Electroencephalography (EEG) based epilepsy diagnosis via multiple feature space fusion using shared hidden space-driven multi-view learning

Frequency-domain representation extraction

Frequency-domain feature representation originates from the significant changes in energy in EEG during epileptic seizures. To extract frequency-domain representation from EEG signals, the Daubechies4 wavelet coefficients are utilized to decompose the original signals into a series of binary wavelets. The frequency band of each Daubechies4 wavelet coefficient is provided in Table 2. By applying these settings, the EEG signals are divided into six distinct frequency bands. An illustrative example of the decomposed signals from group E is depicted in Fig. 2.

Time-domain feature extraction

Time-domain features are the fundamental features in EEG signal processing, primarily extracted by directly observing and calculating relevant characteristics from the raw signal. Their advantages lie in their simplicity of computation and ease of interpretation for researchers. In this study, we employ kernel principal component analysis (KPCA) (Li et al., 2022b) on the raw EEG signals to enable complex nonlinear mapping. Previous research has shown that KPCA features offer discriminative patterns suitable for pattern recognition. An illustration depicting an example of KPCA features from group E can be observed in Fig. 3.

Time-frequency representation extraction

Pure time-domain or frequency-domain feature representations alone cannot comprehensively characterize an EEG signal, and EEG analysis based on the assumption of stationarity is not rigorous. Therefore, researchers have turned their attention to time-frequency analysis methods, such as time-frequency transformations, to re-represent non-stationary EEG signals and extract corresponding features. To capture time-frequency representation, researchers often employ the short-time Fourier transform (STFT) (Li et al., 2022a). STFT allows for the analysis of how the frequency content of a signal changes over time. It can be formulated as follows:

(1) $$F_{time-fre}\left(time,fre\right)={\int }_{-inf}^{+inf}x\left(time\right)g\left(time-u\right)e^{-j2\pi \ast fre\ast time}d\left(time\right).$$

In the context of EEG signal analysis, Eq. (1) represents the transformation of continuous EEG signals, denoted as $x\left(time\right)$, into the time-frequency plane using the function $g\left(time-u\right)$ and a limited width window centered around $u$. This transformation, referred to as $F_{time-fre}\left(time,fre\right)$, provides a means to examine the time-varying nature of the EEG signals, revealing local spectrum discrepancies at different time points. To achieve this, the EEG signals undergo partitioning into several segments of local stationary signals using STFT. Through this process, the time-varying characteristics of the EEG signals are captured, highlighting variations in the spectrum. The extraction of six energy bands as features is accomplished using Eq. (1), which takes into account the observed discrepancies. A visualization of these six energy bands, exemplified by group E, is illustrated in Fig. 4.

Construction of shared hidden feature space

Suppose that $\mathbf{\Omega }\in {\mathit{R}}^{r\times d}$ is an orthogonal matrix subject to $\mathbf{\Omega }{\mathbf{\Omega }}^T=\mathbf{I}\in {\mathit{R}}^{r\times r}$, $f^A=\{{\mathbf{x}}_i^A,y_i|{\mathbf{x}}_i^A\in R^d,i=1,2,\dots ,N\}$ represents one kind of feature space, e.g., time-domain feature space, and $f^B=\{{\mathbf{x}}_i^B,y_i|{\mathbf{x}}_i^B\in R^d,i=1,2,\dots ,N\}$ represents another kind of feature space, then the hidden feature space of $f^A$ and $f^B$ can be generated by ${\mathbf{\Omega }\mathbf{x}}_i^A\in {\mathit{R}}^r$ and ${\mathbf{\Omega }\mathbf{x}}_i^B\in {\mathit{R}}^r$, respectively, where $r$ represents the number of hidden features. To obtain a consistent hidden feature space between ${\mathbf{\Omega }\mathbf{x}}_i^A$ and ${\mathbf{\Omega }\mathbf{x}}_i^B$, it is expected that the difference between them should be minimized as much as possible. Kernel density estimation (KDE), which is one of the non-parametric estimation methods in probability theory, is usually used to estimate the unknown probability density function (Wang, Wang & Chung, 2013). For a training set $X=\{{\mathbf{x}}_i,y_i|{\mathbf{x}}_i\in R^d,i=1,2,\dots ,N\}$, its corresponding kernel density estimation function can be expressed as

