Analysis of EEG signals with and without reflexology using FFT and auto regressive modelling techniques
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《中华医药杂志》英文版
【Abstract】 Reflexology, a natural healing art, was based on the principle that there are reflexes in the feet and hands which correspond to every part of the body. By stimulating and applying pressure to the feet or hands, one could increase circulation and promoting specific bodily and muscular functions. It was known that the reflexology had the positive influence on the brain signals. The electroencephalogram (EEG) was a representative signal containing information about the condition of the brain. Therefore, the EEG signal parameters, extracted and analyzed using computers,were highly useful in diagnostics. This work dealed with a comparative study of FFT by Welch method, Auto Regressive (AR) by Burgs and least squares method on the EEG signal with and without reflexology. The results are tabulated for 30 normal subjects (with and without reflexology).
【Key words】 electroencephalogram;AR Model;FFT;stimulation, reflexology
INTRODUCTION
Computer technology has an important role in structuring biological systems. The explosive growth of high performance computing techniques in recent years with regard to the development of good and accurate models of biological systems has contributed significantly to new approaches to fundamental problems of modelling transient behavior of biological system. The importance of the biological time series analysis, which exhibits typically complex dynamics, has long been recognized in the area of non-linear analysis. Several features of these approaches have been proposed to detect the hidden important dynamical properties of the physiological phenomenon. The nonlinear dynamical techniques are based on the concept of chaos and it has been applied to many areas including the areas of medicine and biology.
Reflexology is the art and science of working on specific reflex points (areas) on the hands, feet, and ears to relax and relieve stress and pain in the body. In clinical terms reflexology is the application of pressure, primarily, but not limited to the feet, hands or ears which causes a physiological response in the body. Reflexology originated in China about 4000 BC. First recorded history of reflexology was in 2330 BC in Egypt: they were masters in setting bones, caring for wounds and treating illness. They recorded the medical practices through drawings of surgical operations and medical treatments. The movement was traced from India to China and then to Japan. Reflexology is a science which is based on the principles that reflexes or areas on the hands, feet and ears of the body relation to internal organs and other structures of the body. It was introduced to the west by Dr. William Fitzgerald Lipsitz,et al., 1992. In reflexology the feet, hands and ears are seen as a perfect microcosm of the body, with somatic replications of all organs, glands and muscles of the body on an area or a reflex point. As a science it is the study of relieving pain and stress in the body through touch using alternating pressure.
Reflexology embraces all body systems and organs, physical: by unlocking, stimulating and improving circulation; mental: through the application of touch and its therapeutic affect on the body; and emotional: the universal connection at the subtle level with client to relieve pain and relax the body. We have used non-linear signal processing parameters to estimate the quantitative difference between the two HRV signals viz: one without reflexology and the other one under reflexological stimulation. Refelexology being a healing art working on the subtle planes of the human body, we are using subtle tools for investigating the same.Siev-Ner etc.have evaluated the effect of reflexology on the subjects with symptoms of multiple sclerosis [Sie-Ner, et al.,2003]. They successfully showed that the specific reflexology treatment (manual pressure on specific points at feet and massage of calf area) alleviates the motor, sensory and urinary symptoms in multiple sclerosis subjects. Frankel has shown the positive effects of reflexology on the baroreceptor reflex sensitivity, blood pressure and sinus arrhythmia [Frankel, 1997]. Bishop et al., have proved that reflexology is an effective method of treating encopresis and constipation patients [Bishop et al., 2003]. Christ Stormer (2005) has clearly explained the healing effect of reflexology on respiratory, circulatory, and cardiac disorders. Most recently, Jennifer et al.have studied the impact of foot massage and guided relaxation following the cardiac surgery [Jennifer et al., 2002].
The electrical activity of a brain measured by Electroencephalogram (EEG) exhibits significant complex behavior with strong non-linear and dynamic properties. This behavior takes the form of EEG activity patterns with different complexities. Considering this, the non-linear dynamics theory may be a better approach than traditional linear methods in characterizing the intrinsic nature of EEG. The study of non-linear dynamics and characterization can contribute to the understanding of the EEG dynamics and underlying brain processes and search for its physiological significance. Hence we chose non-linear analysis tools to investigate the phenomena of reflexology from EEG. The literature on the study of the application of the non-linear dynamics theory to analyze physiological signals, shows that non-linear approaches were used for analysis of heart rate, nerve activity, renal flow, arterial pressure, EEG and respiratory signals [Stein et al., 1992; Hoyer et al., 1997].
