Blind Source Separation by Independent Components Analysis

1
Blind Source Separation by Independent Components
Analysis

• Professor Dr. Barrie W. Jervis
• School of Engineering
• Sheffield Hallam University
• England
• B.W.Jervis_at_shu.ac.uk

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The Problem

• Temporally independent unknown source signals are
  linearly mixed in an unknown system to produce a
  set of measured output signals.
• It is required to determine the source signals.

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• Methods of solving this problem are known as
  Blind Source Separation (BSS) techniques.
• In this presentation the method of Independent
  Components Analysis (ICA) will be described.
• The arrangement is illustrated in the next slide.

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Arrangement for BSS by ICA
y1g1(u1) y2g2(u2) yngn(un)
g(.)
s1 s2 sn
x1 x2 xn
u1 u2 un
A
W
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Neural Network Interpretation

• The si are the independent source signals,
• A is the linear mixing matrix,
• The xi are the measured signals,
• W ? A-1 is the estimated unmixing matrix,
• The ui are the estimated source signals or
  activations, i.e. ui ?? si,
• The gi(ui) are monotonic nonlinear functions
  (sigmoids, hyperbolic tangents),
• The yi are the network outputs.

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Principles of Neural Network Approach

• Use Information Theory to derive an algorithm
  which minimises the mutual information between
  the outputs yg(u).
• This minimises the mutual information between the
  source signal estimates, u, since g(u) introduces
  no dependencies.
• The different u are then temporally independent
  and are the estimated source signals.

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Cautions I

• The magnitudes and signs of the estimated source
  signals are unreliable since
• the magnitudes are not scaled
• the signs are undefined
• because magnitude and sign information is shared
  between the source signal vector and the unmixing
  matrix, W.
• The order of the outputs is permutated compared
  wiith the inputs

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Cautions II

• Similar overlapping source signals may not be
  properly extracted.
• If the number of output channels ? number of
  source signals, those source signals of lowest
  variance will not be extracted. This is a problem
  when these signals are important.

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Information Theory I

• If X is a vector of variables (messages) xi which
  occur with probabilities P(xi), then the average
  information content of a stream of N messages is

bits
and is known as the entropy of the random
variable, X.
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Information Theory II

• Note that the entropy is expressible in terms of
  probability.
• Given the probability density distribution (pdf)
  of X we can find the associated entropy.
• This link between entropy and pdf is of the
  greatest importance in ICA theory.

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Information Theory III

• The joint entropy between two random variables X
  and Y is given by

• For independent variables

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Information Theory IV

• The conditional entropy of Y given X measures the
  average uncertainty remaining about y when x is
  known, and is

• The mutual information between Y and X is

• In ICA, X represents the measured signals, which
  are applied to the nonlinear function g(u) to
  obtain the outputs Y.

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Bell and Sejnowskis ICA Theory (1995)

• Aim to maximise the amount of mutual information
  between the inputs X and the outputs Y of the
  neural network.

(Uncertainty about Y when X is unknown)

• Y is a function of W and g(u).
• Here we seek to determine the W which
  produces the ui ? si, assuming the correct g(u).

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Differentiating
(0, since it did not come through W from X.) So,
maximising this mutual information is equivalent
to maximising the joint output entropy,
which is seen to be equivalent to minimising the
mutual information between the outputs and hence
the ui, as desired.
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The Functions g(u)

• The outputs yi are amplitude bounded random
  variables, and so the marginal entropies H(yi)
  are maximum when the yi are uniformly distributed
  - a known statistical result.
• With the H(yi) maximised, I(Y,X) 0, and the yi
  uniformly distributed, the nonlinearity gi(ui)
  has the form of the cumulative distribution
  function of the probability density function of
  the si, - a proven result.

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Pause and review g(u) and W

• W has to be chosen to maximise the joint output
  entropy H(Y,X), which minimises the mutual
  information between the estimated source signals,
  ui.
• The g(u) should be the cumulative distribution
  functions of the source signals, si.
• Determining the g(u) is a major problem.

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One input and one output

• For a monotonic nonlinear function, g(x),

• Also

• Substituting

(independent of W)
(we only need to maximise this term)
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• A stochastic gradient ascent learning rule is
  adopted to maximise H(y) by assuming

• Further progress requires knowledge of g(u).
  Assume for now, after Bell and Sejnowski, that
  g(u) is sigmoidal, i.e.

• Also assume

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Learning Rule 1 input, 1 output
Hence, we find
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Learning Rule N inputs, N outputs

• Need

• Assuming g(u) is sigmoidal again, we obtain

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• The network is trained until the changes in the
  weights become acceptably small at each
  iteration.
• Thus the unmixing matrix W is found.

