ECG Arrhythmia Classiﬁcation using High Order Spectrum and 2D Graph Fourier Transform

: Heart diseases are in the front rank among several kinds of life threats, due to its high incidence and mortality. Regarded as a powerful tool in the diagnosis of the cardiac disorder and arrhythmia detection, analysis of electrocardiogram (ECG) signals has become the focus of numerous researches. In this study, a feature extraction method based on the bispectrum and 2D graph Fourier transform (GFT) was developed. High-order matrix founded on bispectrum are extended into structured datasets and transformed into the eigenvalue spectrum domain by GFT, so that features can be extracted from statistical quantities of eigenvalues. Spectral features have been computed to construct the feature vector. Support vector machine based on the radial basis function kernel (SVM-RBF) was used to classify di ﬀ erent arrhythmia heartbeats downloaded from the Massachusetts Institute of Technology - Beth Israel Hospital (MIT-BIH) Arrhythmia Database, according to the Association for the Advancement of Medical Instrumentation (AAMI) standard. Based on the cross-validation method, the experimental results depicted that our proposed model, the combination of bispectrum and 2D-GFT, achieved a high classiﬁcation accuracy of 96.2%.


Introduction
According to the World Health Organization (WHO), about one third of the world's deaths are caused by cardiovascular disease every year statistically [1]. Cardiovascular disease has become one of the leading causes of death from non-infectious and non-transmissible diseases in the world [2]. Therefore, prevention and early diagnosis are key for a successful clinical management.
Arrhythmia is an important group of diseases in cardiovascular disease. The diagnosis of arrhythmia mainly depends on the electrocardiogram (ECG). ECG is an important modern medical tool that can record the process of cardiac excitability, transmission, and recovery. It reflect the degree of myocardial cell damage, development process, functional structure of atrium and ventricle [3]. Detection of irregular heartbeats from ECG signals is a significant task for the automatic diagnosis of cardiovascular disease.
The extraction of features from the ECG signal is a key step for ECG recognition, as it allows to greatly enhance and extract the characteristics of the signal. The quality of morphological features extraction in the ECG greatly affects the recognition and classification rate of ECG signals. Generally, the features in the time domain include the amplitude, the RR interval, the duration, and the shape of the clinical components (P-wave, QRS-complex, and T-wave) of ECG [4][5][6]. The features in the transform

Methods
The block diagram of the proposed method for ECG beat classification is depicted in Figure 1. The procedure is divided into five steps: (1) ECG signal pre-processing, (2) bispectrum analysis, (3) 2D-GFT, (4) feature extraction and (5) classification by the SVM-RBF.

Pre-processing
The first step of our algorithm consisted of data pre-processing. ECG signal can be contaminated by several artifacts, such as baseline wander, electromyographic artifacts and power frequency interferences [27,28]. These noises will affect the detection of real components and the classification of signal features. It is essential to figure out the suitable method for preprocessing while retaining the remarkable characteristics.
In the preprocessing stage, we used four steps to reduce noise. First, the DC component is eliminated by subtracting the average value. Then, the baseline drift is reduced with a median filter. The low-pass filter is used to remove power frequency interference and electromyogram (EMG) noise. Finally, the high-pass filter can eliminate some other low-frequency noise. After denoising, the R peak points are subsequently detected. The single heartbeat data are configured by segmenting the signal to be centered on the R peak point.

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The block diagram of the proposed method for ECG beat classification is depicted in Figure 1.

Bispectrum Analysis
After preprocessing, the single heartbeat data is analyzed by Bispectrum. The HOS is one of the robust methods applied for the non-linear signal analysis. It depicts the spectra of third and higher order statistics (i.e., moments and cumulants). In this work, the third-order statistics (bispectrum) of the ECG signals are gathered [29].
The power spectrum of stochastic process is defined as the Fourier transform of auto-correlation function. Similarly, the high-order spectrum is defined as Fourier transform of high-order moment. The high-order spectrum is proposed to offer precise signal analysis and estimation, which may contain more details than low-order spectrum.
The minimum high-order spectrum is the bispectrum, a two-dimensional function of frequency, which is a very helpful tool to detect and quantify quadratic effects in time-series. Denoting x(n) a stationary, zero mean and stochastic processes with third-order cumulant defined as R 3x (τ 1 , τ 2 ) = E[x(n)x(n + τ 1 )x(n + τ 2 )], (1) where τ 1 and τ 2 denote the time shift. E[·] denotes mathematical expectation. Then, the bispectrum of x(n) is given by the expression where ω 1 , ω 2 are two independent frequencies.

