MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 953745 10.1155/2014/953745 953745 Research Article The Application of FastICA Combined with Related Function in Blind Signal Separation Li Dengao Zhao Junmin Liu Hongyan Hao Defeng Bhatnagar Vishal College of Information Engineering Taiyuan University of Technology Taiyuan 030024 China tyut.edu.cn 2014 342014 2014 20 07 2013 07 03 2014 3 4 2014 2014 Copyright © 2014 Dengao Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Blind source separation (BSS) has applications in the fields of data compression, feature recognition, speech, audio, and biosignal processing. Identification of ECG signal is one of the challenges in the biosignal processing. Proposed in this paper is a new method, which is the combination of related function relevance to estimated signal and negative entropy in fast independent component analysis (FastICA) as objective function, and the iterative formula is derived without any assumptions; then the independent components are found by maximizing the objective function. The improved algorithm shorthand for R-FastICA is applied to extract random mixed signals and ventricular late potential (VLP) signal from normal ECG signal; simultaneously the performance of R-FastICA algorithm is compared with traditional FastICA through simulation. Experimental results show that R-FastICA algorithm outperforms traditional FastICA with higher similarity coefficient and separation precision.

1. Introduction

Blind source separation (BSS)  has been applied successfully to extract mixed signals in different fields of data compression , feature recognition , speech, audio, and biosignal processing [4, 5] as a statistical signal method. Literature  showed that the compression ratio was higher through the ICA method than principal component analysis (PCA). The accuracy ratio of feature recognition was 92.1% based on the complex valued Independent Component Analysis in literature . Literature [4, 5] showed that FastICA was an efficient method in the extraction of speech, audio, and biosignal. Neither the source signals nor the structure of mixed matrix is known .

The detection and analysis of VLP generally appearing in the end of QRS wave and extending to ST segment with a series of high frequency and low-rising weak irregular electrical signal are a kind of effective means to predict unexplained asphyxia, sudden cardiac deaths, and so forth . At present, the analysis methods of VLP commonly have had a time domain method, frequency domain method, spectrum scale measurement analysis method, and so forth . Time domain method is not easy to improve detection rate of VLP; frequency domain analysis is limited by frequency resolution; spectrum scale measurement analysis can overcome some limitations in time domain and frequency domain analysis, but it is easily influenced by the selection of analysis time and positioning of QRS wave terminal in extracting judgmental standard parameters of VLP .

To overcome the abovementioned limitation and improve the detection accuracy, it is necessary to put forward a new detection technology. Independent component analysis (ICA) as a branch of BSS is widely applied to this problem in recent years . Traditional FastICA algorithm has obtained several effects in extracting electromyographic and atrial fibrillation signal. VLP compared with normal ECG signal waveform has a relative independence, and ICA algorithm can accurately distinguish relatively independent component from the ECG; it can be used to identify VLP. In addition, many authors have conducted specific research to ICA algorithm. Literature  proposed an adaptive ICA algorithm based on artificial neural network, which reduced the complexity of obtaining the learning matrix and independent component; literature  designed a fast search algorithm directly based on kurtosis as a measure of non-Gauss; literature  analyzed the principle and method of independent component in serial estimation ICA model; the concept of kernel ICA was proposed in literature , where data was analyzed through related analysis combined with mutual information theory; it had obtained better separation effect.

R-FastICA is proposed in this paper, which is the combination of related function and negative entropy as objective function, and the iterative formula is derived; then the independent components are found by maximizing the objective function. The extracted performance of R-FastICA algorithm is compared with traditional FastICA through simulation of random mixed signals and ECG signal with VLP. Experimental results show that R-FastICA algorithm outperforms traditional FastICA with higher similarity coefficient and separation precision.

2. Theory

ICA firstly proposed by Pierre Comon in 1994 is a method for finding the statistical independent components from multidimensional statistical data . The mathematical model without noises can be expressed as follows: (1) X ( t ) = A S ( t ) , where S ( t ) = [ s 1 ( t ) , s 2 ( t ) , , s N ( t ) ] T is a N column vector of source signals and X ( t ) = [ x 1 ( t ) , x 2 ( t ) , , x M ( t ) ] T is M column vector of observed signals. A is a M × N mixed matrix required that M N .

The goal is to extract independent source signals from mixed signals by finding separation matrix W through some assumptions and constraints under the premise of unknown source signal S ( t ) and mixed matrix A , which makes output Y ( t ) an estimation to source signal S ( t ) . That is to say, (2) Y ( t ) = W X ( t ) = W A S ( t ) S ( t ) , where Y ( t ) = [ y 1 ( t ) , y 2 ( t ) , , y N ( t ) ] T is the estimation of source signals.

Generally, the normal ECG and VLP signal can be thought of as statistical independence with each other; thus VLP signal will be extracted through FastICA algorithm .

The basic model is shown in Figure 1.

