function [ p, e ] = qscmvn( m, r, a, cn, b )
%
%  [ P E ] = QSCMVN( M, R, A, CN, B )
%    uses a randomized quasi-random rule with m points to estimate an
%    MVN probability for positive semi-definite covariance matrix r,
%    with constraints a < cn*x < b. If r is nxn and cn is kxn, then
%    a and b must be column k-vectors.
%   Probability p is output with error estimate e.
%    Example usage:
%     >> r = [4 3 2 1;3 5 -1 1;2 -1 4 2;1 1 2 5];
%     >> a = [ -inf 1 -5 ]'; b = [ 3 inf 4 ]';
%     >> cn = [ 1 2 3 -2; 2 4 1 2; -2 3 4 1 ];
%     >> [ p e ] = qscmvn( 5000, r, a, cn, b ); disp([ p e ])
%
%  This function uses an algorithm given in the paper by Alan Genz:
%   "Numerical Computation of Multivariate Normal Probabilities", in
%     J. of Computational and Graphical Stat., 1(1992), 141-149.
%  The primary references for the numerical integration are 
%   "On a Number-Theoretical Integration Method"
%     H. Niederreiter, Aequationes Mathematicae, 8(1972), 304-11, and
%   "Randomization of Number Theoretic Methods for Multiple Integration"
%     R. Cranley and T.N.L. Patterson, SIAM J Numer Anal, 13(1976), 904-14.
%
%   Alan Genz is the author of this function and following Matlab functions.
%          Alan Genz, WSU Math, PO Box 643113, Pullman, WA 99164-3113
%          Email : AlanGenz@wsu.edu
%
% Initialization
%
[ as ch bs clg n ] = chlsrt( r, a, cn, b ); 
ci = phi( as(1) ); dci = phi( bs(1) ) - ci; p = 0; e = 0; 
ns = 12; nv = fix( max( [ m/ns 1 ] ) ); 
% q = 2.^( [1:n-1]'/n) ; % Niederreiter point set generators
ps = sqrt(primes(5*n*log(n+1)/4)); q = ps(1:n-1)'; % Richtmyer generators
%
% Randomization loop for ns samples
%
for i = 1 : ns
  vi = 0; xr = rand( n-1, 1 ); 
  %
  % Loop for nv quasirandom points
  %
  for  j = 1 : nv
    x = abs( 2*mod( j*q + xr, 1 ) - 1 );                        % periodize
    vp = mvndnt( n, as, ch, bs, clg, ci, dci,   x ); 
    vi = vi + ( vp - vi )/j; 
  end   
  %
  d = ( vi - p )/i; p = p + d; 
  if abs(d) > 0 
    e = abs(d)*sqrt( 1 + ( e/d )^2*( i - 2 )/i );
  else
    if i > 1, e = e*sqrt( ( i - 2 )/i ); end
  end
end
%
e = 3*e; % error estimate is 3 x standard error with ns samples.
%
% end qscmvn
%
function p = mvndnt( n, a, ch, b, clg, ci, dci, x )
%
%  Transformed integrand for computation of MVN probabilities. 
%
y = zeros(n-1,1); c = ci; dc = dci; p = dc; li = 2; lf = 1;
for i = 2 : n 
   y(i-1) = phinv( c + x(i-1)*dc ); lf = lf + clg(i); 
   if lf < li, c = 0; dc = 1;
   else, s = ch(li:lf,1:i-1)*y(1:i-1); 
     ai = max( max( a(li:lf) - s ), -9 ); 
     bi = max( ai, min( min( b(li:lf) - s ),  9 ) ); 
     c = phi(ai); dc = phi(bi) - c; p = p*dc; 
   end, li = li + clg(i);
end 
%
% end mvndnt
%
function [ ap, ch, bp, clg, np ] = chlsrt( r, a, cn, b )
%
%  Computes permuted lower Cholesky factor ch for covariance r which 
%   may be singular, combined with contraints a < cn*x < b, to
%   form revised lower triangular constraint set ap < ch*x < bp; 
%   clg contains information about structure of cn: clg(1) rows for 
%   ch with 1 nonzero, ..., clg(np) rows with np nonzeros.
