FAQ/multinorm - CBU statistics Wiki

You can upload content for the page named below. If you change the page name, you can also upload content for another page. If the page name is empty, we derive the page name from the file name.

File to load page content from
Page name
Comment
Flind the wroneg tetters tin eaech wrord

location: FAQ / multinorm

## Algorithms to calculate probabilities associated with a multivariate normal distribution

1. A list of software in MATLAB and/or FORTRAN prepared by Alan Genz
2. A listing of one MATLAB routine written by Genz.
3. A listing of MATLAB routine mvncdf.m from the MATLAB STatistics Toolbox.

Alan Genz has an interesting page of links to FORTRAN and MATLAB code: http://www.math.wsu.edu/faculty/genz/software/software.html.

The following MATLAB code was found at http://alex.strashny.org/b/mvncdf.m, via http://alex.strashny.org/a/Multivariate-normal-cumulative-distribution-function-(cdf)-in-MATLAB.html.

```function [p, err, N] = mvncdf(x, mu, Sigma, errMax, ci, Nmax)
%MVNCDF Multivariate normal cumulative distribution function (cdf).
%   P = MVNCDF(X,MU,SIGMA) computes the multivariate normal cdf
%   with mean vector MU and variance matrix SIGMA at the values in
%   vector X.
%
%   P = MVNCDF(X,MU,SIGMA,ERRMAX,CI,NMAX) uses additional control
%   parameters. The difference between P and the true value of the
%   cdf is less than ERRMAX CI percent of the time. NMAX is the
%   maximum number of iterations that the algorithm makes. By
%   default, ERRMAX is 0.01, CI is 99, and NMAX is 300.
%
%   [P,ERR,N] = MVNCDF(...) also returns the estimated error and the
%

%   Algorithm from Alan Genz (1992) Numerical Computation of
%   Multivariate Normal Probabilities, Journal of Computational and
%   Graphical Statistics, pp. 141-149.

%   Copyright 2005 Alex Strashny (alex@strashny.org)
%   version 1, April 29, 2005

%     This program is free software; you can redistribute it and/or modify
%     the Free Software Foundation; either version 2 of the License, or
%     (at your option) any later version.
%
%     This program is distributed in the hope that it will be useful,
%     but WITHOUT ANY WARRANTY; without even the implied warranty of
%     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
%     GNU General Public License for more details.
%
%     You should have received a copy of the GNU General Public License
%     along with this program; if not, write to the Free Software
%     Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301  USA

m1 = length(x);
m2 = length(mu);
[m3,m4] = size(Sigma);

if m1 ~= m2 | m1 ~= m3 | m1 ~= m4
error('Dimentions of X, MU, and SIGMA must agree.');
end;
m = m1;
x = x(:) - mu(:);

if nargin < 6
Nmax = 300;
end;
if nargin < 5
alph = 2.3;
else
alph = norminv(ci/100);
end;
if nargin < 4
errMax = 0.01;
end;

C = chol(Sigma)';

p = 0; N = 0; varSum = 0;

% d is always zero
f = zeros(m,1);
f(1) = normcdf(x(1) / C(1,1));

y = zeros(m,1);

err = 2 * errMax;
while err > errMax & N < Nmax
w = unifrnd(0,1,m-1,1);
for i = 2:m
y(i-1) = norminv(w(i-1)*f(i-1));
q = 0;
for j = 1:i-1
q = q + C(i,j)*y(j);
end;
f(i) = normcdf((x(i) - q) / C(i,i)) * f(i-1);
end;
N = N + 1;
del = (f(m) - p) / N;
p = p + del;
varSum = (N-2) * varSum / N + del^2;
err = alph * sqrt(varSum);
end;```

The following code is the function mvncdf.m from the MATLAB Statistics Toolbox version 5.3:

