1. The geometry of random fields in astrophysics and brain mapping
Keith Worsley, Farzan Rohani, McGill
Nicholas Chamandy, McGill and Google
Jonathan Taylor, Stanford and Université de Montréal
Jin Cao, Lucent
Arnaud Charil, Montreal Neurological Institute
Frédéric Gosselin, Université de Montréal
Philippe Schyns, Fraser Smith, Glasgow

2. Astrophysics

3. Sloan Digital Sky Survey, data release 6, Aug. ‘07

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

5. Linear model regressors50
100
150
200
250
300
350 -1 1 2
Alternating hot and warm stimuli separated by rest (9 seconds each).
hot
warm
-0.2
0.2
0.4
Hemodynamic response function: difference of two gamma densities
Regressors = stimuli * HRF, sampled every 3 seconds
Time, seconds

6. Brain imaging Detect sparse regions of “activation”
Construct a test statistic “image” for detecting activation.
Activated regions: test statistic > threshold
Choose threshold to control false positive rate to say 0.05, i.e.
P(max test statistic > threshold) = 0.05
Bonferroni??? Too conservative …
False discovery rate??? Not appropriate …

7. Detecting sparse cone alternatives
Test statistics are usually functions of Gaussian fields, e.g. T or F statistics
Let’s take a challenging example: a random field of chi-bar statistics for detecting a sparse cone alternative.
Application: detecting fMRI activation in the presence of unknown latency of the hemodynamic response function (HRF)
Linear model with two regressors of the HRF shifted by +/-2 seconds
Fit by non-negative least-squares; equivalent to a cone alternative -2 +2 -2 +2

8. Chi-bar ¹ Â = m a x Z c o s + i n X = f s : ¹ Â ¸ g R = f Z : ¹ Â ¸ g
Example test statistic: Â = m a x 2 Z 1 c o s + i n
Z1~N(0,1)
Z2~N(0,1) s2 s1
Excursion sets,
Rejection regions, X t = f s : Â g R t = f Z : Â g
Threshold t Z2
Cone
alternative
Search
Region, S Z1
Null

9. Euler characteristic heuristic again
0.5 1
1.5 2
2.5 3
3.5 4 -2 6 8 10
Euler characteristic heuristic again
Excursion sets, Xt
Search Region, S
EC= #blobs - # holes =
Observed H e u r i s t c : P ( m a x 2 S Â ) E C = 5 3 7
Expected
Euler characteristic, EC
Threshold, t E ( C S \ X t ) = D d L
EXACT!

10. L i p s c h t z - K l n g u r v a e ( S ) E C d e n s i t y ½ ( ) ½ (\ X t ) = D d L = S d @ Z s L i p s c h t z - K l n g u r v a e d ( S ) E C d e n s i t y ( )
Steiner-Weyl Tube Formula (1930)
Morse Theory Approach (1995)
Put a tube of radius r about the search
region λS d ( t ) = 1 E f Z g e @ 2 s P r
Tube(λS,r) λS
For Z a Gaussian random field d ( t ) = 1 p 2 @ P Z
For Z a chi-bar random field???
Find volume, expand as a power series
in r, pull off coefficients: j T u b e ( S ; r ) = D X d 2 + 1 L

11. L i p s c h t z - K l n g u r v a e ( S ) o f d A r e a ( T u b ¸ S ;
Tube(λS,r) λS = S d @ Z s p 4 l o g 2 F W H M
Steiner-Weyl Volume of Tubes Formula (1930) A r e a ( T u b S ; ) = D X d 2 + 1 L P i m t E C
Lipschitz-Killing curvatures are just “intrinisic volumes” or “Minkowski functionals”
in the (Riemannian) metric of the variance of the derivative of the process

12. L i p s c h t z - K l n g u r v a e ( S ) o f y S S S d L ( ² ) = 1 ,= S d @ Z s
Edge length × λ
Lipschitz-Killing curvature of triangles L ( ) = 1 , N e d g l n t h 2 p r i m a
Lipschitz-Killing curvature of union of triangles L ( S ) = P + N 1 2

13. Non-isotropic data? Use Riemannian metric of Var(∇Z)
Z~N(0,1)
Z~N(0,1) s2 = S d @ Z s s1
Edge length × λ
Lipschitz-Killing curvature of triangles L ( ) = 1 , N e d g l n t h 2 p r i m a
Lipschitz-Killing curvature of union of triangles L ( S ) = P + N 1 2

