Detecting change in a time-series (Corresp.)

A method is presented which provides a criterion for detecting a change in the structure of a model generating a stochastic sequence. Models that can be represented by a sequence of predictive probability distributions are considered. The method is based on the transformation of the observed sequence
\{x_{n}\}
into a sequence of partial sums of the general innovations, computed for the sequence
\{-\log f(x_{n}|x_{n-1},x_{n-2}, \cdots ,x_{0})\}
. If no change occurs the transformed sequence behaves like a Wiener process, but its mean will exhibit a monotonic growth after the process changes. Based on the properties of this transformation, fixed sample size and sequential tests for the change are constructed. The technique is applied to test for a change in the mean vector in a sequence of (generally dependent) Gaussian random variables, a change of coefficients of an autoregressive process, and a change of distribution in a sequence of discrete independent identically distributed random variables.