Unraveling intermittent features in single-particle trajectories by a local convex hull method

We propose a model-free method to detect change points between distinct phases in a single random trajectory of an intermittent stochastic process. The local convex hull (LCH) is constructed for each trajectory point, while its geometric properties (e.g., the diameter or the volume) are used as discriminators between phases. The efficiency of the LCH method is validated for six models of intermittent motion, including Brownian motion with different diffusivities or drifts, fractional Brownian motion with different Hurst exponents, and surface-mediated diffusion. We discuss potential applications of the method for detection of active and passive phases in the intracellular transport, temporal trapping or binding of diffusing molecules, alternating bulk and surface diffusion, run and tumble (or search) phases in the motion of bacteria and foraging animals, and instantaneous firing rates in neurons.

Figure 1

Illustration of the local convex hull method applied to an intermittent planar trajectory, alternating a “fast” phase of Brownian motion (with $D=1/2$) and a “slow” phase of an Ornstein-Uhlenbeck process (with $D=1/2$ and $k=0.1$; see Sec. 3d), of equal durations $T_1=T_2=40$. Only two LCHs (based on $2\tau +1=41$ points) are shown by shadowed regions that correspond to the fast (dark red) and slow (light green) phases. Arbitrary units are used.

Figure 2

Model 1. (a) A single trajectory of planar Brownian motion alternating a “slow” phase ($D_1=1/2$, dark blue) and a “fast” phase ($D_2=2$, light green), of the mean phase duration $T=100$. Circle and square indicate the starting and ending points. (b) The weighted LCH diameter $S_d\left(n\right)$ with the window size $\tau =10$, applied to this trajectory. Pink shadow highlights the false classification zones. Dashed horizontal line shows the empirical mean $S_d$ over that trajectory. (c,d,e) Recognition score $R$ of the diameter-based discriminator $S_d\left(n\right)$ (c), the volume-based discriminator $S_v\left(n\right)$ (d), and the TA MSD-based discriminator (e) as a function of the mean phase duration $T$. Lines show the results for the case $D_1=1/2, D_2=2$ with three noise levels ${\sigma }_n$: 0 (blue solid), $0.5\sigma$ (red dashed), and $\sigma$ (gray dash-dotted) ($\sigma$ being the empirical standard deviation of increment calculated for each trajectory). Symbols correspond to the case $D_1=1/2, D_2=1$ with the same levels of noise ${\sigma }_n$: 0 (circles), $0.5\sigma$ (crosses), and $\sigma$ (triangles). Thin black solid line shows the hypothetical curve $1-\tau /\left(2T\right)$ for qualitative comparison. Arbitrary units are used.

Figure 3

Model 2. (a) A single trajectory of an intermittent process alternating a “slow” phase of planar Brownian motion without drift (dark blue) and a “fast” phase with drift ($\mu =0.5$, light green), of the mean phase duration $T=100$ (both phases with $D=1/2$). Circle and square indicate the starting and ending points. (b) The weighted LCH diameter $S_d\left(n\right)$ with the window size $\tau =10$ applied to this trajectory. Pink shadow highlights the false classification zones. Dashed horizontal line shows the empirical mean $S_d$ over that trajectory. (c), (d), (e) Recognition score $R$ of the diameter-based discriminator $S_d\left(n\right)$ (c), the volume-based discriminator $S_v\left(n\right)$ (d), and the TA MSD-based discriminator (e) as a function of the mean phase duration $T$. Lines show the results for the case $\mu =0.5$ with three noise levels ${\sigma }_n$: 0 (blue solid), $0.5\sigma$ (red dashed), and $\sigma$ (gray dash-dotted) ($\sigma$ being the empirical standard deviation of increment calculated for each trajectory). Symbols correspond to the case $\mu =0.1$ with the same levels of noise ${\sigma }_n$: 0 (circles), $0.5\sigma$ (crosses), and $\sigma$ (triangles). Arbitrary units are used.

Figure 4

Model 3. (a) A single trajectory of planar fBm, alternating a “slow” phase ($H_1=0.35$, dark blue) and a “fast” phase ($H_2=0.7$, light green), of the mean phase duration $T=100$ and $D_H=1/2$. Circle and square indicate the starting and ending points. (b) The weighted LCH diameter $S_d\left(n\right)$ with the window size $\tau =10$ applied to this trajectory. Pink shadow highlights the false classification zones. Dashed horizontal line shows the empirical mean $S_d$ over that trajectory. (c), (d), (e) Recognition score $R$ of the diameter-based discriminator $S_d\left(n\right)$ (c), the volume-based discriminator $S_v\left(n\right)$ (d), and the TA MSD-based discriminator (e) as a function of the mean phase duration $T$. Lines show the results for the case $H_1=0.35, H_2=0.7$ with three noise levels ${\sigma }_n$: 0 (blue solid), $0.5\sigma$ (red dashed), and $\sigma$ (gray dash-dotted) ($\sigma$ being the empirical standard deviation of increment calculated for each trajectory). Symbols correspond to the case $H_1=0.35, H_2=0.5$ with the same levels of noise ${\sigma }_n$: 0 (circles), $0.5\sigma$ (crosses), and $\sigma$ (triangles). Arbitrary units are used.

