Time series are an important data class that includes recordings ranging from radio emissions, seismic activity, global positioning data, and stock prices to EEG measurements, vital signs, and voice recordings. Rapid growth in sensor and recording technologies is increasing the production of time series data and the importance of rapid, accurate analyses. Time series data are commonly analyzed using time-varying spectral methods to characterize their nonstationary and often oscillatory structure. Current methods provide local estimates of data features. However, they do not offer a statistical inference framework that applies to the entire time series. The important advances that we report are state-space multitaper (SS-MT) methods, which provide a statistical inference framework for time-varying spectral analysis of nonstationary time series. We model nonstationary time series as a sequence of second-order stationary Gaussian processes defined on nonoverlapping intervals. We use a frequency-domain random-walk model to relate the spectral representations of the Gaussian processes across intervals. The SS-MT algorithm efficiently computes spectral updates using parallel 1D complex Kalman filters. An expectation–maximization algorithm computes static and dynamic model parameter estimates. We test the framework in time-varying spectral analyses of simulated time series and EEG recordings from patients receiving general anesthesia. Relative to standard multitaper (MT), SS-MT gave enhanced spectral resolution and noise reduction (>10 dB) and allowed statistical comparisons of spectral properties among arbitrary time series segments. SS-MT also extracts time-domain estimates of signal components. The SS-MT paradigm is a broadly applicable, empirical Bayes’ framework for statistical inference that can help ensure accurate, reproducible findings from nonstationary time series analyses.
Time Series Forecasting with Gaussian Processes Needs Priors