Graph Fourier Transform (GFT)
The Graph Fourier Transform (GFT) is the expansion of a graph signal function in terms of the eigenfunctions of the graph Laplacian matrix and is the foundation of GSP. The GFT is a data conversion method that is similar to the Fourier transform (FT).
After bispectrum analysis for the single heartbeat, the bispectrum results can be regarded as an image. We can execute 2D-GFT on the bispectrum results which is similar to execute 2D-FFT on images.
Superior to classical signal processing, graphs can be applied in irregular domains, so graph signal processing can be used to handle the irregular relationships between the data points [30].
An undirected graph G(V,E,W) usually includes the following three domain components: the vertex set V, the edge set E, and the weighted adjacency matrix W. N is the length of matrix row or column and also the number of vertices. Element w ij = 1 is defined to show there is a connection between nodes i and j, otherwise w ij = 0. The matrix W is the combination of elements w ij .
Graph Laplacian matrix L is defined by where the ith diagonal element in the diagonal matrix L is defined by.
The real symmetric matrix L has a complete set of orthonormal eigenvectors {x l |l = 0, 1, · · · , N − 1 } and the corresponding real eigenvalues λ l , where the eigenvalues λ l are all non-negative.
Vector s ∈ R N can be used to represent the signal of the vertices of a graph, with the ith component s i representing the signal value at the ith vertex in V.
The classical FT expands a continuous function considering the complex exponentials. Corresponding to FT, for a function s ∈ R N , the GFT expands it in terms of the eigenvectors of the graph Laplacian domain. The GFT and inverse GFT of a graph signal s(n) are defined by Equations (5) and (6).ŝ where x l quantifies the dependency of the vertex l. s, x l represents the inner product of s and x l . * denotes convolution operation. From the definitions of the GFT and inverse GFT, the conservation equation of energy can be obtained as Equation (7).
where · 2 2 represents energy, s 2 2 denotes energy of signal, and ŝ 2 2 denotes energy of eigenvalue spectral components.
In the 1D Euclidian Domain, the adjacency matrix is only affected by its precedent and ancient nodes. However, in 2D Euclidian Domain, the adjacency matrix is dominated by its precedent and ancient nodes both in rows and columns [31]. Meanwhile, the diagonal elements in the diagonal matrix are same as in the 1-D domain. For the Nth point data, the length and width of the adjacency matrix in 2D Euclidian Domain are N.
In 2D Euclidian Domain, the index is bound with the row subscript x and column subscript y, The 2D-GFT and inverse 2D-GFT of a graph signal S(i, j) are defined by Equations (9) and (10).

Feature Extraction
After the 2D-GFT, spectral features can be extracted.
Finding the decisive features of the signals is a crucial step in the procedure. In the following subsections, spectral features are explained in detail. They are extracted from the spectral modes obtained using the bispectrum and 2D-GFT. Different spectral features, including spectral flatness, spectral brightness and spectral roll-off, were extracted from divided single heart beat [25].
Spectral flatness measures the distribution of power in all spectral bands. It is the ratio of the geometric mean to the arithmetic mean of the spectrum. In this paper, it has been extended into the two-dimensional case. Thus, the ratio is determined by the arithmetic and geometric mean both in rows and columns.
where |·| is the absolute value. The spectral flatness ranges from 0 to 1. The spectral flatness close to 0 means an extremely narrow band, while close to 1 means extraordinarily flat.
The spectral brightness is the ratio of sum of magnitudes of spectrum above a boundary F to the total sum of all the magnitudes in the spectrum. Once F determined, a big spectral brightness indicates the concentration of power in the band between F and N/2. The definition has also been expanded into a 2-D case, similar to the spectral flatness.
Spectral Roll-off is also a function corresponding to the boundary F. Spectral roll-off can be used as the skewness which measures the non-uniformity of the signal about its mean value. In 2D-domain, the definition can be described as where β represents the coefficient.

