Time domain measures of inter-channel EEG correlations: a comparison of linear, nonparametric and nonlinear measures.

Correlations between ten-channel EEGs obtained from thirteen healthy adult participants were investigated. Signals were obtained in two behavioral states: eyes open no task and eyes closed no task. Four time domain measures were compared: Pearson product moment correlation, Spearman rank order correlation, Kendall rank order correlation and mutual information. The psychophysiological utility of each measure was assessed by determining its ability to discriminate between conditions. The sensitivity to epoch length was assessed by repeating calculations with 1, 2, 3, …, 8 s epochs. The robustness to noise was assessed by performing calculations with noise corrupted versions of the original signals (SNRs of 0, 5 and 10 dB). Three results were obtained in these calculations. First, mutual information effectively discriminated between states with less data. Pearson, Spearman and Kendall failed to discriminate between states with a 1 s epoch, while a statistically significant separation was obtained with mutual information. Second, at all epoch durations tested, the measure of between-state discrimination was greater for mutual information. Third, discrimination based on mutual information was more robust to noise. The limitations of this study are discussed. Further comparisons should be made with frequency domain measures, with measures constructed with embedded data and with the maximal information coefficient.

Statistical validation of mutual information calculations: comparison of alternative numerical algorithms.

Given two time series X and Y , their mutual information, I (X,Y) = I (Y,X) , is the average number of bits of X that can be predicted by measuring Y and vice versa. In the analysis of observational data, calculation of mutual information occurs in three contexts: identification of nonlinear correlation, determination of an optimal sampling interval, particularly when embedding data, and in the investigation of causal relationships with directed mutual information. In this contribution a minimum description length argument is used to determine the optimal number of elements to use when characterizing the distributions of X and Y . However, even when using partitions of the X and Y axis indicated by minimum description length, mutual information calculations performed with a uniform partition of the XY plane can give misleading results. This motivated the construction of an algorithm for calculating mutual information that uses an adaptive partition. This algorithm also incorporates an explicit test of the statistical independence of X and Y in a calculation that returns an assessment of the corresponding null hypothesis. The previously published Fraser-Swinney algorithm for calculating mutual information includes a sophisticated procedure for local adaptive control of the partitioning process. When the Fraser and Swinney algorithm and the algorithm constructed here are compared, they give very similar numerical results (less than 4% difference in a typical application). Detailed comparisons are possible when X and Y are correlated jointly Gaussian distributed because an analytic expression for I (X,Y) can be derived for that case. Based on these tests, three conclusions can be drawn. First, the algorithm constructed here has an advantage over the Fraser-Swinney algorithm in providing an explicit calculation of the probability of the null hypothesis that X and Y are independent. Second, the Fraser-Swinney algorithm is marginally the more accurate of the two algorithms when large data sets are used. With smaller data sets, however, the Fraser-Swinney algorithm reports structures that disappear when more data are available. Third, the algorithm constructed here requires about 0.5% of the computation time required by the Fraser-Swinney algorithm.

Dynamical analysis reveals individuality of locomotion in goldfish.

Goldfish swimming was analysed quantitatively to determine if it exhibits distinctive individual spatio-temporal patterns. Due to the inherent variability in fish locomotion, this hypothesis was tested using five nonlinear measures, complemented by mean velocity. A library was constructed of 75 trajectories, each of 5 min duration, acquired from five fish swimming in a constant and relatively homogeneous environment. Three nonlinear measures, the 'characteristic fractal dimension' and 'Richardson dimension', both quantifying the degree to which a trajectory departs from a straight line, and 'relative dispersion', characterizing the variance as a function of the duration, have coefficients of variation less than 7%, in contrast to mean velocity (30%). A discriminant analysis, or classification system, based on all six measures revealed that trajectories are indeed highly individualistic, with the probability that any two trajectories generated from different fish are equivalent being less than 1%. That is, the combination of these measures allows a given trajectory to be assigned to its source with a high degree of confidence. The Richardson dimension and the 'Hurst exponent', which quantifies persistence, were the most effective measures.

The algorithmic complexity of multichannel EEGs is sensitive to changes in behavior.

Symbolic measures of complexity provide a quantitative characterization of the sequential structure of symbol sequences. Promising results from the application of these methods to the analysis of electroencephalographic (EEG) and event-related brain potential (ERP) activity have been reported. Symbolic measures used thus far have two limitations, however. First, because the value of complexity increases with the length of the message, it is difficult to compare signals of different epoch lengths. Second, these symbolic measures do not generalize easily to the multichannel case. We address these issues in studies in which both single and multichannel EEGs were analyzed using measures of signal complexity and algorithmic redundancy, the latter being defined as a sequence-sensitive generalization of Shannon's redundancy. Using a binary partition of EEG activity about the median, redundancy was shown to be insensitive to the size of the data set while being sensitive to changes in the subject's behavioral state (eyes open vs. eyes closed). The covariance complexity, calculated from the singular value spectrum of a multichannel signal, was also found to be sensitive to changes in behavioral state. Statistical separations between the eyes open and eyes closed conditions were found to decrease following removal of the 8- to 12-Hz content in the EEG, but still remained statistically significant. Use of symbolic measures in multivariate signal classification is described.

Comparative study of embedding methods.

Embedding experimental data is a common first step in many forms of dynamical analysis. The choice of appropriate embedding parameters (dimension and lag) is crucial to the success of the subsequent analysis. We argue here that the optimal embedding of a time series cannot be determined by criteria based solely on the time series itself. Therefore we base our analysis on an examination of systems that have explicit analytic representations. A comparison of analytically obtained results with those obtained by an examination of the corresponding time series provides a means of assessing the comparative success of different embedding criteria. The assessment also includes measures of robustness to noise. The limitations of this study are explicitly delineated. While bearing these limitations in mind, we conclude that for the examples considered here, the best identification of the embedding dimension was achieved with a global false nearest neighbors argument, and the best value of lag was identified by the mutual information function.

Effective normalization of complexity measurements for epoch length and sampling frequency.

The algorithmic complexity of a symbol sequence is sensitive to the length of the message. Additionally, in those cases where the sequence is constructed by the symbolic reduction of an experimentally observed wave form, the corresponding value of algorithmic complexity is also sensitive to the sampling frequency. In this contribution, we present definitions of algorithmic redundancy that are sequence-sensitive generalizations of Shannon's original definition of information redundancy. In contrast with algorithmic complexity, we demonstrate that algorithmic redundancy is not sensitive to message length or to observation scale (sampling frequency) when stationary systems are examined.

Detecting noise in a time series.

A numerical algorithm is presented for estimating whether, and roughly to what extent, a time series is noise corrupted. Using phase-randomized surrogates constructed from the original signal, metrics are defined which can be used to quantify the noise level. A saturation occurs in these metrics at signal to noise ratios (SNRs) of around 0 dB and below, and also at around 20 dB and above. In between these two regions there is a monotonic transition in the value of the metrics from one region to the other corresponding to changes in the SNR. (c) 1997 American Institute of Physics.