Higuchi fractal dimension

Higuchi Fractal Dimension (DF) [7] is calculated directly from the signal, in time domain, without reconstruction of strange attractors in multidimensional phase space. Value of DF is always between 1 and 2 (simple curve has dimension equal 1 and a plane has dimension equal 2). Fractional part of DF shows what fraction of the plane is “filled up” by the curve, so it is a measure of the signal’s complexity. Further description of Higuchi Fractal Dimension is in Klonowski (2007) [8].

DFof the resting state (relaxed with closed eyes) EEG record (about 250 seconds long) was calculated in 0.5 second length time windows and then averaged for each channel. DF of EEG during the match was calculated at time intervals corresponding to the thinking on each chess movement by the player (it gave about 40 values of DF for specific match) and then averaged for each channel. The maps of mean DFfor each channel were constructed. The standard deviation was used as a parameter of statistical significance.

Empirical mode decomposition

Empirical Mode Decomposition (EMD) is an entirely data-driven algorithm which breaks down nonstationary, multicomponent signal into its monocomponents. Such monocomponents are called Intrinsic Mode Functions (IMFs). Each IMF must fulfill the following criteria:

the number of extrema and zeros are equal or their difference is not greater than 1,

the signal has “zero mean” - the mean value of the envelope determined by maxima and the envelope determined by minima is equal 0 at every point.

Further description of EMD is in Huang et al. (1998) [9]. We used modified algorithm called Sliding Window Empirical Mode Decomposition (SWEMD, see [10]) which speeds up the calculation about 10 times when compared with ’classical’ EMD method.

Signal decomposed by SWEMD can be further analyzed by obtaining the marginal Hilbert-Huang Spectrum h h s(f), see equation (17) in [10].

The information about the contribution of a frequency range to the total power of signal can be obtained by integration of the marginal Hilbert-Huang Spectrum with frequency range limits. For example the contribution of the alpha band in EEG can be calculated with:

$$\alpha\, power=\frac{\intop_{8}^{13}hss(f)df}{\intop_{0}^{\infty}hss(f)df} $$

Contribution of each EEG band was calculated using SWEMD and the marginal Hilbert-Huang Spectrum in similar way as Higuchi Fractal Dimension. For the resting state contribution of each band was calculated in 0.5 second length time windows and then averaged for each channel and standard deviation was calculated. Bands’ contributions during the match was calculated at time intervals corresponding to the thinking on each chess movement by the player and then averaged for each channel and standard deviation was obtained.