Classifier
Support vector machine (SVM) is a popular supervised learning method which is widely used in pattern and object recognition, image processing and classification.
SVM can minimizes the empirical classification error by an optimal separating hyperplane, which can be represented by a decision function. In order to formulate the SVM algorithm based on radial basis function, we suppose a training set consists of M samples (x i , y i ), i = 1, · · · , M , where x i denotes the data element and y i represents the corresponding class label. The decision function can be formulated as (14), (15) Where w i is the product of the Lagrange multiplier, b is the bias term, K(x, x i ) is the kernel function, and γ is kernel function parameter.

MIT-BIH Arrhythmia Database
The MIT-BIH Arrhythmia Database [26] is an internationally recognized database to evaluate the effectiveness and feasibility of proposed algorithms in the ECG signal processing field. The database consists of 48 real ECG recordings, including normal and abnormal beats. Each recording taken by two leads lasts for 30 min with a sampling rate of 360 Hz. Only the modified limb lead II (ML II) for each recording is used for the classification task. The recordings used in this paper can be found in the Supplementary Materials. Table 1 lists the detailed heartbeat types. We randomly selected 33 recordings from the remaining 44 records. Q class (unclassified and paced heartbeats) is discarded since this class is marginally represented in MIT-BIH database. Finally, the classification is only realized on other four classes (N, S, V and F) in our study. In our experiment, the cross-validation method and inter-patient scheme are used to measure classification performance.
In the cross-validation test, group of 75,604 heartbeats was randomly partitioned into 5 equal sized subsets (5-fold). From the 5 subsets, one is considered as the test-set for validating the trained model. The training set is constructed with the rest of subsets. This training and validation process is repeated 5 times performing the validation each time with a different subset.
In inter-patient scheme, all records are split into two groups (Training set & Testing set) with a proportion of the whole dataset. Grouping assignment is listed in Table 2.

Evaluation Metrics
To evaluate the classification performance, six classification metrics were used in this study. The metrics are defined by four concepts: true positive (TP), true negative (TN), false positive (FP) and false negative (FN), respectively [32].
• Accuracy Accuracy (Acc) is a relatively comprehensive evaluated guideline, directly reflecting the performance. It is the proportion of test examples classified correctly. •

Error rate
Error rate (Err) is ratio of incorrect predictions over the total number of instances evaluated.
• Sensitivity Sensitivity (Sen), also called Recall, represents the probability that true positive samples can be correctly detected.
• Specificity Specificity (Spe) means the percentage of correctly classified negative instances. •

Positive predictive rate
Positive predictive rate (Ppr), also called Precision, means the proportion of positive samples classified correctly. •

Macro-averaged F-score
Macro-averaged F-score (F − score) means relations between data's positive labels and those given by a classifier based on a per-class average.

Experiments and Analysis
All the experiments were carried out using Matlab R2016a programming environment on a desktop Personal Computer (CW65S, Hasee Computer, Shenzhen, China) with Intel(R) Core i5-4210 CPU (2.60 GHz) and 8GB RAM configuration.

Pre-processing
ECG signal is easy to be interfered by several kinds of noises when collected. The common types of ECG signal noise are power frequency interference, baseline drift, and EMG noise [33]. Among them, EMG noise has a significant impact on ECG signal.
The MIT-BIH noise stress test database [34] offers EMG noise recording when assembling the ECG signal. The noise recordings were made using physically active volunteers and standard ECG recorders, leads, and electrodes; the electrodes were placed on the limbs in positions in which the subjects' ECGs were not visible. The recordings are created by the script nstdbgen-using clean recordings from the MIT-BIH, to which calibrated amounts of noise from record 'em' were added using nst- [26].
According to the requirement of Signal to Noise Ratio (SNR), ECG signal and noise of certain amplitude can be superposed. Here, two groups of ECG data with EMG noise, which SNR is 24 dB and 12 dB, are assembled as shown in Figure 2. There are fluctuations on original signals combined with EMG noises, and waves seems less smoothly with SNR declining.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 23 ECG signal is easy to be interfered by several kinds of noises when collected. The common types 229 of ECG signal noise are power frequency interference, baseline drift, and EMG noise [33]. Among 230 them, EMG noise has a significant impact on ECG signal.

