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Principal Component Analysis in ECG Signal Processing Cheng-Yi Chiang 11/4/2009 Outline • 1. Introduction to PCA • 2. PCA on ECG signal – Interbeat – Intrabeat – Interlead • 3. Adaptive coefficient estimation • 4. Application 1: Myocardial Ischemia 心肌衰弱造成的 局部缺血 • 5. Application 2: Atrial Fibrillation 心室纖震 • 6. Application 3: T-Wave Alternans 交錯性T-Wave 2 Linear Regression Model We got some 2D data… In linear regression model, we want to minimize the sum of d. How about minimizing the sum of d’ or d’2? Question 1 • We got some M dimensional data, x1 , x 2 ..., xn • Question 1: How to find a vector x 0 , such that square error criterion is minimized? Solution to Question 1 • Intuitively, use the mean of all the vectors. Question 2 • Question II: How to find an unit vector e and a set of coefficients {ak }, k 1,2,..., n such that and minimize Solution to Question 2 (I) • Assume we know the unit vector e Partial differentiate J 1 w.r.t ak and set it to be 0. Solution to Question 2 (II) • Assume we know {ak } n S (x k m)(x k m) t k 1 Solution to Question 2 (III) • How to minimize subjected to ? • Use Lagrange multiplier Set to 0 Then we get , e is the eigenvector of S What is ? (I) • Different choice of e corresponds to different value of et Se et e , and thus different J Variance in M dim n n J1 (a1 ,..., an , e ) x k ' x k xk m 2 2 k 1 k 1 Variance in M-1 dim Variance in 1 dim along e From this equation we can see that J is the variance in the M-1 dimension and should be the variance in the dimension along e . What is ? (II) Covariance Matrix Find a set of orthonormal eigenvectors satisfy where Is the Covariance of the Data in the Projected Space Transform X into Y The covariance matrix of Y is The correlation between any two dimension is 0 ! Two Examples Well separated in the projected space Not well-separated in the projected space Notes on PCA (I) • 1. What is the physical meaning of the dimensions of the data. • 2. PCA tends to find a transformation that reduce the correlation between each dimension, and thus exploit the redundancy. • 3. The variance each projected dimension is in represented by • 4. Sometimes it is sufficient to use K<N components to reconstruct / represent the original data. It depends on the fall-off shape of Notes on PCA(II) • All the above theory is applicable to stochastic process. • We have some sample x1 , x2 ..., xn from an zero random process x. • Covariance matrix E{( x m)( x m)t } is estimated by the sample covariance matrix , where PCA on ECG Signal Scenario: 1.We have L leads. 2.We recorded N sample in every beat. 3.The experiment lasted for M beats • Exploit interbeat redundancy • Exploit intrabeat redundancy • Exploit interlead redundancy Interbeat PCA Consider data from one lead. The N dimension is the N samples in one beat Regard M beats as M samples of the process x Estimate the covariance matrix Solve the eigenfunction Then we get the coefficient w after transformation The effective dimension of w can be reduced to k, which depends on Interbeat PCA Example Component in each beat Reduce N-dimension into 2-dimension Variation between beats Intrabeat PCA M dimension corresponds to M beats nth column contains the samples of M beats in time n of each beat Intrabeat PCA Example