Generating artificial Trajectory objects

If you want to generate Trajectory objects based on some statistical constrains, you can use one of the available Generator to construct a list of Trajectory objects.

Random Walk Generator

The RandomWalkGenerator is able to simulate the trajectories of a walker, in an arbitrary number of dimensions, according some probabilistic constrains.

We can import it from generating module as:

from yupi.generators import RandomWalkGenerator

As any other Generator in yupi, you can specify the parameters that define the shape of the trajectories, as well as the number of trajectories to generate:

T = 500     # Total time (number of time steps if dt==1)
dim = 2     # Dimension of the walker trajectories
N = 3       # Number of random walkers
dt = 1      # Time step

The RandomWalkGenerator starts the generation of every trajectory in the origin of the reference frame. Then, iteratively, it computes an increment on each dimension. The increment (also called actions) can be -1, 0 or 1, and it is taken independently on each dimension for each iteration. Additionally, the user can define a list to establish the probabilities of taking each of the available actions:

prob = [[.5, .1, .4],   # x-axis
        [.5,  0, .5]]   # y-axis

Notice that the size of this list should coincide with the desired dimensions of the trajectories being generated, and each element of a list should be a 3-element list describing the probability vector of taking the actions [-1, 0, 1] in that dimension.

Then, we can construct a RandomWalkGenerator with the given variables and call its generate method:

rw = RandomWalkGenerator(T, dim, N, dt, prob)
tr = rw.generate()

In the variable tr we will have a list of N Trajectory objects generated using the given configuration.

The generated trajectories can be inspected using the plot_2d() function:

from yupi.graphics import plot_2d
plot_2d(tr, legend=None)

Langevin Generator

The LangevinGenerator simulates trajectories governed by the Langevin Equation. It allows to produce Trajectory objects that quantitatively emulate several systems.

To use it, we first need to define the general parameters for a generator:

T = 500     # Total time (number of time steps if dt==1)
dim = 2     # Dimension of the walker trajectories
N = 3       # Number of random walkers
dt = 0.5    # Time step

Then, some specific parameters can be set before the generator initialization:

gamma = 1       # Drag parameter
sigma = 0.1     # Scale of the noise pdf

Finally, the generator is created and the trajectories can be generated:

from yupi.generators import LangevinGenerator
lg = LangevinGenerator(T, dim, N, dt, gamma, sigma)
trajectories = lg.generate()

The generated trajectories can be inspected using the plot_2d() function:

from yupi.graphics import plot_2d
plot_2d(trajectories, legend=None)

Although not illustrated in this example, the initial velocities and positions can be specified in the LangevinGenerator creation using the v0 and r0 parameters respectively.

A more complex application of this Generator can be seen in the Example 1.

Diffusing Diffusivity Generator

The DiffDiffGenerator simulates trajectories governed by a diffusion process with fluctuating diffusivity. It allows to produce Trajectory objects that quantitatively emulate different systems.

To use it, we first need to define the general parameters for a generator:

T = 1000   # Total time of the simulation
N = 5      # Number of trajectories
dt = .1    # Time step
dim = 2    # Dimension of the Trajectories

Then, some specific parameters can be set before the generator initialization:

tau = 1         # Relaxation time
sigma = 0.1     # Scale of the noise pdf

The generator is created and the trajectories can be generated:

from yupi.generators import DiffDiffGenerator
dd = DiffDiffGenerator(T, N=N, dt=dt, dim=dim, tau=tau, sigma=sigma)
trajs = dd.generate()

The generated trajectories can be inspected using the plot_2d() function:

from yupi.graphics import plot_2d
plot_2d(trajs, legend=None)

Although not illustrated in this example, the initial positions can be specified in the DiffDiffGenerator creation using the r0 parameter.

A more complex application of this Generator can be seen in the Example 2.

Defining a Custom Generator

A user-defined generator can be easily added by building on top of an abstract class Generator (which is the base of the already implemented generators).