Ilya Sysoev

External low-frequency interference (including interference with a pronounced main frequency) is a common problem in measurements of complex signals, which can affect results of coupling estimation. Since it is impossible to completely remove the interference without affecting the signal itself, the question arises: what distorts the results of coupling estimation to a lesser extent: filtering the interference or ignoring it?
The Granger causality (GC) method is one of the most popular approaches to the detection of directional coupling from observed signals. GC uses predictive empirical models, mostly, linear and nonlinear autoregressive models (recurrence maps). Since the method is highly parametric, its success depends primarily on the parameters of the models and on the properties of the signals. Therefore, the method has to be adapted to the data. In physiology and climatology, most signals have a pronounced time scale, so one of the most important problems is that of adapting the Granger causality method to signals with a selected time scale.
The purpose of this paper is to formulate recommendations for using the Granger causality method for signals with a pronounced temporal scale in the presence of common low-frequency interference. In this paper, we restrict our attention to the case of testing for unilateral coupling and use the recommendations and criteria, developed earlier, for the effectiveness of the method. The sensitivity and specificity of the method are estimated based on surrogate time series. The testing is performed using reference systems of nonlinear dynamics and radiophysics.
It is shown that the loss of sensitivity and specificity of the method decrease nonlinearly with increasing amplitude of the total interference. This dependence varies for different parameters of the method. If the power of interference is several per cent of the signal power, the best results can be achieved by an appropriate choice of parameters of the method rather than by filtering the interference. At a higher noise power, filtering is preferable.

Reconstruction of equations of oscillatory systems from time series is an important problem, since results can be useful in different practical applications, including forecast of future dynamics, indirect measurement of parameters and diagnostics of coupling. The problem of reconstruction of coupling coefficients from time series of ensembles of a large number of oscillators is a practically valid problem. This study aims to develop a method of reconstruction of equations of an ensemble of identical neuron-like oscillators in the presence of time delays in couplings based on a given general form of equations.
The proposed method is based on the previously developed approach for reconstruction of diffusively coupled ensembles of time-delayed oscillators. To determine coupling coefficients, the target function is minimized with least-squares routine for each oscillator independently. This function characterizes the continuity of experimental data. Time delays are revealed using a special version of the gradient descent method adapted to the discrete case.
It is shown in the numerical experiment that the proposed method allows one to accurately estimate most of time delays (∼99%) even if short time series are used. The method is asymptotically unbiased.

The effect of the external measurement noise on characteristics of the Granger causality method was considered for unidirectionally coupled non-linear etalon systems in different oscillation regimes. Coupled maps with the same and different evolution operator in driving and driven systems were studied, as well as coupled flows. The nontrivial dependency of method characteristics was shown in all considered cases for certain parameters and coupling intensity. The reason why this dependency in not monotonous was found out.

The detection of coupling presence and direction between various systems using their time series is a common task in many areas of knowledge. One of the approaches used to solve it is nonlinear Granger causality method. It is based on the construction of forecasting models, so its efficiency defends on selection of model parameters. Two parameters are important for modeling signals with a main time scales: lag that is used for state vector reconstruction and prediction length.
In this paper, we propose two criteria for evaluating performance of the method of nonlinear Granger causality. These criteria allow to select lag and prediction length, that provide the best sensitivity and specificity. Sensitivity determines the weakest coupling method can detect, and specificity refers to the ability to avoid false positive results. As a result of the criteria application to several etalon unidirectionally coupled systems, practical recommendations for the selection of the model parameters (lag and prediction length) were formulated.

A method is proposed for the reconstruction of first-order time-delay systems under external periodic driving from their time series. The method takes into account the structure of the model equation of the system, while constructing the autoregressive model. The proposed method allows one to reconstruct the delay time, the parameter characterizing the system inertial properties, the nonlinear function, and the amplitude and frequency of the external periodic driving. The method efficiency is demonstrated in a numerical experiment by reconstructing a number of different nonautonomous time-delay systems.