Self-similarity index estimation via wavelets for locally self-similar processes

Many naturally occurring phenomena can be effectively modeled using self-similar processes. In such applications, accurate estimation of the scaling exponent is vital, since it is this index which characterizes the nature of the self-similarity. Although estimation of the scaling exponent has been extensively studied, previous work has generally assumed that this parameter is constant. Such an assumption may be unrealistic in settings where it is evident that the nature of the self-similarity changes as the phenomenon evolves. For such applications, the scaling exponent must be allowed to vary as a function of time, and a procedure must be available which provides a statistical characterization of this progression. In what follows, we propose and describe such a procedure. Our method uses wavelets to construct local estimates of time-varying scaling exponents for locally self-similar processes. We establish a consistency result for these estimates. We investigate the effectiveness of our procedure in a simulation study, and demonstrate its applicability in the analyses of a hydrological and a geophysical time series, each of which exhibit locally self-similar behavior.