Despite the many applications of EEG in clinical neuro-physiology [Gevins et al., 1988; Niedermeyer et al., 1993; James et al., 2000; Xu-Sheng et al., 2001; Borel et al., 1985; Agarwal et al., 1998], its visual interpretation is very subjective and does not lend itself to statistical analysis. As a result a number of research groups have proposed methods to quantify the information content of the EEG. Among these are the Fourier transform, the wavelet transform, chaos, entropy, and the subband wavelet entropy [Goel et al., 1996; Niedermeyer et al., 1999; Geocadin et al., 2000; Bezerianos et al., 2001; Rosso et al., 2001]. In addition, the EEG signal modelling is important to achieve a better understanding of the physical mechanisms generating these signals and to identify the causes of EEG signal changes. Furthermore, pattern recognition and classification of EEG abnormalities can be achieved through the analysis of the estimated model parameters. Modelling can also be used for predicting the future neurological outcome and for data compression. Simulation based on EEG signal model can be used to better demonstrate the effectiveness of a certain quantitative analysis method or EEG feature extraction.
Al-Nashash et al have used the neural network method for modelling EEG signals [Al-Nashash et al., 2003]. And also, they have implemented the EEG modelling using adaptive Markov process amplitude [Al-Nashash et al., 2004]. Recently Paul et al., have studied the performance of influence of reflexology on the heart rate variability [Paul et al.,2004]. Similarly, Kannathal et al., have studied the effect of reflexology on the brain signals using nonlinear analysis method [Kannathal et al., 2004a; Kannathal et al., 2004b]. In this work, we have studied the performance of different modelling techniques for the EEG signals with and without reflexology.
MATERIALS
EEG signals are being recorded by using BIOPAC equipment with ACQKNOWLEDGE 3.7.2 as data acquisition software [Biopac, 2001]. Silver-Silver chloride electrodes were applied following the 10-20 system. EEGs are recorded from 30 subjects (15 boys and 15 girls) with age ranging from 20 ± 3 years for 20 minutes. The EEG is recorded at 500 samples/sec with a resolution of 12 bits/sample and for a duration of twenty minutes. After this, the same subject was given reflexological stimulation just below the toes of both feet by a mechanical reflexological device (Massager-scroller type) in the same relaxed sitting for another 20 minutes. And the EEG is recorded during the reflexological stimulation. Then the data is digitally filtered using 1-50 Hz band pass filter. Time gap of 30 minutes is given between any two successive recordings to ensure that a previous state stimulus doesnt affect the subsequent state EEG.
SPECTRAL ANALYSIS
The approaches for spectrum estimation may be generally categorized into one of the two classes. The first includes the classical or non-parametric methods that begin by estimating the autocorrelation sequence rx(k) from a given set of data. The Power spectrum is then estimated by Fourier transforming the estimated autocorrelation sequence. The second class includes the non-classical or parametric approaches, which are based on using a model for the process in order to estimate the power spectrum. Three types of modelling techniques are practically available to estimate the power spectrum.
Non-parametric method (FFT) The Welch method [Welch, 1967] is one of the most popular classical methods to estimate the power spectrum of any given time sequence. The sequences are allowed to overlap and a data window is applied to each sequence. This will produce set of modified periodograms that are to be averaged. The data sequences xi(n) can be represented as
Where iD is the starting point for the ith sequence. Finally we can form K data segments each of length 2M. The resulting modified periodogram is
Where U is the normalization factor for the power in the window function and is selected as
The Welch power spectrum is average of these modified periodograms that is
Parametric method The non parametric method we have seen so far is not optimal, because this method suffers from spectral leakage effects due to windowing. The Spectral leakage leads to masking of weak signal that is present in the data. A parametric (model based) power spectrum estimation method avoids the problem of spectral leakage and provides better frequency resolution than non-parametric or classical methods. The paramedic methods assume the signal to be a stationary random process. This process can be modeled as the output of a filter with white noise input. The filter parameters are obtained from the signal, therefore the name: Parametric Method. There are different ways to obtain the parameters; these methods are classified depending on presence of poles in the z-domain. If there are no poles, one speaks of an MA (Moving Average) Model. In the case of an MA model the filter used in the model is a simple Finite Impulse Response (FIR) characteristic. If there are poles present and all zeros are located in the origin it is an (Auto Recursive AR) Model. This results in an Infinite Impulse Response (IIR) filter characteristic where only the output is used for feedback. A model having poles and zeros freely distributed in the z-domain is called ARMA Model. An ARMA model uses an IIR filter with feedback paths for input and output. According to the Wold theorem [Wold, 1938] it is possible to translate one model into another without any loss. That means the models can be chosen according to simplicity of the algorithm or computational overhead. In the present study we have chosen Yule-Walker and Burgs Method, both PSD estimation models are using AR models.
