The problem of segmentation considered here is: Given a time series {x(t), t = 1,2,...,n}, partition the set of values of t into segments (sub-series, regimes) within which the behavior of x(t) is homogeneous. The segments are considered as falling into several classes.

Examples. (i) An economic time series is to be segmented into the four classes, depression, recession, recovery, and expansion. (ii) An electrocardiogram is to be segmented into rhythmic and arhythmic periods. (iii) A returned radar signal is to be divided into segments corresponding to background, target, background again, another target, etc. (iv) A multiple time series of electroencephalographic and eye-movement measurements on a sleeping person is to be segmented into periods of deep sleep and restless or fitful sleep.

The observation X may be a scalar, vector, or matrix--any element of a linear space, for which the operations of addition and scalar multiplication are defined. (If X is a scalar, operations such as x(t) - cx(t-1), where c is a scalar, are required. If X is a vector or matrix, the operation x(t) - Cx(t-1), where C is a matrix, is required.)

In some applications the definition of the classes involves the possible observed values of X. In this case the classes may be viewed simply as a partition of its value-space. In other cases the definition of the classes is logically independent of the value-space of X.