Jump process for the trend estimation of time series

Abstract

A jump process approach is proposed for the trend estimation of time series. The proposed jump process estimator can locally minimize two important features of a trend, the smoothness and fidelity, and explicitly balance the fundamental tradeoff between them. A weighted average form of the jump process estimator is derived. The connection of the proposed approach to the Hanning filter, Gaussian kernel regression, the heat equation and the Wiener process is discussed. It is found that the weight function of the jump process approaches the Gaussian kernel, as the smoothing parameter increases. The proposed method is validated through numerical applications to both real data analysis and simulation study, and a comparison with the Henderson filter.