Figure 5

Model 4. (a) A single trajectory of planar motion, alternating a “slow” phase ($k=0.1$, dark blue) and a “fast” phase ($k=0$, light green), with $D=1/2$ and the mean phase duration $T=100$. Circle and square indicate the starting and ending points. (b) The weighted LCH diameter $S_d\left(n\right)$ with the window size $\tau =10$ applied to this trajectory. Pink shadow highlights the false classification zones. Dashed horizontal line shows the empirical mean $S_d$ over that trajectory. (c), (d), (e) Recognition score $R$ of the diameter-based discriminator $S_d\left(n\right)$ (c), the volume-based discriminator $S_v\left(n\right)$ (d), and the TA MSD-based discriminator (e) as a function of the mean phase duration $T$. Lines show the results for the case $k=1$ with three noise levels ${\sigma }_n$: 0 (blue solid), $0.5\sigma$ (red dashed), and $\sigma$ (gray dash-dotted) ($\sigma$ being the empirical standard deviation of increment calculated for each trajectory). Symbols correspond to the case $k=0.1$ with the same levels of noise ${\sigma }_n$: 0 (circles), $0.5\sigma$ (crosses), and $\sigma$ (triangles). Arbitrary units are used.

Figure 6

Model 5. (a) A single trajectory of an intermittent process, alternating a “slow” phase of planar Brownian motion (with $D=1/2$, dark blue) and a “fast” phase of two-dimensional exponential flights (with $\ell =3$, light green), of the mean phase duration $T=100$. Circle and square indicate the starting and ending points. (b) The weighted LCH diameter $S_d\left(n\right)$ with the window size $\tau =10$ applied to this trajectory. Pink shadow highlights the false classification zones. Dashed horizontal line shows the empirical mean $S_d$ over that trajectory. (c), (d), (e) Recognition score $R$ of the diameter-based discriminator $S_d\left(n\right)$ (c), the volume-based discriminator $S_v\left(n\right)$ (d), and the TA MSD-based discriminator (e) as a function of the mean phase duration $T$. Lines show the results for the case $\ell =3$ with three noise levels ${\sigma }_n$: 0 (blue solid), $0.5\sigma$ (red dashed), and $\sigma$ (gray dash-dotted) ($\sigma$ being the empirical standard deviation of increment calculated for each trajectory). Symbols correspond to the case $\ell =\sqrt{2}$ with the same levels of noise ${\sigma }_n$: 0 (circles), $0.5\sigma$ (crosses), and $\sigma$ (triangles). Arbitrary units are used.

Figure 7

Model 6. (a) A single trajectory of surface-mediated diffusion in a three-dimensional sphere of radius $R=1$, alternating bulk diffusion (light green) and surface diffusion (dark blue), with $D_{2d}=D_{3d}={10}^{-3}$ and $\lambda ={10}^{-2}$. Circle and square indicate the starting and ending points. (b) The weighted LCH diameter $S_d\left(n\right)$ with the window size $\tau =10$ applied to this trajectory. Pink shadow highlights the false classification zones. Dashed horizontal line shows the empirical mean $S_d$ over that trajectory. (c), (d), (e) Recognition score $R$ of the diameter-based discriminator $S_d\left(n\right)$ (c), the volume-based discriminator $S_v\left(n\right)$ (d), and the TA MSD-based discriminator (e) as a function of the mean duration of the surface phase $T=1/\lambda$. Three curves correspond to three noise levels ${\sigma }_n$: 0, $0.5\sigma$, and $\sigma$ ($\sigma$ being the empirical standard deviation of increment calculated for each trajectory). Arbitrary units are used.

Figure 8

Effect of the window size $\tau$ on the recognition score $R$ for intermittent Brownian motion (Model 1) with $D_1=1/2, D_2=2$, and the mean phase durations $T_1=T_2=40$. (a) The diameter-based discriminator $S_d\left(n\right)$; (b) the volume-based discriminator $S_v\left(n\right)$. Four curves correspond to four noise levels ${\sigma }_n$: 0, $0.5\sigma , \sigma$, and $2\sigma$ ($\sigma$ being the empirical standard deviation of increment calculated for each trajectory). Arbitrary units are used.

Figure 9

Effect of the threshold on the recognition quality for six models (following the enumeration at the beginning of Sec. 3), for the diameter-based discriminator (top 6 plots) and the volume-based discriminator (bottom 6 plots). The left lower corner and the right upper corner correspond to the threshold at the maximal and the minimal values, whereas diamonds indicate the mean value $S_d$ (or $S_v$) used as the threshold throughout the paper. Legend of six curves on each plot is the same as in Figs. 2, 3, 4, 5, 6, 7 for the recognition score (three levels of noise and two sets of model parameters, except for the Model 6).

Figure 10

Effect of unequal phase durations for planar Brownian motion alternating “slow” and “fast” phases (Model 1). Recognition score $R$ of the diameter-based discriminator $S_d\left(n\right)$ (a) and the volume-based discriminator $S_v\left(n\right)$ (b) as a function of the second phase duration $T_2$ while the first phase duration $T_1$ is kept fixed: $T_1=40$. Solid line: the first phase is “slow” ($D_1=1/2$), the second phase is “fast” ($D_2=2$); dashes line: the first phase is “fast” ($D_1=2$), the second phase is “slow” ($D_2=1/2$). Arbitrary units are used.