Patrick McSharry

1
Detection of dynamical transitions in biomedical
signals using nonlinear methods

• Patrick McSharry
• University of Oxford
• Email patrick_at_mcsharry.net
• Internet www.mcsharry.net

2
Overview

• Classification of health and illness from data
• Dynamical transitions
• Linear or nonlinear methods?
• Multi-Dimensional Probability Evolution (MDPE)
• Proof of concept using a toy example
• Problems of data quality
• Demonstration using partial epilepsy

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Big picture
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Linear versus nonlinear models
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Nonlinear dynamics chaos

• Invariant constants
• Dimensions (self similarity, fractal geometry)
• Lyapunov exponents (sensitivity to initial state)
• Entropy (loss of information, unpredictability)
• Difficult to estimate in low-dimensional systems
• Almost impossible to estimate for real world
  dynamics which are non-stationary, with noisy
  observations, often of short duration.

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Magic nonlinear statistics?

• One can still use nonlinear statistics!
• Short windows to overcome non-stationarity
• May still offer more information than linear
  statistics?
• Provide better medical diagnostics?
• Note, however, attractive properties of
  invariance are lost!

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Multi-Dimensional Probability Evolution (MDPE)

• Choose centres from points in the delay
  reconstructed state space
• Create a Voronoi partition
• Obtain a distribution for the learning data
• Compare with distribution from sliding window
• Measure similarity using c2

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Methodology

• Observed time series si
• Delay reconstruction xi1 si-(m-1)t, , si-1,
  si
• Reference set A representing normality
• Choose Nc centres and define a Voronoi partition
  on A
• Count the number of points, noi, in each
  partition
• Similary, for any given window of data, count the
  number of points, ni, in each partition
• If N and No are the total number of points in the
  reference set and window respectively and r
  (N/No)1/2, then

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A tale of two processes

• (i) Linear stochastic AR(1)
• xi1axi ni
• (ii) Deterministic nonlinear
• low-dimensional chaotic
• Skew-Tent map
• yi1yi / b, 0ltyi lt1
• 1-yi/1-b, bltyilt1
• Same power spectrum
• a 2b-1
• Transform, same PDFs

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Mixing the two processes

• Signal composed of linear stochastic and
  nonlinear deterministic proccesses.
• Dynamical changes are invisible to the linear
  statistics, but not to the nonlinear statistic
  (MDPE).

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Influence of noise and length of datset on
detection
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Detecting parameter changes
Data
Parameter
MDPE
Lyapunov exponent
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Heart rate and partial epilepsy
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Conclusions

• MDPE is a simple to understand and intuitive
  nonlinear technique
• No need to use nonlinear invariants!
• MDPE can be better than Lyapunov exponent
• Shown to be able to detect nonlinearity in
  simple examples
• Need more data (number of cases and duration of
  datasets) and problems where nonlinearity is
  expected to demonstrate the method
• Future aircraft engine failure