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The Natural Gradient

• The computation of the inverse matrix

is time-consuming, and may be avoided by
rescaling the entropy gradient by multiplying it
by

• Thus, for a sigmoidal g(u) we obtain

• This is the natural gradient, introduced by
  Amari (1998), and now widely adopted.

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The nonlinearity, g(u)

• We have already learnt that the g(u) should be
  the cumulative probability densities of the
  individual source distributions.
• So far the g(u) have been assumed to be
  sigmoidal, so what are the pdfs of the si?
• The corresponding pdfs of the si are
  super-Gaussian.

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Super- and sub-Gaussian pdfs
Gaussian
Super-Gaussian
Sub-Gaussian

• Note there are no mathematical definitions of
  super- and sub-Gaussians

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Super- and sub-Gaussians

• Super-Gaussians kurtosis (fourth order
  central moment, measures the flatness of the
  pdf) gt 0. infrequent signals of short duration,
  e.g. evoked brain signals.
• Sub-Gaussians kurtosis lt 0 signals
  mainly on, e.g. 50/60 Hz electrical mains
  supply, but also eye blinks.
  

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Kurtosis

• Kurtosis 4th order central moment

• and is seen to be calculated from the current
  estimates of the source signals.
• To separate the independent sources, information
  about their pdfs such as skewness (3rd. moment)
  and flatness (kurtosis) is required.
• First and 2nd. moments (mean and variance) are
  insufficient.

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A more generalised learning rule

• Girolami (1997) showed that tanh(ui) and
  -tanh(ui) could be used for super- and
  sub-Gaussians respectively.
• Cardoso and Laheld (1996) developed a stability
  analysis to determine whether the source signals
  were to be considered super- or sub-Gaussian.
• Lee, Girolami, and Sejnowski (1998) applied these
  findings to develop their extended infomax
  algorithm for super- and sub-Gaussians using a
  kurtosis-based switching rule.

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Extended Infomax Learning Rule

• With super-Gaussians modelled as

and sub-Gaussians as a Pearson mixture model
the new extended learning rule is
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Switching Decision
and the ki are the elements of the N-dimensional
diagonal matrix, K, and

• Modifications of the formula for ki exist, but
  in our experience the extended algorithm has been
  unsatisfactory.

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Reasons for unsatisfactory extended algorithm

• 1) Initial assumptions about super- and
  sub-Gaussian distributions may be too inaccurate.
• 2) The switching criterion may be inadequate.

Alternatives

• Postulate vague distributions for the source
  signals which are then developed iteratively
  during training.
• Use an alternative approach, e.g, statistically
  based, JADE (Cardoso).

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Summary so far

• We have seen how W may be obtained by training
  the network, and the extended algorithm for
  switching between super- and sub-Gaussians has
  been described.
• Alternative approaches have been mentioned.
• Next we consider how to obtain the source signals
  knowing W and the measured signals, x.

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Source signal determination

• The system is

Mixing matrix A
Unmixing matrix W
g(u)
si unknown
xi measured
ui?si estimated
yi

• Hence UW.x and xA.S where A?W-1, and U?S.
• The rows of U are the estimated source signals,
  known as activations (as functions of time).
• The rows of x are the time-varying measured
  signals.

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Source Signals
Channel number
Time, or sample number
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Expressions for the Activations

• We see that consecutive values of u are obtained
  by filtering consecutive columns of x by the same
  row of W.

• The ith row of u is the ith row of w by the
  columns of x.

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Procedure

• Record N time points from each of M sensors,
  where N ? 5M.
• Pre-process the data, e.g. filtering, trend
  removal.
• Sphere the data using Principal Components
  Analysis (PCA). This is not essential but speeds
  up the computation by first removing first and
  second order moments.
• Compute the ui ? si. Include desphering.
• Analyse the results.

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Optional Procedures I

• The contribution of each activation at a sensor
  may be found by back-projecting it to the
  sensor.

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Optional Procedures II

• A measured signal which is contaminated by
  artefacts or noise may be extracted by
  back-projecting all the signal activations to
  the measurement electrode, setting other
  activations to zero. (An artefact and noise
  removal method).

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Current Developments

• Overcomplete representations - more signal
  sources than sensors.
• Nonlinear mixing.
• Nonstationary sources.
• General formulation of g(u).

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Conclusions

• It has been shown how to extract temporally
  independent unknown source signals from their
  linear mixtures at the outputs of an unknown
  system using Independent Components Analysis.
• Some of the limitations of the method have been
  mentioned.
• Current developments have been highlighted.