Burg Method The Burg method for AR spectral estimation is based on minimizing the forward and backward prediction errors while satisfying the Levinson-Durbin recursion [Proakis et al., 1996; Marple, 1987]. In contrast to other AR estimation methods, the Burg method avoids calculating the autocorrelation function, and instead estimates the reflection coefficients directly. The primary advantages of the Burg method are resolving closely spaced sinusoids in signals with low noise levels, and estimating short data records, in which case the AR power spectral density estimates are very close to the true values. In addition, the Burg method ensures a stable AR model and is computationally efficient. The accuracy of the Burg method is lower for high-order models, long data records, and high signal-to-noise ratios (which can cause line splitting, or the generation of extraneous peaks in the spectrum estimate). The spectral density estimate computed by the Burg method is also susceptible to frequency shifts (relative to the true frequency) resulting from the initial phase of noisy sinusoidal signals. This effect is magnified by analyzing short data sequences. Burg Method differs to the Yule-Walker Method in the way the PSD,PBUxx is obtained, as shown in the following Equation:
Where the a)p(k) are again the estimates of the AR parameters obtained from the Levinson-Durbin recursion. And TP6 are reflection coefficients in an equivalent lattice structure, which are chosen to obtain the total least square error.
Least Square Method (LSM) In least squares (LS) estimation, the unknown values of the parameters, β0,β1……, in the regression function, ,are estimated by finding numerical values for the parameters that minimize the sum of the squared deviations between the observed responses and the functional portion of the model. Mathematically, the least (sum of) squares criterion that is minimized to obtain the parameter estimates is
As previously noted,β0,β1…… are treated as the variables in the optimization and the predictor variable values, x1，x2，……are treated as coefficients. To emphasize the fact that the estimates of the parameter values are not the same as the true values of the parameters, the estimates are denoted by ,…….For linear models, the least squares minimization is usually done analytically using calculus. For nonlinear models, on the other hand, the minimization must almost always be done using iterative numerical algorithms.To illustrate, considering the straight-line model:
For this model the least squares estimates of the parameters would be computed by minimizing
Doing this by
1. taking partial derivatives of with respect to β^ 0 and β^ 1
2. setting each partial derivative equal to zero, and
3. solving the resulting system of two equations with two unknowns yields the following estimators for the parameters:
These formulas are instructive because they show that the parameter estimators are functions of both the predictor and response variables and that the estimators are not independent of each other unless x=0. This is clear because the formula for the estimator of the intercept depends directly on the value of the estimator of the slope, except when the second term in the formula for β^ 0 drops out due to multiplication by zero. This means that if the estimate of the slope deviates a lot from the true slope, then the estimate of the intercept will tend to deviate a lot from its true value too. This lack of independence of the parameter estimators, or more specifically the correlation of the parameter estimators, becomes important when computing the uncertainties of predicted values from the model. Although the formulas discussed in this paragraph only apply to the straight-line model, the relationship between the parameter estimators is analogous for more complicated models, including both statistically linear and statistically nonlinear models.
Akaike Information Criterion For both Burg and least square method, the model order was chosen as the one that minimizes the Akaike information criterion (AIC) figure of merit [Akaike, 1969, 1974]:
Where N is the number of data samples and λ^ 2 is the estimated white noise variance. To reduce computational costs, we assumed as optimal the value of P that fulfilled the AIC criterion in the first two epochs.
One of the most important aspects of the use of AR method is the selection of the order p. Much work has been done by various researchers on this problem and many experimental results have been given in literature such as the papers presented by [Anita et al., 2002]. In this work the order of the AR model is taken as: p=20.
RESULTS
The EEG signal with and without reflexology is analyzed using FFT by Welch method and AR method by Burgs method and least square method.
