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

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

Rest Mozart Eminem J. Brown Temporal components (sd, % variance explained) Period: 5.2 16.1 15.6 11.6 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)

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

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 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 µ 2 S ¸ ¶ ¼ 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

? 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

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

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)

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)

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 ¼ © Z ¯ Q + ® µ ³ ´ - f m S ¸ ¶ · Y

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

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

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 ³ @ ´ ¤ £ p b y L z - K g v S P µ x ¸ ¶ ¼ E C \ X ½ & A 3 B ± R j ¡ [ F W 7

Referee report Why bother? Why not just do simulations?

‘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 1 2 3 4 5 6 7 … 3000 fMRI