ICA - Independent Component Analysis

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ICA - Independent Component Analysis
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ICA Concept
Original signals
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ICA Concept
Mixed signals
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ICA Concept
Estimates of original signal
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ICA Concept
The key to estimating the ICA model is
nongaussianity. From Central Limit Theorem
(a classical result in probability theory) A
sum of two independent random variables usually
has a distribution that is closer to gaussian
than any of the two original random variables.
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Preprocessing for ICA

• Suppose x is the mixed signal
• Preprocessing for ICA includes
• Centering Make x a zero-mean variable.
• 2. Whitening Transform x linearly so that it is
  white.
• ( components are uncorrelated and their
  variances
• equal unity )
• 3. Other application-dependent preprocessing

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Whitening Example
Two random sources A and B
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Whitening Example
Two linear mixtures of A and B
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Whitening Example
Whitened linear mixtures
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ICA Algorithm
The projection on both axis is quite Gaussian
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ICA Algorithm
ICA rotates the whitened matrix back to the
original space.
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Summary
ICA performs the rotation by minimizing the
Gaussianity of the data projected on both axes.
It recovers the original sources which are
statistically Independent.
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Applications of ICA
Separation of artifacts in MEG data
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Applications of ICA
Finding hidden factors in financial data
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Applications of ICA
Reducing noise in natural images
original image
corrupted with noise
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Applications of ICA
Reducing noise in natural images
wiener filtered image
recovered image applying sparse code shrinkage
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Applications of ICA
Telecommunications Separation of the user's own
signal from the interfering other users' signals
in CDMA (Code-Division Multiple Access) mobile
communications