Wavelet-based Scaling Indices for Machine Learning: Robust Trimean Estimators

Speaker: Branislav Vidakovic, Georgia Tech/NSF

Abstract: Massive data sets, functional data, and high-frequency sampled processes intrinsically invariant to changes in scale are routinely observed and stored. A catchphrase "scaling is omnipresent" certainly holds for many high frequency time series and high-resolution images. General multiscale domains provide an environment for analyzing, describing, and modeling data that scale, and for unifying several related attributes describing regularity, fractality, multi-fractality, self-similarity, and long memory.

In this talk we focus on the wavelet-based estimation of scaling indices. In particular, we focus on nondecimated and scale-mixing decompositions. They result in a hierarchy of imbedded multiresolution subspaces that lead to a multiscale spectra. Like in the Fourier transforms, where the rate of linear  decay of the log-power spectra over the frequencies characterizes the regularity/smoothness of a time series/image, the slopes in the regression involving the log-averages of squared wavelet coefficients, lead to alternative and arguably more local and stable descriptors of signal/image regularity. Such descriptors, as arguments in ML procedures provide an additional discriminatory power. An asymptotic distribution for a robust estimator of Hurst exponent based on sample quantiles of log-squared wavelet coefficients is derived. Tukey and Gastwirth trimean based estimators are explored and weights for an optimal trimean estimator are derived as well.

We discuss how the proposed estimators perform in problems coming from medical diagnostics where the scaling indices turn out to be informative in tasks of supervised learning. This work is joint with Chen Feng, speaker's graduate student. The talk is aimed at nonspecialists in wavelets, and a broader scientific audience.