Scalar Least Squares Modeling

Abstract

In this chapter we review some of the applications of smoothness priors modeling of time series that can be done by least squares, or equivalently as linear Gaussian modeling. Smoothness priors trend estimation for scalar time series is treated in Section 4.1. There, the smoothness priors constraint is expressed as a k-th order random walk with a normally distributed zero-mean, unknown variance perturbation. The (normalized) variance is a hyperparameter of the prior distribution. This constraint is a time domain constraint on the priors. The concept of frequency domain priors is introduced and exploited in Sections 4.2 and 4.3. There, the fitting of a smoothness priors long AR model for the spectrum estimation of a scalar stationary time series and a smoothness priors model for transfer function estimation between two simultaneously observed time series, are respectively shown. In addition in Section 4.2, the superiority of the smoothness priors long AR model as compared to ordinary AIC AR model order determined spectral analysis, is demonstrated by a Monte Carlo computation of entropy.