Multifractal spectral features enhance classification of anomalous diffusion

Anomalous diffusion processes, characterized by their nonstandard scaling of the mean-squared displacement, pose a unique challenge in classification and characterization. In a previous study [Mangalam et al., Phys. Rev. Res. 5, 023144 (2023)], we established a comprehensive 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. We generate extensive datasets comprising ${10}^6$ trajectories from these five anomalous diffusion models and extract multiple multifractal spectra from each trajectory to accomplish this. Our investigation entails a thorough analysis of neural network performance, encompassing features derived from varying numbers of spectra. We also 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 $p$ varation 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. In summary, our findings underscore the diverse and potent efficacy of multifractal spectral features in enhancing the predictive capacity of machine learning to classify anomalous diffusion processes.

Figure 1

Representative trajectories of the five anomalous diffusion processes for various anomalous exponents [(a) and (c)] and the respective multifractal spectrum [(b) and (d)]. The spectrum in blue (bold) 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 $\left\{\alpha \right(q),f(q\left)\right\}$. $\alpha \left(q\right)$ is the singularity exponent and $f\left(q\right)$ the corresponding singularity dimension as defined in Eqs. (7) and (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.

Figure 6

Confusion matrices showing the accuracy of the anomalous diffusion classification using MFS features and traditional features. The confusion 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. (a) Smaller feature set without any spectrum. (b) Smaller feature set with two multifractal spectra. (c) Extended feature set without any spectrum. (d) Extended feature set with two multifractal spectra.