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67number of pixels minus the number of boundary pixels). The number of degreesof freedom D K in this equation is equal to the number of parameters estimated;in other words, D K is equal to the number of elements in θ, which contains the 4autoregressive paramters, K − 1 mixture proportions, and the density parameters.For a Gaussian mixture, D K = 3K + 3; for a Poisson mixture, D K = 2K + 3.To choose the number of segments, we compute BIC(K) for several valuesof K, starting with K = 1. We then increase K until we find a local maximumin BIC(K); the value of K at this first local maximum is chosen as the optimalnumber of segments. This scheme for choosing K is reasonable when one expects asmall number of segments; it avoids problems with the model failing to hold whenlarge numbers of segments are fitted.Unfortunately, although the results presented in this chapter for using BIC withautoregressive data are valid, the raster scan autoregression model is not good forimage segmentation, as shown in the next section.4.6 Application of the RSA Model to Image DataAlthough it was initially thought that the RSA model might provide a computationallyfast way of performing image segmentation, it turns out that the RSAmodel is a bad fit for image data. This is because a model with too few segmentswill result in a very large R 2 value due to large changes in the mean from segmentto segment. This simple flaw can be illustrated even with a one-dimensionalexample.Figure 4.1 shows two time series. The first was generated by an AR(1) process(autoregressive coefficient 0.95, Gaussian noise with mean 0.5 and standard deviation1), while the second consists of two sequences of independent Gaussian noise(means 5 and 15, both with standard deviation 1). When we fit an AR(1) modelto each of these signals, a high level of autoregressive dependence is evident; the