(2) $$P\left(\mathbf{x}\right)={\displaystyle \frac{1}{N}\sum _{i=1}^N{\delta }^{2}K\left({\displaystyle \frac{\mathbf{x}-{\mathbf{x}}_i}{\delta }}\right),}$$where $\delta$ is the kernel width, $K(\cdot )$ is the kernel function. If the Gaussian kernel function is adopted, then Eq. (2) can be updated as $P\left(\mathbf{x}\right)={\displaystyle \frac{1}{N}\sum _{i=1}^N{\displaystyle \frac{1}{\delta \sqrt{2\pi }}\exp \left(-{\displaystyle \frac{1}{2}{\left({\displaystyle \frac{\mathbf{x}-{\mathbf{x}}_i}{\delta }}\right)}^2}\right).}}$ Therefore, the kernel density estimation of ${\mathbf{\Omega }\mathbf{x}}_i^A$ and ${\mathbf{\Omega }\mathbf{x}}_i^B$ can be expressed as follows when using the Gaussian kernel function, respectively,

(3) $-\frac{\parallel {\mathbf{\Omega }\mathbf{x}\mathbf{-}\mathbf{\Omega }\mathbf{x}}_i^A{\parallel }^2}{2{\delta }^2}{P}_A(\stackrel{\mathbf{~}}{\mathbf{x}}\mathbf{)}=P_A(\mathbf{\Omega }\mathbf{x})=\frac{1}{N\cdot \delta \sqrt{2\pi }}\sum _{i=1}^N\mathrm{e},$

(4) $-\frac{\parallel {\mathbf{\Omega }\mathbf{x}\mathbf{-}\mathbf{\Omega }\mathbf{x}}_i^B{\parallel }^2}{2{\delta }^2}{P}_B(\stackrel{\mathbf{~}}{\mathbf{x}}\mathbf{)}=P_B(\mathbf{\Omega }\mathbf{x})=\frac{1}{N\cdot \delta \sqrt{2\pi }}\sum _{i=1}^N\mathrm{e}.$

In this study, the difference between $P_A\left(\stackrel{\mathbf{~}}{\mathbf{x}}\right)$ and $P_B\left(\stackrel{\mathbf{~}}{\mathbf{x}}\right)$ is measured by the mean square error, that is

(5) $$J=\int {\left(P_A\left(\stackrel{\mathbf{~}}{\mathbf{x}}\right)-P_B\left(\stackrel{\mathbf{~}}{\mathbf{x}}\right)\right)}^2\mathrm{d}\mathbf{x}.$$

By minimizing $J$, the two-view data ${\mathbf{x}}_i^A$ and ${\mathbf{x}}_i^B$ can be made to have the maximum commonality in the shared hidden space, and thus the challenge of excessive variability between samples from different views can be addressed. In order to solve Eq. (6), we suppose that $G\left(\Omega x,\Omega x_i,{\delta }^2\right)=\frac{1}{\delta \sqrt{2\pi }}{\text{e}}^{-\frac{\Omega x-\Omega x_i^2}{2{\delta }^2}}$, then $P_A\left(\stackrel{\mathbf{~}}{\mathbf{x}}\right)$ and $P_B\left(\stackrel{\mathbf{~}}{\mathbf{x}}\right)$ can be updated as $P_A(\stackrel{\mathbf{~}}{\mathbf{x}})=\frac{1}{N}{\sum }_{i=1}^{N}G({\mathbf{\Omega }\mathbf{x}\mathbf{\Omega }\mathbf{x}}_i^A,{\delta }^2)$ and $P_B(\stackrel{\mathbf{~}}{\mathbf{x}})=\frac{1}{N}{\sum }_{i=1}^{N}G({\mathbf{\Omega }\mathbf{x}\mathbf{\Omega }\mathbf{x}}_i^B,{\delta }^2)$. Therefore, Eq. (5) can be computed by $\mathrm{J}=\int P_A\left(\stackrel{\mathbf{~}}{\mathbf{x}}\right)\mathrm{d}x-2\int P_A\left(\stackrel{\mathbf{~}}{\mathbf{x}}\right)P_B\left(\stackrel{\mathbf{~}}{\mathbf{x}}\right)\mathrm{d}x+\int P_B\left(\stackrel{\mathbf{~}}{\mathbf{x}}\right)\mathrm{d}\mathbf{x}$. According to Wang, Wang & Chung (2013), Hansen, Jaumard & Xiong (1994), we have $\int G\left(\mathbf{x},{\mathbf{x}}_i,{\delta }_1^2\right)G\left(\mathbf{x},{\mathbf{x}}_j,{\delta }_2^2\right)dx=G\left({\mathbf{x}}_i,{\mathbf{x}}_j,{\delta }_1^2+{\delta }_2^2\right)$, Therefore, we have the following equations,

