EEG records electrical signals from the cerebral cortex by measuring the electrical activity of a group of neurons. Because the patterns of EEG signals are often superimposed by the electrical signals from various parts of the brain in the direction of the electrode connections, different from ECG waveforms, it is very difficult for EEG waveforms to distinguish PQRST waves directly in the time domain and locate the specific excited region. Therefore, the analysis of brain waves in the frequency domain is slightly more important. The basic structural features of the alpha, beta, theta, and delta waves, which are roughly divided according to their frequency, can be summarized as follows: alpha (8–13 Hz), beta (18–30 Hz), theta (4–7 Hz), and delta (1–3.5 Hz). These bands are closely related to human behaviour, please see Table 1.

Based on the structural features of the four waveforms in the table, we can quickly conclude that by using other feature extraction methods, such as the Fourier transform, we can derive the intensity of the EEG signal at different frequency components, and the existence of these waveforms and their intensity information are obtained.

A real-world epilepsy EEG signal [2] is shown in Fig. 1. Group A and Group B are the EEG signals of people without epilepsy with their eyes opened and closed, respectively. Group C and Group D are the EEG signals of patients with epilepsy (without seizures) when their eyes are opened and closed, respectively. Group E is the brain electrical signal of patients during epileptic seizures. From Fig. 1, it can be seen that the signals of Groups A and B are stable and dense, and the signals of Groups C, D and E have large amplitudes, and the intensity fluctuates from high to low, indicating that the neuron discharge is abnormal.

In addition to the differences in EEG signal waveforms of different subjects in different states, the EEG signals of subjects in a group are also different at different times. Therefore, we use the feature extraction method to extract artificial features of the signal to reduce the amount of data and noise data and facilitate model learning.

As previously mentioned, on the original EEG signal, due to the large amount of data and changeable characteristics, effective feature extraction methods are needed to extract informative features for machine learning to reduce the computing cost. To this end, the traditional STFT, WPD and KPCA methods are enlisted in our study. Numerous traditional algorithms have been used for medical data analysis [12]. Nevertheless, one limitation of many well-established algorithms is that they can work well only when the data distributions of the training and testing data are the same or overall similar. Otherwise, they are commonly inefficient and even invalid. We briefly introduce three time-frequency methods commonly used in signal analysis [13]. 1) Wavelet packet decomposition (WPD). WPD is suitable for the analysis of nonstationary signals. It can be adapted according to the characteristics of the signal. The wavelet basis function can analyse signals well. 2) Short-time Fourier transform (STFT). Taking the STFT feature extraction method as an example, Fig. 2 shows an epilepsy EEG signal after the feature extraction method. The signal of each group is reduced from 1000 dimensions to 6 dimensions, which not only reduces the number of features but also makes the amplitude of the waveform more stable. Compared with Fig. 1, the amplitude decreases from −200∼200 to 0∼40. This greatly reduces the calculation cost. The reserved 6-dimensional features (especially the 6th dimension) have a unique signal structure, which is also conducive to machine learning. 3) Kernel principal component analysis (KPCA). This method introduces the concept of a kernel function into PCA.

In the field of medicine, the interpretability of the employed machine learning models is indispensable. Doctors will not believe in the prediction results of machines unless they can well understand the inference mechanisms of machines. Compared with SVM, KNN, and NB, TSK fuzzy systems have the desired trait of good interpretability.

The maximum mean discrepancy (MMD) is a nonparametric method used to measure the distance between two distributions [14]. Here, we use MMD to measure the distribution difference between the training data (the source domain) and testing data (the target domain). Minimizing the MMD distance facilitates the two distributions being closer, so that the knowledge learned by the model is compatible with both the training and testing data [14].

Given source domain data DSource={xi,yi},i=1,…,nS and target domain data DTarg⁡et={xi},i=1,…,nT, the MMD distance is defined as: (1)distance(DS,DT)=||1nSϕ(xS)−1nTϕ(xT)||H2=1nS2ϕ(xS)2+1nT2ϕ(xT)2−2nSnTϕ(xS)ϕ(xT)where ϕ(⋅) represents the kernel function.

The rules in a back propagation TSK (BP-TSK) fuzzy system are equivalent to the neurons in neural networks. The network structure of the BP-TSK fuzzy system is also similar to that of a neural network, as shown in Fig. 3.

Given training dataset DTrain={xn,yn},n=1,…,N, where x∈Rnxd,d=1,…,D, D is the feature dimension, y∈Rnxc,c=1,…,C, and C is the total number of categories. Assuming that the BP-TSK fuzzy system has R rules, it is constructed as follows: (2)Ruler: IF x1  is Ar,1 and x2  is Ar,2 and ⋯ and xD  is Ar,D THEN yrc(x)=qr,0c+qr,1cx1+⋯+qr,DcxDwhere Ar,d is the membership function, including the antecedent parameters, and qr,dc denotes the consequent parameters of class c.

