4.2 Proposed Multiscale Principal Component Analysis
4.2.2 Multiscale Correlations
It is expected to get higher correlations and more redundancies if multichannel ECG signals are wavelet transformed to multiscale levels. Redundancies can be evaluated using covariance or corre- lation analysis of matrices. In Fig.4.5, correlations between leads with scatter plots are shown atD6 wavelet matrix. It is seen that there are higher mean for correlation values than that found in original data set.
Table 4.1: Mean and variance analysis of multiscale correlation matrices, CSE database, Dataset-M01-033
Matric Original data set A6 D6 D5 D4 D3 D2 D1
Mean 0.6296 0.6361 0.6479 0.8020 0.8160 0.5344 0.3959 0.3684 Variance 0.0982 0.0850 0.0911 0.0352 0.0299 0.0933 0.0937 0.0951
In Table 4.1, mean and variance values for correlation coefficients are shown for multiscale matri- ces resulted from 6 level wavelet decomposition. The mean of correlation coefficients of multichannel data set is found as 0.6296 whereas for multiscale correlation matrices, the mean values are higher for higher order scales. ForA6,D6,D5 and D4 matrices the mean values of correlation coefficients are 0.6361, 0.6479, 0.8020 and 0.8160 respectively. For lower order matrices the mean values are lower. If mean values and variances of multiscale matrices are observed, it is seen that the correla- tions are higher and it suggests that the redundancies may be higher. As a result, it will help efficient implementation of PCA.
In wavelet domain, covariances quantify the correlations between coefficients of different leads for a subband level. Covariance matrix gives all possible correlations between pairs of leads. In Table 4.2, mean values of covariances of different matrices are shown. The mean values are evaluated taking averages of mean of covariances from10 data sets from CSE multilead measurement library [2], with six level wavelet decomposition. Higher mean values inD5,D6 andA6 indicate presence of
−5 0 5
−5 0 5
V5
−5 0 5
V4
−5 0 5
V3
−5 0 5
V2
−2 0 2
V1
−5 0 5
aVF
−20 0 20
aVL
−5 0 5
aVR
−5 0 5
III
−20 0 20
I I
−20 0 20
−5 0 5
Lead−I
V6
−5 0 5
V5
−5 0 5
V4
−5 0 5
V3
−5 0 5
V2
−2 0 2
V1
−5 0 5
aVF
−20 0 20
aVL
−5 0 5
aVR
−5 0 5
III
−20 0 20
I I
Correlations between leads at D
6
Wavelet scale
−20 0 20
V6
Lead−I
0.8896 0.6928
−0.3596 −0.9818 0.9862 0.0091 −0.1494 0.7103 0.7087 0.6714 0.6274 0.3833
0.1760 −0.5092 0.8009 0.4420 −0.5950 −0.8104 −0.8522
0.0678 −0.6284 −0.5829 −0.5351 −0.4820 −0.2166
0.7606 0.7974 0.7705 0.7357 0.5175
0.7560 −0.1271 −0.4470 −0.5124 −0.5706 −0.7037
0.2292 −0.1160 −0.2118 −0.2332
0.9038
0.9960 0.9852
0.9965 0.9379 0.9135 −0.7082 −0.8197 0.9683 −0.3425 −0.3060 0.7968 0.8796 0.8606
−0.9238
−0.8848
−0.1723
−0.9368
−0.2165
−0.1340
−0.1690
0.8734
0.9320 0.7642
0.9102
0.9585
Scatter plots between leads at D
6
Wavelet scale
Figure 4.5: Scatter plots and correlation coefficients between leads atD6wavelet scale. Signals in different ECG leads are scatter plotted. The diagonal plots show the distribution of individual lead. Database used is CSE multilead measurement library, data set M01-033.
.
higher redundancies in these matrices. This may be due to the presence of components of MECG, such as P-waves, T-waves and part of QRS complexes. QRS-complex constitutes with Q-wave, R- wave and S-wave. The duration of Q-wave is expected to be<40 ms. The amplitude may be 25% of the amplitude of the R-wave. For a six level wavelet decomposition, the constituents of QRS-complex may spread overD4,D5 andD6 [159]. Other lower order subband matrices,D1,D2 andD3, show low mean values of covariances. These subbands fall normally outside the normal ECG spectrum for
4.2 Proposed Multiscale Principal Component Analysis
Table 4.2: Mean value of covariances at multiscale matrices of data set M01-001 to M01-010, from CSE multilead measurement library.
Data set D1 D2 D3 D4 D5 D6 A6
M01-001 0.0001 0.0004 0.002 0.031 0.366 0.844 1.563 M01-002 0.0001 0.0015 0.018 0.181 0.576 0.976 1.471 M01-003 0.0002 0.0013 0.012 0.073 0.224 0.260 1.036 M01-004 0.0001 0.0003 0.003 0.061 0.258 0.430 0.316 M01-005 0.0001 0.0004 0.008 0.071 0.257 0.320 0.335 M01-006 0.0001 0.0003 0.003 0.053 0.114 0.157 0.667 M01-007 0.0000 0.0004 0.007 0.131 0.326 0.566 0.420 M01-008 0.0000 0.0003 0.006 0.093 0.274 0.259 0.545 M01-009 0.0002 0.0005 0.004 0.034 0.116 0.299 0.555 M01-010 0.0000 0.0001 0.001 0.006 0.199 0.829 2.068 Average 0.0001 0.0005 0.006 0.073 0.271 0.494 0.898
a six level wavelet decomposition. Also, uncorrelated noise dominates these subbands.