(6) $\int P_A^2(\stackrel{\mathbf{~}}{\mathbf{x}})dx=\frac{1}{N^2}\sum _{i=1}^N\sum _{j=1}^{N}G({\stackrel{\mathbf{~}}{\mathbf{x}}}_i^A,{\stackrel{\mathbf{~}}{\mathbf{x}}}_j^A,2{\delta }^2)=\frac{1}{N}\sum _{i=1}^N[\frac{1}{N}\sum _{j=1}^{N}G({\stackrel{\mathbf{~}}{\mathbf{x}}}_i^A,{\stackrel{\mathbf{~}}{\mathbf{x}}}_j^A,2{\delta }^2)]$

(7) $\int P_B^2(\stackrel{\mathbf{~}}{\mathbf{x}})dx=\frac{1}{N^2}\sum _{i=1}^N\sum _{j=1}^{N}G({\stackrel{\mathbf{~}}{\mathbf{x}}}_i^B,{\stackrel{\mathbf{~}}{\mathbf{x}}}_j^B,2{\delta }^2)=\frac{1}{N}\sum _{i=1}^N[\frac{1}{N}\sum _{j=1}^{N}G({\stackrel{\mathbf{~}}{\mathbf{x}}}_i^B,{\stackrel{\mathbf{~}}{\mathbf{x}}}_j^B,2{\delta }^2)]$

(8) $\int P_A(\stackrel{\mathbf{~}}{\mathbf{x}})P_B(\stackrel{\mathbf{~}}{\mathbf{x}})dx=\frac{1}{N^2}\sum _{i=1}^N\sum _{j=1}^{N}G({\stackrel{\mathbf{~}}{\mathbf{x}}}_i^A,{\stackrel{\mathbf{~}}{\mathbf{x}}}_j^B,2{\delta }^2)$where $\frac{1}{N}{\sum }_{j=1}^{N}G({\stackrel{\mathbf{~}}{\mathbf{x}}}_i^A,{\stackrel{\mathbf{~}}{\mathbf{x}}}_j^A,2{\sigma }^2)$can be taken as another estimation of $P_A\left({\stackrel{\mathbf{~}}{\mathbf{x}}}_i^A\right)$. Therefore, $\int P_A^2\left(\stackrel{\mathbf{~}}{\mathbf{x}}\right)\mathrm{d}x$ can be estimated by $\frac{1}{N}{\sum }_{j=1}^N{P}_A\left({\stackrel{\mathbf{~}}{\mathbf{x}}}_i^A\right)$, and further ${\displaystyle \frac{1}{N}}$. Similarly, $\int P_B^2\left(\stackrel{\mathbf{~}}{\mathbf{x}}\right)\mathrm{d}x$ can be estimated by ${\displaystyle \frac{1}{N}}$. Thus, we finally have $J\approx {\displaystyle \frac{1}{N}+{\displaystyle \frac{1}{N}-{\displaystyle \frac{2}{N^2}G\left({\stackrel{\mathbf{~}}{\mathbf{x}}}_i^A,{\stackrel{\mathbf{~}}{\mathbf{x}}}_j^B,2{\delta }^2\right)}}}$. Therefore, we have the following objective,

(9) $$\begin{array}{l}\arg {\min }_{\Omega }J\approx \arg {\min }_{\Omega }{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^{N}G}}\left({\tilde{x}}_i^A,{\tilde{x}}_j^B,2{\delta }^2\right)\\ s.t.\Omega {\Omega }^T=I_{r\times r}\end{array}$$

