Independent Component Analysis & Blind Source Separation

1. Independent Component Analysis & Blind Source Separation Bob Durrant School of Computer Science University of Birmingham (Slides: Dr Ata Kabán)

2. Overview • Today we learn about • The cocktail party problem - called also ‘blind source separation’ (BSS) • Independent Component Analysis (ICA) for solving BSS • Other applications of ICA / BSS • At an intuitive & introductory & practical level

3. A bit like… ...in the sense of having to find quantities that are not observable directly.

4. Signals, joint density Joint density Signals Amplitude S1(t) Amplitude S2(t) time marginal densities

5. Original signals (hidden sources) s1(t), s2(t), s3(t), s4(t), t=1:T

6. s3 s4 s1 s2 a13 a12 a11 a14 x1 x2 x3 x4 The ICA model xi(t) = ai1*s1(t) + ai2*s2(t) + ai3*s3(t) + ai4*s4(t) Here, i=1:4. In vector-matrix notation, and dropping index t, this is x = A * s

7. This is recorded by the microphones: a linear mixture of the sources xi(t) = ai1*s1(t) + ai2*s2(t) + ai3*s3(t) + ai4*s4(t)

8. The Cocktail Party Problem Determine the source signals, given only the mixtures Also known as the Blind Source Separation (BSS) problem. Ill-posed problem, unless assumptions are made! The most common assumption is that source signals are statistically independent. This means that knowing the value of one of them does not give any information about the others. The methods based on this assumption are called Independent Component Analysis methods. These are statistical techniques of decomposing a complex data set into independent parts. It can be shown that under some reasonable conditions, if the ICA assumption holds, then the source signals can be recovered up to permutation and scaling.

9. Recovered signals

10. Some further considerations • If we knew the mixing parameters aij then we would just need to solve a linear system of equations. • We know neither aij nor si. • ICA was initially developed to deal with problems closely related to the cocktail party problem • Later it became evident that ICA has many other applications too (e.g. recovering underlying components of brain activity from electrical recordings at different locations of the scalp (EEG signals)).

11. Illustration of ICA with 2 signals s2 x2 a1 a2 a2 a1 s1 x1 Original s Mixed signals Joint distribution of the uniformly distributed signals s1 and s2 Joint distribution of the observed mixtures x1 and x2.

12. Illustration of ICA with 2 signals x2 a1 a2 a2 a1 x1 Mixed signals Step2: Rotation Step1: Sphering

13. Illustration of ICA with 2 signals s2 x2 a1 a2 a2 a1 s1 x1 Original s Mixed signals Step2: Rotatation Step1: Sphering

14. Excluded case There is one case when rotation doesn’t matter. This case cannot be solved by basic ICA. Example of non-Gaussian density (-) vs.Gaussian (-.) Seek non-Gaussian sources for two reasons:* identifiability* interestingness: Gaussians are not interesting since the superposition of independent sources tends to be Gaussian …when both densities are Gaussian

15. 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.

16. 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 soughtΛis Lagrange multiplier to enforce that W is an orthogonal transform i.e. a rotation • Solve by fixed point iterations • The effect ofΛ 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.

17. 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)

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

19. Clustering In multi-variate data search for the direction along which the projection of the data is maximally non-Gaussian = has the most ‘structure’

20. Blind Separation of Information from Galaxy Spectra

21. Decomposition using Physical Models Decomposition using ICA

22. Summing Up • Assumption that the data consists of unknown components • Individual signals in a mix • topics in a text corpus • basis-galaxies • Trying to solve the inverse problem: • Observing the superposition only • Recover components • Components often give simpler, clearer view of the data

23. Related resources • http://www.cis.hut.fi/projects/ica/cocktail/cocktail_en.cgi • Demo and links to further info on ICA. • http://www.cis.hut.fi/projects/ica/fastica/code/dlcode.shtml • ICA software in MatLab. • http://www.cs.helsinki.fi/u/ahyvarin/papers/NN00new.pdf • Comprehensive tutorial paper, slightly more technical.