The statistical analysis of surface data

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The statistical analysis of surface data

• Keith Worsley, McGill
• Jonathan Taylor, Stanford
• Robert Adler, Technion

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Isotropic Gaussian random field in 2D
Intrinsic volumes or Minkowski functionals
EC densities
filter
white noise
Z(s)


FWHM
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Volumes of tubesGetting the P-value of Gaussian
fields directly(Siegmund, Sun, 1989, 1993)
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Jonathan Taylors Gaussian Kinematic Formula
(2003) for functions of non-isotropic Gaussian
fields
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Beautiful symmetry
Steiner-Weyl Tube Formula (1930)
Taylor Kinematic Formula (2003)

• Put a tube of radius r about the search region
  ?S and rejection region R

Z2N(0,1)
r
R
r
Tube(?S,r)
Tube(R,r)
?S
Z1N(0,1)
0

• Find volume or probability, expand as a power
  series in r, pull off coefficients

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SurfStat
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Cluster extent, rather than peak height, for
inference (Friston, 1994)

• Choose a lower level, e.g. t3.11 (P0.001)
• Find clusters i.e. connected components of
  excursion set
• Measure cluster extent
• by resels
• Distribution
• fit a quadratic to the
• peak
• Distribution of maximum cluster extent
• Bonferroni on N clusters E(EC).

Z
D1
extent
t
Peak height
s
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MS lesions and cortical thickness (Charil et
al., 2007)

• Idea MS lesions interrupt neuronal signals,
  causing thinning in down-stream cortex
• Data n 425 mild MS patients

5.5
5
4.5
4
Average cortical thickness (mm)
3.5
3
2.5
Correlation -0.568, T -14.20 (423 df)
2
1.5
0
10
20
30
40
50
60
70
80
Total lesion volume (cc)
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Thresholding? Correlation random field

• Correlation between 2 fields at 2 different
  locations,
• searched over all pairs of locations, one in S,
  one in T
• MS data P0.05, ?424, c0.325, T6.48

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References

• Adler, R.J. and Taylor, J.E. (2007). Random
  fields and geometry. Springer.
• Adler, R.J., Taylor, J.E. and Worsley, K.J.
  (2008). Random fields, geometry, and
  applications. In preparation.