However, it is difficult to solve Eq. (9) directly. Thus, Taylor expansion can be used for getting an approximate solution. Hence, we have

(10) $$G\left({\tilde{x}}_i^A,{\tilde{x}}_j^B,2{\delta }^2\right)=\frac{1}{\sqrt{2\pi }\delta }{\text{e}}^{-\frac{\Omega x_i^A-\Omega x_j^{B^2}}{4{\sigma }^2}}\approx \frac{1}{\sqrt{2\pi }\delta }\left(1-{\left(\Omega x_i^A-\Omega x_j^B\right)}^2\right)$$

Therefore, Eq. (9) can be further updated as

(11) $$\arg \underset{\mathrm{\Omega }}{min}\sum _{i=1}^N\sum _{j=1}^N{({\mathbf{\Omega }\mathbf{x}}_i^A-{\mathbf{\Omega }\mathbf{x}}_j^B)}^2,s.t.\mathbf{\Omega }{\mathbf{\Omega }}^T={\mathbf{I}}_{r\times r}$$in Eq. (11), implicit feature transformation matrix $\mathbf{\Omega }$ still cannot be solved directly, but can be solved by gradient descent method. Thus, Eq. (11) can be updated as

(12) $$\begin{array}{l}J=\underset{\Omega }{\text{argmin}}{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N({\left(x_i^A\right)}^T{\Omega }^T\Omega x_i^A+{\left(x_j^B\right)}^T{\Omega }^T\Omega x_j^B-2{\left(x_i^A\right)}^T{\Omega }^T\Omega x_j^B)}}\\ s.t.\Omega {\Omega }^T=I_{r\times r}\end{array}$$

The partial derivative of $J$ w.r.t. $\mathbf{\Omega }$ is

(13) ${\displaystyle \frac{\mathrm{\partial }J}{\mathrm{\partial }\mathbf{\Omega }}=\sum _{i=1}^N\sum _{j=1}^N(2{\mathbf{\Omega }\mathbf{x}}_i^A{\left({\mathbf{x}}_i^A\right)}^{\mathrm{T}}+2{\mathbf{\Omega }\mathbf{x}}_j^B{\left({\mathbf{x}}_j^B\right)}^{\mathrm{T}}-2\mathbf{\Omega }\left({\mathbf{x}}_i^A{\left({\mathbf{x}}_i^A\right)}^{\mathrm{T}}+{\mathbf{x}}_j^B{\left({\mathbf{x}}_j^B\right)}^{\mathrm{T}}\right))}$

Then the transformation matrix $\mathbf{\Omega }$ can be solved by gradient descent method, that is,

(14) $$\mathbf{\Omega }\leftarrow \mathbf{\Omega }-\eta {\displaystyle \frac{\mathrm{\partial }J}{\mathrm{\partial }\mathbf{\Omega }}\left({\mathbf{I}}_{r\times r}-\mathbf{\Omega }{\mathbf{\Omega }}^T\right)=\mathbf{\Omega }-\eta \mathrm{\nabla }\mathbf{\Omega }}$$where $\eta$ is the step size that can be solved by

(15) $\begin{array}{l}\eta ={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N({\left(x_i^A\right)}^{\text{T}}\left({\Omega }^T\nabla \Omega +\nabla {\Omega }^T\Omega \right)x_i^A+{\left(x_j^B\right)}^{\text{T}}\left({\Omega }^T\nabla \Omega +\nabla {\Omega }^T\Omega \right)x_j^B}}\\ -\frac{2{\left(x_i^A\right)}^{\text{T}}\left({\Omega }^T\nabla \Omega +\nabla {\Omega }^T\Omega \right)x_j^B)}{{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N(2{\left(x_i^A\right)}^{\text{T}}\nabla {\Omega }^T\nabla \Omega x_i^A}}}+{\left(x_j^B\right)}^{\text{T}}\nabla {\Omega }^T\nabla \Omega x_j^B-4{\left(x_i^A\right)}^{\text{T}}\nabla {\Omega }^T\nabla \Omega x_j^B)\end{array}$

According to the above analysis and derivation, the algorithm for solving implicit feature transformation matrix $\mathbf{\Omega }$ is described as follows.