The output of the BP-TSK fuzzy system is the weighted results of each rule: (3.1)yc(x)=∑r=1Rfr(x)yrc(x)∑r′=1Rfr′(x)=∑r=1Rf¯r(x)yrc(x)

The membership degree of d-dimensional features in rule r is: (3.2)μAr,d(xd)=exp⁡(−(xd−mr,d)22σr,d2)where mr,d and σr,d represent the centre and standard deviation of the Gaussian kernel function, respectively, which are often obtained using fuzzy C-means (FCM) clustering or other similar algorithms. In the formulation of rules, we assign semantic words at different levels according to the level of the antecedent parameter σr,d in each rule.

Trigger value of rule r: (3.3)fr(x)=∏d=1DμAr,d(xd)where f¯r(x) represents the normalization trigger level of the r-th rule: (3.4)f¯r(x)=fr(x)∑r=1Rfr(x)

The following formula is constructed according to [12]: (4.1)xe=(1dx1,xT)T (4.2)x¯r=f¯r(x)xe (4.3)xg=((x¯1)T,(x¯2)T,…,(x¯r)T)T (4.4)qrc=(qr,1c,qr,2c,…,qr,Dc)T (4.5)qgc=((q1c)T,(q2c)T,…,(qrc)T)T

Then, (3.1) can be expressed as [15]: (4.6)yc=xgqgc

For classification tasks, cross-entropy can be used to constitute the loss function of the BP-TSK fuzzy system through the Softmax function: (5)p(yc(x))=exp⁡(yc(x))∑c=1Cexp⁡(yc(x)) (6)Loss=−∑n=1Nynp(yc(x))

The objective function of the BP-TSK fuzzy system can be formulated for all attributes xgi as follows: (7)minLqgc=−∑n=1Nynln⁡αi+λ12||qgc||2=−∑n=1Nynln⁡exp⁡(zi)∑c=1Cexp⁡(zi)+λ12||qgc||2=−∑n=1Nynln⁡exp⁡(xgiTqgc)∑c=1Cexp⁡(xgiTqgc)+λ12||qgc||2where the former item is an empirical risk item and the latter is a structural risk item. qgc is the subsequent parameter of type c, and λ1 is the regularization parameter controlling the tolerance of the classifier to errors and can be adjusted by cross-validation.

Minimizing the objective function L requires obtaining the partial derivative of qgc and setting the derivative equal to 0: (8)∂L∂qgc=∂L∂α∂α∂z∂z∂qgc=∑iNxgi(exp⁡(xgiTqgc)∑c=1Cexp⁡(xgiTqgc)−yic)+λ1qgc

We can use the AdaBelief optimizer [15] or other optimization methods [16] to update the parameters and obtain the optimal parameters qg∗ after iterations.

The algorithm of the BP-TSK fuzzy system is as follows:

By introducing MMD into the TSK system, the distribution distance can be further defined as: (9)MMD(DS,DT)=||1nS(xgSTqg)−1nT(xgTTqg)||H2=1nS2qgTxgSxgSTqg+1nT2qgTxgTxgTTqg−2nSnTqgTxgSTxgTqg=qgT(1nS2xgSTxgS+1nT2xgTTxgT−2nSnTxgSTxgT)qg

Let (10)M=1nS2xgSTxgS+1nT2xgTTxgT−2nSnTxgSTxgT

Then, (9) can be rewritten as: (11)MMD(DS,DT)=qgTMqg

The objective function of TSK after adding the MMD item is: (12)minLqgc=−∑i=1Nyiln⁡αi+λ12||qgc||2+λ2(qgc)TMqgc

The derivative corresponding to qg is: (13)∂L∂qgc=∑iNxgi(exp⁡(xgiTqgc)∑c=1Cexp⁡(xgiTqgc)−yic)+(λ1E+λ2M)qgc

The label information of the source domain and the target domain can provide more data distribution information, so many experts have begun to study domain adaptation.

After the BP-TSK fuzzy system is added to the MMD item, it can learn the similar characteristics of two-domain data, so the prediction result has a certain degree of reliability. Based on the MMD-BP-TSK fuzzy system, we save the prediction results as pseudolabels. Then, we use pseudolabels to make the data distributions of the two domains more similar according to categories so that the model can learn more refined knowledge features.

Fig. 4 shows the three steps of data distribution processing: a) the unprocessed initial data distribution, b) the data distribution after processing by the MMD method, and c) the data distribution after processing by the SSL-MMD method.

