Efficient sleep classification based on entropy features and a support vector machine classifier

Considering an EEG or EOG sequence $x=\left\{x(i)\!:1\leqslant i\leqslant N \right\}$ of length N (30 s, 3000 points), the fuzzy entropy, fuzzy measure entropy and sample entropy are calculated in this study.

3.1.1. Fuzzy entropy

Fuzzy entropy (FuzzyEn) is the entropy of a fuzzy set, representing the information of uncertainty for a series (Al-Sharhan et al 2001). Due to the fact that FuzzyEn contains vague and ambiguous un-certainties, it is quite different from the classical Shannon entropy because no probabilistic concept is needed to define it, while the Shannon entropy contains the randomness uncertainty (probabilistic) (Al-Sharhan et al 2001). The FuzzyEn is defined using the concept of membership degree and its calculation procedure is explained in Chen et al (2007). The FuzzyEn statistic, the given input value for embedding dimension m, fuzzy power n, and tolerance threshold r, is defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm FuzzyEn}(m,n,r,N)=\ln {{\phi }^{m}}(n,r)-\ln {{\phi }^{m+1}}(n,r)\nonumber \end{align} \tag{ 1 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\phi }^{m}}(n,r)=\frac{1}{N-m}\sum\limits_{i=1}^{N-m}{\left(\frac{1}{N-m-1}\sum\limits_{j=1,j\ne i}^{N-m}{(\exp (-{{{(d_{ij}^{m})}^{n}}}/{r}\;))} \right)}\nonumber \end{align} \tag{ 2 }$$

in which $d_{ij}^{m}=d[X_{i}^{m},X_{j}^{m}]=\underset{k\in (0,m-1)}{\max }\,\left\{\left| x(i+k)-{{x}_{0}}(i)-(x(\,j+k)-{{x}_{0}}(\,j)) \right| \right\}$, indicating the maximum distance between two local sequence segments $X_{i}^{m}$ and $X_{j}^{m}$, $X_{i}^{m}=\{x(i),x(i+1),\ldots ,x(i+m-1)\}-{{x}_{0}}(i)$, $1\leqslant i,j\leqslant N-m$. While ${{x}_{0}}(i)$ indicates the mean value of the sequence $\{x(i),x(i+1),\ldots ,x(i+m-1)\}$.

3.1.2. Fuzzy measure entropy

Fuzzy measure entropy (FuzzyMEn) has been used for heart rate variability analysis in (Liu et al 2013), while this is the first time it has been used from EEG and EOG for sleep stage classification. This represents an evolution of FuzzyEn and is devised to integrate both local and global characteristics and could reflect the entire irregularity of a time series. FuzzyMEn is calculated based on the fuzzy set theory and constructed with the membership degree of a fuzzy function instead of using the '0–1' judgment of the Heaviside function that is typically used in the approximate entropy and sample entropy; it can improve the poor statistical stability in the approximate entropy and sample entropy. The algorithm of FuzzyMEn is inspired by the study of Chen et al (2007), in which the fuzzy sets are introduced to improve the statistical stability. The detailed description of the calculating procedure is given by Liu et al (2013). Similarly to FuzzyEn, FuzzyMEn is defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm FuzzyMEn}(m,n,r,N)={\rm FuzzyLMEn}(m,n,r,N)+{\rm FuzzyFMEn}(m,n,r,N)\nonumber \end{align} \tag{ 3 }$$

in which ${\rm FuzzyLMEn}(m,n,r,N)={\rm FuzzyEn}(m,n,r,N)$ indicates the local fuzzy measure entropy while ${\rm FuzzyFMEn}(m,n,r,N)$ represents the global fuzzy measure entropy. FuzzyFMEn is computed in a similar fashion to FuzzyLMEn:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm FuzzyFMEn}(m,n,r,N)=\ln \phi _{F}^{m}(n,r)-\ln \phi _{F}^{m+1}(n,r)\nonumber \end{align} \tag{ 4 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \phi _{F}^{m}(n,r)=\frac{1}{N-m}\sum\limits_{i=1}^{N-m}{\left(\frac{1}{N-m-1}\sum\limits_{j=1,j\ne i}^{N-m}{(\exp (-{{{(dF_{ij}^{m})}^{n}}}/{r}\;))} \right)}\nonumber \end{align} \tag{ 5 }$$

