Analysis of Trajectory objects

There are several tools you can use to analyze Trajectory objects. To illustrate the capabilities of yupi, let us consider a list of Trajectory objects generated with a Langevin Generator (See dedicated tutorial of Langevin Generator for a more detailed explanation of the parameters.)

from yupi.generators import LangevinGenerator

T = 500      # Total time (number of time steps if dt==1)
dim = 2      # Dimension of the walker trajectories
N = 500      # Number of random walkers
dt = 0.5     # Time step

gamma = 0.5  # Drag parameter
sigma = 0.1  # Scale of the noise pdf

lg = LangevinGenerator(T=T, dim=dim, dt=dt, gamma=gamma, sigma=sigma, seed=0)
trajs = lg.generate(N)

Two-dimensional spatial projections

The most basic analysis tool is the plot of the trajectories in the space. If you have a list of Trajectory objects, like the ones you get from a generator, you can plot them with:

from yupi.graphics import plot_2d
plot_2d(trajs[:10], legend=False)

Notice that we limited to 10 the number of trajectories to plot for the sake of observability, but we will be using the full list of trajectories (traj) all over this tutorial.

Three-dimensional spatial projections

Plotting in three dimensions can be achieved in a similar way. Let us generate 3D trajectories:

from yupi.generators import LangevinGenerator
lg = LangevinGenerator(T=T, dim=3, dt=dt, gamma=gamma, sigma=sigma, seed=0)
trajs3D = lg.generate()

Then, we can plot them using:

from yupi.graphics import plot_3d
plot_3d(trajs3D, legend=False)

Histogram of Velocity

The analysis of the distribution of velocities among all the samples of an ensemble of trajectories is also possible using:

from yupi.stats import SpeedStat
SpeedStat(trajs).plot(bins=50)

Histogram of Turning Angles

The analysis of the distribution of turning angles allows to understand how likely is the moving object to turn to specific directions during its motion. It can be observe with yupi by using:

from yupi.stats import TurningAngleStat
TurningAngleStat(trajs).plot(bins=30)

Mean Squared Displacement

The Mean Square Displacement (MSD) is a typical indicator to classify processes away from normal diffusion. The MSD of a normal diffusive trajectory arises as a linear function of time. To estimate the MSD of a list of Trajectory objects, you can use:

from yupi.stats import MsdTimeAvgStat
MsdTimeAvgStat(trajs, lag=30).plot()

Kurtosis

Another useful quantity is the kurtosis, a measure of the disparity of spatial scales of a dispersal process and also an intuitive means to understand normality. It can be estimated using:

from yupi.stats import KurtosisStat
KurtosisStat(trajs).plot()

Velocity Autocorrelation Function

The Velocity Autocorrelation Function (VACF) gives valuable information about the influence of correlations during a whole trajectory. To compute it and plot the results, you can use:

from yupi.stats import VacfTimeAvgStat
VacfTimeAvgStat(trajs, lag=50).plot()

Power Spectral Density

The Power Spectral Density, or Power Spectrum, indicates the frequency content of the trajectory. The inspection of the PSD from a collection of trajectories enables the characterization of the motion in terms of the frequency components.

from yupi.stats import PsdStat
PsdStat(trajs, lag=150).plot()