spm_P_RF.m

```function [P,p,Em,En,EN] = spm_P_RF(c,k,Z,df,STAT,R,n)
% Returns the [un]corrected P value using unifed EC theory
% FORMAT [P p Em En EN] = spm_P_RF(c,k,Z,df,STAT,R,n)
%
% c     - cluster number
% k     - extent {RESELS}
% Z     - height {minimum over n values}
% df    - [df{interest} df{error}]
% STAT  - Statisical feild
%		'Z' - Gaussian feild
%		'T' - T - feild
%		'X' - Chi squared feild
%		'F' - F - feild
% R     - RESEL Count {defining search volume}
% n     - number of component SPMs in conjunction
%
% P     - corrected   P value  - P(n > kmax}
% p     - uncorrected P value  - P(n > k}
% Em    - expected total number of maxima {m}
% En    - expected total number of resels per cluster {n}
% EN    - expected total number of voxels {N}
%
%___________________________________________________________________________
%
% spm_P_RF returns the probability of c or more clusters with more than
% k voxels in volume process of R RESELS thresholded at u.  All p values
% can be considered special cases:
%
% spm_P_RF(1,0,Z,df,STAT,1,n) = uncorrected p value
% spm_P_RF(1,0,Z,df,STAT,R,n) = corrected p value {based on height Z)
% spm_P_RF(1,k,u,df,STAT,R,n) = corrected p value {based on extent k at u)
% spm_P_RF(c,k,u,df,STAT,R,n) = corrected p value {based on number c at k and u)
% spm_P_RF(c,0,u,df,STAT,R,n) = omnibus   p value {based on number c at u)
%
% If n > 1 a conjunction probility over the n values of the statistic
% is returned
%
% Ref: Hasofer AM (1978) Upcrossings of random fields
% Suppl Adv Appl Prob 10:14-21
% Ref: Friston et al (1993) Comparing functional images: Assessing
% the spatial extent of activation foci
% Ref: Worsley KJ et al 1996, Hum Brain Mapp. 4:58-73
%___________________________________________________________________________
% @(#)spm_P_RF.m	1.4 Karl Friston 01/07/29

% get expectations
%===========================================================================

% get EC densities
%---------------------------------------------------------------------------
D       = max(find(R));
R       = R(1:D);
G       = sqrt(pi)./gamma(([1:D])/2);
EC      = spm_ECdensity(STAT,Z,df);
EC      = EC([1:D]);

% corrected p value
%---------------------------------------------------------------------------
P       = triu(toeplitz(EC'.*G))^n;
P       = P(1,:)';
Em      = (R./G)*P;
EN      = P(1)*R(D);
En      = G(D)*P(1)/(eps+P(D));				% i.e. En = EN/Em;

% get P{n > k}
%===========================================================================

% assume a Gaussian form for P{n > k} ~ exp(-beta*k^(2/D))
% Appropriate for SPM{Z} and high d.f. SPM{T}
%---------------------------------------------------------------------------
D       = D - 1;
if     ~k | ~D

p    = 1;

elseif STAT == 'Z'

beta = (gamma(D/2 + 1)/En)^(2/D);
p    = exp(-beta*(k^(2/D)));

elseif STAT == 'T'

beta = (gamma(D/2 + 1)/En)^(2/D);
p    = exp(-beta*(k^(2/D)));

elseif STAT == 'X'

beta = (gamma(D/2 + 1)/En)^(2/D);
p    = exp(-beta*(k^(2/D)));

elseif STAT == 'F'

beta = (gamma(D/2 + 1)/En)^(2/D);
p    = exp(-beta*(k^(2/D)));

end

% Poisson clumping heuristic {for multiple clusters}
%===========================================================================
P       = 1 - spm_Pcdf(c - 1,(Em + eps)*p);

% set P and p = [] for non-implemented cases
%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
if k > 0 & n > 1
P    = []; p = [];
end
if k > 0 & (STAT == 'X' | STAT == 'F')
P    = []; p = [];
end
%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

% spm_ECdensity
%===========================================================================

function [EC] = spm_ECdensity(STAT,t,df)
% Returns the EC density
%___________________________________________________________________________
%
% Reference : Worsley KJ et al 1996, Hum Brain Mapp. 4:58-73
%
%---------------------------------------------------------------------------

% EC densities (EC}
%---------------------------------------------------------------------------
t     = t(:)';
if      STAT == 'Z'

% Gaussian Field
%-------------------------------------------------------------------
a       = 4*log(2);
b       = exp(-t.^2/2);

EC(1,:) = 1 - spm_Ncdf(t);
EC(2,:) = a^(1/2)/(2*pi)*b;
EC(3,:) = a/((2*pi)^(3/2))*b.*t;
EC(4,:) = a^(3/2)/((2*pi)^2)*b.*(t.^2 - 1);

elseif  STAT == 'T'

% T - Field
%-------------------------------------------------------------------
v       = df(2);
a       = 4*log(2);
b       = exp(gammaln((v+1)/2) - gammaln(v/2));
c       = (1+t.^2/v).^((1-v)/2);

EC(1,:) = 1 - spm_Tcdf(t,v);
EC(2,:) = a^(1/2)/(2*pi)*c;
EC(3,:) = a/((2*pi)^(3/2))*c.*t/((v/2)^(1/2))*b;
EC(4,:) = a^(3/2)/((2*pi)^2)*c.*((v-1)*(t.^2)/v - 1);

elseif  STAT == 'X'

% X - Field
%-------------------------------------------------------------------
v       = df(2);
a       = (4*log(2))/(2*pi);
b       = t.^(1/2*(v - 1)).*exp(-t/2-gammaln(v/2))/2^((v-2)/2);

EC(1,:) = 1 - spm_Xcdf(t,v);
EC(2,:) = a^(1/2)*b;
EC(3,:) = a*b.*(t-(v-1));
EC(4,:) = a^(3/2)*b.*(t.^2-(2*v-1)*t+(v-1)*(v-2));

elseif  STAT == 'F'

% F Field
%-------------------------------------------------------------------
k       = df(1);
v       = df(2);
a       = (4*log(2))/(2*pi);
b       = gammaln(v/2) + gammaln(k/2);

EC(1,:) = 1 - spm_Fcdf(t,df);
EC(2,:) = a^(1/2)*exp(gammaln((v+k-1)/2)-b)*2^(1/2)...
*(k*t/v).^(1/2*(k-1)).*(1+k*t/v).^(-1/2*(v+k-2));
EC(3,:) = a*exp(gammaln((v+k-2)/2)-b)*(k*t/v).^(1/2*(k-2))...
.*(1+k*t/v).^(-1/2*(v+k-2)).*((v-1)*k*t/v-(k-1));
EC(4,:) = a^(3/2)*exp(gammaln((v+k-3)/2)-b)...
*2^(-1/2)*(k*t/v).^(1/2*(k-3)).*(1+k*t/v).^(-1/2*(v+k-2))...
.*((v-1)*(v-2)*(k*t/v).^2-(2*v*k-v-k-1)*(k*t/v)+(k-1)*(k-2));
end```