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Independent Component Analysis & Blind Source Separation. Ata Kaban The University of Birmingham. 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

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Presentation Transcript
overview
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
a bit like
A bit like…

in the sense of having to find quantities that are not observable directly

signals joint density
Signals, joint density

Joint density

Signals

Amplitude

S1(t)

Amplitude

S2(t)

time

marginal densities

the ica model

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

slide7

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)

slide8
The coctail party problem

Called also 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 other.

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.

Determine the source signals, given only the mixtures

some further considerations
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 coctail party problem
  • Later it became evident that ICA has many other applications too. E.g. from electrical recordings of brain activity from different locations of the scalp (EEG signals) recover underlying components of brain activity
illustration of ica with 2 signals
Illustration of ICA with 2 signals

a1

s2

x2

a2

a1

s1

x1

Original s

Mixed signals

illustration of ica with 2 signals1
Illustration of ICA with 2 signals

a1

x2

a2

a1

x1

Mixed signals

Step2: Rotatation

Step1: Sphering

illustration of ica with 2 signals2
Illustration of ICA with 2 signals

a1

s2

x2

a2

a1

s1

x1

Original s

Mixed signals

Step2: Rotatation

Step1: Sphering

excluded case
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

computing the pre processing steps for ica
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

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 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.
application domains of ica
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)

image denoising
Image denoising

Noisy image

Original image

Wiener filtering

ICA filtering

clustering
Clustering

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

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