Algorithms and technical material for Chapter 8
Provided here are 2 descriptions of the methodology used in the MATLAB functions for fitting VCZAR models to irregularly sampled time series.
The document in the file CSSTransitionAlgorithm has the title Numerical integration of the continuous time state space model. It explains how, given such a model, the transition over a discrete time step may be calculated by a formula of numerical integration. The method used carries over the square root structure of the continuous time disturbance variance matrix to that of the discrete time transition with minimal loss of precision. Results are presented for checks on the accuracy and comparative timings of the MATLAB functions which implement the integration.
The document in the file GaussianEstimation has the title Non-linear Gaussian estimation from orthogonal residuals. It describes the methodology used for calculating the score or gradient vector and an approximation of the Hessian matrix of the log-likelihood of a Gaussian model for which orthogonal residuals and their standard deviations are available. It further describes how these may be used in an iterative scheme to locate the model parameters which maximum the likelihood. This methodology is implemented in the MATLAB function VCZARestimate.m and its subsidiary functions, used to fit VCZAR models to irregularly sampled time series.