Random Fields of
Multivariate Test Statistics,
with Applications to
“Shape” Analysis
Keith Worsley,
McGill
Jonathan Taylor,
Stanford and Université de Montréal
Arnaud Charil,
Montreal Neurological Institute
Francesco Tomaiuolo
Fondazione ‘Santa Lucia’, Roma
Are Multiple Sclerosis lesions
related to patient disability?

Data: n = 425 mild MS patients
Disability measured by EDSS and other scores
8
7
6
Correlation = 0.290,
T = 6.23 (423 df)
5
EDSS

4
3
2
1
0
0
10
20
30
40
50
60
Total lesion volume (cc)
70
80
Charil et al,
NeuroImage (2003)
Which Multiple Sclerosis lesions
are related to patient disability?


Repeat for every voxel: standard VBM study
Find lesion density at every voxel:



Correlation(Lesion density, EDSS)


Segment MS lesions: 1=lesion, 0=not
Smooth 10mm
Convert to T statistic (423 df)
Threshold?


Bonferroni? Too conservative
Random field theory …
Â
¹=
0·µ·¼=2
Example test statistic:
Chi-bar
max Z1 cos µ + Z2 sin µ
Z1~N(0,1)
Z2~N(0,1)
s2
3
2
1
0
-1
-2
Excursion sets,
Xt = fs : Â
¹ ¸ tg s1
-3
Threshold
t 4
Rejection regions,
Z2
2
Search
Region,
S
Rt = fZ : Â
¹ ¸ tg
3
2
1
0
Z1
-2
-2
0
2
Euler characteristic heuristic: EC = #blobs - # holes
Search Region, S
Euler characteristic, EC
EC= 1
7
Excursion sets, Xt
6
5
2
1
1
0
10
Observed
8
¸ t)
P(max Â(s)
¹
s2S
6
¼ E(EC) = 0:05
Expected
) t = 3:75
4
2
0
-2
0
0.5
EXACT!
1
1.5
2
E(EC(S \ Xt )) =
X
D
d=0
2.5
3
L (S)½ (t)
d
d
3.5
4
Threshold, t
E(EC(S \ X )) =
t
Beautiful symmetry:
X
D
L (S)½ (t)
d
d
d=0
Steiner-Weyl Tube Formula (1930)
Tube(λS,r)
Taylor Gaussian Tube Formula (2003)
Radius, r
Tube(Rt,r)
Z2
14
µ
¶
@Z
¸ = Sd
p @s
4 log 2
=
FWHM
r
12
2
10
2
r
1.5
1
Rt
0.8
0.6
λS
8
Radius, r
0
1
6
4
0.5
2
Z1
0.4
0.2
-2
L (S)0 EC density ½ (t)
Lipschitz-Killing2 curvature
4 6 8 10 12 14 d
-2
0
2d
jTube(¸S; r)j
L (S)
2
L
2 1 (S)r
¼L0 (S)r2
Area
100
50
0
jTube(¸S; r)j =
X
D
d=0
0
0.5
1
1.5
¡(d=2 + 1)
L
D ¡d
0.3
(S)r d
p
2¼½1 (t)r
½0 (t)
¼½2 (t)r2
0.2
0.1
0
2
Radius of Tube, r
¼d
P(Tube(Rt ; r))
0.4
Probability
150
0
0
0.5
Radius of Tube, r
Adler &
Taylor,
Rarndom
Fields and
Geometry
(2007)
1
P(Tube(Rt ; r)) =
1
X
(2¼)d=2
d!
d=0
½d (t)rd
EC density ½d (t)
of the Â
¹ statistic
Z2~N(0,1)
r
Tube(Rt,r)
Rejection region
Rt
t-r t
Taylor’s Gaussian Tube Formula:
1
X
P (Z1 ; Z2 2 Tube(Rt ; r)) =
Z1~N(0,1)
(2¼)d=2
½d (t)rd
d!
d=0
½0 (t) =
Z
= ½0 (t) + (2¼)1=2 ½1 (t)r + (2¼)½2 (t)r2 =2 + ¢ ¢ ¢
Z 1
=
(2¼)¡1=2 e¡z2 =2 dz + e¡(t¡r)2 =2 =4
t¡r
1
(2¼)¡1=2 e¡z2 =2 dz + e¡t2 =2 =4
t
½1 (t) = (2¼)¡1 e¡t2 =2 + (2¼)¡1=2 e¡t2 =2 t=4
½2 (t) = (2¼)¡3=2 e¡t2 =2 t + (2¼)¡1 e¡t2 =2 (t2 ¡ 1)=8
..
.
Taylor & Worsley, Annals of Statistics, submitted (2007)
Lipschitz-Killing
curvature Ld (S)
r
Tube(λS,r)
λS
µ
¶
@Z
¸ = Sd
p @s
4 log 2
=
FWHM
Steiner-Weyl Volume of Tubes Formula:
Area(Tube(¸S; r)) =
X
D
¼ d=2
L
¡d (S)r d
D
¡(d=2 + 1)
d=0
= L2 (S) + 2L1 (S)r + ¼ L0 (S)r2
= Area(¸S) + Perimeter(¸S)r + EC(¸S)¼r2
