Higher order spectra-based sleep staging


The electroencephalogram (EEG) signals are widely used to study and diagnose sleep disorders. EEG signals are nonlinear and random in nature. Hence, it is very difficult to interpret the disease if any, by visual analysis and linear techniques. So, we have used the nonlinear technique namely higher order spectra (HOS) to extract the hidden information in the sleep EEG signal. Various HOS features were extracted from the sleep stages (sleep 0 to sleep 5) and fed to the Gaussian mixture model (GMM) classifier for automatic identification. Our results show that, the proposed system is able to identify the unknown class with an accuracy of 86%, with a sensitivity and specificity of 89% and 87% respectively. We have also shown unique bispectrum and bicoherence plots for various sleep stages. Keywords: Sleep, EEG, bispectrum, entropy, higher order spectra, bicoherence.


Nowadays, non-linear signal processing has been effectively applied to EEG signals to study the dynamics of the complex underlying behaviour [McCarthy et al., 1990]. These methods are more superior than the traditional linear methods: time domain analysis and power spectral analysis [Mayer-Kress et al., 1987]. Sleep study is a very important branch of medicine, as it helps in the diagnosis of sleep apnea, schizophrenia, depression and other neural abnormalities.

Typical features of the different sleep stages are given below [http://www.dreamviews.com/sleepstages.php]:

During the sleep 0 (awake) stage, the low amplitude EEG is rapidly varying and beta waves are more prominent. The alpha and theta waves are present in sleep 1 stage (drowsiness state). The EEG frequency will be between 6-8 Hz and eyes begin to roll slightly in this state. Sleep 2 (light sleep) state, the peaks of EEG signals become higher (Sleep spindles), theta waves are more predominant, and K-complexes exits. Usually frequency ranges from 8 to 15 Hz and lasts for few minutes during this state. Deep sleep (Sleep 3) state has very low frequency delta and theta waves (2 to 4Hz). In this sleep state, 20 to 50% of EEG signals are delta waves and rest are theta waves. Sleep 4 (deep sleep) stage, the frequency ranges from 0.5 to 2 Hz. Delta and theta waves are more predominant. In this state, more than 50% of EEG signals are delta waves and rest are theta waves. Rapid Eye Movement (REM) is the Sleep 5 state. Beta waves are more predominant in this state and frequency will be greater than 12Hz. Frequent bursts of eye movement and muscular twitches exits in this state.

Sleep is not a rhythmic signal, but is characterized by a cyclic alternating Non-rapid-eye movement (REM) and REM signals [Dijk et. al, 1997; Agnew et. al, 1967; Bes et. al., 1991; Bixler et. al, 1984; Bliwise et. al, 1993; Feinberg, et. al., 1967]. Sleep stages 1, 2,3 and 4 form the Non-REM sleep, REM sleep is a highly activated state of the brain and is accompanied by dreaming.

A rigorous algorithm to separate different sleep stages of the EEG signal in infants based on statistical analysis of the spectral and nonlinear characteristics of the sleep EEG recordings was developed[Alexandra et al., 2009]. Hence, their proposed method, helps to the assess the sleep EEG stages automatically with greater accuracy.

A novel method for automatic detection and classification of sleep stages using a Multichannel Auto Regressive (MAR) model was studied [Zhovna et al., 2008]. In the training phase, vector quantization algorithm and Linde-Buzo-Gray and during classification phase Kullback-Leibler (KL) divergence was used. Their results show a classification accuracy rate of 93.2%.

Method for the detection of sleep stages using four steps: segmentation; parameter extraction; cluster analysis; and classification was proposed using EEG signals [Hese et al., 2001]. The parameters of Hjorth, the harmonic parameters and the relative band energy were fed to modified K-means algorithm. They have shown that, the proposed method was able to find 'similar' segments and automate the detection of sleep stages.

A sleep electroencephalogram (EEG) recognition neural network (SRNN) to detect several important characteristic waves in sleep EEG to diagnose sleep stages was developed[Shimada et al., 2000]. Their experimental results indicate that the proposed NN model was more capable than other conventional methods for detecting characteristic waves.

