LASS: a tool for the local analysis of self-similarity

The Hurst parameter H characterizes the degree of long-range dependence (and asymptotic self-similarity) in stationary time series. Many methods have been developed for the estimation of H from data. In practice, however, the classical estimation techniques can be severely affected by non-stationary artifacts in the time series. In fact, the assumption that the data can be modeled by a stationary process with a single Hurst exponent H may be unrealistic. We focus on practical issues associated with the detection of long-range dependence in Internet traffic data and develop two tools designed to address some of these issues. The first is an animation tool which is used to visualize the local dependence structure. The second is a statistical tool for the local analysis of self-similarity (LASS). The LASS tool is designed to handle time series that have long-range dependence and are long enough that some parts are essentially stationary, while others exhibit non-stationarity, which are either deterministic or stochastic in nature. The tool uses wavelets to analyze the local dependence structure in the data over a set of windows. It can be used to visualize local deviations from self-similar, long-range dependence scaling and to provide reliable local estimates of the Hurst exponents. The tool, which is illustrated by using a trace of Internet traffic measurements, can also be applied to economic time series. We also develop a median-based wavelet spectrum which can be used to obtain robust local or global estimates of the the Hurst parameter that are less susceptible to local nonstationarity. We make the software tools freely available and describe their use in an appendix.