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In this paper, we present some improvements to the convergent cross mapping (CCM) algorithm for detecting causality in uni-directionally connected chaotic systems. The basic concept of the CCM algorithm is that the causal influence of system X on system Y appears as mapping of the neighbouring states in the reconstructed d-dimensional manifold, My, to the neighbouring states in the reconstructed d-dimensional manifold, Mx, and this effect is evaluated using the correlation coefficient between the estimated and observed values of Mx. We proposed a composite indicator of causality as the ratio between the correlation coefficient and the Shannon entropy of the distribution of the residuals between the estimated and observed values of Mx. Application of the proposed approach to four master-slave Rössler and Lorenz systems and real-world data showed that the new algorithm allowed a slight increase in capability to reveal the presence and direction of couplings.

In this paper, we propose a novel algorithm for the detection changes in the spatio-spectral dynamics of a multivariate time series based on a sliced cross-bispectrum. The singular value decomposition is performed on the matrix of the sliced cross-bispectrum at every frequency of the bandwidth, and a 2D spectrum of the eigenvalues is obtained. The changes in the dynamics are translated into differences between the reference and current values of the 2D eigenvalue spectrum. Hellinger divergence as the measure of the distance between eigenvalue spectrum matrices is used. The performance of the proposed approach is tested on model data and applied to multi-channel electroencephalogram recordings from epilepsy patients to detect ictal states.

In this paper, a novel algorithm based on the local projection noise reduction approach is applied to smooth noise for strongly contaminated chaotic oscillators. Specifically, one-dimensional time series are embedded into a high dimensional phase space and the noise level is defined through orthogonal projections of the data points within the neighbourhood of the reference point onto linear subspaces. The current vector of the phase space is denoised by performing two-dimensional discrete stationary wavelet transform (SWT)-based filtering in the neighbourhood of the phase point. Numerical results show that our algorithm effectively recovers continuous-time chaotic signals in heavy-noise environments and outperforms the classical local projection noise reduction approach for simulated data from the Rössler system and Duffing oscillator at signal-to-noise ratios (SNRs) from 15 to 0 dB, either for the real world data – human breath time series.

In this paper, it is shown that the block circulant matrix decomposition technique makes the multivariate singular spectrum analysis (M-SSA) a well suited tool for detecting of changes of the correlation structure in non-stationary multivariate time series in the presence of high observational noise levels. The major drawback of M-SSA, that it operates on a large covariance matrix and becomes computationally expensive, can be avoided by reordering the Toeplitz-block covariance matrix into a block Toeplitz matrix, embedding this into a block circulant matrix and efficiently block-diagonalizing this by the means of the Fast Fourier Transform (FFT) using the well known algorithm. The overall degree of synchronization among multiple-channel signals is defined by the synchronization index (the S-estimator) of the rearranged and truncated eigenvalue spectrum. Throughout the experiment, the high capability of the proposed algorithm to detect the lag-synchronized state under the influence of strong noise is validated with simulated data—a network of time series generated by autoregressive models (AR) and a network of coupled chaotic Roessler oscillators.