Many are familiar with the periodogram which estimates the power contained within a stationary time series as a function of frequency. For certain kinds of non-stationary time series more sophisticated tools exist that estimate frequency characteristics evolving through time. Instead of looking at the frequency properties of time series we examine behaviour operating at different scalesthis can be extremely appropriate for certain classes of time series.
This talk exhibits a complete suite of theory, methodology and software for time-scale modelling and analysis of non-stationary time series. We assume our time series fall into a class of models generated by a particular and flexible wavelet representation. This permits us to develop an evolutionary wavelet spectrum (EWS) that measures the power of a time series at a particular time and scale (or lag). We also introduce a time-dependent autocovariance function that is the inverse wavelet transform of the EWS (in much the same way as the autocovariance of a stationary time series is the inverse Fourier transform of the spectrum). Estimation of the EWS is performed by a smoothed wavelet periodogram which can be shown to have good theoretical properties.
We demonstrate the utility of our new methodology on infant heart rate data recorded by colleagues at the Institute of Child Health, Royal Hospital for Sick Children, Bristol. The time-scale power of these non-stationary time series can be estimated and shows excellent interpretability and can be related to other physiological variables such as sleep-state. Such information is not readily apparent using other established time-frequency approaches. This talk is based on joint work with Rainer von Sachs (Louvain-la-Neuve) and Gerald Kroisandt (Kaiserslautern).