Using Wavelets to Detect Trends

Edgar L. Andreas U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, New Hampshire

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George Treviño Mechanical Engineering–Engineering Mechanics Department, Michigan Technological University, Houghton, Michigan

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Abstract

Wavelets are a new class of basis functions that are finding wide use for analyzing and interpreting time series data. This paper describes a new use for wavelets—identifying trends in time series. The general signal considered has a quadratic trend. The inverted Haar wavelet and the elephant wavelet, respectively, provide estimates of the first-order and second-order coefficients in the trend polynomial. Unlike usual wavelet applications, however, this analysis requires only one wavelet dilation scale L, where L is the total length of the time series. Error analysis shows that wavelet trend detection is roughly half as accurate as least squares trend detection when accuracy is evaluated in terms of the mean-square error in estimates of the first-order and second-order trend coefficients. But wavelet detection is more than twice as efficient as least squares detection in the sense that it requires fewer than half the number of floating-point operations of least squares regression to yield the three coefficients of the quadratic trend polynomial. This paper demonstrates wavelet trend detection using artificial data and then various turbulence data collected in the atmospheric surface layer, and last, provides guidelines on when linear and quadratic trends are “significant” enough to require removal from a time series.

Corresponding author address: Dr. Edgar L Andreas, U.S. Army Cold Regions Research and Engineering Laboratory, Snow and Ice Division, 72 Lyme Road, Hanover, NH 03755-1290.

Email: eandreas@crrel.usace.army.mil

Abstract

Wavelets are a new class of basis functions that are finding wide use for analyzing and interpreting time series data. This paper describes a new use for wavelets—identifying trends in time series. The general signal considered has a quadratic trend. The inverted Haar wavelet and the elephant wavelet, respectively, provide estimates of the first-order and second-order coefficients in the trend polynomial. Unlike usual wavelet applications, however, this analysis requires only one wavelet dilation scale L, where L is the total length of the time series. Error analysis shows that wavelet trend detection is roughly half as accurate as least squares trend detection when accuracy is evaluated in terms of the mean-square error in estimates of the first-order and second-order trend coefficients. But wavelet detection is more than twice as efficient as least squares detection in the sense that it requires fewer than half the number of floating-point operations of least squares regression to yield the three coefficients of the quadratic trend polynomial. This paper demonstrates wavelet trend detection using artificial data and then various turbulence data collected in the atmospheric surface layer, and last, provides guidelines on when linear and quadratic trends are “significant” enough to require removal from a time series.

Corresponding author address: Dr. Edgar L Andreas, U.S. Army Cold Regions Research and Engineering Laboratory, Snow and Ice Division, 72 Lyme Road, Hanover, NH 03755-1290.

Email: eandreas@crrel.usace.army.mil

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