Principal Component Analysis in ECG Signal Processing by hcj


									Principal Component Analysis in
     ECG Signal Processing
         Cheng-Yi Chiang
• 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
• 5. Application 2: Atrial Fibrillation 心室纖震
• 6. Application 3: T-Wave Alternans 交錯性T-Wave

        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 }

           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


       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
  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
• 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

   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
  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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