L (S) = EC(¸S)
= Resels0 (S)
0
p
L (S) = 1 Perimeter(¸S) = 4 log 2 Resels (S)
1
1
2
L (S) = Area(¸S)
= 4 log 2 Resels2 (S)
2
Lipschitz-Killing curvatures are just “intrinisic volumes” or “Minkowski functionals”
in the metric of the variance of the derivative of the process
How to ¯nd Lipschitz-Killing curvature Ld (S)
S
S
S
¸ = Sd
Edge length × λ
FWHM/√(4log2)
12
10
8
6
4
2
.
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..
..
10
µ
@Z
@s
¶
p
=
4 log 2
FWHM
of simplices
L (Lipschitz-Killing
²) = 1, L (¡) curvature
L (N
=
1,
)=1
0
0
0
L (¡) = edge length, L (N) = 1 perimeter
1
2
L1 (N) = area
2
P Lcurvature
P L
Lipschitz-Killing
union
L
² ¡ Pof L
¡ of simplices
N
(S) = P² 0 ( )
¡ 0( ) +
P
L (S) =
L (¡) ¡
L (N)
¡
N 1
L1 (S) = P L 1(N)
2
N 2
0
N
0
( )
p Var(∇Z)
Non-isotropic data? Use Riemannian metric
µ
¶ of
¸ = Sd
Z~N(0,1)
s2
3
@Z
@s
=
4 log 2
FWHM
2
1
0.14
0.12
0
-1
-2
Edge length × λ
FWHM/√(4log2)
12
10
8
6
4
2
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10
0.1
0.08
0.06
-3
s1
of simplices
L (Lipschitz-Killing
²) = 1, L (¡) curvature
L (N
=
1,
)=1
0
0
0
L (¡) = edge length, L (N) = 1 perimeter
1
2
L1 (N) = area
2
P Lcurvature
P L
Lipschitz-Killing
union
L
² ¡ Pof L
¡ of simplices
N
(S) = P² 0 ( )
¡ 0( ) +
P
L (S) =
L (¡) ¡
L (N)
¡
N 1
L1 (S) = P L 1(N)
2
N 2
0
N
0
( )
Estimating Lipschitz-Killing curvature Ld (S)
We need independent & identically distributed random fields
e.g. residuals from a linear model
Z1
Z2
Z3
Z4
Z5
Z6
Z7
Z9 … Zn
Z8
Replace coordinates of the simplices in S⊂RealD by
normalised residuals (Z1,…,Zn) / ||(Z1,…,Zn)|| in Realn
of simplices
L (Lipschitz-Killing
²) = 1, L (¡) curvature
L (N
=
1,
)=1
0
0
0
L (¡) = edge length, L (N) = 1 perimeter
1
2
L1 (N) = area
2
P Lcurvature
P L
Lipschitz-Killing
union
L
² ¡ Pof L
¡ of simplices
N
(S) = P² 0 ( )
¡ 0( ) +
P
L (S) =
L (¡) ¡
L (N)
¡
N 1
L1 (S) = P L 1(N)
2
N 2
0
N
0
Unbiased!
( )
Unbiased!
Taylor & Worsley, JASA (2007)
Which Multiple Sclerosis lesions
are related to patient disability?
L (S) = 0
0
L (S) = 79:1
1
L (S) = 588:6
2
L (S) = 1404:1
3
º = 423
Z 1
½0 (t) =
¢ µ
¶¡(º+1)=2
2
¡ º+1
u
2 ¡ ¢ 1+
du
º
º
1=2
(º¼) ¡
t
2
µ
¶
¡(º ¡1)=2
2
t
½1 (t) = (2¼)¡1 1 +
º
¡
¢
µ
¶¡(º ¡1)=2
º+1
¡
t2
¡3=2
2
½2 (t) = (2¼)
¡ ¢
¡ ¢ t 1+
º
1=2
º
¡ º
2
µ 2¡
¶µ
¶¡(º ¡1)=2
2
º 1 ¡
t
½3 (t) = (2¼)¡2
t2 1
1+
º
º
P(max T (s) ¸ t) ¼ E(EC(S \ Xt )) =
s2S
¡
X
3
d=0
L (S)½ (t) = 0:05; t = 4:47
d
d
MS lesions and cortical
thickness

Idea: MS lesions interrupt neuronal signals,
causing thinning in down-stream cortex
Average cortical thickness (mm)
5.5
5
4.5
4
3.5
3
2.5
Correlation = -0.568,
T = -14.20 (423 df)
2
1.5
0
10
20
30
40
50
Total lesion volume (cc)
60
70
80
Charil et al,
NeuroImage (2007)
MS lesions and cortical
thickness at all pairs of points


Dominated by total lesions and average cortical
thickness, so remove these effects
Cortical thickness, smoothed 20mm



