The Geometry of Random Fields in Astrophysics and Brain Mapping

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The Geometry of Random Fields in Astrophysics and
Brain Mapping

• Keith Worsley, McGill
• Jonathan Taylor, Stanford and Université de
  Montréal
• Robert Adler, Technion
• Frédéric Gosselin, Université de Montréal
• Philippe Schyns, Fraser Smith, Glasgow
• Arnaud Charil, Montreal Neurological Institute

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Astrophysics
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Sloan Digital Sky Survey, data release 6, Aug. 07
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(No Transcript)
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Nature (2005)
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Subject is shown one of 40 faces chosen at
random
Happy
Sad
Fearful
Neutral
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but face is only revealed through random
bubbles

• First trial Sad expression
• Subject is asked the expression
  Neutral
  
• Response
  Incorrect

75 random bubble centres
Smoothed by a Gaussian bubble
What the subject sees
Sad
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Your turn

• Trial 2

Subject response Fearful CORRECT
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Your turn

• Trial 3

Subject response Happy INCORRECT (Fearful)
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Your turn

• Trial 4

Subject response Happy CORRECT
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Your turn

• Trial 5

Subject response Fearful CORRECT
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Your turn

• Trial 6

Subject response Sad CORRECT
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Your turn

• Trial 7

Subject response Happy CORRECT
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Your turn

• Trial 8

Subject response Neutral CORRECT
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Your turn

• Trial 9

Subject response Happy CORRECT
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Your turn

• Trial 3000

Subject response Happy INCORRECT (Fearful)
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Bubbles analysis

• E.g. Fearful (3000/4750 trials)

Trial 1 2 3 4
5 6 7 750
Sum
Correct trials
Thresholded at proportion of correct
trials0.68, scaled to 0,1
Use this as a bubble mask
Proportion of correct bubbles (sum correct
bubbles) /(sum all bubbles)
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Results

• Mask average face
• But are these features real or just noise?
• Need statistics

Happy Sad
Fearful Neutral
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Statistical analysis

• Correlate bubbles with response (correct 1,
  incorrect 0), separately for each expression
• Equivalent to 2-sample Z-statistic for correct
  vs. incorrect bubbles, e.g. Fearful
• Very similar to the proportion of correct bubbles

ZN(0,1) statistic
Trial 1 2 3 4
5 6 7 750
Response 0 1 1 0
1 1 1 1
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Results

• Thresholded at Z1.64 (P0.05)
• Multiple comparisons correction?
• Need random field theory

ZN(0,1) statistic
Average face Happy Sad
Fearful Neutral
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Euler Characteristic Heuristic
Euler characteristic (EC) blobs - holes (in
2D) Excursion set Xt s Z(s) t, e.g. for
neutral face
EC 0 0 -7 -11
13 14 9 1 0
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Heuristic At high thresholds t, the holes
disappear, EC 1 or 0, E(EC) P(max Z
t).
Observed
Expected
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EC(Xt)
0

• Exact expression for E(EC) for all thresholds,
• E(EC) P(max Z t) is extremely accurate.

-10
-20

-4
-3
-2
-1
0
1
2
3
4
Threshold, t
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The result
Lipschitz-Killing curvatures of S
EC densities of Z above t
filter
white noise
Z(s)


FWHM
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Results, corrected for search

• Random field theory threshold Z3.92 (P0.05)
• Bonferroni threshold Z4.87 (P0.05) nothing

ZN(0,1) statistic
Average face Happy Sad
Fearful Neutral
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Theory (1981,1995)
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Proof.
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Steiner-Weyl Tube Formula (1930)
Morse Theory method (1981, 1995)

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

EC has a point-set representation
r
Tube(?S,r)
?S

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

For a Gaussian random field
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Tube(?S,r)
r
?S
Steiner-Weyl Volume of Tubes Formula (1930)
Lipschitz-Killing curvatures are just intrinisic
volumes or Minkowski functionals in the
(Riemannian) metric of the variance of the
derivative of the process
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S
S
S
Edge length ?
Lipschitz-Killing curvature of triangles
Lipschitz-Killing curvature of union of triangles
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Non-isotropic data
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Non-isotropic data
ZN(0,1)
ZN(0,1)
s2
s1
Edge length ?(s)
Lipschitz-Killing curvature of triangles
Lipschitz-Killing curvature of union of triangles
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We need independent identically distributed
random fields e.g. residuals from a linear model

Lipschitz-Killing curvature of triangles
Lipschitz-Killing curvature of union of triangles
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Scale space
Note scaled to preserve variance, not mean
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Beautiful symmetry
Steiner-Weyl Tube Formula (1930)
Taylor Gaussian Tube Formula (2003)

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

Z2N(0,1)
Rt
r
Tube(Rt,r)
Tube(?S,r)
r
?S
Z1N(0,1)
t
t-r

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

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Z2N(0,1)
Rejection region Rt
Tube(Rt,r)
r
Z1N(0,1)
t
t-r
Taylors Gaussian Tube Formula (2003)
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EC densities for some standard test statistics

• Using Morse theory method (1981, 1995)
• T, ?2, F (1994)
• Scale space (1995, 2001)
• Hotellings T2 (1999)
• Correlation (1999)
• Roys maximum root, maximum canonical correlation
  (2007)
• Wilks Lambda (2007) (approximation only)
• Using Gaussian Kinematic Formula
• T, ?2, F are now one line
• Likelihood ratio tests for cone alternatives (e.g
  chi-bar, beta-bar) and nonnegative least-squares
  (2007)

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Accuracy of the P-value approximation
The expected EC gives all the polynomial terms in
the expansion for the P-value.
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Bubbles task in fMRI scanner

• Correlate bubbles with BOLD at every voxel
• Calculate Z for each pair (bubble pixel, fMRI
  voxel) a 5D image of Z statistics

Trial 1 2 3 4
5 6 7 3000
fMRI
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Thresholding? Cross correlation random field

• Correlation between 2 fields at 2 different
  locations,
• searched over all pairs of locations, one in S,
  one in T
• Bubbles data P0.05, n3000, c0.113, T6.22

Cao Worsley, Annals of Applied Probability
(1999)
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MS lesions and cortical thickness

• Idea MS lesions interrupt neuronal signals,
  causing thinning in down-stream cortex
• Data n 425 mild MS patients
• Lesion density, smoothed 10mm
• Cortical thickness, smoothed 20mm
• Find connectivity i.e. find voxels in 3D, nodes
  in 2D with high correlation(lesion density,
  cortical thickness)
• Look for high negative correlations
• Threshold P0.05, c0.300, T6.48

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n425 subjects, correlation -0.568
Average cortical thickness
Average lesion volume
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Summary

• Points are in a low dimensional space, physically
  meaningful
• Smooth (choice of kernel? Scale space ),
  threshold
• Galaxies
• Looking for sheets, strings, clusters of high
  density
• EC is used to measure topology other intrinsic
  volumes (diameter, surface area, volume) are also
  used
• Compare observed EC with expected EC under some
  model (e.g. inflation)
• Bubbles, MS lesions
• Detect sparse high-density clusters with a very
  low signal to noise
• Thresholding gives maximum likelihood estimates
  under certain conditions
• EC is merely a device for getting an extremely
  accurate approximation to the false positive rate
  (P-value of the maximum)
• Brain mapping data
• Detect sparse high-density activations with a
  very low signal to noise
• Riemannian metric is usually unknown
• But we only need to estimate Lipschitz-Killing
  curvature
• Fill with small simplices, work out LKC on each
  component, sum using inclusion-exclusion formula