USING FISHER’S LINEAR DISCRIMINANT TO CLASSIFY FEATURE VECTORS OF ECG SIGNALS Mirjam Jonkman, Aya Matsuyama, Mohamed Elgendi, Friso de Boer School of Engineering. Charles Darwin University Darwin, Austrailia. Phone: +61-(0)-8-89-46-6671 Fax: +61-(0)-8-89-46-6680. E-mail: [email protected] Abstract: In this paper a method is presented to classify normal and abnormal ECG signals. It employs a combination of wavelet decomposition, feature extraction and classification methodology using Fisher’s linear discriminant. It is shown that the method is effective for quantifying the classification of ECG abnormalities. The results indicate that successful classification relies on the first two wavelet approximations. Further decomposition leads to less accurate classification results. It has also been shown that selection of a suitable wavelet is critical. It could be concluded that the bior3.1 wavelet is not suitable for this method of classification of ECG signals. Keywords: ECG, feature vectors, wavelets, classification, Fisher’s linear discriminant 1 1. INTRODUCTION The Electrocardiogram (ECG) is one of the most effective diagnostic tools to detect cardiac diseases. Traditionally skilled physicians analyse ECG recordings in the time-domain. However, ECG recording in the frequency domain has also been studied for subtle pathological conditions which may not always be obvious in the original time domain [1-3]. Signal processing techniques for the information in the frequency domain include Fourier transforms and wavelet transforms. The latter overcomes the important limitation of Fourier transforms, which is uncertainty of the information in time after the transform [4]. The wavelet transform has been applied to the ECG for a wide range of purposes: feature extraction [5-8], feature detection [9-13], noise reduction [14], and data compression [15]. The combined technique of wavelet decomposition and feature extraction was previously applied to an ECG signal to separate normal beats and abnormal beats [7, 8]. However, the separation was not quantified. In this paper, we combine the previously described technique with a classification using Fisher’s linear discriminant, leading to a quantitative classification system. 2. ECG SIGNAL PROCESSING Figure 1 shows the ECG signal processing flow. The ECG signals were first decomposed with the wavelet transform, after which feature vectors were extracted. These feature vectors were used to classify the signal. The details of each stage are described next. 2 Original ECG signals Wavelet Transfer Decomposed ECG signals Beat Classification Feature Extraction Feature Vectors of the ECG signals Figure 1. ECG Signal Processing Flow 2.1 Wavelet Decomposition By applying the wavelet transform, ECG signals were decomposed to the approximate (low frequency components) and detail information (high frequency components) [16], refer to Figure 2. Original ECG signals Wavelet Transfer Approximate Information Detail Information (Low Frequency) (High Frequency) Figure 2. ECG Signal Decomposition All ECG signals were obtained from the Physionet Database [17]. Using the wavelet packet decomposition command ‘wpdec’ in Matlab [18], each ECG signal was decomposed to Level 4. To investigate the suitability of the type of wavelet for ECG signal analysis, several types 3 of wavelets were applied (db2, db4, bior1.3, bior3.1, bior5.5, bior6.8). The start of the QRS complex was defined as the beginning of each beat and normal beats which occurred immediately before or after abnormal beats were removed. 2.2 Feature Extraction Two features were extracted from the decomposed ECG signals, normalised energy and entropy. The purpose of feature extraction is to select and retain relevant information from the original signals. 2.2.1 Normalised Energy The normalised energy at decomposition level n for each beat was calculated as variance E ( j )n 1 N 2 (x i m ) N 1 i 1 (1) (j: beat number, N: number of samples in one beat, i: sample number, n: decomposition level, m: sample mean) This energy of each beat, was then normalised across the decomposition levels, which allows comparison between the decomposed signals in different levels. The normalised energy of the beat j at decomposition level n is defined as: E ( j )n E ( j )norm _ n E (j 2 )1 2 E ( j )2 ... E ( j )n 2 (2) (j: beat number, n: decomposition level) 2.2.2 Entropy In signal processing, the entropy can be viewed as a measure of uncertainty [19]. The classical log energy entropy was used in this study. The entropy of the beat j at decomposition level n was obtained as follows. N 2 Ent ( j )log_ n log(x i ) i 1 (3) (j: beat number, n: decomposition level, N: sample size, i: sample number) 2.2.3 Feature Vectors Each beat of the decomposed signals at each decomposition level 4 now has two features: normalised energy: E ratio ( j )n and entropy: Ent ratio ( j )n . These feature vectors of normal and abnormal beats compose different groups of vector points, known as clusters. We have obtained feature vectors for six different wavelets (db2, db4, bior1.3, bior3.1, bior5.5, bior6.8) and four decomposition levels. 