1. Independent Components Analysis

2. What is ICA?
“Independent component analysis (ICA) is a method for finding underlying factors or components from multivariate (multi-dimensional) statistical data. What distinguishes ICA from other methods is that it looks for components that are both statistically independent, and nonGaussian.”
A.Hyvarinen, A.Karhunen, E.Oja
‘Independent Component Analysis’

3. ICA
Blind Signal Separation (BSS) or Independent Component Analysis (ICA) is the identification & separation of mixtures of sources with little prior information.
Applications include:
Audio Processing
Medical data
Finance
Array processing (beamforming)
Coding
… and most applications where Factor Analysis and PCA is currently used.
While PCA seeks directions that represents data best in a Σ|x0 - x|2 sense, ICA seeks such directions that are most independent from each other.
Often used on Time Series separation of Multiple Targets

4. ICA estimation principles by A. Hyvarinen, A. Karhunen, E
ICA estimation principles by A.Hyvarinen, A.Karhunen, E.Oja ‘Independent Component Analysis’
Principle 1: “Nonlinear decorrelation. Find the matrix W so that for any i ≠ j , the components yi and yj are uncorrelated, and the transformed components g(yi) and h(yj) are uncorrelated, where g and h are some suitable nonlinear functions.”
Principle 2: “Maximum nongaussianity”. Find the local maxima of nongaussianity of a linear combination y=Wx under the constraint that the variance of x is constant.
Each local maximum gives one independent component.

5. ICA mathematical approach from A. Hyvarinen, A. Karhunen, E
ICA mathematical approach from A.Hyvarinen, A.Karhunen, E.Oja ‘Independent Component Analysis’
“Given a set of observations of random variables x1(t), x2(t)…xn(t), where t is the time or sample index, assume that they are generated as a linear mixture of independent components: y=Wx, where W is some unknown matrix. Independent component analysis now consists of estimating both the matrix W and the yi(t), when we only observe the xi(t).”

6. The simple “Cocktail Party” Problem
Mixing matrix A x1 s1
Observations
Sources x2 s2
x = As
n sources, m=n observations

7. Classical ICA (fast ICA) estimation
Observing signals
Original source signal
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8. Two Independent Sources
Motivation
Two Independent Sources
Mixture at two Mics
aIJ ... Depend on the distances of the microphones from the speakers

9. Get the Independent Signals out of the Mixture
Motivation
Get the Independent Signals out of the Mixture

10. ICA Model (Noise Free) Use statistical “latent variables“ system
Random variable sk instead of time signal
xj = aj1s1 + aj2s ajnsn, for all j
x = As
IC‘s s are latent variables & are unknown AND Mixing matrix A is also unknown
Task: estimate A and s using only the observeable random vector x
Lets assume that no. of IC‘s = no of observable mixtures
and A is square and invertible
So after estimating A, we can compute W=A-1 and hence
s = Wx = A-1x

11. Illustration 2 IC‘s with distribution:
Zero mean and variance equal to 1
Mixing matrix A is
The edges of the parallelogram are in the direction of the cols of A
So if we can Est joint pdf of x1 & x2 and then locating the edges, we can Est A.

12. Restrictions si are statistically independent
p(s1,s2) = p(s1)p(s2)
Nongaussian distributions
The joint density of unit variance s1 & s2 is symmetric. So it doesn‘t contain any information about the directions of the cols of the mixing matrix A. So A cann‘t be estimated.
If only one IC is gaussian, the estimation is still possible.

13. Ambiguities Can‘t determine the variances (energies) of the IC‘s
Both s & A are unknowns, any scalar multiple in one of the sources can always be cancelled by dividing the corresponding col of A by it.
Fix magnitudes of IC‘s assuming unit variance: E{si2} = 1
Only ambiguity of sign remains
Can‘t determine the order of the IC‘s
Terms can be freely changed, because both s and A are unknown. So we can call any IC as the first one.

14. ICA Principal (Non-Gaussian is Independent)
Key to estimating A is non-gaussianity
The distribution of a sum of independent random variables tends toward a Gaussian distribution. (By CLT)
f(s1) f(s2) f(x1) = f(s1 +s2)
Where w is one of the rows of matrix W.
y is a linear combination of si, with weights given by zi.
Since sum of two indep r.v. is more gaussian than individual r.v., so zTs is more gaussian than either of si. AND becomes least gaussian when its equal to one of si.
So we could take w as a vector which maximizes the non-gaussianity of wTx.
Such a w would correspond to a z with only one non zero comp. So we get back the si.

