Stochastic source example

If you want to run the following example on your local machine, you are welcome to download the code here and run it (after having pip-installed the package).

Finally we demonstrate how the PECUZAL method deals with non-deterministic input data. Therefore, we create a simple AR(1)-process:

import numpy as np
import matplotlib.pyplot as plt

def ar_process(u0, alpha, p, N):
    '''Generate `N`-sample data from an auto regressive process of order 1 with autocorrelation-parameter
    `alpha` and amplitude `p` for an intial condition value `u0`.
    '''
    x = np.zeros(N+10)
    x[0] = u0
    for i in range(1,N+10):
        x[i] = alpha*x[i-1] + p*np.random.randn()

    return x[10:]

u0 = .2
data = ar_process(u0, .9, .2, 2000)

plt.figure()
plt.plot(data)
plt.title('AR(1) process')
plt.xlabel('sample')
plt.grid()

When we now call the PECUZAL algorithm pecuzal_embedding.pecuzal_embedding()

from pecuzal_embedding import pecuzal_embedding

Y_reconstruct, tau_vals, ts_vals, Ls, eps = pecuzal_embedding(data, econ = True)

we’ll get the following note in the console:

Algorithm stopped due to increasing L-values in the first embedding cycle.. Valid embedding NOT achieved.

The algorithm did not obtain any valid embedding, thus, it values the input data as a non-deterministic source.