3. CLASSIFICATION The purpose of classification is to assign an object to a certain class. Many classification methods have been described [20]. Here we use Fisher's linear discriminant [21]. Fisher’s linear discriminant is particularly useful for discriminating between two classes in a multidimensional space. Since it is based only on the first and second moments of each distribution, it is not a computationally intensive method. A limitation is that it assumes that the two classes are Gaussian with equal covariance [22]. If this is not the case the discriminant may not give the minimum classification error. 3.1 Fisher’s Linear Discriminant Fisher’s linear discriminant is a classification method that projects high-dimensional data onto a line and performs classification in a onedimensional space. The objective of the method is to reduce the dimensionality while preserving as much of the class discriminatory information as possible. Let the feature vectors be denoted by x . Fisher’s linear discriminant is defined as the linear function y w T x that maximizes the criterion function [23] ~ J (w) ~ | 1 2 |2 ~2 (4) ~2 S1 S 2 ~2 ~2 where (S 1 S 2 ) is the total within-class scatter of the projected samples ~ ~ and 1 2 is the difference of the projected means. The projection results in the optimum separation of the two classes in one dimension. 3.2 Training The training phase consisted of applying Fisher’s linear discriminant to a half of the feature vectors for a particular wavelet and decomposition level. It was necessary to take the logarithm of both coordinates of the feature vectors before applying Fisher’s linear discriminant in order to produce an approximately normal distribution. 5 The threshold, which would be used for classification, was defined as the average of the projected means of the classes. 3.3 Testing The testing phase consisted of applying the previously found linear discriminant to a new set of feature vectors. Different wavelets and decomposition levels were applied to a number of ECG recordings. A record was kept of all classified heartbeats and true positive, true negative, false positive and false negative rates were calculated for each patient and for each beat. 4. RESULTS The results indicate that the method described above does indeed result in a good separation of the classes in the testing phase. In fact, all abnormal beats were correctly classified (there were no false negatives). However a number of normal beats were classified incorrectly. This may be caused by a variance difference between the projections of the normal and abnormal beats. Figure 3. Wavelets used for analysis To evaluate the suitability of a variety of wavelets for classifications purposes, results were calculated for each beat (number of incorrectly classified beats) and each patient (number of incorrectly classified patients, one incorrectly classified beat of a patient will result in an incorrectly classified patient). The db2, db4, bior1.3, bior3.1, bior5.5, bior6.8 wavelets were used for the analysis since they are commonly used. Figure 3 shows the shape of the various wavelets involved. 6 Previous research [7, 8] indicates that successful separation of normal and abnormal beats can be developed using the various levels of approximation of the wavelet decomposition and approximation levels were therefore varied from level 1 to level 4. Table 1. False positive classification using the bior1.3 wavelet Wavelet approximation level – bior1.3 wavelet A1 False positive mis-classifications of patients [No] False positive mis-classifications of beats [%] A2 A3 A4 2 0 3 4 0.09% 0.00% 26.13% 22.22% Table 1 shows the result for 9 patients using the bio1.3 wavelet. At the A1 decomposition level two patients had normal beats incorrectly classified as abnormal. However, only 0.09% of the normal beats were incorrectly classified as abnormal beats. Furthermore, it can be seen that approximation levels A3 and A4 do not lead to good results any more as 3 patients for the A3 level and 4 patients in the A4 level were incorrectly classified and about a quarter of all beats were incorrectly classified (or about half of the beats of the incorrectly classified patients). This indicates that A3 and A4 do not have the enough relevant information of the abnormality to accurately classify. Table 2 presents the aggregated results for all wavelets. It shows that, on average, 9.3% of the patients are incorrectly classified using the A1 approximation and increasing to 42.6% using the A4 approximation. A similar trend exists for incorrectly classified beats, increasing from 2.0% to 38.9%. Clearly the A3 and A4 approximations of the signal do not contain enough information to successfully classify ECG signals. 7 Table 2. False positive classification of aggregated results Wavelet approximation level A1 A2 A3 A4 False positive mis-classifications of patients [%] 9.26% 7.41% 33.33% 42.59% False positive mis-classifications of beats [%] 2.02% 2.04% 23.18% 38.89% After further analysis of individual wavelet results, it became apparent that there was a significant difference in classification accuracy between the bior3.1 and the remaining wavelets. The results for the bior3.1 wavelet are shown in Table 3. Table 3. False positive classification using the bio3.1 wavelet Wavelet approximation level - bior3.1 wavelet A1 False positive mis-classifications of patients [No] False positive mis-classifications of beats [%] A2 A3 A4 3 4 4 3 12.03% 12.26% 33.59% 33.33% It can be concluded that the bior3.1 wavelet is unsuitable for ECG classification techniques. 