in which $dF_{ij}^{m}=d[XF_{i}^{m},XF_{j}^{m}]$, indicating the maximum distance between two global sequence segments $XF_{i}^{m}$ and $XF_{j}^{m}$, $XF_{i}^{m}=\{x(i),x(i+1),\ldots ,x(i+m-1)\}-\bar{x}$, $1\leqslant i,j\leqslant N-m$, in which $\bar{x}$ is the mean value of the entire sequence $x=\left\{x(i):1\leqslant i\leqslant N \right\}$.

3.1.3. Sample entropy

The reader is referred to Ge et al (2007) for the computing procedure. For the N length sequence x, given the parameters m and r, the sample entropy (SampEn) can be calculated as follows.

Considering the $N-m+1$ sequence segments ${{X}^{m}}(i)=\{x(i),x(i+1),\ldots ,x(i+m-1)\}$ for $\{\left. i \right|1\leqslant i\leqslant N-m+1\}$, ${{B}_{i}}$ counts the number of pairs with distances smaller than r between ${{X}^{m}}(i)$ and ${{X}^{m}}(\,j)$, where $i\ne j$; and similarly ${{A}_{i}}$ is the number of pairs with distances smaller than r between ${{X}^{m+1}}(i)$ and ${{X}^{m+1}}(\,j)$, where $i\ne j$.

Then, the probability that the distance between ${{X}^{m}}(i)$ and ${{X}^{m}}(\,j)$ is smaller than r is given by $B_{i}^{m}(r)={{{B}_{i}}}/{N-m-1}\;$ , and the density is ${{B}^{m}}(r)=\frac{1}{N-m}\sum\nolimits_{i=1}^{N-m}{B_{i}^{m}(r)}$. Similarly, $A_{i}^{m}(r)$ and ${{A}^{m}}(r)$ can also be calculated in the same fashion. Then, the number of templates that matches in an m-dimensional (or (m  +  1)-dimensional) phase space within the tolerance $r$ can be calculated as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle B(r)={1}/{2}\;(N-m-1)(N-m){{B}^{m}}(r)\nonumber \end{align} \tag{ 6 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle A(r)={1}/{2}\;(N-m-1)(N-m){{A}^{m}}(r).\nonumber \end{align} \tag{ 7 }$$

Finally, SampEn is defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm SampEn}(m,r,N)=-\log \left(\frac{A(r)}{B(r)} \right).\nonumber \end{align} \tag{ 8 }$$

Therefore, SampEn calculates the negative natural logarithm of the conditional probability that an EEG or EOG sequence x of length N, having repeated itself for $m$ samples within the tolerancer $r$, will also repeat itself for $m+1$ samples, with no self-matches (Richman and Moorman 2000).

For classification, the multi-class SVM is applied in this paper for its simplicity and robustness, and it has been widely used in sleep stage classification by other researchers (Adnane et al 2012, Zhu et al 2012, 2014, Surantha et al 2017). We have tried several types of SVMs, and finally the multi-class SVM using the one-against-all class approach is employed for its simplicity and good performance. The main idea of the one-against-all class approach is simple. As it is designed for binary classification, we need to reconstruct the multi-class classifier, and the idea of the one-against-all class approach comes from the one-against-one class approach used by Nakamura et al (2017). For a multi-class classification task, we reconstruct the multi-class classifier by connecting several binary classifiers in series, and each binary classifier only solves the classification of one class against all the others, and the next binary classifier does the same thing, ignoring the classified class by the previous classifier.

