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The electroencephalogram (EEG) signal is known as a nonlinear and complex signal. The EEG signal has very important information about brain activities and disorders which can detect by an accurate Computeraided diagnosis system. The performance of the Computeraided diagnosis system directly depends on using features in the classifiers. In this paper, we proposed nonlinear geometrical features for the classification of EEG signals. The normal, interictal and ictal EEG signals of the Bonn university EEG database are plotted in 2D space by a novel approach and considering their patterns, six features namely: area of the octagon (AOO), circle area (CA), the summation of vectors length (SVL), centroid to coordinate center (CTC), circular radius out of triangles (CRT) and triangle area (TA) are extracted on different aspects of distance in Cartesian space. Based on the Kruskal–Wallis statistical test, all of the features were found statistically significant in the discrimination of normal vs. ictal and interictal vs. ictal EEG signals (pvalue≈0). Also, the edges of 2D projection EEG signals in the ictal group were sharper than normal and interictal groups. Besides, 2D projection of normal and interictal EEG signals has more regular geometrical shapes than the ictal group. Our proposed features were applied as input on support vector machine (SVM) and knearest neighbors (KNN) classifiers which resulted in more than 99% classification accuracy in a tenfold crossvalidation strategy.
ictal EEG signal, geometrical features, computeraided diagnosis, SVM, KNN
Epilepsy is a common brain disorder due to abnormal activity of neurons. It normally affects more than 50 million of the population in the world who most of them are living in developing countries [1]. Doctors and neurologists can evaluate the brain nerve activity by analyzing the Electroencephalography (EEG) signal. In EEG records, epilepsy attacks have manifested themselves with spikes in the EEG signal which can be detected visually by the medical team [2]. Visual inspection of EEG records for the longterm to detect presence of epilepsy attacks by medical team are very boring, time consuming and prone to human error. Therefore, a Computeraided diagnosis system for detection of epilepsy attacks in EEG signals is desirable [3]. For this purpose, for the past two decades, several machine learning methods based on the extraction of features from EEG signals have been developed in the literates. There is a direct relationship between the clinically significant feature extraction from EEG signals and the performance of machine learning which resulted to design an accurate Computeraided diagnosis system. Traditionally, extracted features are categorized into three groups; including extracted features in time (T) domain, frequency (F) domain and timefrequency (TF) domain. T domain features are extracted from EEG signal without any processing method on EEG signal. In other words, these features are evaluating EEG signal behaviors in T domain. T domain features like mean, median, mode, minimum, maximum, skewness, standard deviation, kurtosis, energy, correlation and histograms have been computed from EEG signal, previously [413]. The T domain cannot show many inherent behaviors of the EEG signal. For this reason, the EEG signal transforms from the T domain to the F domain. The spectrum shows the frequency components of EEG signals. Fast Fourier transforms (FFT) algorithm is the most used tool for representing frequency spectrum. Phase slope index, frequency moment signatures and mean frequency have been computed from EEG signals in the F domain [14, 15]. Although FFT is very useful for analyzing the spectrum, there is no relationship between the T domain and the F domain [16]. In other words, extracted features in the F domain are useful when the EEG signal is assumed stationary. But in essence, the EEG signal is complex and nonstationary and it can be assumed stationary only in the shortterm. So, TF approaches like wavelet transform and empirical mode decomposition was decomposed the EEG signals and several features including: entropybased features (log energy, Shannon, threshold, norm), chaoticbased features (fractal dimension, Lyapunov exponent, centered correntropy, information potential) were extracted from EEG subbands as TF domain features [1722].
Expect of these traditional methods, researchers have suggested a few numbers of geometrical methods for the detection of epilepsy attaches by using secondorder difference plot [23] and reconstructed phase space [24]. The EEG signals were decomposed to intrinsic mode functions (IMFs) by empirical mode decomposition, then the computation of the elliptical pattern of the secondorder difference plot of IMFs was considered as a feature [23]. In similar research [24], the elliptical pattern of the reconstructed phase space of IMFs was considered as a feature. In other words, their works [23, 24] can be categorized in TF methods, because EEG signals are decomposed by empirical mode decomposition.
