This article introduces the class of continuous time locally stationary
wavelet processes. Continuous time models enable us to properly provide
scale-based time series models for irregularly-spaced observations for the
first time. We derive results for both the theoretical setting, where we assume
access to the entire process sample path, and a more practical one, which
develops methods for estimating the quantities of interest from sampled time
series. The latter estimates are accurately computable in reasonable time by
solving the relevant linear integral equation using the iterative thresholding
method due to Daubechies, Defrise and De~Mol. We exemplify our new methods by
computing spectral and autocovariance estimates on irregularly-spaced
heart-rate data obtained from a recent sleep-state study.Comment: 33 pages, 12 figure