function [ p, e ] = qsimvnv( m, r, a, b )
% Numerical Computation of Multivariate Normal Probabilities
%
% [ P E ] = QSIMVNV( M, R, A, B )
% uses a randomized quasi-random rule with m points to estimate an
% MVN probability for positive definite covariance matrix r,
% with lower integration limit column vector a and upper
% integration limit column vector b.
% Probability p is output with error estimate e.
% Example:
% >> r = [4 3 2 1;3 5 -1 1;2 -1 4 2;1 1 2 5];
% >> a = -inf*[1 1 1 1 ]'; b = [ 1 2 3 4 ]';
% >> [ p e ] = qsimvnv( 5000, r, a, b ); disp([ p e ])
%
% This function uses an algorithm given in the paper
% "Numerical Computation of Multivariate Normal Probabilities", in
% J. of Computational and Graphical Stat., 1(1992), pp. 141-149, by
% Alan Genz, WSU Math, PO Box 643113, Pullman, WA 99164-3113
% Email: alangenz@wsu.edu
% The primary references for the numerical integration are
% "On a Number-Theoretical Integration Method"
% H. Niederreiter, Aequationes Mathematicae, 8(1972), pp. 304-11, and
% "Randomization of Number Theoretic Methods for Multiple Integration"
% R. Cranley and T.N.L. Patterson, SIAM J Numer Anal, 13(1976), pp. 904-14.
% Cosmetic revisions were made to this code by Tom Davis (tdavis@
% metzgerwillard.com) on 4/23/2007 to improve MATLAB Central File
% Exchange code metrics. See (http://www.math.wsu.edu/faculty/genz/
% software/matlab/qsimvnv.m) for the latest version from the author.
% Alan Genz is the author of this function and following Matlab functions.
%
% Initialization
%
n = length(r); [ ch as bs ] = chlrdr( r, a, b );
ct = ch(1,1); ai = as(1); bi = bs(1);
if ai > -9*ct, if ai < 9*ct, c = phi(ai/ct); else c=1; end, else c=0; end
if bi > -9*ct, if bi < 9*ct, d = phi(bi/ct); else d=1; end, else d=0; end
ci = c; dci = d - ci; p = 0; e = 0;
ns = 8; nv = fix( max( [ m/( 2*ns ) 1 ] ) );
q = 2.^( (1:n-1)'/n) ; % Niederreiter point set generators
%
% Randomization loop for ns samples
%
for i = 1 : ns
% periodizing transformation
xx(:,1:nv) = abs( 2*mod( q*(1:nv) + rand( n-1, 1 )*ones(1,nv), 1 ) - 1 );
vp = mvndns( n, nv, ch, ci, dci, xx, as, bs );
vp = ( mvndns( n, nv, ch, ci, dci, 1-xx, as, bs ) + vp )/2; % symmetrize
d = ( mean(vp) - p )/i; p = p + d; e = ( i - 2 )*e/i + d^2;
end
%
e = 3*sqrt(e); % error estimate is 3 x standard error with ns samples.
return
%
% end qsimvn
%
function p = mvndns( n, nv, ch, ci, dci, x, a, b )
%
% Transformed integrand for computation of MVN probabilities.
%
y = zeros(n-1,nv); c = ci*ones(1,nv); dc = dci*ones(1,nv); p = dc;
for i = 2 : n
y(i-1,:) = phinv( c + x(i-1,:).*dc ); s = ch(i,1:i-1)*y(1:i-1,:);
ct = ch(i,i)*ones(1,nv); ai = a(i) - s; bi = b(i) - s;
c = ones( 1, nv ); d = c;
c( find( ai <= -9*ct ) ) = 0; d( find( bi <= -9*ct ) ) = 0; %#ok
tstl = find( ai > -9*ct & ai < 9*ct ); c(tstl) = phi( ai(tstl)./ct(tstl) );
tstl = find( bi > -9*ct & bi < 9*ct ); d(tstl) = phi( bi(tstl)./ct(tstl) );
dc = d - c; p = p.*dc;
end
return
%
% end mvndns
%
function [ c, ap, bp ] = chlrdr( R, a, b )
%
% Computes permuted lower Cholesky factor c for R which may be singular,
% also permuting integration limit vectors a and b.
%
ep = 1e-10; % singularity tolerance;
%
n = length(R); c = R; ap = a; bp = b; d = sqrt(max(diag(c),0));
for i = 1 : n
if d(i) > 0
c(:,i) = c(:,i)/d(i); c(i,:) = c(i,:)/d(i);
ap(i) = ap(i)/d(i); bp(i) = bp(i)/d(i);
end
end
y = zeros(n,1); sqtp = sqrt(2*pi);
for k = 1 : n
im = k; ckk = 0; dem = 1; s = 0;
for i = k : n
if c(i,i) > eps
cii = sqrt( max( [c(i,i) 0] ) );
if i > 1, s = c(i,1:k-1)*y(1:k-1); end
ai = ( ap(i)-s )/cii; bi = ( bp(i)-s )/cii; de = phi(bi) - phi(ai);
if de <= dem, ckk = cii; dem = de; am = ai; bm = bi; im = i; end
end
end
if im > k
tv = ap(im); ap(im) = ap(k); ap(k) = tv;
tv = bp(im); bp(im) = bp(k); bp(k) = tv;
c(im,im) = c(k,k);
t = c(im,1:k-1); c(im,1:k-1) = c(k,1:k-1); c(k,1:k-1) = t;
t = c(im+1:n,im); c(im+1:n,im) = c(im+1:n,k); c(im+1:n,k) = t;
t = c(k+1:im-1,k); c(k+1:im-1,k) = c(im,k+1:im-1)'; c(im,k+1:im-1) = t';
end
if ckk > ep*k^2
c(k,k) = ckk; c(k,k+1:n) = 0;
for i = k+1 : n
c(i,k) = c(i,k)/ckk; c(i,k+1:i) = c(i,k+1:i) - c(i,k)*c(k+1:i,k)';
end
if abs(dem) > ep
y(k) = ( exp( -am^2/2 ) - exp( -bm^2/2 ) )/( sqtp*dem );
else
if am < -10
y(k) = bm;
elseif bm > 10
y(k) = am;
else
y(k) = ( am + bm )/2;
end
end
else
c(k:n,k) = 0; y(k) = 0;
end
end
return
%
% end chlrdr
%
function p = phi(z)
%
% Standard statistical normal distribution
%
p = erfc( -z/sqrt(2) )/2;
return
%
% end phi
%
function z = phinv(w)
%
% Standard statistical inverse normal distribution
%
z = -sqrt(2)*erfcinv( 2*w );
return
%
% end phinv
