Time and Frequency Domain Models of Scalar Time Series

Abstract

The most important characteristic of a scalar time series—its spectral density—can be estimated through nonparametric and parametric methods; when the time series is long, the results will be reliable for all mathematically proper methods. The parametric (autoregressive (AR) in this book) approach is relatively reliable with short time series and always provides explicit time domain information in the form of stochastic difference equations, which cannot be obtained through nonparametric methods. The AR approach provides an analytical expression for the spectral density estimate, which is statistically reliable when the AR order is chosen through order selection criteria. Five such criteria are used in this book. The efficiency of autoregressive (or maximum entropy) spectral estimation is demonstrated through parametric and nonparametric analyses of five very short time series whose structures are typical for geophysical data. The AR approach provides an equation for time series extrapolation, for finding its natural frequencies and damping coefficients, and for getting quantitative estimates of time series dependence on its past and upon its innovation sequence. The AR approach is strongly recommended for analysis of geophysical time series. A good nonparametric method of spectral analysis is Thomson’s multitaper method.