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The MIT-BIH noise stress test database [34] offers EMG noise recording when assembling the 232 ECG signal. The noise recordings were made using physically active volunteers and standard ECG 233 recorders, leads, and electrodes; the electrodes were placed on the limbs in positions in which the 234 subjects' ECGs were not visible. The recordings are created by the script nstdbgen-using clean 235 recordings from the MIT-BIH, to which calibrated amounts of noise from record ʹemʹ were added 236 using nst- [26].

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According to the requirement of Signal to Noise Ratio (SNR), ECG signal and noise of certain 238 amplitude can be superposed. Here, two groups of ECG data with EMG noise, which SNR is 24dB 239 and 12dB, are assembled as shown in Figure 2. There are fluctuations on original signals combined 240 with EMG noises, and waves seems less smoothly with SNR declining.

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Several steps are conducted in noise reduction. First of all, the DC components are eliminated 242 by subtracting the mean value. Baseline drift can be removed by a 8.33 ms width median filter. An 243 341th order low-pass filter using a Hanning window, whose cut-off frequency is 40Hz, is used to 244 remove power frequency interference and EMG noise. An 341th order high-pass filter using a

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Hanning window, whose cut-off frequency is 0.5Hz, is used to remove some other low-frequency 246 noise. Figure 3 demonstrates the ECG signal before and after denoising.   Several steps are conducted in noise reduction. First of all, the DC components are eliminated by subtracting the mean value. Baseline drift can be removed by a 8.33 ms width median filter. An 341th order low-pass filter using a Hanning window, whose cut-off frequency is 40 Hz, is used to remove power frequency interference and EMG noise. An 341th order high-pass filter using a Hanning window, whose cut-off frequency is 0.5 Hz, is used to remove some other low-frequency noise. Figure 3 demonstrates the ECG signal before and after denoising.

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As shown in Figure 3, after denoising, the P-wave waveform, which is seriously disturbed by 251 EMG noise, is significantly improved. Adaptive differential threshold method [35] is then conducted 252 for QRS wave detection.

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After detecting the R-peak in every QRS complex, the single heartbeat was segmented such that 254 it was centered around the R peak, as shown in Figure 4. The segmentation length is 556ms.    As shown in Figure 3, after denoising, the P-wave waveform, which is seriously disturbed by EMG noise, is significantly improved. Adaptive differential threshold method [35] is then conducted for QRS wave detection.

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After detecting the R-peak in every QRS complex, the single heartbeat was segmented such that it was centered around the R peak, as shown in Figure 4. The segmentation length is 556 ms.

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As shown in Figure 3, after denoising, the P-wave waveform, which is seriously disturbed by 251 EMG noise, is significantly improved. Adaptive differential threshold method [35] is then conducted 252 for QRS wave detection.

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After detecting the R-peak in every QRS complex, the single heartbeat was segmented such that 254 it was centered around the R peak, as shown in Figure 4. The segmentation length is 556ms.

Transforming the ECG Signal into Bispectrum
Set the length of every segment as 128 and 50% as the overlapping rate. Figure 5 shows the bispectrum contour plots of the four types (N, S, V, and F beats) of ECG signals (from recordings No. 100, 232, 208 and 213).   As shown in the Figure 5, due to the symmetrical characteristic of bispectrum, all information can be extracted from one part. Hence, only data in the first quadrant are recorded in this paper. There are no values in the upper triangle of the first quadrant and the lower triangle of the third quadrant of the dual-frequency plane, which corresponds to the undefined term of the bispectrum matrix.
Bispectrums of different beats have different contour shapes, therefore, the bispectrum can be considered as a feature. However, when bispectrum is used as a feature, matrices which contain much redundant information will be generated. So, bispectrum cannot be directly used as a feature of classification, and further feature extraction is needed. To solve this problem, various methods were proposed, including integral bispectrum [35], slice spectrum [29], Principal Component Analysis (PCA) [10], Independent Component Analysis (ICA) [36] and Kernel Fisher Discriminant Analysis (KFDA) [37].

