A Novel Wavelet Based Approach for Time Series Data Analysis
Thomas Meinl · Repository KITopen (Karlsruhe Institute of Technology) · 2011
Time series analysis is still a very wide field of research from both a theoretical point of view as well as amongst practitioners. Among the very first tasks in the analysis procedure is the estimation of long-term trends, that is, the separation of this generally slowly evolving component from any short-term fluctuations. Usually, the trend curve, which in most cases is expected to be smooth, can be extracted by a variety of different methods. However, in many application scenarios the trend must also account for sudden changes. These sudden changes comprise of not only jumps, but also other phenomena like steep slopes and valleys. This challenge constitutes an on-going problem for traditional trend estimation methods. While established filtering techniques either fail to capture these sudden changes accurately or are sensitive to high-amplitude fluctuations, the application of parametric methods is challenging due to the generally unknown trend and the innumerable shapes that these sudden changes can assume. This thesis proposes a trend extraction approach based on wavelet methods. The new algorithm, named local linear scaling approximation (LLSA), is developed by analyzing specific wavelet coefficient step response structures and by transferring these structures onto real signals. This procedure enables the analyst to extract a trend whose smoothness is comparable to the output of linear filtering techniques, while at the same time capturing the details of sudden changes with arbitrary shapes, an area in which usually most nonlinear filters excel. Therefore, LLSA can be seen as a novel approach to bridge the gap between linear and nonlinear filters. The algorithm was developed to be applicable on homogeneous time series without any further requirements on these, and to work with only two additional input parameters, which can also be set in a heuristic manner, yielding a directly implementable and usable method.