Multi-view learning based on shared hidden feature space

After determining the shared hidden space between two views, the extended space can be generated by combining the original space and the shared hidden space. Then, a multi-view classifier based on SVM is designed for multi-view data classification in the extended space. In existing multi-view learning mechanisms, it is generally assumed that each view can provide a classifier containing specific information, and classifiers constructed from different view tend to be consistent. Additionally, since views can provide specific information to each other, the proposed model establishes the objective function by considering the mutual information between two views. In summary, the proposed model, based on SVM, restructures the slack variables on each view, and then narrows the gap between the two views by using the corresponding regularization term. The objective function of multi-view learning based on shared hidden feature space can be formulated as

(16) $\begin{array}{l}arg\underset{w_A,w_B,v_A,v_B,b_A,b_B}{\min }\frac{1}{2}\parallel w_A{\parallel }^2+\frac{1}{2}\parallel w_B{\parallel }^2+\frac{1}{2}\parallel v_A{\parallel }^2+\frac{1}{2}\parallel v_B{\parallel }^2+C^A{\displaystyle \sum _{i=1}^N{\xi }_i^A}+C^B{\displaystyle \sum _{i=1}^N{\xi }_i^B+\lambda \parallel v_A-v_B{\parallel }^2}\\ s.t.y_i(w_A^{\text{T}}\varphi (x_i^A)+v_A^{\text{T}}\varphi (\Omega x_i^A)+b_A)\ge 1-{\xi }_i^A\\ y_i(w_B^{\text{T}}\varphi (x_i^B)+v_B^{\text{T}}\varphi (\Omega x_i^B)+b_B)\ge 1-{\xi }_i^B\\ {\xi }_i^A,{\xi }_i^B\ge 0,i=1,2,\dots ,N\end{array}$where $\lambda$, $C^A$ and $C^B$ are the regularization parameters. Observe that Eq. (16) consists of three parts: the first four terms reflect the outcome risk in the original feature space and the shared hidden space respectively; the second two terms represent the empirical risk; and the third term reflects the difference between the two views in the shared hidden space. The objective function in Eq. (16) strengthens the constraints based on the traditional SVM through the implicit mapping, so that the probability distributions of data from different views in the shared hidden space are as consistent as possible, which can well solve the problem described at the beginning of this study. In order to solve Eq. (16) efficiently, the relevant Lagrangian multipliers are introduced according to the Lagrangian optimization theory, hence Eq. (16) can be converted into the corresponding dual form as follows. The Lagrangian function corresponding to Eq. (16) is

(17) $$\begin{array}{l}L=\frac{1}{2}\parallel w_A{\parallel }^2+\frac{1}{2}\parallel w_B{\parallel }^2+\frac{1}{2}\parallel v_A{\parallel }^2+\frac{1}{2}\parallel v_B{\parallel }^2+C^A{\displaystyle \sum _{i=1}^N{\xi }_i^A}\\ +C^B{\displaystyle \sum _{i=1}^N{\xi }_i^B+\lambda \parallel v_A-v_B{\parallel }^2}\\ +{\displaystyle \sum _{i=1}^N{\alpha }_i^A(1-{\xi }_i^A-y_i(w_A^{\text{T}}\varphi (x_i^A)+v_A^{\text{T}}\varphi (\Omega x_i^A)+b_A))}\\ +{\displaystyle \sum _{i=1}^N{\alpha }_i^B(1-{\xi }_i^B-y_i(w_B^{\text{T}}\varphi (x_i^B)+v_B^{\text{T}}\varphi (\Omega x_i^B)}\\ +b_B))-{\displaystyle \sum _{i=1}^N{\mu }_i^A{\xi }_i^A}-{\displaystyle \sum _{i=1}^N{\mu }_i^B{\xi }_i^B}\end{array}$$where ${\alpha }_i^A\ge 0$, ${\alpha }_i^B\ge 0$, ${\mu }_i^A\ge 0$, and ${\mu }_i^B\ge 0$ are Lagrangian multipliers. By setting the partial derivatives of Lagrangian function $L$ with respect to ${\mathbf{w}}_A$, ${\mathbf{w}}_B$, ${\mathbf{v}}_A$, ${\mathbf{v}}_B$, $b_A$, $b_B$, ${\xi }_i^A$, and ${\xi }_i^B$ to 0, we have