2D-GFT
In order to obtain high separability, the Graph Fourier Transform (GFT) of ECG signals is investigated from the graph spectrum domain. 2D-GFT helps enlarge the nuances among different classes. The impulse amplitude of the eigenvalue spectrum of the graph signal reflects the information of different components of the graph signal. The impulse amplitude of the eigenvalue is inversely proportional to the maximum value of the orthogonal eigenvector, that is, there are different degrees of amplification effects. With the increase of vertex number for graph signal, the maximum of orthonormal eigenvectors of Laplacian matrix will become smaller gradually. Effect of magnifying the difference should be more obvious [30].
The GFT matrix parameter is set to 64. Figure 6 is the whole mesh figure of four types (N, S, V & F beats) in ECG recordings. Figure 7 is top view of part of Figure 6. magnifying the difference should be more obvious [30].

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The GFT matrix parameter is set to 64. Figure 6 is the whole mesh figure of four types (N, S, V As shown in Figure 6, there is also the symmetrical characteristic in 2D-GFT results. In Figure 7,

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The following content introduces the determination of parameters F and  . Acc is used 300 as evaluation metrics. The higher Acc , the better the performance of the parameter.

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In four types of beats (N, S, V and F), we can randomly select 500 samples from each type. The 302 total number of the samples is 2000.

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Since the eigenvalue matrix has symmetry and the values in first columns of the matrix are too 304 small, so the boundary is set from 17, second half of matrix.

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To determine the optimal boundary F in spectral brightness, 15 experiments have been 306 performed on different boundary values. Figure 8(a)

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Similar to the boundary in spectral brightness, with same number of samples, coefficient  in 311 spectral roll-off is also determined by several experiments. As shows in Figure 8  As shown in Figure 6, there is also the symmetrical characteristic in 2D-GFT results. In Figure 7, graphics change greatly with the transformation of types. In the enumerated range, the eigenvalue spectrum of Class V is much more complete than that of Class F in Figure 7. That means the values in Class V are relatively concentrated, Class S and Class N in the middle, then Class F the most decentralized. GFT helps expand the gap among all types.
GFT graph spectra distributions for the bispectrum of four types in ECG records are obviously different in the same region, the features can be extracted according to this characteristic.

Parameter Analysis in Spectral Features
The following content introduces the determination of parameters F and β. Acc is used as evaluation metrics. The higher Acc, the better the performance of the parameter.
In four types of beats (N, S, V and F), we can randomly select 500 samples from each type. The total number of the samples is 2000.
Since the eigenvalue matrix has symmetry and the values in first columns of the matrix are too small, so the boundary is set from 17, second half of matrix.
To determine the optimal boundary F in spectral brightness, 15 experiments have been performed on different boundary values. Figure 8a shows the Acc results under varied boundary from 17 to 31. The recognition rate reaches up to the highest when F is 25, which is 94.7%. There's a slight decline around 25. And Acc is the lowest value when F is 31, that's because most power are concentrated before the boundary, leaving little room for fluctuation.

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On the basis of the bispectrum and 2D-GFT, more distinguishing features can be obtained. The 316 spectral flatness, spectral brightness and spectral roll off are calculated. Results are present in Figure   317 9 and Figure 10. Similar to the boundary in spectral brightness, with same number of samples, coefficient β in spectral roll-off is also determined by several experiments. As shows in Figure 8b, Acc results change with increasing parameters (0.1 to 0.9). Accuracy reaches up to the peak when the coefficient β is 0.3, which is 95.4%. Higher or lower coefficients both lead to the decrease of recognition rate.

Feature Extraction
On the basis of the bispectrum and 2D-GFT, more distinguishing features can be obtained. The spectral flatness, spectral brightness and spectral roll off are calculated. Results are present in Figures 9 and 10.

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off is the key factor to raise recognition rate.

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Data listed in Table 3   336 Figure 10 contains the spectral feature after 2D-GFT. In Figure 10 Figure 9a shows the spectral flatness values mainly range from 0 to 0.18, and the median value of Class N and Class S is lower than Class V and Class F. In Figure 9b, most values are close to 1. There is an obvious distinction between the spectral brightness of Class F and other three classes. Class F signals can be discriminated based on spectral brightness. Because the measure of the right skewness of the signal is the roll-off of the spectrum, most of the energy contained in the spectral coefficients is concentrated to the right of the average. It can be seen from the Figure 9c that the median value of the spectral roll-off of class F is the largest, about 30.6, while the median value of class N is the smallest, about 28.5. Spectral roll-off is the key factor to raise recognition rate.