```function [y,err] = mvncdf(varargin)
%MVNCDF Multivariate normal cumulative distribution function (cdf).
%   Y = MVNCDF(X) returns the cumulative probability of the multivariate
%   normal distribution with zero mean and identity covariance matrix,
%   evaluated at each row of X.  Rows of the N-by-D matrix X correspond to
%   observations or points, and columns correspond to variables or
%   coordinates.  Y is an N-by-1 vector.
%
%   Y = MVNCDF(X,MU,SIGMA) returns the cumulative probability of the
%   multivariate normal distribution with mean MU and covariance SIGMA,
%   evaluated at each row of X.  MU is a 1-by-D vector, and SIGMA is a D-by-D
%   symmetric, positive definite matrix.  MU can also be a scalar value, which
%   MVNCDF replicates to match the size of X.  Pass in the empty matrix for MU
%   to use its default value when you want to only specify SIGMA.
%
%   The multivariate normal cumulative probability at X is defined as the
%   probability that a random vector V, distributed as multivariate normal,
%   will fall within the semi-infinite rectangle with upper limits defined by
%   X, i.e., Pr{V(1)<=X(1), V(2)<=X(2), ... V(D)<=X(D)}.
%
%   Y = MVNCDF(XL,XU,MU,SIGMA) returns the multivariate normal cumulative
%   probability evaluated over the rectangle with lower and upper limits
%   defined by XL and XU, respectively.
%
%   [Y,ERR] = MVNCDF(...) returns an estimate of the error in Y.  For
%   a transformation of the t density, based on methods developed by Drezner
%   and Wesolowsky, and by Genz, as described in the references.  The default
%   absolute error tolerance for these cases is 1e-8.  For four or more
%   dimensions, MVNCDF uses a quasi-Monte Carlo integration algorithm based on
%   methods developed by Genz and Bretz, as described in the references.  The
%   default absolute error tolerance for these cases is 1e-4.
%
%   [...] = MVNCDF(...,OPTIONS) specifies control parameters for the numerical
%   integration used to compute Y.  This argument can be created by a call to
%   STATSET.  Choices of STATSET parameters are:
%
%         'TolFun'      - Maximum absolute error tolerance.  Default is 1e-8
%                         when D < 4, or 1e-4 when D >= 4.
%         'MaxFunEvals' - Maximum number of integrand evaluations allowed when
%                         D >= 4.  Default is 1e7.  Ignored when D < 4.
%         'Display'     - Level of display output.  Choices are 'off' (the
%                         default), 'iter', and 'final'.  Ignored when D < 4.
%
%   Example:
%
%      mu = [1 -1]; Sigma = [.9 .4; .4 .3];
%      [X1,X2] = meshgrid(linspace(-1,3,25)', linspace(-3,1,25)');
%      X = [X1(:) X2(:)];
%      p = mvncdf(X, mu, Sigma);
%      surf(X1,X2,reshape(p,25,25));
%

%   References:
%       Drezner, Z. and G.O. Wesolowsky (1989) "On the Computation of the
%          Bivariate Normal Integral", J.Statist.Comput.Simul., 35:101-107.
%       Drezner, Z. (1994) "Computation of the Trivariate Normal Integral",
%          Mathematics of Computation, 63:289-294.
%       Genz, A. (2004) "Numerical Computation of Recutangular Bivariate
%          and Trivariate Normal and t Probabilities", Statistics and
%          Computing, 14(3):251-260.
%       Genz, A. and F. Bretz (1999) "Numerical Computation of Multivariate
%          t Probabilities with Application to Power Calculation of Multiple
%          Contrasts", J.Statist.Comput.Simul., 63:361-378.
%       Genz, A. and F. Bretz (2002) "Comparison of Methods for the
%          Computation of Multivariate t Probabilities", J.Comp.Graph.Stat.,
%          11(4):950-971.

%   Copyright 2005-2006 The MathWorks, Inc.
%   \$Revision: 1.1.6.1.6.1 \$  \$Date: 2006/07/13 16:54:04 \$

% Strip off an options structure if it's there.
if isstruct(varargin{end})
opts = statset(statset('mvncdf'), varargin{end});
nin = nargin - 1;
else
opts = statset('mvncdf');
nin = nargin;
end

if nin < 1
error('stats:mvncdf:TooFewInputs','Requires at least one input.');

elseif nin < 4 % MVNCDF(XU,MU,SIGMA)
upperLimitOnly = true;
XU = varargin{1};
if ndims(XU)~=2
error('stats:mvncdf:InvalidData','X must be a matrix.');
end
XL = -Inf(size(XU),class(XU));
if nin > 1, mu = varargin{2}; else mu = []; end
if nin > 2, Sigma = varargin{3}; else Sigma = []; end

else % MVNCDF(XL,XU,MU,SIGMA)
upperLimitOnly = false;
XL = varargin{1};
XU = varargin{2};
mu = varargin{3};
Sigma = varargin{4};
if ndims(XU)~=2 || ~isequal(size(XL),size(XU))
error('stats:mvncdf:InvalidData','XL and XU must be matrices and have the same size.');
elseif any(any(XL > XU))
error('stats:mvncdf:InvalidData','XL must be less than or equal to XU.');
end
end

% Get size of data.  Column vectors provisionally interpreted as multiple scalar data.
[n,d] = size(XU);
if d<1
error('stats:mvncdf:TooFewDimensions','X must have at least one column.');
end

% Assume zero mean, data are already centered
if isempty(mu)
XL0 = XL;
XU0 = XU;

% Get scalar mean, and use it to center data
elseif isscalar(mu)
XL0 = XL - mu;
XU0 = XU - mu;

