Locally Self-Similar Processes and Their Wavelet Analysis

• Activity Number: 77
• Type: Contributed
• Date/Time: Monday, August 12, 2002 : 8:30 AM to 10:20 AM
• Sponsor: Section on Physical & Engineering Sciences*
• Author(s): J. Davis*+ and Joseph Cavanaugh and Yazhen Wang
• Affiliation(s): University of Missouri, Columbia and University of Missouri, Columbia and University of Connecticut
• Keywords: long memory ; long-range dependence ; local stationarity ; wavelets ; time series

Abstract

Many natural phenomena can be effectively modeled using self-similar processes. In such applications, accurate estimation of the scaling exponent is essential since it fundamentally characterizes the self-similarity. Estimation of the scaling exponent has been extensively studied, yet previous work has generally assumed that this parameter is constant. Such an assumption may be unrealistic in settings where it is apparent that the nature of the self-similarity evolves as the phenomenon progresses. Serial data arising in these settings is best modeled using locally self-similar processes, which allow for the scaling exponent to change over time.

We outline a method for estimating the time-varying scaling function for a series that exhibits locally self-similar behavior. Our procedure utilizes the discrete wavelet transformation of the series, exploiting an approximate log-linear relationship between the scale and the energy content. We propose a method based on this relationship for constructing confidence bands. The effectiveness of the procedures for estimating the function and assigning error bounds is examined via simulations. We close with an application involving real data.