Here we have implemented solution of nonlinear SDE:
In this context "solution" is understood as obtaining single sample trajectory of desired length and discretization time step.
The numerical solution is obtained by solving Bessel process SDE:
and applying nonlinear transformation:
![x = \left[ \left( \eta - 1 \right) y \right]^\frac{1}{1-\eta}](/akononovicius/pyNSDE/blob/main/figs/transform.png)
to obtain the solutions of the nonlinear SDE above.
Bessel process is solved using Euler-Maruyama method with variable time step.
You could use this library to generate time series, which exhibit pink or 1-over-f noise. Depending on the model and simulation parameters this can be achieved in an arbitrarily broad range of frequencies. Below follows example with simulation results of the calculation with mostly default parameter values.
import numpy as np
import matplotlib.pyplot as plt
from pyNSDE import generate_series
from stats.pdf import MakeLogPdf
from stats.psd import MakeSegLogPsd
# simulation
series = generate_series(1048576, 1e-3, seed=123)
# calculating PDF / PSD
pdf = MakeLogPdf(series)
psd = MakeSegLogPsd(series, fs=1e3)
# creating simple visualization
plt.figure(figsize=(12,3))
plt.subplot(131)
plt.xlabel('t')
plt.ylabel('x(t)')
plt.plot(series[::256], 'r-')
plt.subplot(132)
plt.loglog()
plt.xlabel('x')
plt.ylabel('p(x)')
plt.plot(pdf[:, 0], pdf[:, 1], 'r-')
plt.plot(pdf[:, 0], 2*(pdf[:, 0]**-3), 'k--')
plt.subplot(133)
plt.loglog()
plt.xlabel('f')
plt.ylabel('S(f)')
plt.plot(psd[:, 0], psd[:, 1], 'r-')
plt.plot(psd[20:, 0], 1.5*(psd[20:, 0]**-1), 'k--')
plt.tight_layout()
plt.show()
In this code snippet stats library was cloned from
https://github.com/akononovicius/python-stats.
Earlier implementation of a less flexible program solving the same nonlinear stochastic differential equation is available at https://github.com/JuliusRuseckas/numerical-sde-variable-step.
https://github.com/akononovicius/python-stats library might be useful when analyzing simulated time series.
Couple of scientific review papers specific to the SDE being solved:
- B. Kaulakys and J. Ruseckas, Stochastic nonlinear differential equation generating 1/f noise, Phys. Rev. E 70, 020101 (2004). doi: 10.1103/PhysRevE.70.020101. arXiv:cond-mat/0408507 [cond-mat.stat-mech].
- B. Kaulakys, J. Ruseckas, V. Gontis and M. Alaburda, Nonlinear stochastic models of 1/f noise and power-law distributions, Physica A 365, 217-221 (2006). doi: 10.1016/j.physa.2006.01.017. arXiv:cond-mat/0509626 [cond-mat.stat-mech].
Recent review with variety of applications related to the SDE being solved:
- R. Kazakevičius, A. Kononovicius, B. Kaulakys, V. Gontis. Understanding the nature of the long-range memory phenomenon in socio-economic systems. Entropy 23: 1125 (2021). doi: 10.3390/e23091125. arXiv:2108.02506 [physics.soc-ph].