Multifractal-spectral features enhance classification of anomalous diffusion

TL;DR

This work demonstrates that multifractal spectral features provide robust, discriminative information for classifying trajectories from five anomalous-diffusion models (FBM, SBM, CTRW, ATTM, LW). By computing nine spectra per trajectory via the Chhabra–Jensen method and testing a neural network on MFS features alone and in combination with traditional descriptors, the authors show that MFS features can approach the performance of state-of-the-art trajectory learners when used alone, and can meaningfully boost traditional feature-based classifiers. The results reveal that spectra from multiple timescales (especially two to three spectra) yield substantial gains, while a single spectrum already performs competitively, highlighting the versatility of multifractal descriptors. These findings support the integration of fractal geometry into diffusion-model classification and point to broader applications in cascade-dynamics and multi-particle systems where ergodicity-breaking in standard features can be leveraged for improved model discrimination.

Abstract

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 Lévy 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.

Figures (6)

• Figure 1: Representative trajectories of the five anomalous diffusion processes for various anomalous exponents (a, c) and the respective multifractal spectrum (b, d). The spectrum in blue corresponds to the original time series, while the spectra in gray correspond to five IAAFT surrogates.
• Figure 2: Determining MFS features of anomalous diffusion trajectories. The multifractal spectrum of each trajectory was created by plotting the parametric curve $\{\alpha(q),f(q)\}$. $\alpha(q)$ is the singularity exponent and $f(q)$ the corresponding singularity dimension as defined in Eqs. (\ref{['eq: 7']}, \ref{['eq: 8']}).
• Figure 3: Neural network architecture used for anomalous diffusion classification. A fully connected neural network was used with three hidden layers of size $128$, $64$, and $32$ (or $256$, $128$, and $64$ when using all additional features). The input layer comprised the (normalized) feature vector, with its dimension determined by the number of spectra used ($13$ features per spectrum). It incorporated an additional $26$ or $39$ features for the original or extended sets, respectively. The network then generated model scores for the five diffusion models examined.
• Figure 4: Achieved accuracy (a) and loss (b) for anomalous diffusion classification using features from different numbers of multifractal spectra and feature combinations. The depicted error bars are obtained via subsampling on the test dataset.
• Figure 5: Confusion matrices showing the accuracy of the anomalous diffusion classification using only the MFS features from just $1$ spectrum (a) and $3$ spectra (b), as well as for a state-of-the-art LSTM neural network trained on raw trajectories (c). The matrices show the probability of a ground truth model on the vertical axis to be predicted as one of the models on the horizontal axis.