Two methods of phase reconstructed phase space [23] and secondorder difference plot [24] have been used in the past to represent the signal in two dimensions. The reconstructed phase space requires the estimation of two parameters, namely the time delay and the embedding dimension, which are calculated with mutual information (MI) and false nearest neighbor (FNN) [16, 25], respectively. The disadvantage of the phase space reconstruction method is that it is timeconsuming because the MI and FNN calculations are extremely heavy [24].
The secondorder difference plot method is suitable for showing signal variations because of drawing a twodimensional signal difference [23]. But sometimes signal differentiation cannot give us good information about the behavior and dynamics of the signal itself.
For this reason, in this paper, we propose a novel method for plotting the signal in two dimensions, which, unlike phase space reconstruction, does not require any parameter, so it can quickly plot the signal in two dimensions. In addition, the proposed method shows the signal itself in two dimensions and the difference, unlike the secondorder difference plot method. So, the behavior and dynamics of the signals can be better observed.
In this paper, the new features represent the complex behaviors of EEG signals in 2D space and their proposed patterns can discriminate the EEG signals in normal vs. ictal, and interictal vs. ictal groups better than traditional approaches, without any TF methods [26].
The rest part of the paper is organized as follows: In section 2, geometrical method with used database, proposed 2D projection, area of the octagon (AOO), circle area (CA), the summation of vector length (SVL), centroid to centroid (CTC) and triangle area (TA) are covered. Results are provided in section 3. Finally, the discussion and conclusion of our paper are presented in section 4.
2.1 Used database
Our proposed method has been evaluated by known Bonn university EEG database which is freely downloadable from the http://epileptologiebonn.de web site. There are five subsets of this database called A, B, C, D and E. Each subset has 100 EEG signals in a Text Document format (.txt). The duration of each EEG signal is 23.6 seconds, which was sampled at a rate of 173.61 Hz, so it has 4097 samples. The subset of A and B belong to five healthy subjects in eyes opened and closed conditions, respectively. The subset of C and D belong to five patients who had completed recovered from seizure control after surgery of epileptic locations. The subset E is composed of EEG signals with epileptic seizure activities that are observed in the epileptogenic zone. The signals in A and B subset are considered normal signals, C and D subsets are considered as interictal signals and signals in the E subset are considered as ictal signals. More details about the Bonn university EEG database can be found by Andrzejak et al. [27].
2.2 Proposed 2D projection
In this work, 4096 sample points of EEG signals were selected for evaluating our proposed method. All the calculations of our proposed method are carried out using the MATLAB software, also any tool box not used. In this paper, a geometrical nonlinear method is developed which is directly depended to the number of EEG samples. By assuming that EEG signal be x(n)=[x_{1};x_{2};x_{3};…;x_{n}], the EEG signals should be evolved into 2D projection with the following formula:
$X(m)=\left[x_{1} ; x_{3} ; x_{5} ; \ldots ; x_{4095}\right]$ (1)
$Y(m)=\left[x_{2} ; x_{4} ; x_{6} ; \ldots ; x_{4096}\right]$ (2)
2D Projection $(m)=[X(m), Y(m)]=\left[\left(x_{1}, x_{2}\right)_{1} ;\left(x_{3}, x_{4}\right)_{2} ; \ldots ;\left(x_{n1}, x_{n}\right)_{m}\right]$ (3)
where, n is the lengths of the input signal x(n) and m is the 2D projection points and equal to n/2 (i.e. 2048 points which are mapped on Cartesian space).
Figure 1 shows a sample of 2D projection of normal, interictal and ictal EEG signals. Due to 2D projection of EEG signal pattern, we can compute the significant geometrical features to discrimination of ictal EEG signals from normal and interictal EEG signals.