In this paper, the Gaussian radial basis function (RBF) (Chriskos et al 2017) is selected as the kernel function, as it is shown to perform better than linear or polynomial kernels (Bsoul et al 2011) in the context of sleep classification. In the SVM training phase, tuning of the parameters is necessary for a better classification performance, including choosing the box constraint (C) and $\gamma$. Specifically, C is a tradeoff parameter between regularization and accuracy, which influences the behavior of the support vector selection, and $\gamma$ is an important factor to control the RBF kernel in transmitting data to a new hyperspace. Optimization of the hyperparameters is investigated by following a 10-fold cross-validation that divides the training set into ten subgroups and iteratively minimizes the error using nine training groups and testing against the remaining subgroup. We implement a grid-search by scanning over the range [0.10:0.01:10] for both parameters, and the best C and $\gamma$ are found at 2.97 and 0.74, respectively. These two selected hyperparameters are then used for the leave-one-subject-out cross-validation.

We evaluated the performance of the proposed method using overall accuracy (ACC), Cohen's kappa coefficient ($\kappa$), per-class precision (PR), and per-class recall (RE). With chance agreement removed, $\kappa$ determines the agreement between scorers. The $\kappa$ values between 0.00–0.20, 0.21–0.40, 0.41–0.60, 0.61–0.80 and 0.81–1.00 correspond to slight, fair, moderate, substantial and almost perfect agreement, respectively. The per-class metrics are computed by considering a single class as a positive class and all other classes combined as a negative one. The ACC, PR, RE and $\kappa$ are calculated as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm ACC}=\frac{{{T}_{P}}+{{T}_{N}}}{{{T}_{P}}+{{F}_{P}}+{{T}_{N}}+{{F}_{N}}},{\rm PR}=\frac{{{T}_{P}}}{{{T}_{P}}+{{F}_{P}}},{\rm RE}=\frac{{{T}_{P}}}{{{T}_{P}}+{{F}_{N}}},\kappa =\frac{{{P}_{o}}-{{P}_{e}}}{1-{{P}_{e}}},\nonumber \end{align}$$

where ${{T}_{P}}$ is the number of positive class epochs classified correctly, ${{T}_{N}}$ is the number of negative class epochs identified correctly, ${{F}_{P}}$ is the number of negative class epochs classified incorrectly as a positive class, ${{F}_{N}}$ is the number of positive class epochs identified incorrectly as a negative class, ${{P}_{o}}$ is the observed agreement ratio and ${{P}_{e}}$ is the chance agreement probability (Sors et al 2018). Since we are dealing with a multi-class classification problem in this study, we calculate the class-specific PR and class-specific RE, as shown in tables 4 and 6.

As discussed in section 2, the data is divided into epochs every 30 s, and there are 63 767 epochs in total for all 61 subjects. Following the data preprocessing, FuzzyEn, FuzzyMEn and SampEn are calculated for each epoch of the three channels. For parameter selection, the embedding dimension m is chosen as the default value 2, and different choices of tolerance threshold $r={{r}^{\prime }}*{\rm SD}$ (SD is the standard deviation of each 30 s epoch) and fuzzy power n influence the standard deviation of the calculated entropy. It is worth noting that $n$ and $r$ are the gradient and width of the boundary of the fuzzy function $\mu (d_{ij}^{m},n,r)$, respectively. We test different ${{r}^{\prime }}$ values from 0.05 to 0.2 and different n values from 1 to 5, and for a comprehensive consideration of a steady standard deviation of the calculated entropy and a small computation, we set the tolerance threshold $r=0.15*{\rm SD}$ and fuzzy power $n=2$. In addition, the time lag tau is set to the default value 1. The parameter selection in this study refers to the suggestions in Azami et al (2017) and Richman and Moorman (2000).