%
ep = 1e-10; % singularity tolerance;
%
[n n] = size(r); [m n] = size(cn); ch = cn; np = 0;
ap = a; bp = b; y = zeros(n,1); sqtp = sqrt(2*pi);
c = r; d = sqrt(max(diag(c),0));
for i = 1 : n, di = d(i);
  if di > 0
    c(:,i) = c(:,i)/di; c(i,:) = c(i,:)/di; ch(:,i) = ch(:,i)*di;
  end
end
%
% determine (with pivoting) Cholesky factor for r 
%  and form revised constraint matrix ch
%
for i = 1 : n
  clg(i) = 0; epi = ep*i^2; j = i; 
  for l = i+1 : n, if c(l,l) > c(j,j), j = l; end, end
  if j > i
    t = c(i,i); c(i,i) = c(j,j); c(j,j) = t;
    t = c(i,1:i-1); c(i,1:i-1) = c(j,1:i-1); c(j,1:i-1) = t;
    t = c(i+1:j-1,i); c(i+1:j-1,i) = c(j,i+1:j-1)'; c(j,i+1:j-1) = t';
    t = c(j+1:n,i); c(j+1:n,i) = c(j+1:n,j); c(j+1:n,j) = t;
    t = ch(:,i); ch(:,i) = ch(:,j); ch(:,j) = t;
  end
  if c(i,i) < epi, break, end, cvd = sqrt( c(i,i) ); c(i,i) = cvd;
  for l = i+1 : n
    c(l,i) = c(l,i)/cvd; c(l,i+1:l) = c(l,i+1:l) - c(l,i)*c(i+1:l,i)';
  end
  ch(:,i) = ch(:,i:n)*c(i:n,i); np = np + 1;
end
%
% use right reflectors to reduce ch to lower triangular
%
for i = 1 : min( np-1, m )
  epi = ep*i*i; vm = 1; lm = i;
  %
  % permute rows so that smallest variance variables are first.
  %
  for l = i : m    
    v = ch(l,1:np); s = v(1:i-1)*y(1:i-1); 
    ss = max( sqrt( sum( v(i:np).^2 ) ), epi ); 
    al = ( ap(l) - s )/ss; bl = ( bp(l) - s )/ss; 
    dna = 0; dsa = 0; dnb = 0; dsb = 1;
    if al > -9, dna = exp(-al*al/2)/sqtp; dsa = phi(al); end
    if bl <  9, dnb = exp(-bl*bl/2)/sqtp; dsb = phi(bl); end
    if dsb - dsa > epi
      if      al <= -9, mn =      -dnb; vr =         -bl*dnb;
      elseif  bl >=  9, mn = dna;       vr = al*dna; 
      else,             mn = dna - dnb; vr = al*dna - bl*dnb; 
      end, mn = mn/( dsb - dsa ); vr = 1 + vr/( dsb - dsa ) - mn^2;
    else
      if     al <= -9,   mn = bl; 
      elseif bl >=  9,   mn = al; 
      else,              mn = ( al + bl )/2;  
      end, vr = 0;
    end
    if vr <= vm, lm = l; vm = vr; y(i) = mn; end
  end
  v = ch(lm,1:np);
  if lm > i 
    ch(lm,1:np) = ch(i,1:np); ch(i,1:np) = v;
    tl = ap(i); ap(i) = ap(lm); ap(lm) = tl;
    tl = bp(i); bp(i) = bp(lm); bp(lm) = tl;
  end
  ch(i,i+1:np) = 0; ss = sum( v(i+1:np).^2 ); 
  if ( ss > epi )
    ss = sqrt( ss + v(i)^2 ); if v(i) < 0, ss = -ss; end
    ch(i,i) = -ss; v(i) = v(i) + ss; vt = v(i:np)'/( ss*v(i) );
    ch(i+1:m,i:np) = ch(i+1:m,i:np) - ch(i+1:m,i:np)*vt*v(i:np);
  end
end
%
% scale and sort constraints
%
for i = 1 : m
  v = ch(i,1:np); clm(i) = min(i,np); 
  jm = 1; for j = 1 : clm(i), if abs(v(j)) > ep*j*j, jm = j; end, end 
  if jm < np, v(jm+1:np) = 0; end, clg(jm) = clg(jm) + 1;
  at = ap(i); bt = bp(i); j = i;
  for l = i-1 : -1 : 1
    if jm >= clm(l), break, end
    ch(l+1,1:np) = ch(l,1:np); j = l;
    ap(l+1) = ap(l); bp(l+1) = bp(l); clm(l+1) = clm(l);
  end
  clm(j) = jm; vjm = v(jm); ch(j,1:np) = v/vjm;
  ap(j) = at/vjm; bp(j) = bt/vjm;
  if vjm < 0, tl = ap(j); ap(j) = bp(j); bp(j) = tl; end
end
j = 0; for i = 1 : np, if clg(i) > 0, j = i; end, end, np = j;
%
% combine constraints for first variable
%
if clg(1) > 1 
  ap(1) = max( ap(1:clg(1)) ); bp(1) = max( ap(1), min( bp(1:clg(1)) ) ); 
  ap(2:m-clg(1)+1) = ap(clg(1)+1:m); bp(2:m-clg(1)+1) = bp(clg(1)+1:m);
  ch(2:m-clg(1)+1,:) = ch(clg(1)+1:m,:); 
  clg(1) = 1;  
end
%
% end chlsrt
%
function p = phi(z)
%
%  Standard statistical normal distribution cdf
%
p = erfc( -z/sqrt(2) )/2;
%
% end phi
%
function z = phinv(w)
%
%  Standard statistical inverse normal distribution
%
z = -sqrt(2)*erfcinv( 2*w );
%
% end phinv