Spectral Analysis of the EEG Signals The PSD was calculated for each of the signals using Welch, Burgs and least square method. The first plot in Figure 1(a) shows the PSD obtained by Welch in the Control data without reflexology. The second plot [Figure 1(b)] shows the PSD by Welch method for the same subject with reflexology. In the PSD plots, a square marks indicates the first, second and third prominent peaks. The first, second and thirds prominent frequencies are simply the first second and third maxima of the PSD function respectively. In the past, it shown that, the EEG signal variability decreases with the reflexology [Kannathal et al.,2004; Kannathal et al.,2004]. Hence, in figure 1(a) the peaks have higher value as compared to the figure 1(b). Figure 2 shows the PSDs for the same signals, but the PSD is obtained using a parametric method, namely Burgs method. In this method also, figure 2(b) has less predominant peaks as compared to the figure 2(a) due to the influence of reflexology. A similar pattern can be observed in Figure 3, which shows the PSDs obtained by least square method.
ANOVA Test To explore the relationship between the prominent frequencies and a specific data group the location and power for the first three prominent frequencies were evaluated for all signals. Table 1 shows the values of nine parameters with and without reflexology. The first parameter, P1, represents mean and variance of the power for the first prominent frequency within one data group. The second parameter, F1, represents mean and variance of the frequency location for the first prominent frequency within one data group. The third parameter, P1/F1, is obtained by dividing P1 by F1. The parameters P2 and F2 represent mean and variance of frequency and power for the second prominent frequency within one data group. P2/F2 is again the relative power for the second prominent frequency. The parameters P3, F3 and P3/F3 concern the third prominent frequency and they were estimated in the same way.
Figure 1 (a) PSD estimation using Welch Method without reflexology
Figure 1 (b) PSD estimation using Welch Method with reflexology
Figure 2 (a) PSD estimation using Burg?es method without reflexology
Figure 2 (b) PSD estimation using Burg?es method with reflexology
Figure 3 (a) PSD estimation using least square method without reflexology
Figure 3 (b) PSD estimation using least square method with reflexology
The last column in Table 1 states the p-value for the individual parameter. The P-value was calculated using the one way ANOVA algorithm [Sahai et al., 2000; Jobson, 1991]. The purpose of one-way ANOVA is to find out whether datas from several groups have a common mean. That is, to determine whether the groups are actually different in the measured characteristic.
ROC Test The values for the parameters F1, F2 and F3 in Table 1, Table 2 and Table 3 show that the frequency locations of the prominent frequencies have distinct mean values and low variances. This indicates that the freqeuency location can be used for classification. To support this statement a ROC (Receiver Operating Characteristic) test [Egan, 1975; Jalihal et al., 1994] was performed using the parameters F1, F2 and F3. ROC is a one class classification method, which classifies a target class within a dataset. The dataset consists of target (desired) values or vectors and outliers. Results show that (table 1,2 and 3) are peak frequencies are distinct for various techniques with and without reflexology. Figure 4 show the comparison of P1/F1 result for all the three methods using ROC test. In a ROC curve the true positive rate (Sensitivity) is plotted in function of the false positive rate (100-Specificity) for different cut-off points of a parameter. Each point on the ROC plot represents a sensitivity/specificity pair corresponding to a particular decision threshold. The area under the ROC curve is a measure of how well a parameter can distinguish between two diagnostic groups (diseased/normal). It can be observed that the ROC test shown in Figure 4, yields low values for the area under the curves for FFT and AR method by Burgs technique. Welchs method yields a spiky spectrum, where the first three spikes are mistaken to be the prominent frequencies. The parametric methods perform much better in the ROC test, as shown in Figures 4. The area under the ROC curve for least squares method is more than the other two methods and it performs better than the rest.
Figure 4 Comparison of P1/F1 ratio for various methods by ROC techniques with reflexology
CONCLUSION
The mental state of a person influences the EEG signals. It can be seen from the results that, with reflexology, the power spectral density decreases. Therefore, the variation in the EEG signals decreases, which indicates presence of more low frequency (alpha waves) signals. Different ways to represent the power distribution in the frequency domain where applied and their results are compared. The comparison based on the frequency location of the prominent frequencies shows that parametric methods of frequency estimation are superior to non-parametric methods. From the Figures 1,2 and 3, we can infer that, the peak power points can be clearly seen in least square AR method and is more superior as compared to the other techniques. This is shown by the ROC method also. Hence, we can use these modelling techniques to show the influence of reflexology on EEG signals. And among the techniques, we studied, least squares AR method shows better performance compared to the other methods used.