FastICA principle diagram.

3. Methodology 3.1. R-FastICA Algorithm

Traditional FastICA method is to estimate source signals based on negative entropy.

In order to improve separation precisely, negative entropy combined with related function as objective function is proposed in this paper. The updating formula of R-FastICA algorithm is vector gradient derived by the negative entropy combined with related function. The basic idea of R-FastICA algorithm requires that extracted signals are not only independent but also have high precision. Related function δ is introduced and defined by following formula: (3) δ = lg i = 1 N y i · y i T MSE , where MSE represents the mean square error between source signal and estimated signal, defined as  (4) MSE = i = 1 N ( s i - y i ) ( s i - y i ) T N .

The MSE in type (3) is substituted by type (4), simplified as (5) δ = N lg y · y T ( s - y ) ( s - y ) T .

Source signal is replaced by the average of estimated signal. It is defined by s = y * = ( 1 / p ) j = 1 p y ( t - j ) , where p is an arbitrary integer less than 100 . The type (5) can be simplified as (6) δ = N lg y · y T ( y * - y ) ( y * - y ) T .

BSS model is Y ( t ) = W X ( t ) and X * is the average of X . It can be known that (7) y * = w T X * , where w is a column vector and w T is a row vector.

Combining type (6) and type (7), it can be got that (8) δ = lg w T X · ( w T X ) T ( w T X * - w T X ) ( w T X * - w T X ) T = lg w T X ( X ) T w w T ( X * - X ) ( X * - X ) T w = lg U V = lg U - lg V , where (9) U = w T X ( X ) T w = [ w 1 w 2 w M ] [ x 1 x 2 x M ] [ x 1 x 2 x M ] [ w 1 w 2 w M ] = w 1 2 x 1 T x 1 + w 2 w 1 x 2 T x 1 + + w M 2 x M T x M = i = 1 M j = 1 M w i w j x i T x j . Similarly, (10) V = w T ( X * - X ) ( X * - X ) T w = i = 1 M j = 1 M w i w j ( x i * - x i ) T ( x j * - x j ) .

From the above, δ is a function based on w and X ; the vector gradient can be obtained by the bottom of related function: (11) F ( w , X ) w = ( δ ) w = 1 U U w - 1 V V w .

Vector gradient is defined as (12) g w = [ g w 1 , g w 2 , , g w M ] T .

The gradient of U can be calculated according to type (12): (13) U w = ( i = 1 M j = 1 M w i w j x i T x j ) w = 2 [ j = 1 M w j x 1 T x j , j = 1 M w j x 2 T x j , , j = 1 M w j x N T x j ] = [ x 1 T x 1 x 1 T x 2 x 1 T x M x 2 T x 1 x 2 T x 2 x 1 T x M x M T x 1 x M T x 2 x M T x M ] [ w 1 w 2 w M ] = 2 [ x 1 T x 2 T x M T ] [ x 1 x 2 x M ] [ w 1 w 2 w M ] = 2 X X T w . Similarly, (14) V w = w T ( X * - X ) ( X * - X ) T w w = 2 ( X * - X ) ( X * - X ) T w .

Type (15) is a gradient of related function to w , which can be calculated combining type (13) with type (14): (15) F ( w , X ) w = 2 { X X T w w T X X T - ( X * - X ) ( X * - X ) T w w T ( X * - X ) ( X * - X ) T w } .

In literature , the approximate calculation formula of negative entropy is (16) J ( y ) { E [ G ( y ) ] - E [ G ( v ) ] } 2 .

The objective function is composed by negative entropy and related function including the information between source signal and estimated signal in R-FastICA: (17) φ ( y ) = { E [ G ( y ) ] - [ G ( v ) ] } 2 · δ , where v and y are Gauss random variables with the same covariance (zero mean and unit variance) and G is a nonlinear function selected by distribution form of source signals.

The vector gradient of objective function φ ( y ) to w is as follows: (18) φ ( y ) w = { E [ G ( y ) ] - E [ G ( v ) ] } 2 F ( w , X ) w = 2 { E [ G ( y ) ] - E [ G ( v ) ] } w · F ( w , X ) + { E [ G ( y ) ] - E [ G ( v ) ] } 2 · F ( w , X ) w = 2 { E [ X g ( w T X ) ] - E [ G ( v ) ] } · lg w T X X T w w T ( X * - X ) ( X * - X ) T w + 2 { E [ G ( y ) ] - E [ G ( v ) ] } 2 · E [ X X T w w T X X T w - ( X * - X ) ( X * - X ) T w w T ( X * - X ) ( X * - X ) T w ] .

Type (18) is a new updating formula. The improved algorithm can ensure that the estimated signals are independent and the precision is higher due to the fact that related function δ is relevant to estimated signal.

