Generate stationary correlated time-series

Summary

This Python function creates a time-series (discrete-time random process) with a specific autocorrelation function (ACF) and continuous probability distribution, e.g with predefined probability density function (PDF).

Usage:

from generate_corr_sequence import gen_corr_sequence
gen_corr_sequence(dist_obj=uniform,
                  target_acf=1 - np.minimum(np.arange(0, 100), 100) / 100,
                  L: int = 2 ** 20,
                  seed=None,
                  debug: bool = False,
                  plot_figures_name: str = None)

Arguments

• dist_obj - A continuous distribution object from scipy.stats, default is uniform
• target_acf - A required ACF function, default is a linear function
• L - Number of required samples
• seed - Seed of the random number generator
• debug - Plots PDF and ACF graphs
• plot_figures_name - Filename of the debug figure to be saved, only when debug is True. In no extension is provided, the default is png.

Returns:

• A random sequence with pre-defined ACF and distribution with type numpy.ndarray.

Examples

Default settings with uniform distribution and linear ACF

The example below shows the default settings of the function from the examples/default_settings_example.ipynb file.

# Example usage of the function with default settings
sequence = generate_corr_sequence(debug=True)

Nakagami distribution with Bessel function ACF

The example below is from the examples/nakagami_example.ipynb file.

# Example usage of the function with Nakagami distribution and an autocorrelation function
from generate_corr_sequence import gen_corr_sequence
import numpy as np
from scipy.stats import nakagami
from scipy.special import j0
# %%
m = np.arange(0, 100)
signal = gen_corr_sequence(
    dist_obj=nakagami(nu=1),
    target_acf=np.array(j0(0.1 * np.pi * abs(m))),
    debug=True)

Notes

1. There is no responsibility for the correctness of the results. It may work and it may not - use debug option to check the results.
2. Examples for different distributions (uniform, exponential, Laplace, Rayleigh, triangle, gamma, lognormal, Nakagami) and four different ACFs are provided in the examples/evaluation/ folder. The evaluate_PDFs.ipynb file used for generation of all the ACF figures in the directory.
3. It takes about 2-3 seconds to generate a single default-length sequence.
4. The example of code repeatability is provided in the examples/nakagami_example.ipynb notebook as extension of Nakagami distribution example. The sequence is generated 20 times with ACFs as follows.

References

The algorithm is mainly based on the following papers:

1. Filho, José Cândido Silveira Santos, and Michael Daoud Yacoub. "Coloring Non-Gaussian Sequences." IEEE Transactions on Signal Processing, vol. 56, no. 12, 2008, p. 6.
2. Liu, Bede, et al. "Generation of a Random Sequence Having a Jointly Specified Marginal Distribution and Autocovariance." IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-30, no. 6, 1982, p. 11.

Contributors

Dima Bykhovsky, Netanel Tochilovsky, Alexander Rudyak

License

This project is licensed under the MIT license.

Todo

• custom (non-scipy) distributions support
• higher-speed algorithm for Gaussian distributions
• fix lognormal distribution problem for oscillatory ACFs