Investigation of Mutual Behavior of Stochastic Normally Distributed Processes with Additive Amplitude Randomization

The joint analysis of several signals is essential for better understanding of the principles underlying the complex systems dynamics. We consider three methods for estimating the stability of the relative dynamics of two surrogate processes. The first one is based on calculation of the phase synchronization coefficient S and the second one on estimation of the cross-conditional entropy CE. The third approach uses the average value of the coherence function of the two processes - the coherence coefficient C. We study the sensitivity of these methods in relation to the amplitude randomization between test processes. All methods are applied to analyze two types of normally distributed random stochastic processes, with either short-term correlations characterized by finite correlation time or long-term correlations with theoretically infinite correlation time characterized by Hurst exponents. In our research, we generate two copies of the surrogate process with either short-term or long-term correlations. Then we attribute the additive white noise to one of these copies at first with the uniform distribution and then with the Gaussian distribution and the same variance. Next, we calculate the coefficients that characterize the mutual behavior of the two test processes and estimate their statistical characteristics. It is found that the sensitivity of all methods to Gaussian additive noise is higher than that of uniform one. We show that processes with long-term correlation react more actively to the additive amplitude noise then processes with short-term correlation. The influence of Hurst exponent value for the processes with long-term correlation is expressed for the coefficients S and C. The influence of correlation time is demonstrated for the coefficients S and СЕ. Our results may be useful in investigations of the mutual dynamics of two processes belonging to the considered types. 

1. Bartsch R. P., Liu K. K. L., Bashan A., Ivanov P. Ch. Network Physiology: How Organ Systems Dynamically Interact. PLoS ONE. 2015, vol. 10, no. 11, pp. 1-34.

2. Bartsch R. P., Ivanov P. Ch. Coexisting Forms of Coupling and Phase-Transitions in Physiological Networks. Communications in Computer and Information Science. 2014, vol. 438, pp. 270-287.

3. Marwan N., Donges J. F., Zou Y., Donner R. V., Kurths J. Complex Network Approach for Recurrence Anal-ysis of Time Series. Physics Letters A. 2009, vol. 373, iss. 46, pp. 4246-254.

4. Bashan A., Bartsch R. P., Kantelhardt J. W., Havlin S., Ivanov P. Ch. Network Physiology Reveals Relations between Network Topology and Physiological Function. Nature Communications. 2012, vol. 3, art. 702, pp. 1-9.

5. Ludescher J., Gozolchiani A., Bogachev M. I., Bunde A., Havlin Sh., Schellnhuber H. J. Improved El Nino Fore-casting by Cooperativity Detection. Proc. of the National Academy of Science of the USA. 2013, vol. 110, no. 29, pp. 11742-11745.

6. Pyko N. S., Pyko S. A., Markelov O. A., Uljanitski Yu. D., Bogachev M. I., Mamontov O. V. Two Approaches to Estimating the Relative Dynamic Stability of Physiological Processes. Soft Computing and Measurements (SCM), 2016 XIX IEEE Int. Conf. St. Petersburg, 25-27 May 2016. Available at: http://doi.org.10.1109/SCM.2016.7519684 (accessed: 25 December 2017).

7. Pyko N. S., Pyko S. A., Markelov O. A., Uljanitski Yu. D., Bogachev M. I. Assessment of the Mutual Stability of Two Correlated Stochastic Signals: The Effects of Phase Randomization. 2017 IEEE Conf. of Russian Young Researchers in Electrical and Electronic Engineering (EIConRus), St. Petersburg, 1-3 Febr. 2017. Available at: http://doi.org.10. 1109/EIConRus.2017.7910659 (accessed: 25 December 2017).

8. Pyko N. S., Pyko S. A., Markelov O. A., Uljanitski Yu. D. Assessment of the Mutual Synchronization of Two Stochastic Data Sets: the Effects of Additive and Multiplicative White Noise, Proc. of 2017 20th IEEE Int. Conf. on Soft Computing and Measurements (SCM'2017), St. Petersburg, 24-26 May 2017. Available at: http://doi.org.10. 1109/SCM.2017.7970495 (accessed: 25 December 2017).

9. Porta A., Baselli G., Lombardi F., Cerutti S. Conditional Entropy Approach for the Evaluation of the Coupling Strength. Biological Cybernetics. 1999, vol. 81, no. 2, pp. 119-129.