1. Maxima of discretely sampled random fields
ISI Platinum Jubilee, Jan 1-4, 2008
Maxima of discretely sampled random fields
Keith Worsley,
McGill
Jonathan Taylor,
Stanford and Université de Montréal

2. Bad design:
2 mins rest
2 mins Mozart
2 mins Eminem
2 mins James Brown

3. Rest Mozart Eminem J. Brown
Temporal components (sd, % variance explained)
Period: seconds 1
0.41, 17%
Component 2
0.31, 9.5% 3
0.24, 5.6% 50
100
150
200
Frame
Spatial components 1 1
0.5
Component 2
-0.5 3 -1 2 4 6 8 10 12 14 16 18
Slice (0 based)

4. fMRI data: 120 scans, 3 scans hot, rest, warm, rest, …
First scan of fMRI data
1000
Highly significant effect, T=6.59
890
hot
500
880
rest
870
warm
100
200
300
No significant effect, T=-0.74
820
hot
T statistic for hot - warm effect
rest
800 5
warm
100
200
300
Drift
810
800 -5
790
T = (hot – warm effect) / S.d.
~ t110 if no effect
100
200
300
Time, seconds

5. 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)>0 at a sparse set of points;
Z(s1), Z(s2) are spatially correlated.
To control the false positive rate to ≤α we want a good approximation to α = P{maxS Z(s) ≥ t}:
Bonferroni (1936)
Random field theory (1970’s)
Discrete local maxima (2005, 2007)

6. Bonferroni S is a set of N discrete points The Bonferroni P-value is
P{maxS Z(s) ≥ t} ≤ N × P{Z(s) ≥ t}
We only need to evaluate a univariate integral
Conservative

7. Random field theory I f Z ( s ) i c o n t u w h e m d a r p G ¯ l F W
white noise
filter = *
FWHM I f Z ( s ) i c o n t u w h e m d a r p G l F W H M x P 2 S E C 1 = z + D 4 g A 3 V :
EC0(S)
Resels0(S)
EC1(S)
Resels1(S)
EC2(S)
Resels2(S)
Resels3(S)
EC3(S)
Resels (Resolution elements)
EC densities

8. ? 105 simulations, threshold chosen so that P{maxS Z(s) ≥ t} = 0.05
0.1
105 simulations, threshold chosen
so that P{maxS Z(s) ≥ t} = 0.05
0.09
0.08
Bonferroni
Random field theory ?
0.07
0.06
P value
0.05
0.04
0.03 2
0.02 -2
0.01
Z(s) 1 2 3 4 5 6 7 8 9 10
FWHM (Full Width at Half Maximum) of smoothing filter
FWHM

9. 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)) = ρ(s1-s2),
Then the IMP-BON P-value is E(#upcrossings)
P{maxS Z(s) ≥ t} ≤ N × P{Z(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

10. 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 Z’s ≤ Z(s)}
Conservative, very accurate
If Z(s) is stationary, with
Cor(Z(s1),Z(s2)) = ρ(s1-s2),
Then the DLM P-value is E(#discrete local maxima)
P{maxS Z(s) ≥ t} ≤ N × P{Z(s) ≥ t and neighbour Z’s ≤ Z(s)}
We only need to evaluate a (2D+1)-variate integral …
Z(s2) ≤
Z(s-1)≤ Z(s) ≥Z(s1) ≥
Z(s-2)

11. Discrete local maxima: “Markovian” trick
If ρ is “separable”: s=(x,y),
ρ((x,y)) = ρ((x,0)) × ρ((0,y))
e.g. Gaussian spatial correlation function:
ρ((x,y)) = exp(-½(x2+y2)/w2)
Then Z(s) has a “Markovian” property:
conditional on central Z(s), Z’s on
different axes are independent:
Z(s±1) ┴ Z(s±2) | Z(s)
So condition on Z(s)=z, find
P{neighbour Z’s ≤ z | Z(s)=z} = ∏dP{Z(s±d) ≤ z | Z(s)=z}
then take expectations over Z(s)=z
Cuts the (2D+1)-variate integral down to a bivariate integral
Z(s2) ≤
Z(s-1)≤ Z(s) ≥Z(s1) ≥
Z(s-2)

12. T h e r s u l t o n y i v c a ½ b w j x g , = 1 ; : D . F G P Á ( z )2 Z Q + - f m S Y

13. 105 simulations, threshold chosen so that P{maxS Z(s) ≥ t} = 0.05
0.1
105 simulations, threshold chosen
so that P{maxS Z(s) ≥ t} = 0.05
0.09
0.08
Bonferroni
Random field theory
0.07
0.06
P value
0.05
Discrete local maxima
0.04
0.03 2
0.02 -2
0.01
Z(s) 1 2 3 4 5 6 7 8 9 10
FWHM (Full Width at Half Maximum) of smoothing filter
FWHM

14. Comparison Bonferroni (1936) Discrete local maxima (2005, 2007)
Conservative
Accurate if spatial correlation is low
Simple
Discrete local maxima (2005, 2007)
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 (1970’s)
Approximation based on assuming S is continuous
Accurate if spatial correlation is high
Elegant
Easily extended to non-Gaussian, non-isotropic random fields

15. Random field theory: Non-Gaussian non-iostropic( s ) = Z 1 ; : n i a u c t o . d G r m e l N , 2
< D w h V @ p b y L z - K g v S P x E C \ X
& A 3 B R j [ F W 7

16. Referee report
Why bother?
Why not just do simulations?

17. ‘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 …
fMRI