A multiwavelet-based time-varying model identification approach for time–frequency analysis of EEG signals
Introduction
Electroencephalography (EEG) is a significant non-invasive technique for clinical diagnosis, as well as for scientific research of brain function. Electroencephalograms commonly contain a wealth of information of brain activity during certain task activities or particularly mental processes. Careful analysis of the EEG records can provide valuable insight into the widespread brain disorder. Recently, EEG has been widely employed to investigate clinical neuropathology for medical diagnosis [1], [2]. Since EEG signals contain rich information about brain functions, many spectral analysis approaches can be adopted to analyze and detect EEG signals in brain abnormalities, and further to use this analytical information in medical researches and clinical diagnosis [3].
Generally, two main classes of time–frequency analysis (TFA) approaches have been proposed for EEG signals with non-stationary frequency contents. The first is non-parametric spectral estimation methods. Among these, the most common transform for TFA is the short-time Fourier transform (STFT), which takes the fast Fourier transform (FFT) of successive and overlapping windows of a signal. However, the STFT has two main disadvantages. Firstly, the time frequency resolution of the STFT cannot simultaneously obtain a high frequency resolution and accurately the timing of any changes in frequency [4]. Secondly, the spectrum obtained by using the STFT method is characterized by a main lobe with a width proportional to the side-lobe leakage, which can prevent the distinction between spectral peaks or mask low amplitude spectral peaks. The choice of a proper taper can decrease the side-lobes amplitude, which is at the expense of decreasing the amplitude and widening the main lobe that results in reduction of the spectral resolution [5]. Other time–frequency transforms including the continuous wavelet transform (CWT) can achieve better time–frequency estimates for nonstationary signals by applying a short window at high frequency and a long window at low frequency [6], [7]. However, they have degraded frequency resolution for high-frequency contents and degraded time resolution for low-frequency components, and also displayed spectral smearing due to the finite size of their operator [4].
Unlike the non-parametric spectral methods, the power spectral density (PSD) estimates based time-varying autoregressive (TVAR) is a parametric TFA approach. Parametric spectral analysis methods can usually achieve higher frequency resolution than non-parametric approaches provided that an appropriate (e.g., TVAR) model should be obtained [6]. The TVAR model is often utilized as an efficient tool to reveal the dynamics of nonstationary signals such as biomedical signals, due to its simplicity and effectiveness [8], [9]. However, a continuing challenge is that a sufficiently precise TVAR model should be established. Typically, two main classes of methods have been applied to estimate the TVAR model, namely, adaptive Kalman filter algorithm and basis function expansion and regression method.
The adaptive Kalman filter algorithm is an online approach that is amongst the most popular parametric models and identification approaches [6]. The KF method is computationally efficient in estimating time-varying (TV) coefficients for TVAR model provided that the system is slowly varying, compared to the algorithm׳s convergence time. For the TV parameters that change fast enough, the adaptive algorithm cannot handle and track systems that vary rapidly, because the speed of convergence of the algorithms is not fast enough. However, the disadvantages in the typical adaptive method is that the conventional KF is high sensitive to the selection of model parameters and thus leads to a large estimation variance or a lag tracking capability for TV parameters [10]. Aboy et al. [11] reported that the applications of KF algorithm were mainly focused on eye open/closed EEG data with stable time–frequency characteristics. However, as discussed in [6], the KF algorithm is still not a well-accepted option for spectral analysis of highly nonstationary EEG signals [6].
Basis function expansion and regression method has the advantage of excellent capability of tracking parameters changing over time. This method expands the TV parameters in the TVAR model into a finite sequence of pre-determined basis functions such as Fourier series [12], Legendre or Chebyshev polynomials [13], [14], wavelets [14], [15], [16], [17]. In fact, selecting an appropriate set of basis functions is essential for the accurate parameter estimation of this approach. Each family of basis functions have its own unique tractability and accuracy, for instance, numerical experiments by Chon et al. [15] showed that the Chebshev or Legendre polynomials is effective for parameters that change smoothly or slowly with time, Walsh functions, however, are effective for TV parameters that have sharp variations or piecewise stationary changes. To approximate the jump or step coefficient functions in the TVAR model, Chen et al. [18] demonstrated that Haar wavelet are superior to Legendre basis function. In order to alleviate the dilemma on selection of basis function, Li et al. [19] was applied to combine cardinal B-splines wavelet function with a block least mean squares algorithm, both rapid and slow changes of TV parameters can be effectively tracked. Zhang et al. [6] showed that a set of local polynomial modeling (LPM) method within a data window having variable bandwidth can accurately identify and estimate TV parameters in the TVAR model, and further illustrate that the LPM approach can achieve a more accurate time–frequency representation of nonstationary EEG signals.