The box plots of the three entropies is given in figure 2, from which it is clearly shown that the entropy values change as the sleep grows deeper and the entropy of the stage W is generally higher than all other sleep stages. The FuzzyEn and SampEn of the three channels show a similar trend in that the entropy values decrease slightly as the sleep grows deeper from wake to REM. However, the values of FuzzyMEn are relatively smaller and the REM stage is higher than the others for both EEG channels. In contrast, the FuzzyMEn of the EOG channel shows a different trend in that the value of the S2 stage is higher than other sleep stages while that of REM is the lowest.

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To test the differences in the entropy features, the independent samples t-test is performed between every possible stage-pair for three entropy values of the three channels. However, since we are performing 15 stage-pair tests in total, some tests may result in statistically significant differences just by chance. Thus, we compensate for this random effect by performing Bonferroni's corrections, in which a new threshold of 0.0033 is given by dividing 0.05 by 15. Then, it would be more dependable to be considered as significant if the p-value is still lower than this new threshold.

The results are given in table 3, in which the p-value of most stage-pairs is 0.000 (p  <  0.0033), meaning significant differences between the corresponding pairs. Meanwhile, it is worth noting that the p-value in bold with ^* annotated indicates the stage-pair that have no significant differences, including the FuzzyMEn of the EOG between W–S4; the FuzzyEn of Pz-Oz between S2–S3, S2–S4 and S3–S4; the FuzzyEn of Fpz-Cz between S1–S3; the SampEn of Fpz-Cz between S1–S2, S1–S3, S1–S4 and S3–S4; and the SampEn of Pz-Oz between S1–S3, S1–S4 and S3–S4. The independent samples t-test has shown significant differences between most possible stage-pairs, which establishes the foundation for sleep stage scoring based on entropy features.

Non-independent training and testing sets are commonly used in sleep stage scoring, and like Zhu et al (2014) and Hassan (2015), we thus first give the performance of the SC-En&SVM for the non-independent scheme.

The 10-fold cross-validation is performed for $C=2-6$ with the non-independent training and testing set. In this study, ten experiments are performed for each $C$, and in each experiment a random 90% of the 63 767 epochs are chosen as the training data and the remaining 10% epochs as the testing data. Table 4 presents the average RE, PR, ACC and $\kappa$ of ten experiments for each $C$. It is noted that we only present the standard deviation of ACC and $\kappa$ (±standard deviation) in the last two rows, ignoring those of RE and PR for a simple and clear table. It shows that the non-independent scheme results in a good performance; nevertheless, we believe that the independent scheme is more suitable for practical evaluation in clinical diagnosis. Thus, we test our method using these two schemes.

Table 4. The average RE, PR, accuracy and $\kappa$ of the 10-fold cross-validation.

	$C=2$ RE(%)/PR(%)	$C=3$ RE(%)/PR(%)	$C=4$ RE(%)/PR(%)	$C=5$ RE(%)/PR(%)	$C=6$ RE(%)/PR(%)
W	95.31/86.88	96.75/89.84	95.71/89.63	96.63/87.52	95.74/87.52
S1	97.35/99.11	91.61/99.05	85.58/94.75	38.91/47.97	34.61/46.63
S2				89.67/88.84	90.14/88.91
S3			92.67/82.79	87.32/86.43	75.74/63.34
S4					78.45/92.05
REM		93.53/74.25	90.27/79.97	88.57/76.98	87.22/76.64
ACC	97.02  ±  0.58	92.74  ±  1.32	89.08  ±  0.90	86.02  ±  1.06	83.94  ±  1.61
$\kappa$	0.89  ±  0.02	0.85  ±  0.01	0.84  ±  0.01	0.80  ±  0.01	0.78  ±  0.02

We further evaluate our method using the leave-one-subject-out cross-validation scheme for 61 subjects and the average evaluation criteria, i.e. ACC, PR, RE and $\kappa$, are presented to verify the reliability of the proposed SC-En&SVM. Specifically, we perform 61 experiments in total for each classification ($C$, $C=2,3,4,5,6$) so that each recording can be tested, and in each experiment, we select one individual subject as the testing subject and the remaining subjects constitute the training set.

