A New Approach for Automatic Sleep Staging: Siamese Neural Networks

2.1 Dataset and data preparation

The PhysioNet Sleep EDF database [32] is widely adopted in the research of automatic sleep staging [33]. The dataset contains 61 nocturnal polysomnography records of 42 people, which are sampled at the rate of 100Hz. The records include EEG, EOG, and EMG signals, as well as event markers. The dataset was established on two studies: sleep cassette (SC) and sleep telemetry (ST). The former investigates the effect of age on healthy people, and the latter investigates the effect of temazepam on sleep.

The recordings were evaluated by sleep staging experts in 30s epochs, according to the R&K standard [34]. During the staging phase, the following labels were used: W, N1, N2, N3, N4, REM, Movement, and Unknown. Each EEG recording was acquired by Fpz-Cz and Pz-Oz electrodes. The recordings from Fpz-Cz electrodes were utilized, because they were crisper than those from Pz-Oz electrodes in the Sleep-EDF database [12, 35]. Firstly, the Movement and Unknown data were eliminated from the dataset. Next, the N3 stage was merged with the N4 stage by the AASM standard, reducing the total number of stages from 6 to 5 (W, N1, N2, N3 / N4, REM).

2.2 Overlap technique

The overlap method is adopted more widely and more successfully than the other strategies [36-39], owing to the following advantages: the method is simple to use and reproduce; the current training set can be expanded several times, reducing the size of each training sample; the resulting trained network will have a better translational invariance. For these reasons, this paper chooses the overlap method [38]. Firstly, each epoch belonging to the same class was combined, producing a long signal. Next, the long signal was processed by overlapping rectangular windows of a certain duration [39]. This procedure is depicted in Figure 1.

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Figure 1. Data augmentation by overlap technique

2.3 SNNs

Traditional deep networks need hundreds of labeled data in each class to realize classification. Take a dataset with three labels, i.e., cars, planes, and birds, for example. If only trained by images in the three classes, the neural network cannot work effectively on a new class, e.g., trucks. Then, lots of truck images must be added to the dataset to retain the network. However, the addition and retaining are often time-consuming and costly [40]. Thus, SNNs have been developed to solve the classification of unbalanced data.

Every SNN consists of two identical neural networks, each of which can learn the hidden representation of an input vector [25]. The two networks are identical in that they share the same setup, including parameters and weights. The data belonging to the same class or two different classes are imported to the two networks. Then, the SNN produces two vectors that represent the two input data in lower dimensions. The distance between the two vectors is calculated by a distance measurement method. The greater the distance, the less similarity between the two input data. For this reason, a purely empirical threshold is determined for comparison. The distance between the two eigenvectors varies with the distance measurement methods, for each method has a unique equation. Therefore, the optimal threshold, that is, the threshold leading to the highest accuracy on the training set, depends on the specific method for distance measurement [41].

To implement all the above processes, the SNN needs to be trained through pairwise learning. Therefore, the cross-entropy loss function must be replaced with the comparative loss function [42]:

$L(y, d)=\frac{1}{2}(y \times \mathrm{d}+(1-y) \times \max \{m-d, 0\})$          (1)

where, d is the distance between the two input eigenvectors; y is the binary output; m is the margin. If the input eigenvectors are dissimilar, they cannot contribute to the loss function, unless their distance is within the margin.

The SNN can work on different distance measurement methods. Nevertheless, not every method ensures the good performance of the network. Thus, it is very important to know which method does better in a specific scenario, and choose the most suitable method for distance measurement. For example, the performance of the Euclidean distance method decreases with the growing data size, while the cosine distance method increases with the size of the dataset. In addition, the threshold should be adjusted according to the selected method. For this purpose, the SNN must be pre-trained, and the most suitable distance measurement method and threshold must be selected. The architecture of the SNN is illustrated in Figure 2.

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Figure 2. Architecture of SNN

This paper calls the Adam optimizer to iteratively update the network weights based on the training data. This optimization technique was selected to replace the traditional SNN training algorithm of stochastic gradient descent. As shown in Figure 3, the Adam optimizer uses two identical CNNs, which are responsible for acquiring the eigenvectors. Moreover, the comparative loss function was adopted to evaluate the ability of the SNN to differentiate between the two data.