Recently, a considerable attention has been received for system modeling and signal processing using wavelet transform approach [16], [17], [20], [21]. Wavelet is a function localized both in time and frequency domain [15], [21], which can be used to represent an abrupt variation or a local function vanishing outside a short time interval adaptively [22]. Wavelets have been successfully applied to physiological signal processing and analysis [23], [24], as well as being widely used in many other fields including signal processing and system identification [20]. Physiological data often show both fast and slow variations [25]. In order to reduce the difficulties on selection of basis functions, in this paper, a novel TV modeling method is proposed by using a multiwavelet-based basis function expansion scheme, where properties of different types of wavelets are employed and combined in a form of multi-resolution decompositions. Specifically, the meaning of the term multiwavelet is twofold. Firstly, the time-varying coefficients in the AR model are approximated by different types of wavelet basis functions, namely, the time-dependent parameter estimation involves multiple wavelets. Secondly, these wavelet basis functions are combined in a form of multi-resolution wavelet decompositions. The time-dependent coefficients are expanded by using a finite number of multiwavelet basis functions that are well suited for approximating general nonstationary signals [16]. Thus, the proposed multiwavelet-based scheme can make the modeling algorithm more flexible and track both slow and fast variations rapidly.
The proposed TVAR identification method based multiwavelet basis function expansion may be used in various applications for analyzing biomedical signals. In this study, we will focus on application to TFA of EEG signals. Time–frequency analysis methods have been commonly and systematically applied in EEG study, and a desirable TFA approach for EEG study should have both good time resolution and good frequency resolution, simultaneously. However, the performance of conventional TFA approaches are generally highly dependent on the choice of model parameters or the convergence deficiency of traditional TFA techniques. Therefore, higher frequency resolution is obtained at the expense of undesirable low time resolution and vice versa. The proposed multiwavelet-based expansion method provides a good alternative for the TFA of highly nonstationary processes because it can achieve a good tradeoff between time and frequency resolution by employing multiresolution wavelet. The simulated results do indeed show that the new modeling framework using multiwavelet expansions can capture various time–frequency components of an artificial EEG signal more clearly and accurately than conventional power spectra estimation methods like CWT, STFT and classical adaptive AR methods; this is usually very useful for feature extraction from EEG signals in both the time and frequency domains.
The rest of the paper is organized as follows. In Section 2, the multiwavelet-based expansion and parameter estimation method for TVAR models are introduced. The simulation examples with the multiwavelet-based TFA method and conventional TFA techniques are presented in Section 3. The multiwavelet-based expansion TFA approach is further extended to analyze the real EEG data in Section 4. Final conclusions are given in Section 5.
Section snippets
Identification of the TVAR model
In the TVAR method, data can be modeled as output of a causal, all-pole, discrete filter whose input is white noise. The pth order TVAR model is formulated as belowwhere (, is the total number of samples) is the time instant or sampling index of the signal , is the order of the TVAR model, represents the TVAR coefficients, are delayed samples of the signal, and is assumed to be a sequence of independent and normal distributed
Simulation examples
Prior to applying the proposed multiwavelet-based modeling approach to a real EEG recording analysis, different TFA approaches including CWT, STFT, KF, Chebshev polynomials and B-splines wavelets based TDS estimation, are compared using the simulated EEG signals. An artificial EEG signal that consists of the sum of four distinct sinusoidal components with different frequencies and durations was considered. The signal was defined as below:
Application: time–frequency analysis of real EEG signals
The proposed TVAR modeling approach is used to analyze the real EEG data. As an example, an EEG recording was considered to illustrate the application of the proposed TVAR modeling framework. The EEG time sequence of 1042 data points is shown in Fig. 3, recorded for 6 s, with a sampling rate of 173.61 Hz [43], [44]. This recording is for a sort of seizure activity of a patient. A detail description can be found in [43]. The time–frequency characteristics of the real EEG data have been revealed by
Conclusions
A novel multiresolution wavelet expansion scheme for the identification of TVAR models with emphasis on its application to time–frequency analysis of EEG data has been presented in this study, where the associated time-dependent parameters are approximated using basis functions of multiresolution B-splines wavelet. In most existing TV parametric models, the basis functions used for expanding the TV parameters are global, while the basis functions investigated in the proposed modeling method are
Acknowledgments
This work was supported by the National Natural Science Foundation of China (61403016), Specialized Research Fund for the Doctoral Program of Higher Education (20131102120008), Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and the Fundamental Research Funds for the Central Universities, as well as by Beihang University Innovation & Practice Fund for Graduate (YCSJ-02-2015-12).
