Mathematical Programming for Piecewise Linear Representation of Discrete Time Series

Abstract

Piecewise linear representation (PLR) of a time series arises in variety of disciplines in data mining. Unlike most PLR methods who separate a discrete time series into a few discontinuous line segments, \( \ell 1 \) trend filtering method is one of the few PLR methods who generates continuous line segment representations. However, the approximation errors of \( \ell 1 \) trend filtering seldom reach its minimum. In this paper, we propose a binary integer programming model to produce a continuous PLR of time series with the least approximation error, and therefore it is well suitable to analyzing time series with an underlying piecewise linear trend. We describe the motives of the proposed method and give some illustrative examples. The improvement in approximation error is demonstrated by some experiments on some real-world time series datasets.