Improved moment scaling estimation for multifractal signals

A fundamental problem in the analysis of multifractal processes is to estimate the scaling exponent K(q) of moments of different order q from data. Conventional estimators use the empirical moments < html >< body > mu < span style='margin-left: -.6em; vertical-align: super;'>boolean AND </span ></body ></html >(q)(r)=aY vertical bar epsilon(r)(tau)vertical bar(q)aY (c) of wavelet coefficients epsilon(r)(tau), where tau is location and r is resolution. For stationary measures one usually considers 'wavelets of order 0' (averages), whereas for functions with multifractal increments one must use wavelets of order at least 1. One obtains < html >< body > K < span style='margin-left: -.6em; vertical-align: super;'>boolean AND </span ></body ></html >(q) as the slope of log(< html >< body > mu < span style='margin-left: -.6em; vertical-align: super;'>boolean AND </span ></body ></html >(q)(r)) against log(r) over a range of r. Negative moments are sensitive to measurement noise and quantization. For them, one typically uses only the local maxima of vertical bar epsilon(r)(tau)vertical bar (modulus maxima methods). For the positive moments, we modify the standard estimator < html >< body > K < span style='margin-left: -.6em; vertical-align: super;'>boolean AND </span ></body ></html >(q) to significantly reduce its variance at the expense of a modest increase in the bias. This is done by separately estimating K(q) from sub-records and averaging the results. For the negative moments, we show that the standard modulus maxima estimator is biased and, in the case of additive noise or quantization, is not applicable with wavelets of order 1 or higher. For these cases we propose alternative estimators. We also consider the fitting of parametric models of K(q) and show how, by splitting the record into sub-records as indicated above, the accuracy of standard methods can be significantly improved.