A primer on modeling EEG activity using timefrequency transformations

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A primer on modeling EEG activity using
time-frequency transformations

• Petros Xanthopoulos

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Outline of the presentation

• EEGs and frequency
• Frequencies- The EEG alphabet
• Examples of EEG waves
• Fourier transform
• Example 1Stationary signal
• Example 2Non Stationary signal
• STFT transform
• Demo
• Wavelet transform
• Cohen class transformations
• Wigner ville energy transformation
• How to select features
• Further work - collaboration
• Where to start

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Frequencies- The EEG alphabet

• Activity in certain EEG bands are associated with
  specific pathologies-activity (e.g Delta waves
  are often associate d with some ciate d with some
  young encephalopathies, beta waves are associated
  with active concentration)
• EEG waves are indicative of neural network
  oscillations
• Energy of frequency bands can be used as features
  for classifiers.

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Examples of EEG waves
One second of EEG activity
Delta waves (0.1-3Hz)
Beta waves (13-30Hz)
Theta waves (3-8Hz)
Gamma waves (30- 45Hz)
Alpha waves (8-13Hz)
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Fourier transform (FT)

• Fourier transform expands a time signal x(t) into
  a sum (or integral for continuous time) of
  infinite waves
• It can be thought of a transform from
  time-domain to frequency domain. FT is
  formally defined by
• 
  
• Used in many areas because the concept of
  frequency and periodic signals is shared in many
  science domains.
• When we talk about Power spectrum of a signal
  we mean the amplitude of the Fourier transform

Fourier transform FT
Inverse Fourier transform IFT
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Example 1 Stationary signal
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Example 1 Stationary signal
Signal in time domain
Power spectrum
By looking at the Power spectrum of the signal we
can recognize three frequency Components (at
2,10,20Hz respectively).
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Example 2 non stationary signals
Consider two linearly modulated sinusoids
(chirps). The first with increasing frequency
and the second with decreasing.
In this case we have two nonstationary signals in
time with identical FTs. Confusion Arise and
Power spectrum cannot is not very useful.
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Time frequency transformation

• Since FT discards time information its useful
  only for stationary signals (signals of constant
  frequency). For non stationary signal analysis we
  use time frequency transformations
• The aim is to determine the function
  based only on the initial signal information

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Short Time Fourier Transform (STFT)

• The first step towards TF representation was done
  in 1946 by D.Gabor. Formally defined by
• Simple idea Perform Fourier transform by using a
  time moving window (Not necessarily rectangular-
  might also be Gaussian or Hann window).
• Drawbacks
• If time window is small (better time resolution)
  frequency resolution gets worse
• If time window is large enough (better frequency
  resolution) time resolution gets worse.
• Unavoidable due to Heisenbergs principle of
  uncertainty.

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Example -Demo
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Wavelet transform

• Wavelet transform is the next logical step. We
  make use of varying length window. WT is a time
  to time-scale transformation. We can move to time
  frequency domain using

? is the mother wavelet function.
Fourier transform
Wavelet transform
Analogy with FT and STFT
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Some mother wavelet function examples
There are literately infinite mother wavelet
functions satisfying Different properties. The
challenge Is to find the right one for a given
application
Meyer
Mortlet
Mexican Hat
biorthogonal
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Cohen class transformations

• In contrast with the linear TF transformations
  (which decompose the signal into atoms) the
  purpose of the energy distributions is to
  distribute the energy of the signal over the two
  description variables (time and frequency).
• Where is the energy density
  function of signal x. In addition the following
  two marginal properties should be satisfied
• In addition if the time and covariance property
  is satisfied then the transformation belongs to
  the Cohens class transformation.

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The Wigner-Ville distribution

• It belongs to Cohens class. It is defined as
• Or equivalently
• This distribution satisfies the Cohens class
  properties plus a large number of other desirable
  properties (see also Time frequency toolbox
  tutorial).

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Example chirp signal (WV vs STFT)
Wigner-Ville distribution
STFT
signal
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Example chirp signal (WV vs WT)
Wavelet transform (Mexican hat)
Wigner-Ville distribution
signal
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How to select features

• TF plots are representations of higher dimension
  that help us define better features for our
  classifiers.
• Highly application dependent (e.g. might be
  interested in specific frequency band).
• Some potential features
• average frequency at each time point,
• maximum frequency at each time point
• The maximum amount of energy for a specific time
  point.
• Area of connected frequency components
  (application in epileptic seizures)

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A feature selection example (thanks to Onur)
Smoothed pseudo Wigner-Ville time frequency
representation
Feature No 1 Maximum energy at every time point
Feature No 2 Frequency that appears maximum
energy at each time point
Time domain signal
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Further work - collaboration

• Anyone who attempts classification on EEG data or
  intracranial data.
• Might be interesting to find other features
  beyond instantaneous frequency or maximum
  frequency (MD doctor expertise could help).

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Where to start

• Fourier transform, STFT (Implemented in Matlab).
  To get an idea here are two links from wikipedia
• FT http//en.wikipedia.org/wiki/Fourier_transform
  
• STFT http//en.wikipedia.org/wiki/Short-time_Four
  ier_transform
• Wavelets
• Matlab wavelet toolbox documentation. Also
  available online http//www.mathworks.com/access/h
  elpdesk/help/toolbox/wavelet/
• A practical Guide to wavelet analysis
  (http//atoc.colorado.edu/research/wavelets/
  toolbox guide)
• Cohen class transformations
• Time frequency toolbox (http//tftb.nongnu.org/ -
  Very good introduction to energy
  transformations).
• Includes implementations of various TF
  transformations.
• Cohen, L. Time-frequency distributions-a review
  Proceeding of the IEEE Jul 1989 Volume 77, 
  Issue 7 pp 941-981 (http//ieeexplore.ieee.org/xp
  l/freeabs_all.jsp?isnumber1321arnumber30749typ
  eref )

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Je finis par trouver sacre le desorde de mon
esprit Arthur Rimbaud Une saison en Enfer
At last, I began to consider my mind's disorder
a sacred thing Arthur Rimbaud One season in
hell