# Real-Time Localization of Epileptogenic Foci EEG Signals: An FPGA-Based Implementation
## 2 Methodology, Materials and Methods
### Feature Reduction
#### 2.4.1 Log Energy Entropy  
Entropy is a concept handling predictability and randomness, with higher values of entropy always being related to a lesser system order and more randomness. The entropy of an EEG channel is a measure of uncertainty, where the EEG signals are considered a random variable. The log energy entropy of a x EEG signal is defined as [40]:  
\[H_{LogEn}(x)=-\sum_{i=0}^{N-1}{(log_{2}(p_{i}(x)))^{2}}, \tag{1}\]  
\begin{table}
\begin{tabular}{l l} \hline
**EEG Feature** & **Description** \\ \hline Log energy entropy (LogEn) [30] & Non-normalized energy based entropy \\ Median frequency (MDF) [31] & Division of the EEG power spectrum into two regions \\ Mean frequency (MNF) [31] & Mean normalized frequency of the power spectrum \\ Katz fractal dimension (KFD) [31] & Index characterizing the fractal pattern complexity \\ Lower quartile 1 (Q1) [31] & 25\% of the EEG signal \\ Upper quartile 3 (Q3) [31] & 75\% of the EEG signal \\ Inter quartile range (IQR) [31] & Difference between Q3 and Q1 \\ Semi inter quartile deviation (SID) [31] & Measure of spread \\ Skewness (Sk) [32] & Measure of the degree of symmetry \\ Kurtosis (Kr) [32] & Measure of tailedness of the probability distribution \\ Root mean square (RMS) [33] & Root mean square of the EEG signal \\ Band power (PB) [33] & Average power of the EEG signal (0 to \(f_{s}/2\)) \\ Zero crossing (ZC) [33] & Number of times that the signal changes of sign \\ Complexity (Comp) [33] & Hjorth parameter \\ Mobility (Mob) [33] & Hjorth parameter \\ Activity (Act) [33] & Hjorth parameter \\ Spurious free dynamic range (SFDR) [34] & Length along a EEG signal \\ Curve length (CL) [34] & Length along a EEG signal \\ Teager energy (TE) [34] & Non linear energy \\ Variance (Var) [34] & Variance of the EEG signal \\ Standard deviation (Std) [34] & Standard deviation of the signal \\ Mean (Mean) [34] & Mean of the EEG signal \\
1 st derivative variance (Var1) [34] & Variance of the first derivative \\
1 st derivative standard deviation (Std1) [34] & Standard deviation of the first derivative \\
1 st derivative mean (Mean1) [34] & Mean of the first derivative \\
2 nd derivative variance (Var2) [34] & Variance of the second derivative \\
2 nd derivative standard deviation (Std2) [34] & Standard deviation of the second derivative \\
2 nd derivative mean (Mean2) [34] & Mean of the second derivative \\ Derivative variance ratio (RatioVar) [36] & Ratio of derivative respect absolute of derivative variances \\ Power (P) [35] & Power of the signal window \\
1 st difference (1d) [35] & Feature extraction and selection for emotion recognition from eeg \\ Normalized 1 st difference (N1d) [35] & Normalization of 1 st difference \\
2 nd difference (2d) [35] & Feature extraction and selection for emotion recognition from eeg \\ Normalized 2 nd difference (N2d) [35] & Normalization of 2 nd difference \\ Normalized Length Density (NLD) [35] & Quantifies self-similarities \\ Higher order crossings (HOC) [38] & Describes the oscillatory pattern of a signal \\ Band power (Pu) [35] & Spectral power in certain spectral band \\ Recursive Effective Efficiency (REE) [35] & Energy ratio of spectral bands \\ Relative Energy Ratio (rE) [37] & Relative energy between bands \\ \hline \end{tabular}
\end{table}
Table 1: List of extracted features.  
\(p(x)\) being the probability density function. With this entropy calculated regarding the signal power spectrum as a probability distribution, the log energy spectral entropy is obtained.