Retrospective technology of segmentation and classification for GARCH models based on the concept of the $ \epsilon $-complexity of continuous functions

Abstract

We consider a retrospective segmentation and classification problem for GARCH models. Segmentation is the partition of a long time series into homogeneous fragments. A fragment is homogeneous if only one mechanism generates it. The points of "concatenation" of homogeneous segments we call (by analogy with the term used in the stochastic literature) points of disorder or change-points. We call classification the separation of two relatively short time series generated by different mechanisms. By classification, we mean the way in which two groups of time series with unknown generating mechanism (in particularly, generated by GARCH models) can be distinguished, and the new time series can be assigned to the class. Our model free technology is based on our concept of $ \epsilon $-complexity of individual continuous functions. This technology does not use information about the time series generation mechanism. We demonstrate our approach on time series generated by GARCH models. We present simulations and real data analysis results confirming the effectiveness of the methodology.