14. E s t i m a n g L p c h z - K l u r v e ( S ) d R e p l a c o r d i n1 2 3 4 5 6 7 8 9 n E s t i m a n g L p c h z - K l u r v e d ( S )
We need independent & identically distributed random fields
e.g. residuals from a linear model …
Lipschitz-Killing curvature of triangles R e p l a c o r d i n t s f h g S
< 2 b y m u Z j ; = ( 1 : ) L ( ) = 1 , N e d g l n t h 2 p r i m a
Lipschitz-Killing curvature of union of triangles L ( S ) = P + N 1 2
Taylor & Worsley, JASA (2007)

15. Beautiful symmetry: L i p s c h t z - K l n g u r v a e ( S ) E C d e\ X t ) = D d L
Beautiful symmetry: = S d @ Z s L i p s c h t z - K l n g u r v a e d ( S ) E C d e n s i t y ( )
Steiner-Weyl Tube Formula (1930)
Taylor Gaussian Tube Formula (2003)
Put a tube of radius r about the search region λS and rejection region Rt:
Z2~N(0,1) Rt r
Tube(λS,r)
Tube(Rt,r) r λS
Z1~N(0,1)
t-r t
Find volume or probability, expand as a power series in r, pull off coefficients: j T u b e ( S ; r ) = D X d 2 + 1 L P ( T u b e R t ; r ) = 1 X d 2 !

16. E C d e n s i t y ½ ( ) o f h ¹ Â a c P ( Z ; T u b e R r ) = X ¼ ! ½
Z2~N(0,1) E C d e n s i t y ( ) o f h  a c
Rejection region Rt
Tube(Rt,r) r
Z1~N(0,1)
t-r t
Taylor’s Gaussian Tube Formula (2003) P ( Z 1 ; 2 T u b e R t r ) = X d ! + z 4 3 8 .
Taylor & Worsley, Annals of Statistics, submitted (2007)

17. General cone alternativesZ = ( 1 ; : n ) N I 2 . W w i s h o H v C f a u U S p r g k R c q l B m x j
& d y , T K 9 7 P X F b E A

18. Proof, n=3:

19. Gaussian random field in 3DZ ( s ) N ; 1 i a n o t r p c G u d m e l , 2
< 3 w h = V @ E C S \ : g z + D A v F W H M x 4 . L ( S ) 1 2 3 ( t ) 1 2 3
Lipschitz-Killing
curvatures of S
EC densities
of Z
filter
FWHM

20. The accuracy of the EC heuristicZ ( s ) N ; 1 i a n o t r p c G u d m e l , 2
< 3 w h = V @
> P x S E C \ : g + O z D A L ( S ) 1 2 3 ( t ) 1 2 3
Lipschitz-Killing
curvatures of S
EC densities
of Z
The expected EC gives all the polynomial
terms in the expansion for the P-value.

21. What is ‘bubbles’?

22. Nature (2005)

23. Subject is shown one of 40 faces chosen at random …
Happy
Sad
Fearful
Neutral

24. … but face is only revealed through random ‘bubbles’
First trial: “Sad” expression
Subject is asked the expression: “Neutral”
Response: Incorrect
75 random bubble centres
Smoothed by a
Gaussian ‘bubble’
What the
subject sees
Sad

25. Your turn …
Trial 2
Subject response:
“Fearful”
CORRECT

26. Your turn …
Trial 3
Subject response:
“Happy”
INCORRECT
(Fearful)

27. Your turn …
Trial 4
Subject response:
“Happy”
CORRECT

28. Your turn …
Trial 5
Subject response:
“Fearful”
CORRECT

29. Your turn …
Trial 6
Subject response:
“Sad”
CORRECT

30. Your turn …
Trial 7
Subject response:
“Happy”
CORRECT

31. Your turn …
Trial 8
Subject response:
“Neutral”
CORRECT

32. Your turn …
Trial 9
Subject response:
“Happy”
CORRECT

33. Your turn …
Trial 3000
Subject response:
“Happy”
INCORRECT
(Fearful)

34. E.g. Fearful (3000/4=750 trials):
Bubbles analysis
E.g. Fearful (3000/4=750 trials):
Trial
… + 750
= Sum
Correct
trials
Thresholded at
proportion of
correct trials=0.68,
scaled to [0,1]
Use this
as a
bubble
mask
Proportion of correct bubbles
=(sum correct bubbles)
/(sum all bubbles)

35. Happy Sad Fearful Neutral
Results
Mask average face
But are these features real or just noise?
Need statistics …
Happy Sad Fearful Neutral

