Slepian wavelet variances for regularly and irregularly sampled time series

We discuss approximate scale-based analysis of variance for Gaussian time series based upon Slepian wavelets. These wavelets arise as eigenfunctions of an energy maximization problem in a pass band of frequencies. Unlike the commonly used Daubechies wavelets, Slepian wavelets have the ability to accommodate both regularly and irregularly sampled data. For regularly sampled Gaussian time series, we derive statistical theory for Slepian-based wavelet variances and show that it is comparable to Daubechies-based variances. For irregularly sampled time series data, we derive a corresponding statistical theory for Slepian-based wavelet variances. We demonstrate its use on X-ray fluctuations from a binary star system and on a light curve from the variable star Z UMa.