Figure 1. 2D projection of the normal, interictal and ictal EEG signals
2.3 Area of octagon (AOO)
It is clear from Figure 1 that the 2D projection of EEG signals has geometrical patterns. It motivates us to compute the area of geometrical patterns as discrimination features. So, seems that the area of the octagon (AOO) of the 2D projection of EEG signals can give useful diagnostic feature. The AOO of 2D projection can be computed as follows [28]:
$M_{y^{2}}=(1 / M) \sum_{m=0}^{M1}[Y(m)]^{2}$ (4)
$M_{x y}=(1 / M) \sum X(m) Y(m)$ (5)
$M_{s u m^{2}}=M_{X^{2}}+M_{Y^{2}}$ (6)
$M_{p r d^{2}}=M_{X^{2}} \times M_{Y^{2}}$ (7)
$D=\left[M_{s u m^{2}}4\left(M_{p r d^{2}}M_{X Y^{2}}\right)\right]^{1 / 2}$ (8)
$D i a=\left[3\left(M_{s u m^{2}}+D\right)\right]^{1 / 2}$ (9)
The AOO is obtained from Figure 2 as given below.
Figure 2. The octagon for AOO calculation
In the above Figure 2 From ΔABC, as all sides are equal in an octagon,
$S^{2}=A C^{2}+B C^{2}$ (10)
Therefore
$A C=S / \sqrt{2}$ (11)
where, ‘S’ is the sides of the octagon. In general, Diameter of the octagon is given by
Diameter $=S+2(S \sqrt{2})$ (12)
since $S=$Diameter $/(1+\sqrt{2})$ (13)
Area of octagon $=2(1+\sqrt{2}) \times S^{2}$ (14)
On substituting ‘S’ value in AOO
$A O O=\frac{2}{1+\sqrt{2}} \times D i a^{2}$ (15)
2.4 Circle Area (CA)
It can observe from Figure 1 that 2D projection of ictal EEG signal occupies more area than normal and intellectual groups. It mentioned to us that area computation can give useful diagnostic feature. Every two successive points on 2D projection, make a vector and each vector can make a circle with diameter D which equal to the vector length. In this work, the summation of circle areas using consecutive vectors is computed as a feature. The summation of circle areas using consecutive vectors is defined as:
$C A=\sum_{i=1}^{n3} \frac{\pi}{2}\left(\left(x_{i+2}x_{i}\right)^{2}+\left(x_{i+3}x_{i+1}\right)^{2}\right)$ (16)
where, n denote to the samples of input EEG signal x(n); also, (x_{i}, x_{i}_{+1}), (x_{i}_{+2}, x_{i}_{+3}), indicated to coordinate the two successive points on the 2D projection of EEG signals. Besides, CA is a summation of the circle area using consecutive vectors. The summation area of brown circles in Figure 3 is considered as a feature.
Figure 3. Illustration of CA as s feature
2.5 Summation of vectors length (SVL)
For quantification EEG signal amplitude changes in the time domain, we can compute the summation of successive vector lengths which make by the consecutive points on the 2D projection of EEG signals. In other words, we calculate the summation of consecutive vector length as a feature. It's defined as follows:
$S V L=\sum_{i=1}^{n3} \sqrt{\left(x_{i+2}x_{i}\right)^{2}+\left(x_{i+3}x_{i+1}\right)^{2}}$ (17)
where, n denote to the samples of input EEG signal x(n); also, (x_{i}, x_{i}_{+1}), and (x_{i}_{+2}, x_{i}_{+3}) indicated to coordinate the two successive points on 2D projection of EEG signals which make a vector. Besides, SVL is the summation of successive vector lengths. The summation distance of a, b, c, d, e, f, g and h vector length in Figure 4 are considered as a feature.