% Get vector mean, and use it to center data
elseif isvector(mu)
[n2,d2] = size(mu);
if d2 ~= d % has to have same number of coords as X
error('stats:mvncdf:InputSizeMismatch',...
'MU must be a row vector with the same number of columns as X.');
elseif n2 == 1 % mean is a single row, rep it out to match data
XL0 = XL - repmat(mu,n,1);
XU0 = XU - repmat(mu,n,1);
elseif n2 == n
% if X and mu are column vectors and lengths match, provisionally
% interpret this as multivariate data
XL0 = XL - mu;
XU0 = XU - mu;
else % sizes don't match
error('stats:mvncdf:InputSizeMismatch',...
'MU must be a row vector with the same number of columns as X.');
end
else
error('stats:mvncdf:InvalidMu', 'MU must be a row vector.');
end

% Assume identity covariance, data are already standardized
if isempty(Sigma)
% Special case: if Sigma isn't supplied, then interpret X
% and MU as row vectors if they were both column vectors
if d == 1
XL0 = XL0';
XU0 = XU0';
[n,d] = size(XU0);
end
Sigma = eye(d);

else
% Special case: if Sigma is supplied, then use it to try to interpret
% X and MU as row vectors if they were both column vectors.
if (d == 1)
if size(Sigma,1) == n
XL0 = XL0';
XU0 = XU0';
[n,d] = size(XU0);
elseif ~isscalar(mu)
error('stats:mvncdf:InputSizeMismatch',...
'MU must be a row vector with the same number of columns as X.');
end
end

% Make sure Sigma is a valid covariance matrix
sz = size(Sigma);
if sz(1) ~= sz(2)
'SIGMA must be a square matrix.');
elseif ~isequal(sz, [d d])
error('stats:mvncdf:InputSizeMismatch',...
'SIGMA must be a square matrix with size equal to the number of columns in X.');
else
[T,err] = statchol(Sigma,0);
if err ~= 0
'SIGMA must be symmetric and positive definite.');
end
end
end

% Standardize Sigma and X to correlation if necessary
s = sqrt(diag(Sigma));
Rho = Sigma ./ (s*s');
XL0 = XL0 ./ repmat(s',n,1);
XU0 = XU0 ./ repmat(s',n,1);

% Call the appropriate integration routine for the umber of dimensions.
if d == 1
y = normcdf(XU0,0,1) - normcdf(XL0,0,1);
if nargout > 1
err = NaN(size(y),class(y));
end

elseif d <= 3
tol = opts.TolFun; if isempty(tol), tol = 1e-8; end
if d == 2, rho = Rho(2); else rho = Rho([2 3 6]); end
if upperLimitOnly
if d == 2
y = bvncdf(XU0, rho, tol);
else
y = tvncdf(XU0, rho, tol);
end
else % lower and upper limits
% Compute the probability over the rectangle as sums and differences
% of integrals over semi-infinite half-rectangles.  For degenerate
% rectangles, force an exact zero by making each piece exactly zero.
equalLims = (XL0==XU0);
XL0(equalLims) = -Inf;
XU0(equalLims) = -Inf;
y = zeros(n,1,superiorfloat(XL0,XU0,Rho));
for i = 0:d
k = nchoosek(1:d,i);
for j = 1:size(k,1)
X = XU0; X(:,k(j,:)) = XL0(:,k(j,:));
if d == 2
y = y + (-1)^i * bvncdf(X, rho, tol/4);
else
y = y + (-1)^i * tvncdf(X, rho, tol/8);
end
end
end
end
if nargout > 1
err = repmat(cast(tol,class(y)),size(y));
end

elseif d <= 25
tol = opts.TolFun; if isempty(tol), tol = 1e-4; end
maxfunevals = opts.MaxFunEvals;
verbose = find(strcmp(opts.Display,{'off' 'final' 'iter'})) - 1;
y = zeros(n,1,superiorfloat(XL0,XU0,Rho));
err = zeros(n,1,class(y));
for i = 1:n
[y(i),err(i)] = mvtcdfqmc(XL0(i,:),XU0(i,:),Rho,Inf,tol,maxfunevals,verbose);
end

else
error('stats:mvncdf:DimensionTooLarge',...
'Number of dimensions must be less than or equal to 25.');
end

end

%----------------------------------------------------------------------
function p = bvncdf(b,rho,tol)
% CDF for the bivariate normal.
%
% Implements the unnumbered equation between (3) and (4) in Section 2.2 of
% Genz (2004), integrating in terms of theta between asin(rho) and +/- pi/2,

