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.

Keywords:

• Change point detection
• Complexity
• Time series

Subject Classifications:

• 62L10
• 62M10
• 68Q30

ACKNOWLEDGMENTS

The authors thank Professor Wojbor Woyczynski for his valuable comments. They also thank an Associate Editor and the anonymous referees for the time they devoted to the article.

Notes

Recommended by Nitis Mukhopadhyay

Color versions of one or more of the figures in the article can be found online at http://www.tandfonline.com/lsqa.