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The classification refers to distinguishing different types of arrhythmias. In this section, SVM-

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RBF classifier was used to measure classification performance in cross-validation and inter-patient 352 scheme. The penalty coefficient of SVM-RBF is 5 and the parameter gamma is 8.

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The whole amount of the heartbeats used for cross-validation is 75604. Numbers of different 356 types of heartbeat are shown in Table 5.  Table 3 are confidence interval (CI = 95%) of three kinds of features. For instance, the probability of a selected sample's spectral flatness falling in interval [0.0101, 0.0106] is 95%. Compared with other classes, class N has the closest upper and lower bounds, which means little fluctuation.  Figure 10 contains the spectral feature after 2D-GFT. In Figure 10a, median spectral flatness values are between 0.35 to 0.36 for Class N, 0.36 to 0.37 for Class S, 0.34 to 0.35 for Class V and 0.31 to 0.32 for Class F. Compared with Figure 9a, the value in Figure 10a is more distinguishable. In Figure 10b, median spectral brightness values of three classes are near 0.53, only value of class V is above it. The boxplot of Figure 10c is more concentrated than that of Figure 9c, the median spectral roll-off values of three classes are 25, 24, 25 and 26 with less outliers (+) outside the top and bottom edges. Table 4 contains the confidence intervals of four classes after bispectrum and 2D-GFT. Similarly to those in Table 3, the indexes of class N are the most stable.

Classification Results
The classification refers to distinguishing different types of arrhythmias. In this section, SVM-RBF classifier was used to measure classification performance in cross-validation and inter-patient scheme. The penalty coefficient of SVM-RBF is 5 and the parameter gamma is 8.

Cross-validation Test
The whole amount of the heartbeats used for cross-validation is 75,604. Numbers of different types of heartbeat are shown in Table 5.  Table 6 is the average confusion matrix of classification results after merely bispectrum. Take N beats in Table 6 as an example, 89.95% beats are correctly classified on average, 10.05% beats are misclassified into Class S, Class V and Class F.  Table 7 shows the metrics values measuring the classification performance associated to each class, including average value and standard deviation (SD). The method gives a high value of correct rate in executing arrhythmia classification, which is above 89.3%. Among all classes, standard deviation keeps below 0.245. Take F beats in Table 7 as an example, the Acc is 96.9%, the Err is 3.1%, and the SD is 0.002.  Table 8 is the average confusion matrix of classification results after bispectrum and 2D-GFT. Also take N beats as an example, 96.28% beats are correctly classified on average, 3.72% beats are misclassified into Class S, Class V and Class F. Compared with the corresponding ones in Table 6, data in Table 8 get a remarkable improvement, about 5% for each class.  Table 9 shows the metrics values after bispectrum+2D GFT. Also take F beats as an example, the Acc is 98.8%, the Err is 1.2%, and the SD is 0.1%. Compared with Table 7, the indicators in Table 9 are better.  Tables 10 and 11 is a more direct presentation of the proposed method compared with the results in the experiments conducted by [38][39][40][41][42][43][44]. All papers in Tables 10 and 11 use the cross-validation method. The methods mentioned in Table 10 all use feature-extraction-pattern, the ones in Table 11 are not. The recordings used in other references listed in Tables 10 and 11 are all from MIT-BIH database. However, the records and segments used in the training set and test set in each literature are not exactly the same. For example, a total of 46 records were used by Hui Yang et al. [38], except recordings No.102 and No.104 which have no lead II information. Huang et al. [43] used 14 recordings for classification, and for each type of the ECG signal, a segment of 10 seconds were selected.  [39]. Paweł Pławiak [40] used the best evolutionary neural system based on SVM classifier to classify the data of 17 categories, then obtained an average accuracy of 90.20%. The methods proposed by Zhang et al. [41] and Zhang et al. [42] in previous years were Multi-lead fused method and ECG Morphology, respectively, achieving an accuracy of 86%. Huang et al. proposed a combination of STFT-Based Spectrogram and Convolutional Neural Network to achieve an accuracy of 99.0% [43]. Shu Lih Oh et al. [44], got an accuracy of 98.1% when combined the automated system of CNN and LSTM to analyze ECG signals. In this paper, we propose a method of combining 2D-GFT with bispectrum, which achieves 96.2% accuracy.
In our results, the average accuracy of merely bispectrum is only 87.8%, lower than most of methods listed in this Table 10. But when combined with 2D-GFT, the proposed method yields a higher average accuracy of this experiment to 96.2%, which suggests the method is a promising one in cross-validation. Table 12 contains the detailed values of the proposed method in this section. After 2D-GFT, all metrics are improved. Especially, Sen is improved by more than 30%. By our designed method, the Sen, Spe and Ppr of the S beats are 71.5%,99.3% and 82.6%, those of the V beats are 76.7%, 99.2% and 86.9%, those of the F beats are 46.2%, 99.9% and 91.0%, respectively. The time cost of each method is listed in Table 13 which represents the time for one heartbeat to process. The running time of bispectrum combined with 2D-GFT is longer because of the higher algorithmic complexity and better performance of detection, which matches our anticipation. The 2D-GFT is very time consuming because of varied parameters. The second experiment is based on inter-patient scheme, detailed numbers are listed in Table 14.  Table 15 shows the confusion matrix of the classification results on the testing set in inter-patient scheme. 1371, 1643 and 296 Class N samples are misclassified into other three classes separately. Class N is still the main misclassification destination for other three classes. Results quoted in Table 16 are the comparison of the results conducted by Afkhami et al. [7], Zhang et al. [42], Ye et al. [45] and the proposed method. The overall accuracy is improved to 92.2%, higher than any other related literature. Besides, Spe and Ppr get a remarkable improvement in this experiment. For instance, the specific of S beats is 99.7%, much higher than the methods proposed by Afkhami et al. [7] and Ye et al. [45], and 5.8% higher than method proposed by Zhang et al. [42]. It demonstrates that the proposed method in this paper is a reliable tool in detecting the arrhythmia signals.