Lesion density, smoothed 10mm
Find partial correlation(lesion density, cortical
thickness) removing total lesion volume


linear model: CT-av(CT) ~ 1 + TLV + LD, test for LD
Repeat of all voxels in 3D, nodes in 2D


Subtract average cortical thickness
~1 billion correlations, so thresholding essential!
Look for high negative correlations …
Thresholding?
Correlation random field

Correlation between 2 fields at 2 different
locations, searched over all pairs of locations

one in R (D dimensions), one in S (E dimensions)
µ
¶ X
D X
E
¼
L (R) L (S) ½ (c)
P max Correlation > c
d
e
d;e
R;S
d=0 e=0
X
(d ¡ 1)!e!2n¡d¡e¡2 X ¡
¡d¡e¡1
¡
¡
¡
n
½d;e (c) =
( 1)k cd+e 1 2k (1 c2 ) 2 +k
+1
¼ d+e
2
k
i;j
¡( n¡d + i)¡( n¡e + j)
2
2
¡
¡
¡
¡
¡
i!j!(k i j)!(n 1 d e + i + j + k)!(d ¡ 1 ¡ k ¡ i + j)!(e ¡ k ¡ j + i)!
Cao & Worsley, Annals of Applied Probability (1999)

MS lesion data: P=0.05, c=0.300, T=6.46
Cluster extent rather than
peak height (Friston, 1994)



fit a quadratic
Y
to the peak:
L (cluster)
»c
®

D

Peak
height

Choose a lower level, e.g. t=3.11 (P=0.001)
Find clusters i.e. connected components of excursion
set
Z
D=1
Measure cluster
L
extent
extent by D
t
Distribution:
k
Distribution of maximum cluster extent:

s
Bonferroni on N = #clusters ~ E(EC).
Cao, Advances
in Applied
Probability (1999)
How do you choose the threshold
t for defining clusters?

If signal is focal i.e. ~FWHM of noise



If signal is broad i.e. >>FWHM of noise





Choose a high threshold
i.e. peak height is better
Choose a low threshold
i.e. cluster extent is better
Conclusion: cluster extent is better for detecting
broad signals
Alternative: smooth data with filter that matches
signal (Matched Filter Theorem)… try range of filter
widths … scale space search … correct using
random field theory … a lot of work …
Cluster extent is easier!
Multivariate linear models for
random field data

Y(s) = X B(s) + E(s), test H0: B(s)=0, s in S
nxq

nxp pxq
nxq
EC densities known (<2007) for:
q=1
p=1
p>1
T
F
q>1 Hotelling’s
T2

Which to choose?
Wilks’ Λ ?
Pillai’s trace ?
Roy’s max root ?
but not
these!
Roy’s union-intersection
principle (1954)


Make it into a univariate linear model by multiplying
by vector qx1
v
Y(s)v = X B(s)v + E(s)v, H0: B(s)v=0
nx1



P
µ
nxp px1
nx1
Calculate usual F-statistic F(s,v)
Maxv in unit q-sphere V F(s,v) = Roy’s maximum root
Now it is an F-field in search region S ⊗ V and we
already know¶ the EC density of the F-field:
max F (s; v) ¸ t
s2S;v 2V
¼
D+q
X
d=0
L (S
d
V )½F (t) =
d
X
D
d=0
L (S)
d
q
X
L (V )½F
k
d+k
(t)
k=0
Taylor & Worsley, Annals of Statistics (2007)
Maximum canonical correlation
Let X(s), s 2 S ½ <M , and Y (t), t 2 T ½ <N be matrices of Gaussian
random ¯elds with p and q columns and the same number º of rows. De¯ne the
maximum canonical correlation random ¯eld as
u0 X(s)0 Y (t)v
C(s; t) = max
u;v (u0 X(s)0 X(s)u
v0Y
(t)0 Y
(t)v)1=2
;
the maximum of the canonical correlations between X and Y , de¯ned as the
singular values of (X 0 X)¡1=2 X 0 Y (Y 0 Y )¡1=2 .
P
µ
max C(s; t) ¸ c
s2S;t2T
¶
¼
1
2
X
M
i=0
L (S)
i
X
N
L (T )
j
j=0
p
X
L (U )
k
k=0
q
X
L (V )½
(c)
l
i+k;j+l
l=0
where U is the unit sphere in <p , V is the unit sphere in <q .
¡
¢
k+1 ¼ k ¡ p+1
2
L (U ) =
³ 2
´2
k
¡
¡
k! p 1 k !
2
if p ¡ 1 ¡ k is even, and zero otherwise, k = 0; : : : ; p ¡ 1.
Now available in
stat_threshold.m
Deformation Based Morphometry
(DBM) (Tomaiuolo et al., 2004)





n1 = 19 non-missile brain trauma patients, 3-14
days in coma,
n2 = 17 age and gender matched controls
Data: non-linear vector (q=3) deformations
needed to warp each MRI to an atlas standard
Locate damage: find regions where deformations
are different, hence shape change
Is damage connected? Find pairs of regions with
high canonical correlation.
Worsley et al. NeuroImage (2004)