Dr. Yang Li obtained his Ph.D. degree in Automatic Control and Systems Engineering at Sheffield University in September 2011. After one year of postdoctoral research in the Department of Computer and Biomedical Engineering at the University of North Carolina at Chapel Hill, Dr. Li joined Beihang University as an Associate Professor in the Department of Automation Sciences and Electrical Engineering starting in February, 2013. His main research area is involved system identification and modeling
References (48)
- et al.
Comparison of STFT and wavelet transform methods in determining epileptic seizure activity in EEG signals for real-time application
Comput. Biol. Med.
(2005) - et al.
Automatic recognition of epileptic EEG patterns via extreme learning machine and multiresolution feature extraction
Expert Syst. Appl.
(2013) - et al.
Efficient feature selection and linear discrimination of EEG signals
Neurocomputing
(2013) - et al.
Bayesian nonstationary source separation
Neurocomputing
(2008) - et al.
Adaptation and tracking in system-identification – a survey
Automatica
(1990) - et al.
EEG signal analysis using FB expansion and second-order linear TVAR process
Signal Process.
(2008) - et al.
Time-varying model identification for time-frequency feature extraction from EEG data
J. Neurosci. Methods
(2011) - et al.
Recovering EEG brain signals: artifact suppression with wavelet enhanced independent component analysis
J. Neurosci. Methods
(2006) - et al.
An adaptive filtering approach for electrocardiogram (ECG) signal noise reduction using neural networks
Neurocomputing
(2013) - et al.
A comparative study of wavelet families for EEG signal classification
Neurocomputing
(2011)
Short-term time series algebraic forecasting with internal smoothing
Neurocomputing
Auditory event-related dynamics of the EEG spectrum and effects of exposure to tones
Electroencephalogr. Clin. Neurophysiol.
Epileptic seizure detection using DWT based fuzzy approximate entropy and support vector machine
Neurocomputing
Identification of nonlinear time-varying systems using an online sliding-window and common model structure selection (CMSS) approach with applications to EEG
Int. J. Syst. Sci.
Time-varying autoregressive model for spectral analysis of microseismic experiments and long-period volcanic events
Geophys. J. Int.
Resolution and uncertainty in spectral decomposition
First Break
Local polynomial modeling of time-varying autoregressive models with application to time–frequency analysis of event-related EEG
IEEE Trans. Biomed. Eng.
Application of wavelet energy and Shannon entropy for feature extraction in gearbox fault detection under varying speed conditions
Neurocomputing
Adaptive AR modeling of nonstationary time series by means of Kalman filtering
IEEE Trans. Biomed. Eng.
Adaptive modeling and spectral estimation of nonstationary biomedical signals based on Kalman filtering
IEEE Trans. Biomed. Eng.
Statistical Digital Signal Processing and Modeling
Identification of Time-Varying Process
Multiple time-varying dynamic analysis using multiple sets of basis functions
IEEE Trans. Biomed. Eng.
Time-varying parametric modelling and time-dependent spectral characterisation with applications to EEG signals using multiwavelets
Int. J. Model. Identif. Control
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Dr. Yang Li obtained his Ph.D. degree in Automatic Control and Systems Engineering at Sheffield University in September 2011. After one year of postdoctoral research in the Department of Computer and Biomedical Engineering at the University of North Carolina at Chapel Hill, Dr. Li joined Beihang University as an Associate Professor in the Department of Automation Sciences and Electrical Engineering starting in February, 2013. His main research area is involved system identification and modeling for complex nonlinear processes: NARMAX methodology and applications; nonlinear and nonstationary signal processing; intelligent computation and data mining, parameter estimation and model optimization, sparse representation etc.
Mei-Lin Luo received his bachelor degree from Beihang University in 2014. Now he is a graduate student in Department of Automation Science and Electrical Engineering at Beihang University, Beijing, China. His main research interests include system identification, signal processing and pattern recognition.
Dr. Ke Li received his Ph.D. degree from Beihang University, Beijing,China, in 2008. His current research interests include networked control system, control methods of thermal engineering, intelligent control algorithms, brain–machine interface and machine learning.