solve the problem exactly by using a minimum cut. By varying
$\alpha$ we can obtain solutions for various $k$. If we have a constraint $k$
for which we cannot find appropriate $\alpha$, we demonstrate a simple
algorithm that yields $O(\sqrt{n})$ approximation guarantee by connecting the
problem to a minimum $k$-union problem. This connection also implies that we
cannot approximate the problem better than $O(n^{1/4})$ under some (mild)
assumptions. Finally, we consider a variant where instead of minimizing the
area we minimize the maximum width. Here, we demonstrate a simple
2-approximation algorithm and show that we cannot achieve better approximation
guarantee.