In Fig. 4a, when the model has learned the data with labels, it is difficult for the model to adjust its decision boundary when the distribution is different from the previous data distribution, that is, when there is a chaotic data distribution.

In Fig. 4b, we can see that the chaotic data distribution has been improved, and the model can find the appropriate decision boundary between the two domains.

In Fig. 4c, a pseudotag is added on the basis of Fig. 4b to guide the model to determine a more appropriate decision boundary.

The objective function of SSL-MMD is: (14)SSL−MMD(DS,DT)=∑c=1C||∑i=1nSwic(xgi,STqg)−∑j=1nTwjc(xgj,TTqg)||H2where {wic=yi/nSwjc=y^j/nT, yi is the known source domain label. y^i is an unknown target domain label, which is replaced by a pseudolabel.

Let (15)MC=1nS2wicTxgSxgSTwic+1nT2wjcTxgTxgTTwjc−1nSnTwicTxgSxgTTwjc−1nSnTwjcTxgTxgSTwic

Similarly, (14) can be rewritten as: (16)SSL−MMD(DS,DT)=(qgc)TMCqgc

The objective function of SSL-MMD-BP-TSK is: (17)minLqgc=−∑i=1Nyiln⁡αi+λ12||qgc||2+λ2(qgc)TMqgc+λ3(qgc)TMCqgc

The derivative corresponding to qgc is: (18)∂L∂qgc=∑iNxgi(exp⁡(xgiTqgc)∑c=1Cexp⁡(xgiTqgc)−yic)+(λ1E+λ2M+λ3MC)qgc

We use the idea of deep learning to try to transfer the knowledge learned in the previous round to the next round. Therefore, the concept of a semi-supervised inheritance (SSL-I) mechanism is introduced to increase the information interaction between different domains. Therefore, we proposed a TSK fuzzy system for cross-domain learning, which includes SSL-MMD and SSL-I mechanisms.

Unlike the SSL-MMD method, which reduces the difference in data distributions, the SSL-I method makes full use of pseudolabels to allow the model to learn additional feature representations. The SSL-I method merges the existing tags and pseudo tags with the source domain data and target domain data, respectively, to construct a new dataset.

Experiments show that pseudolabels can be used for effective feature representation, allowing the model to learn deeper knowledge.

Suppose a given dataset is Dtrain(xi,yi), Dtest(xj); after learning by the MMD-BP-TSK fuzzy system, the prediction result is obtained. The prediction result is used as a pseudolabel y^ of the target domain data to reconstruct the datasets ExDSource{xi,yi},xi∈ℜn×(d+yi),i=1,…,NS and ExDTarg⁡et{xj,y^j},xj∈ℜn×(d+y^j),j=1,…,NT.

where yi and y^j are labels in the form of one-hot vectors. The new dataset is learned again through the SSL-MMD-BP-TSK fuzzy system. The structure of the CD-BP-TSK fuzzy system is shown in Fig. 5.

The proposed new classification algorithm is summarized as follows:

In this section, we build datasets in different scenarios to comprehensively test our methods. The experiment was completed in the following working environment: Processor: Intel(R) Core(TM) i5-8500 CPU@3.00 GHz; Memory: 12.0 GB; Software: MATLABR2021A.

1) Build the datasets

The datasets are divided into two categories. As shown in Table 2, one is a dataset with a similar data distribution, and the other is a dataset with a chaotic data distribution.

2) Performance index

In classification problems, accuracy is used to judge whether a model’s performance is excellent.

1) Comparison algorithm

We compared 6 traditional machine learning algorithms and 3 semi-supervised learning algorithms, as shown in Table 3.

Among the compared methods, traditional methods, such as SVM, NB, KNN, and LDA, are set according to [17,18], and the semi-supervised method, S4VM, is set according to [19]. The rule number R in the TSK fuzzy system is selected from {5, 6, 7, 8, 9, 10}, and the regularization parameter λ1,λ2,λ3 is obtained from {10−3,10−2,10−1,0,101,102,103}. The number of iterations of the BP-TSK fuzzy systems is selected from {1000, 1500, 2000, 2500, 3000}.

2) Experimental results

The accuracies of each algorithm on the dataset after the three feature extraction methods are shown in Tables 4–6. Obviously, all methods can achieve good accuracy for datasets with similar distributions. However, the accuracies of the three semi-supervised learning methods are much higher than those of the other algorithms. Although all data in the dataset are taken from the same defined group, there is still a deviation in the data distribution.

Second, for datasets with chaotic data distributions, the performance of traditional methods decreases significantly, and the minimum accuracy is only 51.4%, which is because they cannot be classified correctly. There is also a case in which the traditional algorithm is effective for datasets with chaotic data distributions, which reflects that although scenes with chaotic data distributions are constructed according to different defined groups, the results are not scientific.

