ISI Platinum Jubilee, Jan 1-4, 2008
Maxima of discretely
sampled random fields
Keith Worsley,
McGill
Jonathan Taylor,
Stanford and Université de Montréal
Bad design:
2 mins rest
2 mins Mozart
2 mins Eminem
2 mins James Brown
Rest
Mozart
Eminem
J. Brown
Temporal components
Component
Period:
5.2
16.1
(sd, % variance explained)
15.6
11.6
seconds
1
0.41, 17%
2
0.31, 9.5%
3
0.24, 5.6%
0
50
100
Frame
Spatial components
150
200
1
Component
1
0.5
2
0
-0.5
3
0
2
4
6
8
10
12
Slice (0 based)
14
16
18
-1
fMRI data: 120 scans, 3 scans hot, rest, warm, rest, …
First scan of fMRI data
1000
Highly significant effect, T=6.59
500
hot
rest
warm
890
880
870
0
0
100
200
300
No significant effect, T=-0.74
820
hot
rest
warm
T statistic for hot - warm effect
5
800
0
100
0
100
0
-5
T = (hot – warm effect) / S.d.
~ t110 if no effect
200
Drift
300
810
800
790
200
Time, seconds
300
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)
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
Random
field theory
Z(s)
white noise
=
filter
*
FWHM
If Z (s) is cont inuous whit e noise smoot hed wit h an isot ropic Gaussian ¯lt er of
Full Widt h at Half Maximum FWHM
µ
¶
Z
1
1
P max Z (s) ¸ t ¼ E C(S)
e¡ z 2 =2 dz
(2¼) 1=2
s2 S
EC (S)
t
Resels0(S)
Resels1(S)
Resels2(S)
Resels3(S)
Resels (Resolution elements)
Diamet er(S)
e¡ t 2 =2
FWHM
2¼
Area(S) 4 log 2
1
+
te¡ t 2 =2
2 FWHM 2 (2¼) 3=2
Volume(S) (4 log 2) 3=2
+
(t 2 ¡ 1)e¡
3
(2¼) 2
FWHM
+ 2
0
(4 log 2) 1=2
EC1(S)
EC2(S)
t 2 =2 :
EC3(S)
EC densities
0.1
105 simulations, threshold chosen
so that P{maxS Z(s) ≥ t} = 0.05
0.09
0.08
Random field theory
Bonferroni
0.07
?
P value
0.06
0.05
0.04
2
0.03
0
0.02
-2
0.01
0
Z(s)
0
1
2
3
4
5
6
7
8
FWHM (Full Width at Half Maximum) of smoothing filter
9
10
FWHM
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
Discrete local maxima
Z(s)
t
Cor(Z(s1),Z(s2)) = ρ(s1-s2),
s
s-1 s
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

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
Z(s2)
≤
Z(s-1)≤ Z(s) ≥Z(s1)
Cor(Z(s1),Z(s2)) = ρ(s1-s2),
≥
Z(s-2)
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 …
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)

Z(s2)
≤
Z(s-1)≤ Z(s) ≥Z(s1)
≥
Z(s-2)
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

T he result only involves t he correlat ion ½d between adjacent voxels along
each lat t ice axis d, d = 1; : : : ; D . First let t he Gaussian density and uncorrect ed
P values be
Z
p
1
2
Á(z) = exp(¡ z =2)= 2¼; ©(z) =
Á(u)du;
z
respect ively. T hen de¯ne
1
Q(½; z) = 1 ¡ 2©(hz) +
¼
where
® = sin¡
³p
1
Z
®
exp(¡
1 h2 z2 =sin2
2
0
r
´
(1 ¡ ½2 )=2 ;
h=
µ)dµ;
1¡ ½
:
1+ ½
T hen t he P-value of t he maximum is bounded by
µ
P
¶
max Z (s) ¸ t
s2 S
Z
· jSj
t
1
YD
Q(½d ; z) Á(z)dz;
d= 1
where jSj is t he number of voxels s in t he search region S. For a voxel on
t he boundary of t he search region wit h just one neighbour in axis direct ion d,
replace Q(½; z) by 1 ¡ ©(hz), and by 1 if it has no neighbours.
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
P value
0.06
0.05
Discrete local maxima
0.04
2
0.03
0
0.02
-2
0.01
0
Z(s)
0
1
2
3
4
5
6
7
8
FWHM (Full Width at Half Maximum) of smoothing filter
9
10
FWHM
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 (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
Random field theory:
Non-Gaussian non-iostropic
If T (s) = f (Z 1 (s); : : : ; Z n (s)) is a funct ion³ of i.i.d.
Gaussian random ¯elds
´
Z i (s) » Z (s) » N(0; 1), s 2 < D , wit h V @Z ( s) = ¤ D £ D (s), t hen replace
@s
resels by Lipschitz-K illing curvature L d (S; ¤ ):
µ
¶
P max T (s) ¸ t
s2 S
XD
¼ E(E C(S \ f s : T (s) ¸ tg)) =
L d (S; ¤ )½d (t);
d= 0
where ½d (t) is t he same EC density for t he isot ropic case wit h ¤ (s) = I D £ D
(Taylor & Adler, 2003). Bot h Lipschit z-K illing curvat ure L d (S; ¤ ) and EC
density ½d (t) are de¯ned implicit ly as coe± cient s of a power series expansion of
t he volume of a t ube as a funct ion of it s radius. In t he case of Lipschit z-K illing
curvat ure, t he t ube is about t he search region S in t he Riemannian met ric
induced by ¤ (s); in t he case of t he EC densit ies, t he t ube is about t he reject ion
region f ¡ 1 ([t; 1 )) and volume is replaced by Gaussian probability. Fort unat ely
t here are simple ways of est imat ing Lipschit z-K illing curvat ure from sample dat a
(Taylor & Worsley, 2007), and simple ways of calculat ing EC densit ies.
Referee report

Why bother?

Why not just do simulations?
‘Bubbles’ task in fMRI scanner

Correlate bubbles with BOLD at every voxel:
Trial
1
2
3
4
5
6
7 …
3000
1
0.5
0
fMRI
10000
0

Calculate Z for each pair (bubble pixel, fMRI voxel) – a 5D “image”
of Z statistics …