A New Multivariate Method

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A New Multivariate Method

• for comparing two n-dimensional distributions

By Jim Loudin and Hannu Mietinen
PhyStat 2003 9/10/03
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Overview

• Review of Kernel Density Estimation
• Optimizing kernel width
• Our new method
• Some Monte Carlo results
• Conclusions

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A review of KDE

• THE IDEA center a gaussian around each data
  point and take the sum

• One parameter h (width of the gaussians)
• Conceptually very simple (n-dimensional scatter
  plots)

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Optimizing h

• h is optimized by minimizing the difference
  squared between the smooth surface and the lego
  plot

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Examples of D-histograms
Since D(x)fa/(fafb), if fa and fb are
approximately equal, we expect a peak at
roughly .5
If they are not the same, we expect a wide or
noisy histogram with peaks at 1 and/or 0.
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The K-S statistic
d max distance between CDFs
Significance(d0)probability dgtd0, where d0 is
the observed value of d
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Some Significance Curves
A nice result concerning significance curves is
that they are nearly independent of gaussian width
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Some Simple Results

• We test our method on gaussian distributions of
  different widths.

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Conclusions

• We have a new method of comparing n-dimensional
  data sets/distributions
• It has tested well on simple gaussian
  distributions
• Future tests will involve distributions of more
  complicated structure
• We are in the process of applying this to
  Extrasolar Planetary Candidate data