The CNNs, a specific form of linear operation, are a basic neural network that employs convolution instead of matrix multiplication in at least one layer [43]. Each CNN consists of blocks that are added one after the other to learn complex features, and each block extracts the features from the data on the previous blocks. During the operation, the convolution layer (Layer C) learns simple features, and uses nonlinear activation functions to learn increasingly complex features. Then, the pooling layer (Layer P) brings the key information in the data to the foreground. Figure 3 shows the structure of a CNN in the SNN.

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Figure 3. Structure of a CNN in our SNN

To realize a fair comparison with the SNN, the softmax function was added to the last layer of each CNN in the SNN, so as to perform the traditional classification.

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Figure 4. Flow of the proposed system

Figure 4 shows the flow of the proposed system, which compares the CNN model with the SNN model created with each distance measurement method on augmented and non-augmented data. Firstly, seven different distance measurement methods were used separately in the SNN, using the sleep staging dataset, and the method with the best performance was determined. Then, the best-performing SNN model was compared with the CNN model on the same dataset.

2.4 Distance measures

2.4.1 Euclidean distance

In artificial intelligence, Euclidean distance is the most widely metric of the distance between two points [44]. Figure 5 depicts the calculation of Euclidean distance. Euclidean distance can be calculated by the Pythagorean theorem:

$D(x, y)=\sqrt{\sum_{i=1}^{n}\left(x_{i}-y_{i}\right)^{2}}$          (2)

where, x and y are the Cartesian coordinates of each point.

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Figure 5. Euclidean distance between 2 points

2.4.2 Manhattan distance

As shown in Figure 6, Manhattan distance might perform worse than Euclidean distance, for the failure to give the shortest distance. But some scholars found this measure outperforming Euclidean distance [45]. Manhattan distance can be calculated without any diagonal movement:

$D(x, y)=\sqrt{\sum_{i=1}^{n}\left|x_{i}-y_{i}\right|}$       (3)

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Figure 6. Manhattan distance between 2 points

2.4.3 Jaccard distance

The Jaccard distance statistically evaluates the similarity between two clusters. As shown in Figure 7, the intersection of the two sets can be identified by dividing with the total number of elements. If the two sets are the same, the intersection is 1; if the two sets have no common feature, the intersection is 0.

To calculate the Jaccard distance, it is necessary to subtract the Jaccard index from 1, for the distance is inversely proportional to similarity. The Jaccard distance between two points can be calculated by:

$D(A, B)=1-\frac{|A \cap B|}{|A \cup B|}$        (4)

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Figure 7. Jaccard distance between 2 samples

2.4.4 Cosine distance

The cosine angle between two vectors in a multidimensional space is a yardstick of the similarity between these vectors. If the two vectors have the same orientation, the cosine similarity is 1; if the two vectors have diametrically opposite orientations, the cosine similarity is -1. Note that cosine similarity only considers the direction of the vectors, without accounting their magnitude [46]. As shown in Figure 8, the cosine distance can be calculated by subtracting 1 from cosine similarity:

$D(x, y)=1-\cos (\theta)=1-\frac{\sum_{i=1}^{n} x_{i} y_{i}}{\sqrt{\sum_{i=1}^{n} x_{i}{ }^{2}} \sqrt{\sum_{i=1}^{n} y_{i}{ }^{2}}}$              (5)

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Figure 8. Cosine similarity between 2 samples

2.4.5 Canberra distance

The Canberra distance numerically measures the separation between two points in a vector space. If the coordinates of both samples are close to zero, the Canberra distance will be sensitive to tiny changes [47]. Mathematically, this distance measure can be defined as:

$D(x, y)=\sum_{i=1}^{n} \frac{\left|x_{i}-y_{i}\right|}{\left|x_{i}\right|+\left|y_{i}\right|}$         (6)

2.4.6 Bray-Curtis distance

Bray-Curtis distance is not technically a metric, as it does not provide the triangle inequality property. But it is a common way to measure the difference between samples. If the coordinates of both samples are close to zero, this measure is meaningless. Mathematically, this distance measure can be defined as:

