On the trend, detrending, and variability of nonlinear and nonstationary time series

*Center for Ocean-Land-Atmosphere Studies, 4041 Powder Mill Road, Suite 302, Calverton, MD 20705;
^†Research Center for Adaptive Data Analysis, National Central University, Chungli 32054, Taiwan, Republic of China;
^‡Ocean Sciences Branch, Code 614.2, National Aeronautics and Space Administration Goddard Space Flight Center, Wallops Flight Facility, Wallops Island, VA 23337; and
^§Division of Interdisciplinary Medicine and Biotechnology, Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, MA 02215

Edited by Inez Y. Fung, University of California, Berkeley, CA, and approved July 27, 2007 (received for review February 9, 2007)

Abstract

Determining trend and implementing detrending operations are important steps in data analysis. Yet there is no precise definition of “trend” nor any logical algorithm for extracting it. As a result, various ad hoc extrinsic methods have been used to determine trend and to facilitate a detrending operation. In this article, a simple and logical definition of trend is given for any nonlinear and nonstationary time series as an intrinsically determined monotonic function within a certain temporal span (most often that of the data span), or a function in which there can be at most one extremum within that temporal span. Being intrinsic, the method to derive the trend has to be adaptive. This definition of trend also presumes the existence of a natural time scale. All these requirements suggest the Empirical Mode Decomposition (EMD) method as the logical choice of algorithm for extracting various trends from a data set. Once the trend is determined, the corresponding detrending operation can be implemented. With this definition of trend, the variability of the data on various time scales also can be derived naturally. Climate data are used to illustrate the determination of the intrinsic trend and natural variability.

Footnotes

^¶To whom correspondence should be addressed. E-mail: zhwu{at}cola.iges.org
Author contributions: N.E.H. designed research; Z.W., N.E.H., S.R.L., and C.-K.P. performed research; N.E.H. contributed new reagents/analytic tools; Z.W., N.E.H., and S.R.L. analyzed data; and Z.W., N.E.H., S.R.L., and C.-K.P. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
↵ ‖ A dyadic filter bank is a collection of band-pass filters that have a constant band-pass shape (e.g., a Gaussian distribution) but with neighboring filters covering half of or double the frequency range of any single filter in the bank. The frequency ranges of the filters can be overlapped. For example, a simple dyadic filter bank can include filters covering frequency windows such as 50 to 120 Hz, 100 to 240 Hz, 200 to 480 Hz, etc.
Abbreviations:

EMD,

Empirical Mode Decomposition;

GSTA,

global surface air temperature anomaly;

IMF,

intrinsic mode function.