Higher-Order Phase-Space Reconstruction for Detection of Epileptic Electroencephalogram

Article History: Received: 11 January 2021; Accepted: 27 February 2021; Published online: 5 April 2021 Abstract: In this paper, the authors propose a new technique for the classification of seizures, non-seizures, and seizure-free EEG signals based on non-linear trajectories of EEG signals. The EEG signals are decomposed using the EMD technique to obtain intrinsic mode functions (IMFs). The phase space of these IMFs is then reconstructed using a novel technique of higherorder dimensions (3D, 4D, 5D, 6D, 7D, 8D, 9D, and 10D). The existing techniques of seizure detection have deployed 2D & 3D phase–space reconstruction only. The Euclidean distance of all higher-order PSR is used as a feature to classify seizures, non-seizures, and seizure-free EEG signals. The performance of the proposed method is analyzed on the Bonn University database in which 7D reconstructed phase space classification accuracy of 99.9% has been achieved both using Random Forest classifier and J48 decision tree.


Introduction
Epilepsy is an acute nervous disorder occurring due to episodes of neuronal discharges within the brain. Around 0.8% of the world population is suffering from epilepsy. A very high percentage of epileptic patients belong to middle-income countries and therefore, unable to receive the required medication [1]. The uncertainty and random occurrence associated with seizures are the main cause of causality. This uncertainty can be reduced by devising systems for automated detection & prediction of epileptic seizures. The symptom of epilepsy is a subject of debate for various surgeons. Epilepsy is a crucial problem and posing a burden in various countries across the world including South Africa, the US, Pakistan, etc. In Pakistan, this rate is 0.99 % [2].
The analysis of EEG signals involves feature extraction and classification using time-domain, frequencydomain & time-frequency domain methods. The dynamics of EEG signals can be captured more accurately by non-linear signals. In the literature, there exist various non-linear techniques [3][4][5][6][7][8] mostly based on phase-space to detect seizures. The most common and highly explored methods based on non-linear signal analysis include principal component analysis, Lyapunov exponent, etc. Lyapunov exponent has shown promising results in its ability to characterize the dynamics of the pre-ictal state by measuring the divergence in trajectories of the dynamical system. The first-order derivative along with phase-space reconstruction using Euclidean distance had been used in [9]. In [10,11] the classification of ictal and seizure-free EEG signals using a second-order difference plot (SODP) has been used. Recently, permutation entropy [12] has been computed to distinguish pre-ictal and normal EEG signals. In [13] the convolutional neural-network-based approach is used to detect Nocturnal Frontal Lobe Epilepsy (NFLE). The challenges in accurate seizure detection and prediction are enormous. In [14], seizure onset has been identified using synchrony measures from different epileptogenic zones. The brain is the most complex system in the human body that has adaptive, nonlinear, and dynamic characteristics. It can be considered as a self-organizing spatially embedded network at different temporal and spatial scales. It is difficult to analyze such a system due to the formation of complex dynamic structures. Hence space reconstruction technique or embedding time series is used to represent the state space of a dynamic system [7]. The latest technique [15] on phase space elliptic density has used a meta-heuristic optimization method for the detection of epileptic EEG signals. The time-scale decomposition, discrete wavelet transform (DWT),phasespace reconstruction, and neural networks have been utilized to discriminate normal, ictal, and inter-ictal EEG signals [16]. The most common procedure adopted for nonlinear dynamical analysis is the time-delay approach. Hence, in this paper phase space is reconstructed for seizure and non-seizure EEG signals using 2D, 3D,4D,5D ,6D,7D,8D,9D,10D phase space plots.

EEG Database
The study has been performed on the Bonn University EEG database publicly available and described in [17]. The data has been recorded using a sampling frequency of 173.61 Hz for 23.6 sec using 100 single-channel EEG electrodes. It has five subsets: Z, O, N, and F for non-seizures and S for seizures, each consisting of 100 singlechannel EEG signals.