The correlation dimensions calculated for a polysomnography signal [Kobayashi et al., 1999]. The correlation dimensions decreased from the 'awake' stage to sleep stages 1-3 and increased during rapid eye movement (REM) sleep. In each sleep cycle, the correlation dimensions decreased for slow wave sleep, and increased for REM sleep.

The emergence of functional clusters related to the spontaneous brain activity during sleep was investigated[Dimitriadis et al., 2009]. Their results suggest a pivotal role for the functional coupling during the different stages and indicate interesting dynamic characteristics like its variable hemispheric asymmetry and the isolation between anterior and posterior cortical areas during REM.

Respiration data during REM sleep were analyzed using the correlation integral and the slope of its log-log plot, representing the correlation dimension [Pilgram et al., 1995]. The dynamics of a signal and its degrees of freedom was characterized by the correlation integral and by the correlation dimension, respectively. Their result indicate that highly irregular- breathing patterns during REM sleep can be described by a deterministic system, and the physiological significance of this finding was also discussed.

Clinical diagnosis of obstructive sleep apnea syndrome (OSAS) requires 1) assessment of relevant clinical features, and 2) objective demonstration of abnormal breathing and sleep disturbance, for which the gold standard is attended, in-laboratory polysomnography (PSG). Sleep disturbance is inferred from sleep stages, obtained by manually scoring electro-encephalography (EEG), electro-oculography (EOG) and electro-myography (EMG) montages obtained from PSG [Silber et al., 2007].

However, increasing public awareness of OSAS and resource constraints have restricted access to attended, in-laboratory PSG in many countries [Flemons et al., 2004], and there is therefore significant interest from professional bodies and sleep experts in simple and low-cost portable monitors [Portable monitoring, 2007].

To our knowledge there are very few portable monitors capable of providing full sleep staging information, i.e. distinguishing between Wake, Rapid Eye Movement (REM) sleep, and the different levels of Non-REM (NREM) sleep. Those that do require all the five sleep staging signals, making them cumbersome and requiring considerable technical expertise to operate in a home environment. Simplified systems employing actimetry are available, but they can only distinguish between Wake and Sleep, but cannot indicate the specific stage of Sleep.

A simple system capable of providing full sleep staging information is therefore of potential clinical value. Accordingly, we present an automated system that distinguishes between Wake, REM, and the different levels of NREM sleep, using only two channels of EEG. In this paper, we have analyzed and classified the different sleep stages using higher order spectra features and GMM classifier.

material and Methods

Subjects and Laboratory Procedures

Data from two cohorts were used for this study. Twenty-five subjects were recruited randomly from patients attending the Sleep Disorders Clinic at St Vincent's University Hospital, Ireland, for evaluation of suspected OSAS (21 M 4 F, age 50 ± 10 years, BMI 31.6 ± 4.0 kg/m², AHI 24.1 ± 20.3). The study was approved by the Hospital's Ethics Committee, and all subjects provided written, informed consent. The subjects underwent standard overnight, attended PSG. An experienced sleep technologist subsequently performed sleep staging according to the Rechtschaffen and Kales rules [Rechtschaffen et al., 1968] and annotated the respiratory events.

In another study at University College Dublin, 14 subjects with no known medical conditions were recruited from the general population (12 M 2 F, age 27 ± 4 years, BMI 25 ± 4 kg/m2). The study was approved by the University's Ethics, and all subjects provided written, informed consent. Standard sleep staging signals were recorded overnight using a set of Grass amplifiers (Astro-Med Inc, USA) while subjects obtained their usual sleep. Sleep staging was subsequently performed using the Somnolyzer 24x7 system [Anderer et al., 2005] using the Rechtschaffen and Kales rules [Rechtschaffen et al., 1968]. A recent validation study using the large Siesta database indicated that the Somnolyzer achieves similar inter-rater reliability with a human scorer as between two human scorers [Anderer et al., 2005].