Figure 4. Illustration of SVL as s feature
2.6 Centroid to Coordinate Center (CTC)
Every three successive points in the 2D plan make a triangle. So, points on the 2D projection of EEG signals can evolve into successive triangles. The distance between the center of the triangle and coordinate center can give useful diagnostic considering to the pattern of EEG signals in 2D space, because of ictal EEG signals that have occupied more area than normal and interictal EEG signals. The coordinates of the center of a triangle with (x_{i}, x_{i}_{+1}), (x_{i}_{+2}, x_{i}_{+3}) and (x_{i}_{+4}, x_{i}_{+5}) edges can be computed as follow [34]:
$X_{c}=\frac{x_{i}+x_{i+2}+x_{i+4}}{3}$ (18)
$Y_{C}=\frac{x_{i+1}+x_{i+3}+x_{i+5}}{3}$ (19)
where, (X_{C}, Y_{C}) denote to the triangle center coordinate. So the summation of centroid to coordinate center (CTC) distance of successive triangles can be defined as follows:
$C T C=\sum_{i=1}^{n} \sqrt{\left[X_{C}\right]^{2}+\left[Y_{C}\right]^{2}}$ (20)
where, u denote the number of successive triangles on 2D projection of EEG signals that is equal to 1023 which make with 2047 points.
The summation length of orange vectors in Figure 5 is considered as a feature.
Figure 5. Illustration of CTC as s feature
2.7 Circular radius out of triangles (CRT)
We said that each three successive points on coordinate plane make a triangle. We can draw a circle with radius r out of the each of triangle. In this paper, we are computed the summation of radius of the circles out of successive triangles as a feature.
Assuming that (x_{i}, x_{i}_{+1}), (x_{i}_{+2}, x_{i}_{+3}) and (x_{i}_{+4}, x_{i}_{+5}) be three edges of a triangle on 2D space, then three sides of triangle are calculated as follow:
$a=\sqrt{\left(x_{i+2}x_{i}\right)^{2}+\left(x_{i+3}x_{i+1}\right)^{2}}$ (21)
$b=\sqrt{\left(x_{i+4}x_{i+2}\right)^{2}+\left(x_{i+5}x_{i+3}\right)^{2}}$ (22)
$c=\sqrt{\left(x_{i}x_{i+4}\right)^{2}+\left(x_{i+1}x_{i+5}\right)^{2}}$ (23)
where, a, b and c denote to three sides of first triangle. So, the triangle area can be calculated by the Heron principle as follow:
$\operatorname{area}=\sqrt{\frac{(a+b+c)}{2}+\frac{(a+b+c)}{2}+\frac{(ab+c)}{2}+\frac{(a+bc)}{2}}$ (24)
Finally, the radius of circle out of first triangle is calculated as:
$r=\frac{a \times b \times c}{4 \times a r e a}$ (25)
In this work, the summation of the circular radius out of the successive triangles is computed as a feature as follow:
$C R T=\sum_{i=1}^{u} r_{i}$ (26)
where, u denote the number of successive triangles on the 2D projection of EEG signals and it is equal to 1023 which makes with 2047 points. The CTC and SVL are similar to CRT with this difference, that SVL and CTC quantify the variability of amplitude in the time domain, but CRT quantifies the selfsimilarity of the EEG signals on 2D projection plan with more flexibility than crosscorrelation [8]. Although the SVL and CTC evaluate the variability of amplitude, SVL is sensitive to CTC. On the other hand, CTC can be more useful than SVL in ictal detection when EEG signal has baseline noise.
Figure 6. Illustration of CRT as s feature
The summation raduis of brown circles intriangles in Figure 6 is considered as a feature.