15. Measures of Non-Gaussianity
We need to have a quantitative measure of non-gaussianity for ICA Estimation.
Kurtotis : gauss=0 (sensitive to outliers)
Entropy : gauss=largest
Neg-entropy : gauss = 0 (difficult to estimate)
Approximations
where v is a standard gaussian random variable and :

16. Data Centering & Whitening
x = x‘ – E{x‘}
But this doesn‘t mean that ICA cannt estimate the mean, but it just simplifies the Alg.
IC‘s are also zero mean because of:
E{s} = WE{x}
After ICA, add W.E{x‘} to zero mean IC‘s
Whitening
We transform the x’s linearly so that the x~ are white. Its done by EVD.
x~ = (ED-1/2ET)x = ED-1/2ET Ax = A~s
where E{xx~} = EDET
So we have to Estimate Orthonormal Matrix A~
An orthonormal matrix has n(n-1)/2 degrees of freedom. So for large dim A we have to est only half as much parameters. This greatly simplifies ICA.
Reducing dim of data (choosing dominant Eig) while doing whitening also help.

17. Computing the pre-processing steps for ICA
0) Centring = make the signals centred in zero
xi  xi - E[xi] for each i
1) Sphering = make the signals uncorrelated. I.e. apply a transform V to x such that Cov(Vx)=I // where Cov(y)=E[yyT] denotes covariance matrix
V=E[xxT]-1/2 // can be done using ‘sqrtm’ function in MatLab
xVx // for all t (indexes t dropped here)
// bold lowercase refers to column vector; bold upper to matrix
Scope: to make the remaining computations simpler. It is known that independent variables must be uncorrelated – so this can be fulfilled before proceeding to the full ICA

18. Computing the rotation step
Aapo Hyvarinen (97)
Computing the rotation step
This is based on an the maximisation of an objective function G(.) which contains an approximate non-Gaussianity measure.
Fixed Point Algorithm
Input: X
Random init of W
Iterate until convergence:
Output: W, S
where g(.) is derivative of G(.), W is the rotation transform sought Λ is Lagrange multiplier to enforce that W is an orthogonal transform i.e. a rotation
Solve by fixed point iterations
The effect of Λ is an orthogonal de-correlation
The overall transform then to take X back to S is (WTV)
There are several g(.) options, each will work best in special cases. See FastICA sw / tut for details.

19. Application domains of ICA
Blind source separation (Bell&Sejnowski, Te won Lee, Girolami, Hyvarinen, etc.)
Image denoising (Hyvarinen)
Medical signal processing – fMRI, ECG, EEG (Mackeig)
Modelling of the hippocampus and visual cortex (Lorincz, Hyvarinen)
Feature extraction, face recognition (Marni Bartlett)
Compression, redundancy reduction
Watermarking (D Lowe)
Clustering (Girolami, Kolenda)
Time series analysis (Back, Valpola)
Topic extraction (Kolenda, Bingham, Kaban)
Scientific Data Mining (Kaban, etc)

20. Image denoising Noisy image Original image Wiener filtering
ICA filtering

21. Noisy ICA Model x = As + n x~= (E{xxT} - Σ)-1/2 x x~ = Bs + n~.
A ... mxn mixing matrix
s ... n-dimensional vector of IC‘s
n ... m-dimensional random noise vector
Same assumptions as for noise-free model, if we use measures of nongaussianity which are immune to gaussian noise.
So gaussian moments are used as contrast functions. i.e.
however, in pre-whitening the effect of noise must be taken in to account:
x~= (E{xxT} - Σ)-1/2 x
x~ = Bs + n~.

22. Exercise (part 1, Updated Nov 10)
How would you calculate efficiently the PCA of data where the dimensionality d is much larger than the number of vector observations n?
Download the Wisconsin Data from the UC Irvine repository, extract PCAs from the data, test scatter plots of original data and after projecting onto the principal components, plot Eigen values
(c) Tralvex Yeap. All Rights Reserved

23. Ex1. Part 2 to ninbbelt@gmail.com subject: Ex1 and last names
Given a high dimensional data, is there a way to know if all possible projections of the data are Gaussian? Explain
- What if there is some additive Gaussian noise?

24. Ex1. (cont.)
2. Use Fast ICA (easily found in google)
Choose your favorite two songs
Create 3 mixture matrices and mix them
Apply fastica to de-mix

25. Ex1 (cont.) Discuss the results
What happens when the mixing matrix is symmetric
Why did u get different results with different mixing matrices
Demonstrate that you got close to the original files
Try different nonlinearity of fastica, which one is best, can you see that from the data

26. References Feature extraction (Images, Video)
Aapo Hyvarinen: ICA (1999)
ICA demo step-by-step
Lots of links
object-based audio capture demos
Demo for BBS with „CoBliSS“ (wav-files)
Tomas Zeman‘s page on BSS research
Virtual Laboratories in Probability and Statistics