Principal Component Analysis in ECG Signal Processing by hcj

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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
• 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:
2.We recorded N sample in every beat.
3.The experiment lasted for M beats

• Exploit interbeat redundancy
• Exploit intrabeat 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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