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Principal Component Analysis in ECG Signal Processing

VIEWS: 3 PAGES: 20

									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

								
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