Maxima of discretely sampled random fields

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Maxima of discretely sampled random fields
ISI Platinum Jubilee, Jan 1-4, 2008

• Keith Worsley,
• McGill
• Jonathan Taylor,
• Stanford and Université de Montréal

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Bad design 2 mins rest 2 mins Mozart 2 mins
Eminem 2 mins James Brown
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Rest Mozart Eminem J. Brown
Temporal components (sd, variance
explained)
Period 5.2 16.1
15.6 11.6 seconds
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0.41, 17
2
0.31, 9.5
Component
3
0.24, 5.6
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Frame
Spatial components

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0.5
Component
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-0.5
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-1
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Slice (0 based)
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fMRI data 120 scans, 3 scans hot, rest, warm,
rest,
First scan of fMRI data
1000
Highly significant effect, T6.59
hot
890
rest
880
500
870
warm
0
100
200
300
No significant effect, T-0.74
0
820
hot
T statistic for hot - warm effect
rest
800
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warm
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Drift
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810
800
-5
790
T (hot warm effect) / S.d. t110 if no
effect
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Time, seconds
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Three methods so far

• The set-up
• S is a subset of a D-dimensional lattice (e.g.
  voxels)
• Z(s) N(0,1) at most points s in S
• Z(s) N(µ(s),1), µ(s)gt0 at a sparse set of
  points
• Z(s1), Z(s2) are spatially correlated.
• To control the false positive rate to a we want
  a good approximation to a PmaxS Z(s) t
• Bonferroni (1936)
• Random field theory (1970s)
• Discrete local maxima (2005, 2007)

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Bonferroni

• S is a set of N discrete points
• The Bonferroni P-value is
• PmaxS Z(s) t N PZ(s) t
• We only need to evaluate a univariate integral
• Conservative

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Random field theory
filter
white noise
Z(s)


FWHM
EC0(S)
Resels0(S)
EC1(S)
Resels1(S)
EC2(S)
Resels2(S)
Resels3(S)
EC3(S)
Resels (Resolution elements)
EC densities
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0.1
105 simulations, threshold chosen so that PmaxS
Z(s) t 0.05
0.09
0.08
Random field theory
Bonferroni
?
0.07
0.06
0.05
P value
0.04

2
0.03
0
0.02
-2

0.01
Z(s)
0
0
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FWHM (Full Width at Half Maximum) of smoothing
filter
FWHM
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Improved Bonferroni (1977,1983,1997)
Efron, B. (1997). The length heuristic for
simultaneous hypothesis tests

• Only works in 1D Bonferroni applied to N events
• Z(s) t and Z(s-1) t i.e.
• Z(s) is an upcrossing of t
• Conservative, very accurate
• If Z(s) is stationary, with
• Cor(Z(s1),Z(s2)) ?(s1-s2),
• Then the IMP-BON P-value is E(upcrossings)
• PmaxS Z(s) t N PZ(s) t and Z(s-1)
  t
• We only need to evaluate a bivariate integral
• However it is hard to generalise upcrossings to
  higher D

Discrete local maxima
Z(s)
t
s
s-1
s
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Discrete local maxima

• Bonferroni applied to N events
• Z(s) t and Z(s) is a discrete local
  maximum i.e.
• Z(s) t and neighbour Zs Z(s)
• Conservative, very accurate
• If Z(s) is stationary, with
• Cor(Z(s1),Z(s2)) ?(s1-s2),
• Then the DLM P-value is E(discrete local maxima)
• PmaxS Z(s) t N PZ(s) t and neighbour
  Zs Z(s)
• We only need to evaluate a (2D1)-variate
  integral

Z(s2)

Z(s-1) Z(s) Z(s1)

Z(s-2)
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Discrete local maxima Markovian trick

• If ? is separable s(x,y),
• ?((x,y)) ?((x,0))
  ?((0,y))
• e.g. Gaussian spatial correlation function
• ?((x,y)) exp(-½(x2y2)/w2)
• Then Z(s) has a Markovian property
• conditional on central Z(s), Zs on
• different axes are independent
• Z(s1) - Z(s2) Z(s)
• So condition on Z(s)z, find
• Pneighbour Zs z Z(s)z ?dPZ(sd) z
  Z(s)z
• then take expectations over Z(s)z
• Cuts the (2D1)-variate integral down to a
  bivariate integral

Z(s2)

Z(s-1) Z(s) Z(s1)

Z(s-2)
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(No Transcript)
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0.1
105 simulations, threshold chosen so that PmaxS
Z(s) t 0.05
0.09
0.08
Random field theory
Bonferroni
0.07
0.06
0.05
P value
Discrete local maxima
0.04

2
0.03
0
0.02
-2

0.01
Z(s)
0
0
1
2
3
4
5
6
7
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9
10
FWHM (Full Width at Half Maximum) of smoothing
filter
FWHM
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Comparison

• Bonferroni (1936)
• Conservative
• Accurate if spatial correlation is low
• Simple
• Discrete local maxima (2005, 2007)
• Conservative
• Accurate for all ranges of spatial correlation
• A bit messy
• Only easy for stationary separable Gaussian data
  on rectilinear lattices
• Even if not separable, always seems to be
  conservative
• Random field theory (1970s)
• Approximation based on assuming S is continuous
• Accurate if spatial correlation is high
• Elegant
• Easily extended to non-Gaussian, non-isotropic
  random fields

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Random field theory Non-Gaussian non-iostropic
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Referee report

• Why bother?
• Why not just do simulations?

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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