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1 Department of ECE, School of Engineering, Ngee Ann Polytechnic, Singapore 599489
2 Department of Electrical Engineering, NITC, Calicut, India
3 School of Mechanical and Aerospace Engineering, College of Engineering, Nanyang Technological University, 50, Nanyang Avenue, Singapore 639798
*Corresponding to mykng@ntu.edu.sg
(Editor Guo Hui-ling)ARTICLES(Kannathal N, Rajendra Ach)
【Key words】 electroencephalogram;AR Model;FFT;stimulation, reflexology
INTRODUCTION
Computer technology has an important role in structuring biological systems. The explosive growth of high performance computing techniques in recent years with regard to the development of good and accurate models of biological systems has contributed significantly to new approaches to fundamental problems of modelling transient behavior of biological system. The importance of the biological time series analysis, which exhibits typically complex dynamics, has long been recognized in the area of non-linear analysis. Several features of these approaches have been proposed to detect the hidden important dynamical properties of the physiological phenomenon. The nonlinear dynamical techniques are based on the concept of chaos and it has been applied to many areas including the areas of medicine and biology.
Reflexology is the art and science of working on specific reflex points (areas) on the hands, feet, and ears to relax and relieve stress and pain in the body. In clinical terms reflexology is the application of pressure, primarily, but not limited to the feet, hands or ears which causes a physiological response in the body. Reflexology originated in China about 4000 BC. First recorded history of reflexology was in 2330 BC in Egypt: they were masters in setting bones, caring for wounds and treating illness. They recorded the medical practices through drawings of surgical operations and medical treatments. The movement was traced from India to China and then to Japan. Reflexology is a science which is based on the principles that reflexes or areas on the hands, feet and ears of the body relation to internal organs and other structures of the body. It was introduced to the west by Dr. William Fitzgerald Lipsitz,et al., 1992. In reflexology the feet, hands and ears are seen as a perfect microcosm of the body, with somatic replications of all organs, glands and muscles of the body on an area or a reflex point. As a science it is the study of relieving pain and stress in the body through touch using alternating pressure.
Reflexology embraces all body systems and organs, physical: by unlocking, stimulating and improving circulation; mental: through the application of touch and its therapeutic affect on the body; and emotional: the universal connection at the subtle level with client to relieve pain and relax the body. We have used non-linear signal processing parameters to estimate the quantitative difference between the two HRV signals viz: one without reflexology and the other one under reflexological stimulation. Refelexology being a healing art working on the subtle planes of the human body, we are using subtle tools for investigating the same.Siev-Ner etc.have evaluated the effect of reflexology on the subjects with symptoms of multiple sclerosis [Sie-Ner, et al.,2003]. They successfully showed that the specific reflexology treatment (manual pressure on specific points at feet and massage of calf area) alleviates the motor, sensory and urinary symptoms in multiple sclerosis subjects. Frankel has shown the positive effects of reflexology on the baroreceptor reflex sensitivity, blood pressure and sinus arrhythmia [Frankel, 1997]. Bishop et al., have proved that reflexology is an effective method of treating encopresis and constipation patients [Bishop et al., 2003]. Christ Stormer (2005) has clearly explained the healing effect of reflexology on respiratory, circulatory, and cardiac disorders. Most recently, Jennifer et al.have studied the impact of foot massage and guided relaxation following the cardiac surgery [Jennifer et al., 2002].
The electrical activity of a brain measured by Electroencephalogram (EEG) exhibits significant complex behavior with strong non-linear and dynamic properties. This behavior takes the form of EEG activity patterns with different complexities. Considering this, the non-linear dynamics theory may be a better approach than traditional linear methods in characterizing the intrinsic nature of EEG. The study of non-linear dynamics and characterization can contribute to the understanding of the EEG dynamics and underlying brain processes and search for its physiological significance. Hence we chose non-linear analysis tools to investigate the phenomena of reflexology from EEG. The literature on the study of the application of the non-linear dynamics theory to analyze physiological signals, shows that non-linear approaches were used for analysis of heart rate, nerve activity, renal flow, arterial pressure, EEG and respiratory signals [Stein et al., 1992; Hoyer et al., 1997].
