Methods for selecting

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Methods for selecting extreme events in
multivariate data
Alberto Bernacchia
Dip.di Fisica E.Fermi, Universita di Roma La
Sapienza
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Summary

• How to deal with extremes and multivariate data?
• Extreme Value Theory is well established for
  univariate data.
• For multivariate data, there is no finite
  parameterization of extreme value distributions
  (Coles 2001)
• One possible approach is dimensionality
  reduction Principal Component Analysis (PCA) is
  a widely used technique, but is mostly affected
  by the center rather than the extremes of the
  density
• Recently, NonLinear PCA (NLPCA) has been applied
  to geophysical data (Hsieh 2001), but results are
  neither robust nor reproducible
• I define a new method for detecting extreme
  events (only) of multivariate densities,
  eliminating the drawbacks of NLPCA

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Non Linear PCA
Kramer (1991)

• It determines low-dimensional nonlinear
  representations of data
• The projection over the nonlinear manifold is
  continuous
• It reduces to PCA by tuning a control parameter

Hsieh (2001)
Feed-forward neural network
Bottleneck
Cost function
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Non Linear PCA

• It determines low-dimensional nonlinear
  representations of data
• The projection over the nonlinear manifold is
  continuous
• It reduces to PCA by tuning a control parameter

Hsieh (2001)
ENSO Sea Surface Temperatures
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Non Linear PCA

• It determines low-dimensional nonlinear
  representations of data
• The projection over the nonlinear manifold is
  continuous
• It reduces to PCA by tuning a control parameter

Hsieh (2001)
Lorenz attractor
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Drawbacks

• In order to be computationally accessible, it
  requires a preliminary dimensionality reduction,
  leaving part of the information

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Drawbacks

• Despite lowering the computational cost, the
  global solution (minimum) is often not
  accessible, and results cannot be rigorously
  reproduced

Local minima
Lorenz attractor
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Drawbacks

• The solution strongly depends on two parameters,
  leaving opened a fundamental ambiguity

Christiansen (2005)
NH wintertime geopotential height
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Summary 2

• NonLinear PCA (NLPCA) determines a
  low-dimensional (highly informative) nonlinear
  fit to data, but is ambiguous and frail
• I want to find a robust dimensionality reduction
  method which accounts not for the entire body of
  data, but just for the extremes, i.e. a method
  able to detect a subspace of data where the
  extreme events are dense

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The Generating Function for detecting extremes
Estimated Generating Function
Working assumption the vectors maximizing G, at
fixed modulus, are the directions of the extreme
events
modulus
unit vector
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The Generating Function for detecting extremes
development in cumulants of the projection
variance
skewness
kurtosis
(mean0)
small y ? PCA
few data points contribute the sum
large y ? Finite-size effects
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The Generating Function for detecting extremes
The value of y is fixed by a tolerance
finite-size error e2
For uncorrelated Normal data, it is given by
For exponentially correlated Normal data, one
must solve
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An application El Nino Southern Oscillation
Sea Surface Temperatures (monthly anomalies),
1949-1999
27 x 9 grid (step 5o)
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An application El Nino Southern Oscillation
Exponential correlations over short periods
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An application El Nino Southern Oscillation
set e2 0.1
El Nino
La Nina
y 2.1
two solutions
Note the method is applied to the entire space
and not just to 1PC and 2PC
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An application El Nino Southern Oscillation
Comparison with the NLPCA solution
La Nina
El Nino
Hsieh (2001)
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An application El Nino Southern Oscillation
El Nino
NLPCA, Hsieh (2001)
Maximum of Generating Function
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An application El Nino Southern Oscillation
La Nina
NLPCA, Hsieh (2001)
Maximum of Generating Function
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Generalized Extreme Value fit of projected data
60 blocks, 10 data points each
El Nino
x -0.150.21
La Nina
x -0.300.13
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Conclusions

• Interesting directions in the space of data are
  obtained by the local maxima of the (biased)
  generating function in n dimensions
• These directions are supposed to point towards
  the extreme events of the underlying (if any)
  stationary probability distribution
• The method is computationally cheap, and has no
  free parameter (once the tolerance error is
  fixated)
• La Nina is characterized by a finite lower bound
  of temperatures.
• El Nino seems to have a finite tail, but the
  result is weakly significant

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Application to Lorenz attractor
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Application to Lorenz attractor
set e2 0.1
y ?
two solutions
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Application to Lorenz attractor
set e2 0.1
y 3.7
two solutions
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Drawbacks

• For nearly isotropic data, it fits poorly even
  normal distributions

Christiansen (2005)
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