n = size(b,1);
if rho == 0
p = cast(prod(normcdf(b),2), superiorfloat(b,rho));
else
if rho > 0
p1 = normcdf(min(b,[],2));
p1(any(isnan(b),2)) = NaN;
else
p1 = normcdf(b(:,1))-normcdf(-b(:,2));
p1(p1<0) = 0; % max would drop NaNs
end
if abs(rho) < 1
loLimit = asin(rho);
hiLimit = sign(rho).*pi./2;
p2 = zeros(size(p1),class(p1));
for i = 1:n
b1 = b(i,1); b2 = b(i,2);
if isfinite(b1) && isfinite(b2)
c1 = ((b1-b2).^2) ./ 2;
c2 = b1.*b2;
if c1 > 0 || c2 ~= 0 % b1 ~= b2 or b1 == b2 but ~= 0
else % b1 == b2 == 0
% get this case exactly
p2(i) = hiLimit - loLimit;
end
else
% This piece is zero if either limit is +/- infinity.  If
% either is NaN, p1 will already be NaN.
end
end
else
p2 = zeros(class(p1));
end
p = cast(p1 - p2./(2.*pi), superiorfloat(b,rho));
end

function integrand = bvnIntegrand(theta)
% exp(-(b1.^2 + b2.^2 - 2*b1*b2*sin(theta))/(2*cos(theta).^2));
sintheta = sin(theta);
cossqtheta = cos(theta).^2;
nz = (cossqtheta > 0);
if all(nz)
expon = (c1 + c2.*(1-sintheta))./cossqtheta;
else % some theta == +/- pi
expon = -Inf(size(theta)); % cos^2 -> 0 faster than numerator can
expon(nz) = (c1 + c2.*(1-sintheta(nz)))./cossqtheta(nz);
end
integrand = exp(-expon);
end
end

%----------------------------------------------------------------------
function p = tvncdf(b,rho,tol)
% CDF for the trivariate normal
%
% Implements equation (14) in Section 3.2 of Genz (2004), integrating each
% term in (14) separately in terms of theta between 0 and asin(rho_j1), using

n = size(b,1);

% Find a permutation that makes rho_32 == max(rho)
[dum,imax] = max(abs(rho));
if imax == 1 % swap 1 and 3
rho_21 = rho(3); rho_31 = rho(2); rho_32 = rho(1);
b = b(:,[3 2 1]);
elseif imax == 2 % swap 1 and 2
rho_21 = rho(1); rho_31 = rho(3); rho_32 = rho(2);
b = b(:,[2 1 3]);
else % imax == 3
rho_21 = rho(1); rho_31 = rho(2); rho_32 = rho(3);
% b already in correct order
end

% CDF for the normal distribution.
Phi = @(z) 0.5 * erfc(-z ./ sqrt(2));

p1 = Phi(b(:,1)).*bvncdf(b(:,2:3),rho_32,tol/3);

if abs(rho_21) > 0
loLimit = 0;
hiLimit = asin(rho_21);
rho_j1 = rho_21;
rho_k1 = rho_31;
p2 = zeros(size(p1),class(p1));
for i = 1:n
b1 = b(i,1); bj = b(i,2); bk = b(i,3);
if isfinite(b1) && isfinite(bj) && ~isnan(bk)
c1 = ((b1-bj).^2) ./ 2;
c2 = b1.*bj;
else
% This piece is zero if either limit is +/- infinity.  If
% either is NaN, p1 will already be NaN.
end
end
else
p2 = zeros(class(p1));
end

if abs(rho_31) > 0
loLimit = 0;
hiLimit = asin(rho_31);
rho_j1 = rho_31;
rho_k1 = rho_21;
p3 = zeros(size(p1),class(p1));
for i = 1:n
b1 = b(i,1); bj = b(i,3); bk = b(i,2);
if isfinite(b1) && isfinite(bj) && ~isnan(bk)
c1 = ((b1-bj).^2) ./ 2;
c2 = b1.*bj;
else
% This piece is zero if either limit is +/- infinity.  If
% either is NaN, p1 will already be NaN.
end
end
else
p3 = zeros(class(p1));
end

p = cast(p1 + (p2 + p3)./(2.*pi), superiorfloat(b,rho));

function integrand = tvnIntegrand(theta)
sintheta = sin(theta);
cossqtheta = cos(theta).^2;
nz = (cossqtheta > 0);
if all(nz)
% exp(-(b1.^2 + bj.^2 - 2*b1*bj*sin(theta))/(2*cos(theta).^2));
expon = (c1 + c2.*(1-sintheta))./cossqtheta;
else % some theta == +/- pi
expon = -Inf(size(theta)); % cos^2 -> 0 faster than numerator can
expon(nz) = (c1 + c2.*(1-sintheta(nz)))./cossqtheta(nz);
end
sinphi = sintheta .* rho_k1 ./ rho_j1;
numeru = bk.*cossqtheta - b1.*(sinphi - rho_32.*sintheta) ...
- bj.*(rho_32 - sintheta.*sinphi);
denomu = sqrt(cossqtheta.*(cossqtheta -sinphi.*sinphi ...
- rho_32.*(rho_32 - 2.*sintheta.*sinphi)));
integrand = exp(-expon) .* Phi(numeru./denomu);
end
end```