Despite the many applications of EEG in clinical neuro-physiology [Gevins et al., 1988; Niedermeyer et al., 1993; James et al., 2000; Xu-Sheng et al., 2001; Borel et al., 1985; Agarwal et al., 1998], its visual interpretation is very subjective and does not lend itself to statistical analysis. As a result a number of research groups have proposed methods to quantify the information content of the EEG. Among these are the Fourier transform, the wavelet transform, chaos, entropy, and the subband wavelet entropy [Goel et al., 1996; Niedermeyer et al., 1999; Geocadin et al., 2000; Bezerianos et al., 2001; Rosso et al., 2001]. In addition, the EEG signal modelling is important to achieve a better understanding of the physical mechanisms generating these signals and to identify the causes of EEG signal changes. Furthermore, pattern recognition and classification of EEG abnormalities can be achieved through the analysis of the estimated model parameters. Modelling can also be used for predicting the future neurological outcome and for data compression. Simulation based on EEG signal model can be used to better demonstrate the effectiveness of a certain quantitative analysis method or EEG feature extraction.
Al-Nashash et al have used the neural network method for modelling EEG signals [Al-Nashash et al., 2003]. And also, they have implemented the EEG modelling using adaptive Markov process amplitude [Al-Nashash et al., 2004]. Recently Paul et al., have studied the performance of influence of reflexology on the heart rate variability [Paul et al.,2004]. Similarly, Kannathal et al., have studied the effect of reflexology on the brain signals using nonlinear analysis method [Kannathal et al., 2004a; Kannathal et al., 2004b]. In this work, we have studied the performance of different modelling techniques for the EEG signals with and without reflexology.
MATERIALS
EEG signals are being recorded by using BIOPAC equipment with ACQKNOWLEDGE 3.7.2 as data acquisition software [Biopac, 2001]. Silver-Silver chloride electrodes were applied following the 10-20 system. EEGs are recorded from 30 subjects (15 boys and 15 girls) with age ranging from 20 ± 3 years for 20 minutes. The EEG is recorded at 500 samples/sec with a resolution of 12 bits/sample and for a duration of twenty minutes. After this, the same subject was given reflexological stimulation just below the toes of both feet by a mechanical reflexological device (Massager-scroller type) in the same relaxed sitting for another 20 minutes. And the EEG is recorded during the reflexological stimulation. Then the data is digitally filtered using 1-50 Hz band pass filter. Time gap of 30 minutes is given between any two successive recordings to ensure that a previous state stimulus doesnt affect the subsequent state EEG.
SPECTRAL ANALYSIS
The approaches for spectrum estimation may be generally categorized into one of the two classes. The first includes the classical or non-parametric methods that begin by estimating the autocorrelation sequence rx(k) from a given set of data. The Power spectrum is then estimated by Fourier transforming the estimated autocorrelation sequence. The second class includes the non-classical or parametric approaches, which are based on using a model for the process in order to estimate the power spectrum. Three types of modelling techniques are practically available to estimate the power spectrum.
Non-parametric method (FFT) The Welch method [Welch, 1967] is one of the most popular classical methods to estimate the power spectrum of any given time sequence. The sequences are allowed to overlap and a data window is applied to each sequence. This will produce set of modified periodograms that are to be averaged. The data sequences xi(n) can be represented as
Where iD is the starting point for the ith sequence. Finally we can form K data segments each of length 2M. The resulting modified periodogram is
Where U is the normalization factor for the power in the window function and is selected as
The Welch power spectrum is average of these modified periodograms that is
Parametric method The non parametric method we have seen so far is not optimal, because this method suffers from spectral leakage effects due to windowing. The Spectral leakage leads to masking of weak signal that is present in the data. A parametric (model based) power spectrum estimation method avoids the problem of spectral leakage and provides better frequency resolution than non-parametric or classical methods. The paramedic methods assume the signal to be a stationary random process. This process can be modeled as the output of a filter with white noise input. The filter parameters are obtained from the signal, therefore the name: Parametric Method. There are different ways to obtain the parameters; these methods are classified depending on presence of poles in the z-domain. If there are no poles, one speaks of an MA (Moving Average) Model. In the case of an MA model the filter used in the model is a simple Finite Impulse Response (FIR) characteristic. If there are poles present and all zeros are located in the origin it is an (Auto Recursive AR) Model. This results in an Infinite Impulse Response (IIR) filter characteristic where only the output is used for feedback. A model having poles and zeros freely distributed in the z-domain is called ARMA Model. An ARMA model uses an IIR filter with feedback paths for input and output. According to the Wold theorem [Wold, 1938] it is possible to translate one model into another without any loss. That means the models can be chosen according to simplicity of the algorithm or computational overhead. In the present study we have chosen Yule-Walker and Burgs Method, both PSD estimation models are using AR models.