2.8 Triangle area (TA)
It is clear from Figure 1 that the scatter of points of EEG signals on the 2D projection plane is more on y=x and y=x lines. For quantifying the scatter of points on y=x line, we computed the shortest distance of each point from 45degree line (shD) and for quantifying the scatter of points on y=x line, distance of each point from coordinate center (crD) is computed. To combination of the shD and crD, we computed the area of a triangle which the shD and crD are two sides of it. With having the two sides of a rightangled triangle, we can compute the third side by the Pythagorean Theorem. The shD and crD of (x_{i}, x_{i}_{+1}) point on EEG signals on 2D space can be defined as:
$s h D=\frac{\leftx_{i}x_{i+1}\right}{\sqrt{2}}$ (27)
$c r D=\sqrt{x_{i}^{2}+x_{i+1}^{2}}$ (28)
where, n denote to the samples of input EEG signal x(n). So, with having the two sides of triangle, the third side is calculated as follow:
$c r D^{2}=s h D^{2}+L^{2}$ (29)
Finally the triangle area can be calculated as follow:
$S=\frac{\operatorname{sh} D \times L}{2}$ (30)
In this work the summation area of triangles is used as a feature. It defined for 2D projection of EEG signals as follow:
$T A=\sum_{i=1}^{n} S_{i}$ (31)
where, n denote to the samples of input EEG signal x(n); also, TA is the summation of triangle areas. The summation area of blue triangles in Figure 7 is considered as a feature.
Figure 7. Illustration of TA as s feature
2.9 Classifiers
For classification of geometrical features, we employed two wellknown classifiers, namely: support vector machine (SVM) and knearest neighbors (KNN) [3, 29, 30]. In the binary SVM classifier, input features are mapped to hyper dimensional space using kernel function; then a hyperplane with a maximum margin (optimal margin) is separated the features in two classes [29, 30]. KNN is a simple classifier with fast implementation [3]. In binary KNN classifier, first the training data of both classes are sorted. Then, the test data can be classified in the first or second group with considering to its K closed training data [3, 17]. That way, the test data belongs to classes that have more members among K closed neighbors (i.e. K closed test data). The kernel function and distance metric have a direct relationship in the correct classification of test data in SVM and KNN classifiers, respectively. In this work, radial basis function (RBF) and cityblock are applied as kernel function and distance metric in SVM and KNN classifiers, respectively. Figure 8 and Figure 9 are explained the SVM and KNN algorithms in the binary classification by assuming that two features from each group are extracted. It can be understood form Figure 8 that the optimal margin (best hyperplane) can separate two classes better than the other margin. In Figure 9, the first, second, third and fourth circles around test data determine the one, seven, fourteen and twentyone closed training data. The test data belongs to classes A, B, A and B with assuming that K is 1, 7, 14 and 21, respectively.
Figure 8. Illustration of the SVM algorithm as a used classifier
Figure 9. Illustration of the KNN algorithm as a used classifier
In this paper, we proposed novel nonlinear geometrical features in order to discrimination of EEG signals in two classification tasks, including: normal vs. ictal (i.e. AB vs. E) and interictal vs. ictal (i.e. CD vs. E). Pvalue evaluates the ability of features in order to discriminate between the two classes [3]. In the present study, Kruskal–Wallis statistical test is applied to pvalue computation [3, 1519, 22, 24, 28, 30]. The kruskalwallis function in MATLAB is used for pvalue computation. Features with less than 0.001 the pvalue can be considered as clinically significant features in designing a Computeraided diagnosis system [3133]. In other word, lesser pvalue indicates to better ability of features in the discrimination of groups [3, 16, 30]. In this work, we computed the pvalues for normal vs. ictal and interictal vs. ictal groups. Also, the box plot is employed in showing the minimum, first quartile, median, third quartile, and maximum values of the extracted features. The mean and standard deviation of features are written in the Table 1 and resulted pvalues are written in the Table 2. Also, box plots of features are shown in Figure 10.