Conclusions
The aim of the conducted research is to develop a new methodology that enables the efficient classification of cardiovascular disease. The ECG signals can be tool for the medical practitioners to detect various types of cardiac arrhythmia because abnormal heart electrical activity can cause irregular morphology changes which would be checked through ECG signals.
In this paper, we proposed an ECG arrhythmia analysis method based on the combination of bispectrum and 2D-GFT. In the proposed method, we use high-order spectra to detect the basic features and then execute 2D-GFT on the result. And extract 2D spectral features. Support vector machine based on the RBF kernel are employed in the classification procedure. The experimental results depicted that our proposed model achieved a high classification accuracy of 96.2%. The novelty of this approach is to regard the bispectrum results as an image and apply 2D-GFT to the ECG signal. Compared with the previous references, such as Acharya [10] and Zhelin [17], our advantage is that we can achieve effective classification using only three features, which greatly improves the classification efficiency. Furthermore, the methods used by Acharya et al. [10] and Zhelin et al. [17] only made binary classification, while the proposed method can achieve multi-class categorization. Compared with Zhang et al. [41], who also used three features for classification, the accuracy of our method is 10% higher than theirs.
ECG recordings of four arrhythmia types, shared by the MIT-BIH arrhythmia database, were used for evaluations of the algorithm performance. We used the cross-validation method for classification. Compared with bispectrum analysis, after 2D-GFT, all metrics are improved. By our designed method, the sensitivity, specificity and positive predictive rate of the S beats are improved 30.5%, 0.3% and 8.3%, those of the V beats are improved 26.9%, 9.7% and 9.7%, those of the F beats are improved 27.2%, 0.4% and 36.4%, respectively. Comparisons with peer works prove a marginal progress in heart arrhythmia classification performance. This study proves that the proposed method is an excellent model for the diagnosis of heart diseases based on ECG signals.
In comparison with some references, the data pre-processing process in this paper is comparative to recent approaches, which may affect the classification effect of the ECG signals. Many methods have been proposed to remove noise, such as wavelet transform [46], empirical mode decomposition [47],