(18) $${\mathbf{w}}_A=\sum _{i=1}^N{\alpha }_i^A{y}_i\varphi \left({\mathbf{x}}_i^A\right),{\mathbf{w}}_B=\sum _{i=1}^N{\alpha }_i^B{y}_i\varphi \left(x_i^B\right),$$

(19) ${\mathbf{v}}_A={\displaystyle \frac{1+2\lambda }{1+4\lambda }\sum _{i=1}^N{\alpha }_i^A{y}_i\varphi \left({\mathbf{x}}_i^A\right)+{\displaystyle \frac{2\lambda }{1+4\lambda }\sum _{i=1}^N{\alpha }_i^B{y}_i\varphi \left({\mathbf{x}}_i^B\right),}}$

(20) $${\mathbf{v}}_B={\displaystyle \frac{1+2\lambda }{1+4\lambda }\sum _{i=1}^N{\alpha }_i^B{y}_i\varphi \left({\mathbf{x}}_i^B\right)+{\displaystyle \frac{2\lambda }{1+4\lambda }\sum _{i=1}^N{\alpha }_i^A{y}_i\varphi \left({\mathbf{x}}_i^A\right),}}$$

(21) $\sum _{i=1}^N{\alpha }_i^A{y}_i=0,\sum _{i=1}^N{\alpha }_i^B{y}_i=0,$

(22) $$C_A={\alpha }_i^A+u_i^A,C_B={\alpha }_i^B+u_i^B$$

By submitting Eqs. (18–22) to Eq. (16), we have the dual problem of Eq. (24), which can be defined as

(23) $$\underset{\tilde{\alpha }}{\arg max}-{\displaystyle \frac{1}{2}{\tilde{\alpha }}^{\mathrm{T}}\tilde{\alpha }+{\tilde{\alpha }}^{\mathrm{T}}1.}s.t.{\tilde{\alpha }}^{\mathrm{T}}\mathit{f}=0,\mathit{f}={\left[{\mathit{y}}^{\mathrm{T}},{\mathit{y}}^{\mathrm{T}}\right]}^{\mathrm{T}}{\tilde{\alpha }}_{i}0,\mathrm{\forall }i$$where

(24) $$\tilde{\alpha }={\left[{\alpha }_1^A,{\alpha }_2^A,\dots ,{\alpha }_N^A,{\alpha }_1^B,{\alpha }_2^B,\dots ,{\alpha }_N^B\right]}^{\mathrm{T}},$$

(25) $${\mathit{K}}_A=K\left(x^A,x^A\right)\mathit{y}{\mathit{y}}^{\mathrm{T}}+{\displaystyle \frac{1+2\lambda }{1+4\lambda }K\left(\mathbf{\Omega }{x}^A,\mathbf{\Omega }{x}^A\right)\mathit{y}{\mathit{y}}^{\mathrm{T}}}$$

(26) $${\mathit{K}}_B=K\left(x^B,x^B\right)\mathit{y}{\mathit{y}}^{\mathrm{T}}+{\displaystyle \frac{1+2\lambda }{1+4\lambda }K\left(\mathbf{\Omega }{x}^B,\mathbf{\Omega }{x}^B\right)\mathit{y}{\mathit{y}}^{\mathrm{T}}}$$

(27) $${\mathit{K}}_{AB}={\displaystyle \frac{2\lambda }{1+4\lambda }K\left(\mathbf{\Omega }{x}^A,\mathbf{\Omega }{x}^B\right)\mathit{y}{\mathit{y}}^{\mathrm{T}}}$$

(28) $$K=\left[\begin{array}{cc}{K}_A& K_{AB}\\ K_{AB}& K_B\end{array}\right]$$

(29) $$\mathit{y}={\left[y_1,y_2,\dots ,y_N\right]}^{\mathrm{T}}$$and $\mathit{K}$ is the kernel function. It is obvious that the optimization of Eq. (23) can be considered as a QP problem, which can be solved according to Deng et al. (2013). The decision function of the proposed model in this study is defined as