We combine the predicted sleep stages from all 61 subjects and compute the performance metrics compared with the annotated labels from experts for all recordings. Table 5 shows the confusion matrix obtained from the leave-one-subject-out cross-validation for the 5-state classification ($C=5$). For simplicity, we only include the confusion matrix for $C=5$, in which S3 and S4 are merged, because the 5-state classification is an issue of common concern with the comprehensive comparison. The numbers in bold indicate the number of epochs that are correctly classified, while the others represent the incorrectly classified epochs. The last two columns in each row indicate per-class performance metrics computed from the confusion matrix.

Table 6 gives the average accuracy and $\kappa$ of 61 subjects for each $C$. Examples of the predicted sleep stages compared with the expert annotation are given in figure 3 (SC^*) and figure 4 (ST^*), where the blue line corresponds to the expert annotation, the red line corresponds to the predicted sleep stages and the black line on the bottom indicates the misclassified epochs.

Table 6. The average accuracy and $\kappa$ for each $C$.

	$C=2$	$C=3$	$C=4$	$C=5$	$C=6$
Average accuracy for 61 individual subjects (%)	94.15  ±  0.95	85.06  ±  0.8	80.96  ±  1.12	78.68  ±  0.93	75.98  ±  1.05
Average kappa for 61 individual subjects	0.81  ±  0.01	0.74  ±  0.01	0.72  ±  0.02	0.71  ±  0.01	0.67  ±  0.02

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Table 7 shows a comprehensive comparison of the classification accuracy with eight state-of-the-art methods. We classify these methods into two groups: non-independent and independent training and testing sets. As presented in section 4, the non-independent training and testing scheme is the method that randomly selects epochs from all subjects to construct the testing set, while the independent one only selects epochs from one individual subject to construct the testing set.

Table 7. Comparisons with other sleep stage scoring methods using the same dataset, sleep-EDF, across the overall accuracy and $\kappa$ coefficient for $C=5$.

Methods	Extracted features	Classifier	Subjects & channels	Accuracy	$\kappa$
Non-independent training and testing
Zhu et al (2014)	14 963 epochs; difference visibility graph (DVG) & horizontal visibility graph (HVG)	SVM	8 recordings, & Pz-Oz	88.9%	—
Hassan (2015)	15 188 epochs; four standard statistics features & five spectral features	Bootstrap aggregating (bagging)	8 recordings, & Pz-Oz	86.53%	—
SC-En&SVM	63 767 epochs; FuzzyEn, FuzzyMEn, SampEn	One-against-all SVM	61 recordings, & Fpz-Cz, Pz-Oz, EOG	86.02%	0.80
Independent training and testing
Rodríguez-Sotelo et al (2014)	40 826 epochs; multiscale entropy	Artificial neural network (ANN)	20 recordings, & Fpz-Cz, Pz-Oz	69%	0.42
Rodríguez-Sotelo et al (2014)	40 826 epochs; approximation entropy	ANN	20 recordings, & Fpz-Cz, Pz-Oz	74%	0.54
Sanders (2014)	9830 epochs; average spectral power, preferential frequency band & cross-frequency-coupling (CFC) method	Linear discriminant analysis (LDA)	10 recordings, & Fpz-Cz	75%	—
Tsinalis et al (2016a)	37 022 epochs; wavelet transform (WT) coefficients	Stacked sparse auto-encoders neural network (NN)	20 recordings, & Fpz-Cz	78.9%	—
Tsinalis et al (2016b)	37 022 epochs; raw EEG input	Convolutional neural network (CNN)	20 recordings, & Fpz-Cz	74.8%	0.65
Supratak et al (2017)	41 950 epochs; time invariant features	Deep learning	20 recordings, & Pz-Oz	79.8%	0.72
SC-En&SVM	63 767 epochs; FuzzyEn, FuzzyMEn, SampEn	One-against-all SVM	61 recordings, & Fpz-Cz, Pz-Oz, EOG	78.68%	0.71