EEG Analysis

For each record, the two standard sleep staging EEG signals (C4/A1 and C3/A2, 128 Hz) were analysed. Each EEG signal was divided into consecutive two-second epochs that overlapped by 1 second. Higher order spectra techniques were then applied to each epoch to obtain various features (details provided in subsequent sections). However, sleep staging was performed in thirty-second, non-overlapping epochs. Therefore, feature values in each thirty-second epoch were averaged to obtain a single value per sleep stage (thirty-second duration).

Higher Order Spectra Analysis

HOS techniques were applied to real signal processing problems in 1970s, and subsequently was applied to various areas such as economics, speech, seismic data processing, plasma physics, optics, bio-signal and medical imaging. The second order statistics deals with mean value and variance . They are defined by expectation operation where "a" is the result of a random process.

Sleep EEG signal is analyzed using different higher order spectra that are spectral representations of higher order moments or cumulants of a signal. The features related to the third order statistics of the signal, namely the bispectrum were extracted. The Bispectrum is the Fourier transform of the third order correlation of the signal and is given by

B(f1,f2) = E[X(f1)X(f2)X*(f1+f2)] (1)

Where X(f) is the Fourier transform of the signal x(nT) and E[.] stands for the expectation operation. The expectation operation is the average over an ensemble of realizations of a random signal. For deterministic signals, the relationship holds without an expectation operation with the third order correlation being a time-average. For deterministic sampled signals, X(f) is computed as the discrete Fourier transform (DFT) with f may be normalized by the Nyquist frequency to be between 0 and 1.

Higher order spectral features

Different bispectral entropies and bicoherence plots have been proposed for cardiac arrhythmia [Chua et. al., 2008] and normal, epileptic and background epileptic EEG signals [Chua et. al., 2009]. Bispectral entropies [Chua et. al., 2008] were derived to find the rhythmic nature of the HRV from bispectrum plots. The formulae for these bispectral entropies are as follows:

The normalization in the equations above ensures that entropy is calculated for a parameter that lies between 0 and 1 (as required of a probability) and hence the entropies (Ent1 and Ent2) computed are also between 0 and 1.

Figure 1 Non-redundant region of computation of the bispectrum for real signals. Features are calculated from this region.Frequencies are shown normalized by the Nyquist frequency.

In this work, we used features related to moments [Zhou et. al 2008] and the weighted centre of bispectrum (WCOB) [Zhang et. al. 1998] to characterize these plots. The features related the moments are:

The sum of logarithmic amplitudes of the bispectrum:

The sum of logarithmic amplitudes of diagonal elements in the bispectrum:

The first-order spectral moment of amplitudes of diagonal elements in the bispectrum:

These features (H1, H2, and H3) can be used to classify mental tasks from EEG signals.

The WCOB [Zhang et. al. 1998] is given by:

Blocks of 1024 samples, corresponding to 256 seconds used for computing the bispectrum. These blocks were taken from each EEG data record with an overlap of 512 point (i.e 50%).

Automated Sleep Stage Classification using Gaussian Mixture Model

In this work, we have used the Gaussian Mixure Model (GMM) classifier for classification. It is briefly explained below:

Gaussian Mixture Model (GMM)

GMM is a statistical model which comprises a number of Gaussian functions. It is used to approximate a continuous probability density function from a multi-dimensional features. The Gaussian mixture distribution is given by:

where (') denotes the vector transpose. Maximum Likelihood (ML) parameter estimation criterion was used determine the solution for parameters of GMM. The model parameters are evaluated through training such that they maximize the likelihood of the observations using Expectation-Maximization (E-M) algorithm [Reynolds et. al.1995, Seo et. al. 2001, Bilmes 1998].

Using K-means algorithm the initial estimates of the parameters were obtained from a sample of the training data. Randomly chosen initial means and unit variances for the covariance matrix was used in this work. The diagonal covariance matrices was used, because it was found to be more computationally efficient and performs better than the full covariance matrix [Bilmes 1998; Nelwamondo et al., 2006].


