Adaptive Filtering and Change Detection Fredrik Gustafsson Copyright © 2000 John Wiley & Sons, Ltd ISBNs: 0-471-49287-6 (Hardback); 0-470-84161-3 (Electronic) Off-line approaches 4.1 Basics 89 4.2 Segmentationcriteria 91 4.2.1 ML change timesequenceestimation 91 4.2.2 Information based segmentation 92 4.3 On-line local search for optimum 94 4.3.1 Local treesearch 94 4.3.2 A simulation example 95 4.4 Off-line global search for optimum 98 4.4.1 Local minima 98 101 4.4.2 An MCMC approach 4.5 Changepointestimation 102 4.5.1 The Bayesian approach 103 4.5.2 Themaximum likelihood approach 104 4.5.3 Anon-parametricapproach 104 4.6 Applications 106 4.6.1 Photon emissions 106 4.6.2 Altitude sensor quality 107 4.6.3 Rat EEG 108 4.1 Basics This chapter surveys off-line formulations of single and multiple change point estimation Although the problemformulation yields algorithms that process data batch.wise, many important algorithms have natural on-line implementations and recursive approximations This chapter is basically a projection of the more general results in Chapter to the case of signal estimation There are, however some dedicated algorithms for estimating one change point offline that apply to the current case of a scalar signal model In the literature of mathematical statistics this areais known as change point estimation In segmentation the goal is to find a sequence kn = ( k ~k2 kn) of time indices where both the numbern and the locations Ici are unknown such that Off-line amroaches 90 the signal can be accurately described as piecewise constant, i.e., is a good description of the observed signal yt The noise variance will be denoted E(ei) = R The standard assumption is that et E N(0, R ) , but there are other possibilities Equation (4.1) will be thesignal model used throughout this chapter, but it should be noted that an important extension to the case where the parameter is slowly varying within each segment is possible with minor modifications However, equation (4.1) illustrates the basic ideas One way to guarantee that thebest possible solution is found is to consider all possible segmentations kn, estimate the mean in each segment, and then choose the particular kn that minimizes an optimality criteria, h kn = arg n>l,O 80, where 81 > 80, where 81 # 80, where 81 # 80, where 190 is unknown 130 = is known... (change point) is given by p3 ,if = argmax I