Title: 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.