Figure 10. Box plot of features
Clearly can be understood from Table 1, the mean values of the extracted features in normal, interictal and ictal EEG signals are significantly different with together. Also, it can be observed from Table 2 that the most of the pvalues are very near to zero. Since the pvalues for all properties are very close to zero, this means that the extracted property has a high ability to separate the two groups. In addition, in Table 2 the mean value of the features in the ictal signal is significantly larger than the mean value of the features in the normal and intractal signal. Given the low pvalue values in Table 1, it can be said that all the features can be well defined in the classification task defined as normal vs. ictal, and interictal vs. ictal. This can also be obtained by analyzing Table 1, since the mean value in all features extracted from ictal signals is significantly larger than normal and Interictal signals. This can also be seen by examining the feature plots in Figure 10, since the box plot of all features for ictal signals is significantly larger than the box features for normal and interictal signals.
Table 1. Mean and standard deviation of features
Feature 
Normal 
Interictal 
Ictal 
AOO 
4.94E+06 ±3.41E+06 
7.74E+06 ±2.09E+07 
1.53E+08 ±1.32E+08 
CA 
2.07E+07 ±1.77E+07 
9.28E+06 ±1.98E+07 
7.25E+08 ±8.88E+08 
SVL 
9.05E+04 ±3.61E+04 
4.91E+04 ±2.71E+04 
4.26E+05 ±2.66E+05 
CTC 
1.22E+05 ±4.58E+04 
1.39E+05 ±8.64E+04 
6.38E+05 ±2.94E+05 
CRT 
4.79E+09 ±8.23E+09 
1.06E+10 ±8.68E+10 
4.93E+12 ±7.60E+12 
TA 
8.41E+05 ±6.08E+05 
6.85E+05 ±1.55E+06 
2.62E+07 ±2.65E+07 
Table 2. Resulted pvalues
Feature 
Normal vs. Ictal 
Interictal vs. Ictal 
AOO 
3.78E45 
7.81E43 
CA 
1.29E42 
3.00E44 
SVL 
1.24E39 
2.56E44 
CTC 
4.0E45 
8.43E43 
CRT 
1.34E43 
1.16E43 
TA 
4.11E44 
5.82E41 
The box plot of the features is illustrated in Figure 10, in which the ictal EEG signals revealed higher AOO, CA, SVL, CTC, CRT and TA values than normal and interictal EEG signals (notice that expect of CRT box plot, the other box plots are illustrated the logarithmic values of the features). These reasons tell us that these nonlinear geometrical features can leaded us in designing an accurate Computeraided diagnosis system. Hence, SVM and KNN classifiers are employed for designing a Computeraided diagnosis system.
These reasons tell us that these three nonlinear geometrical features can leaded us in designing an accurate Computeraided diagnosis system. Hence, SVM and KNN classifiers are employed for classification and designing a Computeraided diagnosis system.
By assuming the EEG signals in ictal and nonictal groups (i.e. normal and interictal be nonictal in normal vs. ictal and interictal vs. ictal classification task, respectively), then the output of classifier is in one of the following conditions [3, 17, 24]:
TP (true positive): when ictal EEG signal is identified as ictal EEG signal.
TN (true negative): when nonictal EEG signal is identified as nonictal EEG signal.
FP (false positive): when nonictal EEG signal is identified as ictal EEG signal.
FN (false positive): when ictal EEG signals is identified as non ictal EEG signal.
With these parameters, accuracy (ACC), sensitivity (SEN) and specificity (SPE) can be defined as follow [3, 17, 24]:
$A C C=\frac{T P+T N}{T P+T N+F P+F N} \times 100$ (33)
$S E N=\frac{T P}{T P+F N} \times 100$ (34)
$S E N=\frac{T N}{T N+F P} \times 100$ (35)
In general, kfold crossvalidation [17, 24] is used to break the data into train and test to be used by the classifier. Actually, for the kfold crossvalidation, k time, k 1 subset is used to train, and one resume subset is used to test the classifier. Finally, the mean value of ACC, SEN and SPE parameters are reported. Most of the articles that we wanted to compare our method with them used crossvalidation technique [10, 11, 13, 2224]. For this reason, we also used 10fold crossvalidation to analyze the results under almost equal conditions. The objective parameters for SVM and KNN classifiers with geometrical features are written in the Table 3 and Table 4, respectively.