Phase Space Representation
According to earlier studies, the phase space technique can be used to extract features for the classification of healthy subjects and epileptic patients. The phase space reconstruction method was proposed by Takens et al. [18]. The auto-regressive modeling of the phase space has been utilized in literature [19] for the classification of 3-class EEG-based motor imagery. In a nonlinear system, the time-series signal is converted to a phase space series signal using the phase space reconstruction method. The nonlinear systems tend to gravitate towards specific regions in phase space known as attractors. Attractors are usually represented using phase space plots of the nonlinear system. The dynamics of a nonlinear EEG signal can be visualized through these phase space trajectories which are computer-generated plots of EEG. The geometrical phase space EEG patterns are plotted when a point traces a trajectory in space. These patterns describe the state of the system as it evolves. The degrees of chaos or complexity can be quantified in terms of dimensionality [20]. N-dimensional phase space consists of d-independent variables. For the phase space reconstruction, the time-delay method is frequently used [18] viz. for the 1-D time series, other dimensions may be obtained by delayed counterparts of the observed variable. Let F[n] is the signal to be analyzed which can be represented as a vector in space as: where n=1, 2, 3… N-(d-1) t, d is called the embedding dimension, and t is the lag factor.
In this method choosing proper values of time -delay and embedding dimension is very important. If timedelay (t) is chosen very small, then coordinates of vector X[n] are nearly equal and the reconstructed phase space vectors are too close to obtain dynamic information of the signal. On the other hand, for a too-large value of timedelay, the reconstructed phase space vectors are too far apart. Hence original phase reconstruction is largely dependent on the choosing correct value of time-delay. Also, it is important to choose a sufficient embedding dimension in phase space to represent the dynamics of the system. The criterion for select the embedding dimension is based on properties of the "non-cutting" trajectory and reversibility of the reconstructed attractors. In this embedded dimension d is chosen 2 and 3 for visualization simplicity and time-lag =1 was used for the construction of an EEG attractor.
The 2D phase space plot can be obtained between two variables (xn, xn+1) while a 3D phase space plot is plotted between xn and its delayed versions by one & two represented as xn+1& xn+2 respectively [21]. A new vector is created in space by time-delayed values of scalar measurements.

Feature extraction for EEG Signals 2.3.1 Area computation of elliptical pattern for 2D PSR
The IMFs of EEG signal obtained after decomposition is oscillatory. Hence the PSR of IMFs is also elliptical. 2D PSR is a plot between the two vectors Vk & Vk+1 [22 ]. The 2D PSR plot of the first four IMFs computed for the 95% confidence area has been shown in Fig.1. It is depicted from the figure that the area of elliptical patterns differs for each signal. The larger area for seizure EEG helps in discriminating it from normal EEG.

Computation of Euclidean Distance & IQR for higher dimension PSR (3D-10D)
3D PSR plot is used to represent the dynamics of the system. It is plotted between three vectors delayed by one unit each i.e. rk,rk+1,rk+2. The distance between origin and the point (rk,rk+1,rk+2) is defined as Euclidean distance and computed as ED=√(r_k^2+r_(k+1)^2+r_(k+2)^2 ) (3) Fig.2. shows the Euclidean distance plot for 3D PSR. It can be seen that the length of Euclidean distance is different for each class of EEG signals.
IQR of Euclidean distance IQR, based on Euclidean distance is the measure of the variability of data. It is the difference between the first & third quartiles based on dividing the dataset into quartiles. Mathematically, IQR can be expressed as: IQR=Q3-Q1 (4)

Results & Discussion
In this approach, the five sets of the Bonn University EEG database are used. In the first stage, empirical mode decomposition of EEG signals is performed to extract intrinsic mode functions. The first four IMFs are used for feature extraction in phase space. The 2D phase space is an elliptical pattern and the area of the ellipse is computed for all EEG signals. For higher-order phase space, Euclidean distance is computed for all EEG signals. The Euclidean distance extracted for all EEG signals is given as input to JRIP, J48 & Random Forest classifier. The comparative performance of classifiers has been done in Table 1 for 3D PSR.  Table 1 shows performance measures for 3D-10D PSR. It can be observed that the Random Forest classifier gives maximum classification accuracy for 3D PSR while for 4D PSR 99.9% classification accuracy has been achieved using an RF classifier. The graphical comparison of performance has been done in Fig 3.
TN → true negative, the number of epochs that are marked as non-seizure by both neurologist and algorithm. FN → false negative, the number of seizure epochs that are misclassified by the algorithm, i.e. recognized as non-seizure but actually, they are seizures.
FP → false positive, the number of non-seizures epochs which are misclassified by the algorithm, i.e. recognized as seizure but actually, they are non-seizures.

Conclusion
Due to the widely accepted fact about the chaotic nature of the brain, its non-linear dynamics have been studied using phase-space representation of higher dimensions. This method intends to improve the classification accuracy of seizure, seizure-free, and non-seizure EEG signals. The novelty of this approach is the use of higherorder PSRs for the detection of epileptic seizures. The first four IMFS are obtained using the EMD technique and 2D PSR plot elliptical area is computed. However, for higher-order PSR, IQR of Euclidean distance is computed. For classification JRIP, the J48 decision tree & Random Forest classifier are used to compare the accuracy, sensitivity, and specificity of different EEG signals. It has been found that the Random Forest Classifier and J48 decision tree perform better as compared to JRIP (Table 1). The accuracy achieved by using 10 fold crossvalidation in 4D,5D 6D, 7D, 8D, 9D & 10D phase space reconstruction is higher as compared to the existing methods based on 2D & 3D PSR.