Quickest Detection of Changes in the Generating Mechanism of a Time Series via the ε-Complexity of Continuous Functions

Abstract

A novel methodology for the quickest detection of abrupt changes in the generating mechanisms (stochastic, deterministic, or mixed) of a time series, without any prior knowledge about them, is developed. This methodology has two components: the first is a novel concept of the ε-complexity and the second is a method for the quickest change point detection (Darkhovsky, 2013). The ε-complexity of a continuous function given on a compact segment is defined. The expression for the ε-complexity of functions with the same modulus of continuity is derived. It is found that, for the Hölder class of functions, there exists an effective characterization of the ε-complexity. The conjecture that the ε-complexity of an individual function from the Hölder class has a similar characterization is formulated. The algorithm for the estimation of the ε-complexity coefficients via finite samples of function values is described. The second conjecture that a change of the generating mechanism of a time series leads to a change in the mean of the complexity coefficients, is formulated. Simulations to support our conjectures and verify the efficiency of our quickest change point detection algorithm are performed.