Using Wavelets to Detect Trends

Abstract

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 thr...