MEDL and MEDLA: Methods for Assessment of Scaling by Medians of Log-Squared Nondecimated Wavelet Coefficients

Abstract: High-frequency measurements and images acquired from various sources in the real world often possess a degree of self-similarity and inherent regular scaling. When data look like a noise, the scaling exponent may be the only informative feature that summarizes such data. Methods for the assessment of self-similarity by estimating Hurst exponent often involve analysis of rate of decay in a spectrum defined in various multiresolution domains. When this spectrum is calculated using discrete non-decimated wavelet transforms, due to increased autocorrelation in wavelet coefficients, the estimators of $H$ show increased bias compared to the estimators that use traditional orthogonal transforms. At the same time, non-decimated transforms have a number of advantages when employed for calculation of wavelet spectra and estimation of Hurst exponents: the variance of the estimator is smaller, input signals and images could be of arbitrary size, and due to the shift-invariance, the local scaling can be assessed as well. We propose two methods based on robust estimation and resampling that alleviate the effect of increased autocorrelation while maintaining all advantages of non-decimated wavelet transforms. The proposed methods extend the approaches in existing literature where the logarithmic transformation and pairing of wavelet coefficients are used for lowering the bias. In a simulation study we use fractional Brownian motions with a range of theoretical Hurst exponents. For such signals for which "true" $H$ is known, we demonstrate bias reduction and overall reduction of the mean-squared error by the two proposed estimators. For fractional Brownian motions, both proposed methods yield estimators of $H$ that are asymptotically normal and unbiased.