The criterion. As with all time series modeling, it is necessary that we be able to

predict the data a step in advance in the future given previous data, which means that

some measure of the prediction error has to remain small. Some adaptive systems use the

instantaneous error ep(n) as error measure. Unfortunately, for our application ep(n) cannot

be used since it is shown that the sequence ep is an insufficient statistic [12], which means

that different experts could produce identical error statistics (or there is a many to one

mapping from different regimes to identical prediction error statistics). This also explains

why segmentation cannot be performed by only tracking the data with one adaptive

system and monitoring ep(n). Besides, in cases where the return maps of several regimes

tend to be close and where the data is noisy, using the instantaneous prediction error as

measure for the performance of the experts will probably cause the segmentation to

exhibit spurious switching (or false alarm), even when it is known that the data switches

less frequently.

 As can be seen in Figure 2-2 on a trivial example, if the segmentation is performed

using ep(n) as criterion, even though Model 1 is not a good predictor at all for the data it

is still seen as the best predictor in areas A and B. In order to avoid this kind of "rattling"

from a model to another, a solution that brings memory to the performance criterion is

introduced [1].

 The criterion has to be a measure of an average performance at that point in time,

so an good criterion would be the expected value of the square error E[ep2]. With the

useful lie of assuming that the system is ergodic, it can be expressed by the mean square

error (MSE).