In conclusion, it is a very common situation that there are differences in data distributions. Although datasets are similar in different domains, there may still be differences in the data distribution. Furthermore, in datasets with chaotic data distributions, there may still be small differences in the data distribution.

It can be seen from Tables 4–6 that the average accuracy of the proposed CD-BP-TSK-FS method reaches more than 95%, and the highest accuracy is 99.9% among the three feature extraction methods. This proves that the new method can solve the problem of performance degradation caused by different data distributions. 3)

Statistical significance analysis

We use the newly proposed method and other methods to perform Friedman [20,21] and Holm tests [22,23] to observe whether there are statistically significant differences. The results are shown in Tables 7 and 8. CD-BP-TSK-FS ranks first in the Friedman test. For the Holm test, p < 0.05 represents a significant difference between our method and other methods. The experimental results show that, in addition to the comparison with the S4VM method under the KPCA characteristics, CD-BP-TSK-FS has statistically significant differences. Although there is no significant difference in performance with S4VM, our method is based on a fuzzy system, and its own explanatory nature can compensate for this shortcoming.

Fuzzy rules can explain the execution process within the entire model through rules [24,25]. However, if the number of rules is too large, we need to define many semantics to describe the rules. This will reduce the interpretation ability of the fuzzy system. Although our method is based on backpropagation, it does not require a large number of rules to be superimposed, and only 5–10 rules are needed to obtain good accuracy. Therefore, our method can completely replace rules with semantics to achieve interpretable functions.

The AE-AE dataset under STFT feature extraction is used to illustrate the execution process of the fuzzy system. The antecedent and subsequent parameters are shown in Table 9.

First, we use “High, Low, Middle, Middle High (MH), and Middle Low (ML)” to describe the strength of each feature element. Then, we use these strength terms in the IF-part to describe the strength of each feature element and use the corresponding linear expression in the Then-part to obtain the value. Finally, the final decision value of the model can be obtained. The decision-making process of the five rules is as follows:

1^st rule:

IF: d1, d2, d3, d4, d5, and d6 are High,

THEN: f1(x)=[0.005−0.074x1−0.034x2−0.035x3−0.027x4−0.010x5−0.001x6,0.045+0.006x1−0.006x2+0.038x3+0.022x4+0.010x5+0.001x6]

2^nd rule:

IF: d1, d2, d3, d4, d5, and d6 are Low,

THEN: f2(x)=[0.477+0.191x1−0.184x2−0.061x3+0.367x4+0.000x5+0.004x6,0.014−0.343x1−0.077x2−0.284x3+0.066x4+0.000x5−0.004x6]

3^rd rule:

IF: d1, d2, d3, d4, d5, and d6 are MH,

THEN: f3(x)=[−0.030−0.030x1−0.030x2−0.029x3−0.026x4−0.005x5−0.001x6,0.030+0.030x1+0.030x2+0.029x3+0.026x4+0.005x5+0.001x6]

4^th rule:

IF: d1, d2, d3, d4, d5, and d6 are ML,

THEN: f4(x)=[0.264−0.268x1−0.016x2−0.192x3+0.337x4−0.489x5+0.009x6,0.226+0.163x1−0.134x2+0.023x3−0.270x4+0.489x5−0.009x6]

5^th rule:

IF: d1, d2, d3, d4, d5, and d6 are Middle,

THEN: f1(x)=[−0.105+0.020x1+0.021x2+0.037x3+0.012x4+0.000x5+0.000x6,0.188+0.072x1+0.329x2−0.091x3−0.128x4+0.000x5+0.000x6]

For ease of understanding, Fig. 6 shows the decision flow diagram of the model, where ‘⊕’ represents the value of the cumulative rules and ‘Max’ represents the principle of “winner takes all”.

Taking the AC-AC dataset with STFT characteristics as an example, we show the time consumption of different optimization methods in the TSK fuzzy system. The results are shown in Table 10. In the experiment, the number of rules is set to 5, and the number of iterations in the BP solution method is set to 1000. The least-squares method is the fastest to optimize, while the back-propagation method requires more time. However, for medical data identification, it is acceptable that a more accurate diagnosis requires more time.

Since the CD-BP-TSK-FS model is an optimization method using gradient descent to obtain the optimal parameters after processing, more time is required for the parameter optimization process. Furthermore, the comparison method uses the least square method, and they can obtain the analytic solution directly, which makes them faster.

In terms of time consumption, compared with the other methods mentioned above, our method is indeed at a disadvantage, but this is also an unavoidable problem of the backpropagation optimization method. We will carry out future work to address this problem and to find a new optimization method to reduce the optimization time and improve the efficiency of the model.