It might also be the signal component st of the measurement yt = st +ut. The measurements zt consist of the measured outputs yt and, when appropriate, the inputs ut. | Adaptive Filtering and Change Detection Fredrik Gustafsson Copyright 2000 John Wiley Sons Ltd ISBNs 0-471-49287-6 Hardback 0-470-84161-3 Electronic Part V Theory Adaptive Filtering and Change Detection Fredrik Gustafsson Copyright 2000 John Wiley Sons Ltd ISBNs 0-471-49287-6 Hardback 0-470-84161-3 Electronic 12 Evaluation theory . Filter . Filter . Performance . Monte Carlo . . MCMC and Gibbs . Evaluation of change . . The ARL . Performance . The MDL . Auto-tuning and . Filter evaluation . Filter definitions Consider a general linear filter or estimator at iZt it i tl as illustrated in Figure . Typically the estimated quantity x is either the parameter vector 0 in a parametric model or the state x in a state space model. It might also be the signal component st of the measurement yt st vf. The measurements zt consist of the measured outputs yt and when appropriate the inputs Uf. The basic definitions that will be used for a linear filter are A time-invariant filter has cttj at for all t and i. A non-causal filter has ti 0. This is used in smoothing. If ti 0 the filter is causal. 428 Evaluation theory Estimator Xl Figure . An estimator takes the observed signal zt and transforms it to estimates xt- A causal filter is HR Infinite Impulse Response if 2 oo otherwise it is FIR Finite Impulse Response . xt is a k-step ahead prediction if ti k 0. Most filters use all past data so 2 00 and the notation xt t-is sometimes useful for highlighting that the estimator is a predictor t 0 a smoother ti 0 or a filter ti 0 . . Performance measures The estimator gives a so called point estimate of x. For evaluation we are interested in the variability between different realizations One such measure is the covariance .