Burg Method The Burg method for AR spectral estimation is based on minimizing the forward and backward prediction errors while satisfying the Levinson-Durbin recursion [Proakis et al., 1996; Marple, 1987]. In contrast to other AR estimation methods, the Burg method avoids calculating the autocorrelation function, and instead estimates the reflection coefficients directly. The primary advantages of the Burg method are resolving closely spaced sinusoids in signals with low noise levels, and estimating short data records, in which case the AR power spectral density estimates are very close to the true values. In addition, the Burg method ensures a stable AR model and is computationally efficient. The accuracy of the Burg method is lower for high-order models, long data records, and high signal-to-noise ratios (which can cause line splitting, or the generation of extraneous peaks in the spectrum estimate). The spectral density estimate computed by the Burg method is also susceptible to frequency shifts (relative to the true frequency) resulting from the initial phase of noisy sinusoidal signals. This effect is magnified by analyzing short data sequences. Burg Method differs to the Yule-Walker Method in the way the PSD,PBUxx is obtained, as shown in the following Equation:
Where the a)p(k) are again the estimates of the AR parameters obtained from the Levinson-Durbin recursion. And TP6 are reflection coefficients in an equivalent lattice structure, which are chosen to obtain the total least square error.
Least Square Method (LSM) In least squares (LS) estimation, the unknown values of the parameters, β0,β1……, in the regression function, ,are estimated by finding numerical values for the parameters that minimize the sum of the squared deviations between the observed responses and the functional portion of the model. Mathematically, the least (sum of) squares criterion that is minimized to obtain the parameter estimates is
As previously noted,β0,β1…… are treated as the variables in the optimization and the predictor variable values, x1，x2，……are treated as coefficients. To emphasize the fact that the estimates of the parameter values are not the same as the true values of the parameters, the estimates are denoted by ,…….For linear models, the least squares minimization is usually done analytically using calculus. For nonlinear models, on the other hand, the minimization must almost always be done using iterative numerical algorithms.To illustrate, considering the straight-line model:
For this model the least squares estimates of the parameters would be computed by minimizing
Doing this by
1. taking partial derivatives of with respect to β^ 0 and β^ 1
2. setting each partial derivative equal to zero, and
3. solving the resulting system of two equations with two unknowns yields the following estimators for the parameters:
These formulas are instructive because they show that the parameter estimators are functions of both the predictor and response variables and that the estimators are not independent of each other unless x=0. This is clear because the formula for the estimator of the intercept depends directly on the value of the estimator of the slope, except when the second term in the formula for β^ 0 drops out due to multiplication by zero. This means that if the estimate of the slope deviates a lot from the true slope, then the estimate of the intercept will tend to deviate a lot from its true value too. This lack of independence of the parameter estimators, or more specifically the correlation of the parameter estimators, becomes important when computing the uncertainties of predicted values from the model. Although the formulas discussed in this paragraph only apply to the straight-line model, the relationship between the parameter estimators is analogous for more complicated models, including both statistically linear and statistically nonlinear models.
Akaike Information Criterion For both Burg and least square method, the model order was chosen as the one that minimizes the Akaike information criterion (AIC) figure of merit [Akaike, 1969, 1974]:
Where N is the number of data samples and λ^ 2 is the estimated white noise variance. To reduce computational costs, we assumed as optimal the value of P that fulfilled the AIC criterion in the first two epochs.
One of the most important aspects of the use of AR method is the selection of the order p. Much work has been done by various researchers on this problem and many experimental results have been given in literature such as the papers presented by [Anita et al., 2002]. In this work the order of the AR model is taken as: p=20.
RESULTS
The EEG signal with and without reflexology is analyzed using FFT by Welch method and AR method by Burgs method and least square method.
Spectral Analysis of the EEG Signals The PSD was calculated for each of the signals using Welch, Burgs and least square method. The first plot in Figure 1(a) shows the PSD obtained by Welch in the Control data without reflexology. The second plot [Figure 1(b)] shows the PSD by Welch method for the same subject with reflexology. In the PSD plots, a square marks indicates the first, second and third prominent peaks. The first, second and thirds prominent frequencies are simply the first second and third maxima of the PSD function respectively. In the past, it shown that, the EEG signal variability decreases with the reflexology [Kannathal et al.,2004; Kannathal et al.,2004]. Hence, in figure 1(a) the peaks have higher value as compared to the figure 1(b). Figure 2 shows the PSDs for the same signals, but the PSD is obtained using a parametric method, namely Burgs method. In this method also, figure 2(b) has less predominant peaks as compared to the figure 2(a) due to the influence of reflexology. A similar pattern can be observed in Figure 3, which shows the PSDs obtained by least square method.