We should note that the comparison with other methods is very difficult for different EEG databases, different numbers of EEG channels and different numbers of sleep epochs. Based on this consideration, we include in this review all studies that report using the same EEG database as in this study, and they all used 30 s as a sleep epoch, and different kinds of features and classifiers are applied for sleep stage scoring. It is noted that most of the studies only use eight or 20 recordings from the sleep-EDF database, while to the best of our knowledge, this study is the first attempt to classify 2–6 sleep stages of 30 s epochs using both EEG and EOG channels with FuzzyMEn features, especially using all 61 recordings. Furthermore, we only give the comparison for $C=5$, because most of the reviewed methods only report their results for the 5-state classification. We have 63 767 epochs in this study, much more than the referred studies, as shown in table 7. Thus, the results of this study are statistically reliable and clinically credible.

Compared with the methods in both groups, it can be seen that our method achieves a similar or slightly better classification accuracy compared with the state-of-the-art methods. Furthermore, the $\kappa$ coefficient shows substantial agreements (0.71) between the sleep experts and our method. Considering that we use three common and feasible statistical features and a simple but efficient classifier that can be easily implemented in hardware, the classification performance is quite satisfying and competitive.

S1 stage discrimination is always the most challenging for automatic sleep stage classification tasks, because S1 is a transition phase between the change from wakefulness to other sleep stages. In this study, the most misclassified pair of sleep stages is S1–W, more than 29% of the S1 stage epochs are misclassified as W, as table 5 shows. The main reason for this problem, we think, is the similarity in the characteristic EEG frequency patterns between W and S1, as described in Iber (2007). Specifically, both stages are characterized by the low voltage mixed 2–7 Hz and alpha activity. Furthermore, S1–REM is another easily misclassified stage-pair, with 26% of the S1 stage epochs being misclassified as REM. The characteristic EEG frequency patterns of both stages are also similar. In addition, S2 is the next stage that S1 is easily confused with (about 21%). The classification between S1 and S2 depends on the transition patterns that partly rely on the detection of arousals, body movement and slow eye movements, which are very difficult to capture (Tsinalis et al 2016a). Moreover, another reason that S1 could not be well detected is that it has the lowest number of samples fed into the classifier resulting in a failure of the learning process to this stage. Also, the possibilities that experts assign wrong labels to S1 in clinical diagnosis may also contribute to this problem.

The FuzzyEn and SampEn, as figure 2 illustrates, show a similar variation trend for the three channels and six sleep stages, while the FuzzyMEn is smaller on the whole and holds relatively more significant differences among different stages. The three features all show the highest value for stage W and basically have the least value for S4. This is because W is the most active stage and it is characterized by the presence of continuous alpha waves, faster frequency rhythms, along with frequent eye movements, while S4 is the deepest sleep stage, characterized by the existence of delta waves in more than 50% of the epochs.

It can be seen from table 5 that the poorest performance is noted for stage S1, with RE less than 30%, while the RE for other stages is significantly better, with the range between 78.03% to 93.72%. Most of the S1 epochs are misclassified as W, REM and S2. In addition, the S2 stage is easily misclassified as SWS (slow wave sleep, including S3 and S4) and REM. At the same time, SWS and REM are also easily misclassified as S2. It can also be seen that the confusion matrix is basically symmetric via the diagonal line (except in the pairs of S1–W and S1–REM). This indicates that the misclassifications are less likely to be due to the problem of data imbalance.

Finally, we should also point out the limitations of the proposed method, which we will focus on improving in our future work. On one hand, as a supervised learning process, the robustness of our method still needs to be tested for other EEG databases, and especially applied to different electrode positions to understand the best channels for sleep classification. Moreover, we only tested healthy people in the current study; the classification performance for people with sleep disorders needs to be tested in the future. In addition, we will also further investigate the correlations between features and the performance may be improved by feature selection. Lastly, even though our classification performance is encouraging, the overall classification accuracy and per-class performance metrics still need improvement, especially when applied to clinical diagnosis.