Figure 2,3,4,5, 6 and 7 shows the bispectral and bicoherence plots for the various sleep states (Sleep 0 to Sleep 5). It can be observed that these plots are unique for each class and however, stage 0 and sleep 5 resemble a lot. This seems to suggest that at state 5 there are more non-linear interaction of higher frequency component as seem from the graphs where the bispectrum extend further away from the central of the graphs.

Table 1 shows the features derived from HOS analysis. These features were subjected to ANOVA test and the result has shown to have very good p-values (p < 0.001). It can be seen from the Table 1 that, the different entropies (Ent1 and Ent2 ) are high for Sleep0, Sleep 1 and Sleep 5 due to more variability (high frequency). Sleep2, Sleep 3 and Sleep 4 stages have lower values of entropies due to lower variability (low frequency). Similarly, other parameters mAmp, Wcobx, Wcoby, H1, H2 and H3 show similar trend for different sleep stages.

We have selected four HOS features namely, Ent1, mAmp, Wcobx and H1 as the input to the GMM classifier for classification. Table 2 shows the classification results and Table 3 shows sensitivity, specificifity, and positive predictive accuracy of our proposed system. Our results show that, system is able to identify the unknown class accurately (86%), with a sensitivity and specificity of 89% and 87% respectively.


The sleep data analysis was carried out using non-linear parameters: correlation dimension, fractal dimension, largest Lyapunov entropy, approximate entropy, Hurst exponent, phase space plot and recurrence plots [Acharya et al., 2005]. Range of values for different non-linear parameters was proposed and they also proposed unique recurrence plots for different sleep stages.

Rapid eye movements (REM) was automatically detected by applying using the Discrete Wavelet Transform (Haar wavelet function) to each 8-s segment of electrooculogram (EOG) data for 30 min of 8 h of normal sleep[Tsuji et al., 2000].By shifting the phase of the analysing wavelet by pi/4 of the function, the proposed method was able to detect 96% of REM.

Second order Daubechies mother wavelet coefficients for the EEG epochs (64 data) were extracted from EEG signals for the training the neural network and to classify sleep spindles (SS), rapid eye movement (REM) sleep and awake (AWA) sates[Sinha, 2008]. The ANN architecture used (64-14-3) was found effective in differentiating the EEG power spectra from different sleep-wake states (96.84% in SS, 93.68% in REM sleep, 95.52% in AWA state).

Wavelet packet coefficients and artificial neural networks were used to classify the sleep stages using EEG signals [Ebrahimi et al., 2008]. Their results show that they were automatically classify four sleep stages with a specificity of 94.4 ± 4.5%, a of sensitivity 84.2+3.9% and an accuracy of 93.0 ± 4.0%.

An automatic algorithm based on the peripheral arterial tone (PAT) signal to differentiate between light and deep sleep stages was developed[Bresler et al., 2008].The algorithm was based on 14 features extracted from two time series of PAT amplitudes and inter-pulse periods (IPP). The sensitivity, specificity and agreement of the automatic algorithm to detect 30 s epochs of light and deep sleep stages were 66%, 89%, 82% and 65%, 87%, 80% for the training and validation sets, respectively. In our work, we have used novel HOS features to extract the hidden complexities in the sleep EEG signals. We have selected four HOS features namely, Ent1, mAmp, Wcobx and H1 as the input to the GMM classifier for classification. Our results show that, system is able to identify the unknown class accurately with a sensitivity and specificity of 89% and 87% respectively.


Sleep EEG signals are highly non-linear and non-stationary in nature. Hence, they are to be analyzed by non-linear dynamics methods. Hence, we have used HOS analysis to extract the salient features from the sleep stages and fed to the GMM classifier classification. Our results show that, our proposed method is able to identify the sleep stage with an accurately of 86% with a sensitivity and specificity of 89% and 87% respectively. However, with more and diverse training data this accuracy can be further increased.


The authors are grateful to Professor Walter T McNicholas of St Vincent's University Hospital and Professor Conor Heneghan of University College Dublin for providing the sleep data for this study.


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