The sigma value of RBF is varied from 0.1 to 1 by 0.1 step and number of k is varied from 1 to 9 by 2 step which the optimal ACC value for SVM and KNN classifier is obtained then sigma is chosen to 0.1 and number of k is chosen to 5, respectively. Besides, the ACC in Table 2 and 3 are resulted by one feature (i.e. feature vector has one array). With according to the resulted ACC, it can be observed that SVM and KNN classifier is better in order to normal vs. ictal and interictal vs. ictal classification, respectively.
Hence, the feature vector length in normal vs. ictal classification task is increased to two by combination of CTC and the other features in SVM classifier. The resulted parameters are written in Table 5, in which the combination of CTC with other features are resulted to perfect average classification ACC of 99.13%.
Table 3. Performance of geometrical features with SVM classifier
Feature vector 
Normal vs. Ictal 

ACC (%) 
SEN (%) 
SPE (%) 

CTC 
99.33 
99 
99.50 
AOO 
98.33 
98 
98.5 
CA 
98 
97 
93.5 
TA 
95.66 
95 
96 
SVL 
94.33 
93 
95 
CRT 
93.66 
81 
100 
Average. value 
96.55 
93.83 
97.08 
Feature vector 
Interictal vs. Ictal 

ACC (%) 
SEN (%) 
SPE (%) 

AOO 
95.66 
100 
94 
CA 
97.66 
99 
97 
TA 
96.33 
99 
95 
SVL 
97.33 
97 
97.5 
CTC 
95.33 
93 
93.50 
CRT 
92.66 
81 
98.50 
Average. value 
95.82 
94.83 
95.91 
Table 4. Performance of geometrical features with KNN classifier
Feature vector 
Normal vs. Ictal 

ACC (%) 
SEN (%) 
SPE (%) 

SVL 
93 
89 
95.50 
CA 
93.66 
90 
96.5 
CTC 
99 
98 
99.50 
TA 
96 
91 
98.50 
AOO 
98.67 
98.00 
99.00 
CRT 
94.33 
90 
97.50 
Average. value 
95.77 
92.66 
97.75 
Feature vector 
Interictal vs. Ictal 

ACC (%) 
SEN (%) 
SPE (%) 

SVL 
97.66 
97 
98 
CA 
97.33 
98 
97.5 
TA 
96.33 
96 
96.50 
CTC 
95.66 
98 
94.50 
AOO 
95.67 
96.00 
95.50 
CRT 
96 
96 
96.50 
Average. value 
96.44 
96.83 
96.41 
Table 5. Performance of SVM classifier in Normal vs. Ictal classification task with two features
Feature vector 
Normal vs. Ictal 

ACC (%) 
SEN (%) 
SPE (%) 

CTC, AOO 
99 
99 
99.50 
CTC,CA 
99 
98 
99.50 
CTC, CRT 
99.33 
99 
99.50 
CTC,SVL 
99.33 
98 
100 
CTC,TA 
99 
98 
99.50 
Average. value 
99.13 
98.40 
99.60 
In the same way, ACC of KNN classifier in interictal vs. ictal classification task can be improved. For this propose, arrays of feature vector in KNN classifier are increased to two by a combination of SVL with the other features. Performance of KNN classifier with this feature vector are written in Table 6.