3.2. Assessment Method

A familiar measure of separation performance is the similarity coefficient defined as  (19) ε i j = ε ( y i , s j ) = | i , j = 1 N y i ( t ) s j ( t ) | i = 1 N y i 2 ( t ) j = 1 N s j 2 ( t ) .

When y i = c s j , separation effect is ideal; when y i and s j are mutual independent, ε i j = 0 ; generally, similarity coefficient matrix is used to measure extracted performance.

The composite scattering plot  is a measure to describe corresponding relationship between the source signal and the extracted signal, where we can see that not only an extracted signal is the recovery of a source signal, but also the phase of source signal and extracted signal is the same or opposite.

4. Simulation

In order to indicate the performance of R-FastICA compared with traditional FastICA, the following simulations were conducted.

Taking random signals as an example in the first simulation, R-FastICA method was proved to be effective. In the second simulation, taking ECG signal with VLP as an example, the original ECG signal without noises was from MIT/BIH database and VLP signal was generated through stacking sine waves with different frequency and amplitude .

4.1. Source Separation of the Random Signals

In this simulation, the source signal s 1 ( t ) was sinusoidal signal and s 2 ( t ) was random noises, whose sampling number was 2000. Signals s 1 ( t ) and s 2 ( t ) were shown in Figure 2, as well as mixed signals x 1 ( t ) and x 2 ( t ) that were shown in Figure 3.

The source signals.

The mixed signals.

The mixed signals were extracted through R-FastICA and traditional FastICA in Figures 4 and 5, respectively.

Extracted signals with R-FastICA.

Extracted signals with traditional FastICA.

In the experiment of extracting random signals, the comparison of similarity coefficient matrix between RFastICA and FastICA algorithm was shown in Table 1.

The comparison of similarity coefficient matrix.

R-FastICA FastICA
Similarity coefficient matrix - 1.0000 0.0001 0.0003 - 1.0000 0.9996 0.0003 0.0005 0.9997

Extracted signal y 1 ( t ) was the estimation of source signal s 1 ( t ) and the phase was opposite in Figure 6 and the same in Figure 7. Extracted signal y 2 ( t ) was the estimation of source signal s 2 ( t ) and the phase was opposite in Figure 6 and the same in Figure 7.

The composite scattering plot with R-FastICA algorithm.

The composite scattering plot with FastICA algorithm.

From the above experiments, we could see that R-FastICA algorithm outperforms traditional FastICA with higher similarity coefficient and separation precision.

4.2. Source Separation of the VLP Signal

In this simulation, the source signal s 1 ( t ) was VLP signal generated by program and s 2 ( t ) was ECG signal from MIT/BIH database, whose sampling number was 1600. The source signals s 1 ( t ) and s 2 ( t ) , as well as mixed signals x 1 ( t ) and x 2 ( t ) , were shown in Figures 8 and 9, respectively.

The source signals.

The observed signals.

The mixed signals were extracted through R-FastICA and traditional FastICA in Figures 10 and 11, respectively.

Extracted signals with R-FastICA.

Extracted signals with FastICA.

In the experiment of extracting VLP signal, the comparison of similarity coefficient matrix between RFastICA and FastICA algorithm was shown in Table 2.

The comparison of similarity coefficient matrix.

R-FastICA FastICA
Similarity coefficient matrix 1.0000 0.0002 0.0003 0.9999 0.9995 - 0.0006 0.0004 - 0.9994

Extracted signal y 1 ( t ) was the estimation of source signal s 1 ( t ) and the phase was the same in Figure 12 and the opposite in Figure 13. Extracted signal y 2 ( t ) was the estimation of source signal s 2 ( t ) and the phase was the same in Figures 12 and 13.

The composite scattering plot with R-FastICA.

The composite scattering plot with FastICA.

From the above experiments, the performance of R-FastICA is superior to the FastICA obviously with higher similarity coefficient and high separation precision.

5. Conclusion

In this paper, R-FastICA algorithm and FastICA algorithm were adapted to extract random signals and to separate VLP signal from ECG signal. We believed that our study produced two important results. Firstly, we proposed a new method through the combination of related function and negative entropy and separated independent components by maximizing new objective function in the experiments. On the other hand, the experiments showed that R-FastICA method outperformed traditional FastICA method with higher similarity coefficient and high separation precision.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This project is supported by The General Object of National Natural Science Foundation (no. 61371062), Youth Science Foundation Project of National Natural Science Foundation (no. 61303207), Ministry of Education in 2012 Colleges and Universities by the Specialized Research Fund for the Doctoral Program of Joint Funding Subject (no. 20121402120020), Shanxi Province Science and Technology Development Project, Industrial Parts (no. 20120321024-01), Shanxi International Science and Technology Cooperation Project (no. 2012081031), Science and Technology Activities Project of Study Abroad Returnees in Shanxi Province in 2012 (Funded by Shanxi province human resources and social security hall), and Research Project supported by Shanxi scholarship council of China (no. 2013-032).

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