(30) $$f\left(x\right)={\displaystyle \frac{1}{2}({\mathbf{w}}_A^{\mathrm{T}}\varphi \left({\mathbf{x}}^A\right)+v_A^{\mathrm{T}}\varphi \left(\mathbf{\Omega }{\mathbf{x}}^A\right)+b_A+{\mathbf{w}}_B^{\mathrm{T}}\varphi \left({\mathbf{x}}^B\right)+v_B^{\mathrm{T}}\varphi \left(\mathbf{\Omega }{\mathbf{x}}^B\right)+b_B)}$$

The algorithm of multi-view learning based on shared hidden feature space can be obtained, as shown in Algorithm 2. From Algorithm 2, we can find that the time complexity is mainly contributed by steps 1, 3 and 4. The time complexity of Algorithm 1 is $O\left(Nrd+r^2\right).$ The time complexity of step 3 is $O\left({\left(r+d\right)}^2\right)$. The time complexity of step 4 is $O\left(N^2\right)$. Therefore, the time complexity of Algorithm 2 is $O\left(Nrd+r^2+{\left(r+d\right)}^2+N^2\right).$

Settings

To observe the merits of the proposed model, k-nearest neighbor (KNN) (Liu & Liu, 2016), support vector machine (SVM) (Liu & Liu, 2016), SVM2K (Farquhar et al., 2005), multi-view L2-SVM (MV-L2-SVM) (Huang, Chung & Wang, 2016), and alternative multi-view MED (AMVMED) (Chao & Sun, 2015) are introduced for comparison studies. Accuracy is used as the evaluation indicator in this study. SVM, SVM2K, MV-L2-SVM, and 2V-SVM-SH are all trained using a Gaussian kernel for experimentation. For all methods, ten-fold cross-validation (CV) is used to determine the optimal parameters. Table 3 provides the specific parameters and ranges used for each method. All experiments are conducted on a PC with a 16-core CPU with a clock speed of 3.40 GHz and 32 GB of memory. The programming environment was Matlab R2016a.

To construct a two-view learning scenario, based on “Data”, three feature extraction methods, namely wavelet packet decomposition (WPD), short-time Fourier transform (STFT) and kernel principal component analysis (KPCA) are adopted, to extract time-frequency features, frequency-domain features and time-domain features from the original EEG signals, as shown in Fig. 2. Finally, 12 datasets are constructed, as shown in Table 4.

Experimental results and analysis

The experimental results are reported in Table 5. We can see from Table 5 that the proposed model wins the best performance on most datasets. Only on DS5, DS9, the proposed model performs worse than SVM-2K and MV-L2-SVM. The advantages of the proposed model indicate the promising ability of the shared hidden space. From the promising results, it can be found that by constructing the expanded space and utilizing the information of both the shared hidden space and the original space for learning, thereby fully utilizing the relevant information of samples within and across views, the proposed model effectively solves the problem that the difference between samples of the same class from different views is greater than the difference between samples of different classes from the same view. The experimental results also indicate the power of KDE which is used to construct the shared hidden space.

Statistical analysis

We use the Friedman test (Zimmerman & Zumbo, 1993; Sakamoto et al., 2015) to conduct a statistical analysis of the experimental results on all methods across all datasets. The Friedman test is a non-parametric testing method that can be used to analyze whether there are significant differences in performance among multiple methods on multiple datasets. The principle is to first obtain the average ranking of each method’s performance on all datasets, and then compare whether these rankings are the same. If they are the same, it indicates that all methods have the same performance, otherwise it suggests that there are significant differences in performance among all methods. If there are significant differences among all methods, we further use a Holm post-hoc hypothesis test to specifically analyze which methods and our proposed algorithm have significant differences. From Fig. 5, we see that 2V-SVM-SH wins the best ranking result. The p-values embedded in Fig. 5 computed by Friedman test hint that there are significant differences among different models. From Table 6, it can be seen that all hypothesis is rejected except the proposed model vs AMVMED and the proposed model vs SVM-2K. These results indicate that the proposed model performs significantly better than KNN-A, KNN-B, SVM-B, SVM-A and MV-L2-SVM. Although the hypothesis of the proposed model vs AMVMED and the proposed model vs SVM-2K is not reject, the low p-value of the proposed model vs AMVMED and the proposed model vs SVM-2K also indicates the reveal the competition of the proposed model.