Table 6. Performance of KNN classifier in Interictal vs. Ictal classification task with two features
Feature vector 
Normal vs. Ictal 

ACC (%) 
SEN (%) 
SPE (%) 

SVL,CTC 
97.33 
99 
96.50 
SVL,CA 
97.33 
98 
97.5 
SVL,TA 
96.67 
97 
97 
SVL,CRT 
96.33 
96 
97 
SVL,AOO 
96 
96 
96 
Average. value 
96.73 
97.20 
96.80 
The KNN classifier resulted in average 97.06% of ACC with two features. Also, to improve the ACC of the classifier, we investigated the combination of several geometrical features under 10fold crossvalidation conditions, which with three AOO, CA, SVL and CTC features vector, and reached a high of 99% ACC.
Epilepsy is a common brain disorder due to abnormal activity of neurons. It normally affects more than 50 million of the population in the world who most of them are living in developing countries [1]. Although doctors and neurologists can detect the of epilepsy attaches by visual analyzing of Electroencephalography (EEG), but visual inspection of EEG records for the longterm are very boring, time consuming and prone to human error. Hence, an accurate Computeraided diagnosis system for detecting the presence of epilepsy attaches in EEG signals is desirable. In this paper, six novel nonlinear geometrical features namely: area of octagon (AOO), circle area (CA), summation of vectors length (SVL), centroid to coordinate center (CTC), circular radius out of triangles (CRT) and triangle area (TA) were proposed to detection of epilepsy attach in EEG signal.
For this propose, these features were employed to discrimination of ictal EEG signals from normal and interictal EEG signals in two classification tasks including: normal vs. ictal, and interictal vs. ictal.
Ictal, interictal and normal EEG signals were selected from wellknown Bonn university database which has 200 normal (A and B subsets), 200 interictal (C and D subsets) and 100 ictal (E subset) EEG signals [27]. We showed that our 2D projection method can be used as a useful approach to illustration of complexity of EEG signals. Considering to Figure 1 the 2D projection of ictal EEG signals was occupied more space than normal and interictal EEG signals; this result has been reported in previous studies by drawing the EEG signals on 2D plane using second order difference plot [23] and phase space reconstructed [24]. Also, the edges of 2D projection EEG signals in ictal group were sharper than normal and interictal groups. Ray [2] said that ictal is manifesting itself with spikes in the EEG signal. For this reason, corresponding to Figure 1, the ictal EEG signals on the 2D projection were revealed by sharp edges. It can be observed from Figure 1, that 2D projection of normal and interictal EEG signals have more regular geometrical shapes. It may be due to synchronous response of brain neurons, which gives rise ictal parts in EEG signals. Support vector machine (SVM) and knearest neighbors (KNN) classifiers were employed for evaluation of accuracy (ACC), sensitivity (SEN) and specificity (SPE) of the proposed geometrical features. Corresponding to the Table 1 and 2, features were extremely severely statistically significant. Our proposed method resulted to more than 99% classification ACC in both classification task. We have compared our proposed method with existing studies on the same database in Table 7.
Table 7. Comparison of proposed method with the exiting work
Reference Year 
Used cross validation 
Used Subsets 
ACC (%) 
[9] (2012) 
No 
A vs. E B vs. E C vs. E D vs. E 
93.55 82.88 88.00 79.94 
[10] (2014) 
10fold 
A vs. E B vs. E C vs. E D vs. E 
99 97 98 93 
[11] (2011) 
10fold 
A vs. E B vs. E C vs. E D vs. E 
99.69 96.78 97.69 93.91 
[12] (2010) 
No 
A, B vs. E C, D vs. E 
96 94 
[20] (2014) 
No 
C, D vs. E 
95.33 
[23] (2014) 
10fold 
C, D vs. E 
97.75 
[24] 2015 
10fold 
C, D vs. E 
98.67 
[13] (2016) 
10fold 
A, B vs. E C, D vs. E 
99.18 95.15 
[19] 2019) 
No 
A vs.E C vs. E D vs. E C, D vs. E 
98.10 95.50 93.80 95.10 
[22] (2017) 
10fold 
C, D vs. E 
97.5 
Proposed method 
10fold 
A, B vs. E C, D vs. E 
99.26 99 
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