The statistical analysis
of surface data
Keith Worsley, McGill
Jonathan Taylor, Stanford
Robert Adler, Technion
Isotropic Gaussian random field in 2D
If Z (s) » N(0; 1) ¡is an¢ isot ropic Gaussian random ¯eld, s 2 < 2 ,
wit h ¸ 2 I 2£ 2 = V @Z ,
@s
µ
¶
P max Z (s) ¸ t ¼ E(E C(S \ f s : Z (s) ¸ tg))
s2 S
Z
1
1
= E C(S) £
e¡ z 2 =2 dz
(2¼) 1=2
t
¸
1
+
Perimet
er(S)
£
e¡ t 2 =2
Intrinsic volumes or
EC
2
2¼
Minkowski functionals
densities
¸2
+ Area(S) £
te¡ t 2 =2
(2¼) 3=2
If Z (s) is whit e noise convolved
wit h an isot ropic Gaussian
Z(s)
¯lt er of Full Widt h at Half
Maximum
FWHM t hen
p
¸ = 4 log 2 :
white noise
=
filter
*
FW HM
FWHM
Volumes of tubes:
Getting the P-value of Gaussian fields directly
(Siegmund, Sun, 1989, 1993)
Approximat e t he Gaussian ¯eld by a K arhunen-Loµ
eve expansion in t erms of
basis functions bj (s) wit h independent Gaussian coe± cient s Z j » N(0; 1):
Xm
Z (s) ¼
bj (s)Z j = (b(s) 0U ) jjZjj;
j=1
where b(s) = (b1 (s); : : : ; bm (s)) 0, Z = (Z 1 ; : : : ; Z m ) 0, and jjZjj » Âm is independent of U = Z=jjZjj » Uniform on t he unit m-sphere ° m . Condit ional on
jjZjj,
³
´
p
f U : Z (s) ¸ tg = Tube b(s); 1 ¡ t 2 =jjZjj 2 ½ ° m
T herefore
µ
¯
P max Z (s) ¸ t ¯ jjZjj
s2 S
¶
=
Vol(Tube)
;
Vol(° m )
so it comes down t o a problem in geomet ry (Hot elling, Weyl, 1939). Takemura
& K uriki (2000) showed t hat t he ¯rst D + 1 t erms are t he same as E(E C).
Jonathan Taylor’s Gaussian Kinematic Formula
(2003) for functions of non-isotropic Gaussian
fields
Let s 2 S ½ < D .
Let Z (s) = (Z 1 (s); : : : ; Z n (s)) be iid smoot h Gaussian random ¯elds.
Let T (s) = f (Z (s)), e.g. Â2 , T , F st at ist ics.
Let X = f s : T (s) ¸ tg be t he excursion set inside S.
Let R = f z : f (z) ¸ tg be t he reject ion region of T .
T hen
XD
E(E C(S \ X t )) =
L d (S)½d (R t ):
d= 0
XD
Beautiful
symmetry:
E(E C(S \ X )) =
L d (S)½d (R)
d= 0
Lipschit z-K illing curvat ure L d (S)
Steiner-Weyl Tube Formula (1930)
EC density ½d (R)
Taylor Kinematic Formula (2003)
• Put a tube of radius r about the search region λS and rejection region R:
Z2~N(0,1)
14
r
12
µ
¶
@Z 10
¸ = Sd
@s 8
λS
R
r
Tube(λS,r)
Tube(R,r)
0
6
Z1~N(0,1)
4
2
2
4
6
8 10 12 14
• Find volume or probability, expand as a power series in r, pull off coefficients:
X1 (2¼) d
XD
¼d
2
P(Tube(R; r )) =
½d (R)r d
V(Tube(¸ S; r )) =
L D ¡ d (S)r d
d!
¡ ( d + 1)
d= 0
2
d= 0
SurfStat
F Fit linear mixed e®ect s model Y n £ 1 » N(X n £ p ¯p£ 1 ; V n £ n (µq£ 1 ))
by ReML using Fisher scoring, separat ely at each vert ex
b ¡ 1=2 (Y ¡ X b̄)
F Find whit ened residuals: r n £ 1 = V ( µ)
F Find normalized residuals: u n £ 1 = r =jjr jj
F For each vert ex on surface, use u as coordinat es of t he vert ices in < n
F Find \ lengt h" of each edge
F Find \ area" and \ perimet er" of each t riangle
F Lb0 = # vert ices - # edges + # t riangles
F Lb1 = sum of \ lengt hs" - 1/ 2 sum of \ perimet ers"
F Lb2 = sum of \ areas"
c 0b̄) and it s average
F For a cont rast c, ¯nd t he T st at ist ic T = c0b̄=Sd(c
d
d Var(c
d
0b̄) 2 =Var(
0b̄))
e®ect ive (Sat t ert hwait e) df º = 2Var(c
F P(max T ¸ t) ¼ Lb0 ½0 (t) + Lb1 ½1 (t) + Lb2 ½2 (t), where t he EC densit ies are
µ
½0 (t) = P(T ¸ t);
½1 (t) = (2¼) ¡
¡
½2 (t) = (2¼) ¡
3
2
¡
¢
¡ º+1
¢1=2 2 ¡ ¢
º
¡ º
2
2
1
µ
t 1+
¶
t 2 ¡ º ¡2 1
1+
;
º
¶ ¡ ( º ¡ 1) =2
t2
º
Cluster extent, rather than peak height, for
inference (Friston, 1994)

Choose a lower level, e.g. t=3.11 (P=0.001)

Find clusters i.e. connected components of excursion set

Measure cluster
extent
L D (clust
er)
by resels
Z
D=1
extent

L D (clust er) » c

t
®
k
Distribution of maximum cluster extent:
 Bonferroni on N = #clusters ~ E(EC).
Peak
height
Distribution:
 fit a quadratic to the
peak:
Y
s
MS lesions and cortical thickness
(Charil et al., 2007)

Idea: MS lesions interrupt neuronal signals, causing thinning in
down-stream cortex
Data: n = 425 mild MS patients
5.5
Average cortical thickness (mm)

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
Thresholding? Correlation random field
Correlation between 2 fields at 2 different locations,
searched over all pairs of locations, one in S, one in T:

µ
P
¶
max C(s; t) ¸ c ¼ E(E Cf s 2 S; t 2 T : C(s; t) ¸ cg)
s2 S;t 2 T
dim
X ( S) dim(
X T)
=
L i (S)L j (T )½i j (C ¸ c)
i= 0
½i j (C ¸ c) =
2º ¡
2¡ h (i
j=0
¡ 1) =2c
¡ 1)!j ! b( h X
¼h =2+ 1
Xk Xk
(¡ 1) k ch ¡
k= 0
1¡ 2k (1 ¡
c2 ) ( º ¡
1¡ h ) =2+ k
l= 0 m= 0
¡ ( º ¡ i + l)¡ ( º ¡ j + m)
2
2
l!m!(k ¡ l ¡ m)!(º ¡ 1 ¡ h + l + m + k)!(i ¡ 1 ¡ k ¡ l + m)!(j ¡ k ¡ m + l)!

MS data: P=0.05, ν=424, c=0.325, T=6.48
References

Adler, R.J. and Taylor, J.E. (2007). Random fields and geometry. Springer.

Adler, R.J., Taylor, J.E. and Worsley, K.J. (2008). Random fields, geometry,
and applications. In preparation.