Multifractal-spectral features enhance classification of anomalous
  diffusion
Henrik Seckler Ralf Metzler Damian G. Kelty-Stephen Madhur Mangalam
Anomalous diffusion processes pose a unique challenge in classification and
characterization. Previously (Mangalam et al., 2023, Physical Review Research
5, 023144), we established a framework for understanding anomalous diffusion
using multifractal formalism. The present study delves into the potential of
multifractal spectral features for effectively distinguishing anomalous
diffusion trajectories from five widely used models: fractional Brownian
motion, scaled Brownian motion, continuous time random walk, annealed transient
time motion, and L\'evy walk. To accomplish this, we generate extensive
datasets comprising $10^6$ trajectories from these five anomalous diffusion
models and extract multiple multifractal spectra from each trajectory. Our
investigation entails a thorough analysis of neural network performance,
encompassing features derived from varying numbers of spectra. Furthermore, we
explore the integration of multifractal spectra into traditional feature
datasets, enabling us to assess their impact comprehensively. To ensure a
statistically meaningful comparison, we categorize features into concept groups
and train neural networks using features from each designated group. Notably,
several feature groups demonstrate similar levels of accuracy, with the highest
performance observed in groups utilizing moving-window characteristics and
$p$-variation features. Multifractal spectral features, particularly those
derived from three spectra involving different timescales and cutoffs, closely
follow, highlighting their robust discriminatory potential. Remarkably, a
neural network exclusively trained on features from a single multifractal
spectrum exhibits commendable performance, surpassing other feature groups. Our
findings underscore the diverse and potent efficacy of multifractal spectral
features in enhancing classification of anomalous diffusion.