ANOVA Test To explore the relationship between the prominent frequencies and a specific data group the location and power for the first three prominent frequencies were evaluated for all signals. Table 1 shows the values of nine parameters with and without reflexology. The first parameter, P1, represents mean and variance of the power for the first prominent frequency within one data group. The second parameter, F1, represents mean and variance of the frequency location for the first prominent frequency within one data group. The third parameter, P1/F1, is obtained by dividing P1 by F1. The parameters P2 and F2 represent mean and variance of frequency and power for the second prominent frequency within one data group. P2/F2 is again the relative power for the second prominent frequency. The parameters P3, F3 and P3/F3 concern the third prominent frequency and they were estimated in the same way.
Figure 1 (a) PSD estimation using Welch Method without reflexology
Figure 1 (b) PSD estimation using Welch Method with reflexology
Figure 2 (a) PSD estimation using Burg?es method without reflexology
Figure 2 (b) PSD estimation using Burg?es method with reflexology
Figure 3 (a) PSD estimation using least square method without reflexology
Figure 3 (b) PSD estimation using least square method with reflexology
The last column in Table 1 states the p-value for the individual parameter. The P-value was calculated using the one way ANOVA algorithm [Sahai et al., 2000; Jobson, 1991]. The purpose of one-way ANOVA is to find out whether datas from several groups have a common mean. That is, to determine whether the groups are actually different in the measured characteristic.
ROC Test The values for the parameters F1, F2 and F3 in Table 1, Table 2 and Table 3 show that the frequency locations of the prominent frequencies have distinct mean values and low variances. This indicates that the freqeuency location can be used for classification. To support this statement a ROC (Receiver Operating Characteristic) test [Egan, 1975; Jalihal et al., 1994] was performed using the parameters F1, F2 and F3. ROC is a one class classification method, which classifies a target class within a dataset. The dataset consists of target (desired) values or vectors and outliers. Results show that (table 1,2 and 3) are peak frequencies are distinct for various techniques with and without reflexology. Figure 4 show the comparison of P1/F1 result for all the three methods using ROC test. In a ROC curve the true positive rate (Sensitivity) is plotted in function of the false positive rate (100-Specificity) for different cut-off points of a parameter. Each point on the ROC plot represents a sensitivity/specificity pair corresponding to a particular decision threshold. The area under the ROC curve is a measure of how well a parameter can distinguish between two diagnostic groups (diseased/normal). It can be observed that the ROC test shown in Figure 4, yields low values for the area under the curves for FFT and AR method by Burgs technique. Welchs method yields a spiky spectrum, where the first three spikes are mistaken to be the prominent frequencies. The parametric methods perform much better in the ROC test, as shown in Figures 4. The area under the ROC curve for least squares method is more than the other two methods and it performs better than the rest.
Figure 4 Comparison of P1/F1 ratio for various methods by ROC techniques with reflexology
CONCLUSION
The mental state of a person influences the EEG signals. It can be seen from the results that, with reflexology, the power spectral density decreases. Therefore, the variation in the EEG signals decreases, which indicates presence of more low frequency (alpha waves) signals. Different ways to represent the power distribution in the frequency domain where applied and their results are compared. The comparison based on the frequency location of the prominent frequencies shows that parametric methods of frequency estimation are superior to non-parametric methods. From the Figures 1,2 and 3, we can infer that, the peak power points can be clearly seen in least square AR method and is more superior as compared to the other techniques. This is shown by the ROC method also. Hence, we can use these modelling techniques to show the influence of reflexology on EEG signals. And among the techniques, we studied, least squares AR method shows better performance compared to the other methods used.
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1 Department of ECE, School of Engineering, Ngee Ann Polytechnic, Singapore 599489
2 Department of Electrical Engineering, NITC, Calicut, India
3 School of Mechanical and Aerospace Engineering, College of Engineering, Nanyang Technological University, 50, Nanyang Avenue, Singapore 639798
*Corresponding to mykng@ntu.edu.sg
(Editor Guo Hui-ling)ARTICLES(Kannathal N, Rajendra Ach)