$D(x, y)=\frac{\sum_{i=1}^{n}\left|x_{i}-y_{i}\right|}{\sum_{i=1}^{n}\left(x_{i}+y_{i}\right)}$         (7)

2.4.7 KLD

The KLD formulates the distance between two probability distributions. Like Bray-Curtis distance, the LKD is not a metric, because it does not satisfy the triangle inequality property. By the KLD, the ratios of the two distributions to each other at each point are taken, and made equal to the sum of the logarithms of the ratios. If the two distributions are the same, the distance is 0; otherwise, the distance is a positive real number.

$D(p \| q)=\sum_{i=1}^{n} p\left(x_{i}\right) \times\left(\log p\left(x_{i}\right)-\log q\left(x_{i}\right)\right)$            (8)

where, $q(x)$ is the approximation; $p(x)$ is the true distribution.

As shown in Figure 2, SNNs were created, the input data were encoded, and the distance between the eigenvectors were measured, using seven different methods: Euclidean, Manhattan, Jaccard, Cosine, Canberra, Bray-Curtis, and KLD. Then, the optimal threshold was calculated empirically for each method, giving the highest classification accuracy. Specifically, numbers in the range of 0-1 were tested on the training dataset with 0.025 intervals, and the value giving the highest accuracy was accepted as the optimal threshold. The results obtained are shown in Table 2.

Table 2. Optimal thresholds

According to the classification results (Table 3) on the data in five different classes (0: W, 1: N1, 2: N2, 3: N3, 4: N4), the SNN with Bray-Curtis achieved the best performance. The classification results of this SNN are reported in Table 4.

Table 3. Classification results with different distance measurement methods

Table 4. Binary classification results with Bray-Curtis

Table 5. Results on augmented dataset

Table 6. Binary classification results with cosine distance

As shown in Table 4, the SNN with Bray-Curtis obtained the best performance, when 0 and 3 stages were given together to the network. Then, the data were approximately doubled using the overlapping technique, and the classification results of the SNN with different distance measures are shown in Table 5. In this case, the best performance corresponded to the cosine distance measure. The classification results of the SNN with cosine distance are given in Table 6.

Finally, the results obtained by one of the identical parallel CNNs in our SNN were compared with those of the traditional classification method (subsection 2.3). The traditional method was evaluated on the same datasets as the SNN, an augmented set and a non-augmented set. The former set is about two times the size of the latter set. As shown in Table 7, the classification performance on augmented set was better than that on non-augmented set. Tables 8 and 9 provide the details on the binary classification results on augmented set.

Table 7. Classification results obtained using the CNN

Table 8. Binary classification results obtained using the CNN

Table 9. Results obtained using the CNN on augmented dataset

During the analysis of Tables 3 and 5, it should be noted that the SNNs with distance measures other than KLD and Manhattan achieved close results, because neural networks are stochastic algorithms. In other words, the similarity of the results cannot reveal the superiority between the techniques, but suggest that these techniques take advantage of stochasticity, such as the random initialization of weights. This means the same network can produce different results despite being trained on the same data.

Furthermore, although the SNN outperformed the CNN, the network did not perform better in the binary classification of all sleep stages. Compares Table 4 and 6 with Tables 8 and 9, it is evident that the CNN outshines the SNN in 0 vs 1, 1 vs 2, 2 vs 3, and 2 vs 4.

Table 10. Comparison between our SNN and previous state-of-the-art results

Table 10 compares the proposed SNN with the existing methods. The highest values are shown in bold font. Overall, it can be said that data augmentation improves the performance of both SNN and CNN.

In conclusion, the SNNs using binary classification methods are much better than traditional methods. Since EEG signals can be easily obtained from the forehead with a dry electrode, our method bodes well for developing low-cost and portable high-performance devices in the future. Meanwhile, the SNN system must be supported by the hold-out technique to solve the high requirements of random-access memory (RAM). If the memory issue can be solved in future, the results can be evaluated through the k-fold cross-validation. In addition, more approaches can be explored, and the system performance can be compared with the system proposed here.