Spectral analysis of time series with latent and irregular times
Many standard methods of time series analysis assume that observation times are both known and regularly-spaced. Regular sampling and known observation times are cornerstones of methods such as autoregressive/moving-average models and spectral estimation using the Fast Fourier Transform. When the measurement process is controlled by the experimenter, these assumptions can be largely met by design. However, there are cases in which the measurement process is not under complete experimental control, and the observation times are either irregular or unknown. For example, many data sets in astronomy have irregular sampling due to the effects of orbital geometry and interfering processes such as celestial bodies or atmospheric processes. In paleoclimate studies, time series data may consist of core samples of known depth, but unknown age. Extending common time series analysis methods to these types of data is a challenge. This thesis makes three key contributions. The first is a new Bayesian method for inferring the chronology -- or age versus depth relationship -- of a core taken from a sedimentary record. The second is an approximate multitaper statistic (mtLS) for irregularly sampled time series. The third is a Bayesian model for the spectrum inspired by the previous work of Mann and Lees (1996) and Thomson et al. (2001), who separate the spectrum into noise and signal components. Together, the three contributions are used for spectral inference of time series obtained from one peat core and three lake cores. The approach quantifies and includes uncertainty from both the chronology and the time series process. In addition, the application of the mtLS statistic as an estimator for the spectrum of an irregularly sampled time series in astronomy is presented.