Based on above investigations of correlations between leads and relative subband energies at wavelet scales for multichannel ECG signals, we propose multiscale principal component analysis as a feasible method. Fig.4.6 shows the block diagram of the proposed multiscale PCA for multichannel ECG signals. Signals of different leads are subjected to amplitude normalization and mean removal.
In wavelet transform block signals are wavelet transformed using 9/7 biorthogonal wavelet filters. The coefficients are arranged in different subbands and multiscale matrices are formed at block shown as multiscale subband matrices. These matrices represent the transformed data at different wavelet scales. For a six level wavelet decomposition there is a single matrix at approximation level asA6and there are six matrices at detail scales asD6,D5,D4,D3,D2,D2andD1. PCA is applied on multiscale matrices at multiscale principal analysis block with proper selection of eigenvalues. The output of this block is PCA transformed data at different scales. Next to this, subband signals are reconstructed to form reconstructed subband matrices. At inverse wavelet transform block reconstruction of ECG signals are performed from recovered wavelet coefficients. The block shown in dotted line is optional.
It can be used to perform quality control MSPCA.
At each wavelet scale, PCA based processing of subband data may be developed using wavelet subband matrices, AL and Dj. Due to Multiresolution Decomposition (MRD), diagnostically impor- tant PQRST morphologies appear at different scales. To get advantage of multivariate nature and
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Figure 4.6: Block diagram of the proposed MSPCA. The block and lines shown with dotted line is optional and not implemented in this work. These may be incorporated to get quality control MSPCA for data reduction.
.
multiscale nature of ECG signal, it is necessary to develop a wavelet weighted multiscale PCA. This may lead to preserve the diagnostic components at selective subbands. Covariance gives the spare- ness of the wavelet coefficients at a particular scale. Evaluation of covariance matrices from mean removed data help find the correlations of data between dimensions (channels) of data. For ap- proximation subband and details subband matrices, the covariance matrices from mean removed multiscale multivariate matrices are evaluated as
CA
L = 1
(n−1)([AL][AL]T) (4.6)
CD
j = 1
(n−1)([Dj][Dj]T) (4.7)
4.2 Proposed Multiscale Principal Component Analysis
wherenis the number of ECG channels.
These square symmetric covariance matrices exploit all the possible correlations at different wavelet scales between coefficients of all channels. The diagonal terms are variances of coeffi- cients and off diagonal terms are correlation between coefficients of all possible pairs of channels at a particular wavelet scale. The main objective is to minimize redundancy measured by covariance and to maximize the signal measured by variance [161]. Higher covariance values indicate higher redundancies. To solve this, eigen-decomposition of above covariance matrices can be performed. If covariance matrix is diagonalized, eigenvectors and eigenvalues at different scales can be obtained.
The eigenvalue decomposition of these covariance matrices results CA
LVA
L =VA
LΛA
L (4.8)
CD
jVD
j =VD
jΛD
j (4.9)
where VA
L, VD
j and ΛA
L, ΛD
j are the eignevectors and eigenvalues for matrices of approxima- tion and details subband matrices respectively. Eigenvector matrices VA
L andVD
j diagonalize the covariance matrixCA
L and matricesCD
j as VA
LCA
LV−1
AL =ΛA
L (4.10)
VD
jCD
jV−1
Dj =ΛD
j (4.11)
ΛA
L and ΛD
j are the diagonal matrices with eigenvalues as diagonal elements. Eigenvectors and eigenvalues appear in pairs. Eigenvalues are arranged in descending order and accordingly the corresponding eigenvectors. The eigenvectors with corresponding higher eigenvalues produce the principal components. Thus these orthonormal eigenvectors represent the signals in the direction of maximum variances. The reduction of dimension depends on the number of eigenvalues selected.
Ordered eigenvalues in approximation and details subband matrices are
λAL1, λAL2,· · ·, λALn (4.12) λDj
1, λDj
2,· · · , λDjn (4.13)
The eigenvectors selected based on significant eigenvalues for multivariate data set will help form
feature vector’s matricesFALandFDj for approximation and details scales respectively. FALandFDj
are constructed by selected significant eigenvectors from all eigenvectors and forming a matrix with eigenvectors in the column.
The feature vector matrices play important role for data reduction. In conventional PCA, eigenval- ues are arranged in descending order and accordingly the corresponding eigenvectors. Similarly. in MSPCA the new data set can be derived out of feature vectors and original mean removed data sets as
RA
L =AL×FAL (4.14)
RD
j =Dj×FDj (4.15)
whereRA
L is dimension reduced data set for approximation andRD
j is dimension reduced data set for details subbands. To get the reconstructed subband matrices from the dimension reduced data set, following matrix operation is performed as
d AL=RA
L×FATL (4.16)
c Dj =RD
j×FDTj (4.17)
wheredAL is the reconstructed approximation subband matrix andDcj are the reconstructed de- tails subband matrices. To get back the reconstructed signal matrix,S, inverse wavelet transform is performed.