36. Very similar to the proportion of correct bubbles:
Statistical analysis
Correlate bubbles with response (correct = 1, incorrect = 0), separately for each expression
Equivalent to 2-sample Z-statistic for correct vs. incorrect bubbles, e.g. Fearful:
Very similar to the proportion of correct bubbles:
Z~N(0,1)
statistic
Trial …
Response …

37. Happy Sad Fearful Neutral
Results
Thresholded at Z=1.64 (P=0.05)
Multiple comparisons correction?
Need random field theory …
Z~N(0,1)
statistic
Average face
Happy Sad Fearful Neutral

38. Euler Characteristic = #blobs - #holes
Excursion set {Z > threshold} for neutral face
EC =
Heuristic:
At high thresholds t,
the holes disappear,
EC ~ 1 or 0,
E(EC) ~ P(max Z > t).
Exact expression for E(EC) for all thresholds,
E(EC) ~ P(max Z > t) is extremely accurate.

39. The result I f Z ( s ) » N ; 1 i a n o t r p c G u d m ¯ e l , 2 <; 1 i a n o t r p c G u d m e l , 2
< 3 w h = V @ P x S E C \ : g z + D A v F W H M 4 . L ( S ) 1 2 3 ( t ) 1 2 3
Lipschitz-Killing
curvatures of S
EC densities
of Z
filter
FWHM

40. Results, corrected for search
Random field theory threshold: Z=3.92 (P=0.05)
Bonferroni threshold: Z=4.87 (P=0.05) – nothing
Z~N(0,1)
statistic
Average face
Happy Sad Fearful Neutral

41. 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

42. Thresholding? Cross correlation random field
Correlation between 2 fields at 2 different locations,
searched over all pairs of locations, one in S, one in T:
Bubbles data: P=0.05, n=3000, c=0.113, T=6.22 P m a x s 2 S ; t T C ( ) c E f : g = d i X j L n h 1 ! + b k l
Cao & Worsley, Annals of Applied Probability (1999)

43. MS lesions and cortical thickness
Idea: MS lesions interrupt neuronal signals, causing thinning in down-stream cortex
Data: n = 425 mild MS patients
Lesion density, smoothed 10mm
Cortical thickness, smoothed 20mm
Find connectivity i.e. find voxels in 3D, nodes in 2D with high correlation(lesion density, cortical thickness)
Look for high negative correlations …
Threshold: P=0.05, c=0.300, T=6.48

44. n=425 subjects, correlation = -0.56810 20 30 40 50 60 70 80
1.5 2
2.5 3
3.5 4
4.5 5
5.5
Average cortical thickness
Average lesion volume

45. Discussion: modeling
The random response is Y=1 (correct) or 0 (incorrect), or Y=fMRI
The regressors are Xj=bubble mask at pixel j, j=1 … 240x380=91200 (!)
Logistic regression or ordinary regression:
logit(E(Y)) or E(Y) = b0+X1b1+…+X91200b91200
But there are only n=3000 observations (trials) …
Instead, since regressors are independent, fit them one at a time:
logit(E(Y)) or E(Y) = b0+Xjbj
However the regressors (bubbles) are random with a simple known distribution, so turn the problem around and condition on Y:
E(Xj) = c0+Ycj
Equivalent to conditional logistic regression (Cox, 1962) which gives exact inference for b1 conditional on sufficient statistics for b0
Cox also suggested using saddle-point approximations to improve accuracy of inference …
Interactions? logit(E(Y)) or E(Y)=b0+X1b1+…+X91200b91200+X1X2b1,2+ …

46. Three methods so far The set-up:
S is a subset of a D-dimensional lattice (e.g. pixels);
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)

47. Simulations (99999) Bonferroni Random field theory1 2 3 4 5 6 7 8 9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
P value
FWHM (Full Width at Half Maximum) of smoothing filter -2
Simulations (99999)
Bonferroni
Random field theory
Discrete local maxima
Z(s)

48. Discrete local maxima Bonferroni applied to events:
{Z(s) ≥ t and Z(s) is a discrete local maximum} i.e.
{Z(s) ≥ t and neighbour Z’s ≤ Z(s)}
Conservative
If Z(s) is stationary, with
Cor(Z(s1),Z(s2)) = ρ(s1-s2),
all we need is
P{Z(s) ≥ t and neighbour Z’s ≤ Z(s)}
a (2D+1)-variate integral
Z(s2) ≤
Z(s-1) ≤ Z(s) ≥ Z(s1) ≥
Z(s-2)

49. “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} = ПdP{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)

50. 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

51. Comparison Bonferroni (1936) 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 (but no proof!)
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