5. CONCLUSIONS The combination of wavelet decomposition and feature extraction, using normalised energy and entropy with classification using Fisher's linear discriminant is an effective method for quantifying the classification of wavelet analysis of ECG abnormalities. In particular it has been shown that the classification relies on information present in level A1 and A2 of the wavelet decomposition and that A3 and A4 do not include this information any more. Furthermore, it has been shown that selection of a suitable wavelet is critical to the success of classification. It was shown that bior3.1 is not suitable for this method of classification of ECG signals. 8 REFERENCES [1] J. Millet-Roig, J. J. Lopez-Soriano, A. Mocholf, R. Ruiz-Granell, and F. J. Chorro, "Study of frequency and time domain parameters extracted by means of wavelet transform applied to ECG to distinguish between VF and other arrhythmias," presented at Computers in Cardiology 1998, 1998. [2] A. Spaargaren and M. J. English, "Analysis of the signal averaged ECG in the time-frequency domain," presented at Computers in Cardiology 1999, 1999. [3] G. Sierra, P. Morel, P. Le Guyader, F. Trellez, R. Nadeau, and P. Savard, "Frequency analysis of the signal-averaged ECG of postinfarction patients for prediction of cardiac death," presented at Engineering in Medicine and Biology society, 1997. Proceedings of the 19th Annual International Conference of the IEEE, 1997. [4] C. Valens, "A Really Friendly Guide to Wavelets," vol. 2003, 1999. [5] G. K. Prasad and J. S. Sahambi, "Classification of ECG arrhythmias using multiresolution analysis and neural networks," presented at TENCON 2003. Conference on Convergent Technologies for Asia-Pacific Region, 2003. [6] B. Castro, D. Kogan, and A. B. Geva, "ECG feature extraction using optimal mother wavelet," presented at Electrical and Electronics Engineers in Israel, 2000. The 21st IEEE Conference, Tel-Aviv, Israel, 2000. [7] A. Matsuyama and M. Jonkman, "The Application of Wavelet and Feature Vectors to ECG Signals," presented at IEEE, Tencon 2005, Melboune, Australia, 2005. [8] A. Matsuyama and M. Jonkman, "The Application of wavelet and feature vectors to ECG signals," Australian Physical & Engineering Sciences in Medicine, vol. 29, pp. 14-17, 2006. [9] S. M. Szilagyi, L. Szilagyi, and L. David, "Comparison between neural-networkbased adaptive filtering and wavelet transform for ECG characteristic points detection," presented at Engineering in Medicine and Biology society, 1997. Proceedings of the 19th Annual International Conference of the IEEE, 1997. [10] T. Stamkopoulos, K. Diamantaras, N. Maglaveras, and M. Strintzis, "ECG analysis using nonlinear PCA neural networks for ischemia detection," Signal Processing, IEEE Transactions on [see also Acoustics, Speech, and Signal Processing, IEEE Transactions on], vol. 46, pp. 3058-3067, 1998. [11] D. Bojanic and D. B. Popovic, "QRS detection from an ongoing ECG recordings by using dyadic wavelets," presented at EMBEC 02, Vienna, 2002. [12] F. Schlegelmilch, M. Helbig, V. Natchkova, G. Ivanova, K. Schellhorn, V. Detschew, D. Mandler, and G. GrieBbach, "A Multi-Channel Approach for Online-Identification of QRS-Regions Using Adaptive Recursive Estimators," presented at EMBEC 02, Vienna, 2002. [13] A. Mousa and A. Yimaz, "Neural network detection of ventricular late potentials in ECG signals using wavelet transform extracted parameters," presented at Engineering in Medicine and Biology Society, 2001. Proceedings of the 23rd Annual International Conference of the IEEE, 2001. [14] L. Senhadji, J. J. Bellanger, G. Carrault, and J. L. Coatrieux, "Wavelet Analysis of E.C.G. Signals," presented at the Twelfth Annual International Conference of IEEE, Philadelphia, 1990. [15] I. Provaznik and J. Kozumplik, "Wavelet Transform in Electrocardiography - Data Compression," International Journal of Medical Informatics, vol. 45, pp. 111-128, 1997. 9 [16] P. O. Ogunbona, M. Milliss, F. De Boer, and M. Fernandes, "Classification of Gas Metal Arc Welds Using Wavelets," presented at Engineering Mathematics and Applications Conference, Adelaide, Australia, 1998. [17] "PhysioBank," vol. 2004: Physionet. [18] M. Misiti, Y. Misiti, G. Oppenheim, and J.-M. Poggi, Wavelet Toolbox User's Guide: Mathwork, 2000. [19] S. Tong, A. Beserianos, J. Paul, Y. Zhu, and N. Thakor, "Nonextensive entropy measure of EEG following brain injury from cardiac arrest," Physica A, pp. 619628, 2002. [20] Anil K. Jain, Robert P.W Duin, and Jianchang Mao. Statistical Pattern Recognition: A Review. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 22 No. 1, 2000. [21] Fisher, R.A. The Use of Multiple Measurements in Taxonomic Problems. Annals of Eugenics, 7: 179-188 (1936) [22] T. Cooke and M. Peake, “The Optimal Classification Using a Linear Discriminant for Two Point Classes having Known Mean and Covariance,” IEEE Transactions On Pattern Analysis And Machine Intelligence, Vol. 24, No. 2 2002 [23] R.O. Duda, P.E. Hart and D.G. Stork, Pattern Classification, New York: John Wiley & Sons, 2001. 10
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