Statistical testing approach for fractional anomalous diffusion classification

Taking advantage of recent single-particle tracking data, we compare the popular standard mean-squared displacement method with a statistical testing hypothesis procedure for three testing statistics and for two particle types: membrane receptors and the G proteins coupled to them. Each method results in different classifications. For this reason, more rigorous statistical tests are analyzed here in detail. The main conclusion is that the statistical testing approaches might provide good results even for short trajectories, but none of the proposed methods is “the best” for all considered examples; in other words, one needs to combine different approaches to get a reliable classification.

Figure 1

Trajectories used for analysis (blue color) for two particle types: receptors (left panel) and G proteins (right panel) with minimal trajectories length $N=50$. The omitted trajectories are marked with gray.

Figure 2

Cumulative distribution $\widehat{F}\left({\beta }_n\right)$ for the power exponent ${\beta }_n$ of ${\text{MSD}}_{\tau }$ calculated with Monte Carlo simulations. In the upper left panel, the distribution of ${\beta }_n$ under $H_0$, i.e., free diffusion, is plotted for different trajectories lengths $n$. In the remaining panels, the distribution of ${\beta }_n$ with $n=1000$ for different alternatives, i.e., fractional Brownian motion (top right panel), the Ornstein-Uhlenbeck process (bottom left panel), and directed Brownian motion (bottom right panel), is plotted. Note that for fBM $H=0.5$ corresponds to the free-diffusion case (i.e., $H_0$), for the Ornstein-Uhlenbeck process $\lambda =0$ corresponds to the free-diffusion case, while for dBM $v=0$. The critical regions are marked with yellow rectangles.

Figure 3

Cumulative distribution function $\widehat{F}\left(T_n\right)$ for $T_n$ calculated with Monte Carlo simulations. In the left panel, the distribution of $T_n$ under $H_0$, i.e., free diffusion, is plotted for different trajectory lengths $n$. In the remaining panels, the distribution of $T_n$ with $n=1000$ for different alternatives, i.e., fractional Brownian motion (top right panel), the Ornstein-Uhlenbeck process (bottom left panel), and directed Brownian motion (bottom right panel), is plotted. Note that for fBM, $H=0.5$ corresponds to the free-diffusion case (i.e., $H_0$), for the Ornstein-Uhlenbeck process $\lambda =0$ corresponds to the free-diffusion case, while for dBM $v=0$. The critical regions are marked with yellow rectangles.

Figure 4

Cumulative distribution function $\widehat{F}\left({\gamma }_n^1\right)$ for the power exponent of $p$-variation ${\widehat{V}}_n^1$ calculated with Monte Carlo simulations. In the top left panel, the distribution under $H_0$, i.e., free diffusion, is plotted for different trajectory lengths $n$. In the remaining panels, the distribution of ${\gamma }_n^1$ with $n=1000$ for different alternatives, i.e., fractional Brownian motion (top right panel), the Ornstein-Uhlenbeck process (bottom left panel), and directed Brownian motion (bottom right panel), is plotted. Note that for fBM $H=0.5$ corresponds to the free-diffusion case (i.e. $H_0$), for the Ornstein-Uhlenbeck process $\lambda =0$ corresponds to the free-diffusion case, while for dBM $v=0$. The critical regions are marked with yellow rectangles.

Figure 5

Cumulative distribution function $\widehat{F}\left({\gamma }_n^4\right)$ for the power exponent of $p$-variation ${\widehat{V}}_n^4$ calculated with Monte Carlo simulations. In the top left panel, the distribution under $H_0$, i.e., free diffusion, is plotted for different trajectories lengths $n$. In the remaining panels, the distribution of ${\gamma }_n^4$ with $n=1000$ for different alternatives, i.e., fractional Brownian motion (top right panel), the Ornstein-Uhlenbeck process (bottom left panel), and directed Brownian motion (bottom right panel), is plotted. Note that for fBM $H=0.5$ corresponds to the free-diffusion case (i.e., $H_0$), for the Ornstein-Uhlenbeck process $\lambda =0$ corresponds to the free-diffusion case, while for dBM $v=0$. The critical regions are marked with yellow rectangles.

Figure 6

Classification results for G proteins obtained with different methods: standard MSD with $c=0.1$ (upper left panel), MSD test (upper right panel), MAX test (lower left panel), and $p$-VAR test with $p=1$ (lower right panel). Trajectories classified as free diffusion are plotted with black, as subdiffusion with yellow, and as superdiffusion with red.

Figure 7

Classification results for receptors obtained with different methods: standard MSD with $c=0.1$ (upper left panel), MSD test (upper right panel), MAX test (lower left panel), and $p$-VAR test with $p=1$ (lower right panel). Trajectories classified as free diffusion are plotted with black, as subdiffusion with yellow, and as superdiffusion with red.

Figure 8

Box plots of the classification results obtained with different methods from 300 simulated trajectories driven by the fractional Brownian motion with different $H$ parameters: 100 for the free-diffusion case ($H=0.5$), 100 for subdiffusion ($H=0.4$), and 100 for superdiffusion ($H=0.6$). Classification procedure was repeated 100 times. The lengths of the simulated trajectories were set as 50 (upper panel) or 500 (bottom panel). The dashed thick line indicates the correct classification value of $1/3$.

Figure 9

Box plots of the classification results obtained with different methods from 300 simulated trajectories driven by the fractional Brownian motion with different $H$ parameters: 100 trajectories for the free-diffusion case ($H=0.5$), 100 for subdiffusion ($H=0.1$), and 100 for superdiffusion ($H=0.9$). Classification procedure was repeated 100 times. The lengths of the simulated trajectories were set as 50 (upper panel) or 500 (bottom panel). The dashed thick line indicates the correct classification value of $1/3$.

Figure 10

Box plots of the classification results obtained with different methods from 300 simulated trajectories: 100 trajectories for the free-diffusion case (driven by Brownian motion), 100 for subdiffusion (driven by the Ornstein-Uhlenbeck process with $\lambda =0.01$), and 100 for superdiffusion (driven by the directed Brownian motion with $v=0.2$). Classification procedure was repeated 100 times. The lengths of the simulated trajectories were set as 50 (upper panel) or 500 (bottom panel). The dashed thick line indicates the correct classification value of $1/3$.

Figure 11

Box plots of the classification results obtained with different methods from 300 simulated trajectories: 100 trajectories for the free-diffusion case (driven by Brownian motion), 100 for subdiffusion (driven by the Ornstein-Uhlenbeck process with $\lambda =0.5$), and 100 for superdiffusion (driven by the directed Brownian motion with $v=0.8$). Classification procedure was repeated 100 times. The lengths of the simulated trajectories were set as 50 (upper panel) or 500 (bottom panel). The dashed thick line indicates the correct classification value of $1/3$.