105 lines
3.0 KiB
Matlab
105 lines
3.0 KiB
Matlab
%
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% prodmom.m Date: 4/29/2006
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% This Matlab program computes the product moment of X_{i_1}^{nu_1}X_{i_2}^{nu_2}...X_{i_m}^{nu_m},
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% where X_{i_j} are elements from X ~ N(0_n,V).
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% V only needs to be positive semidefinite.
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% V: variance-covariance matrix of X
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% ii: vector of i_j
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% nu: power of X_{i_j}
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% Reference: Triantafyllopoulos (2003) On the Central Moments of the Multidimensional
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% Gaussian Distribution, Mathematical Scientist
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% Kotz, Balakrishnan, and Johnson (2000), Continuous Multivariate
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% Distributions, Vol. 1, p.261
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% Note that there is a typo in Eq.(46.25), there should be an extra rho in front
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% of the equation.
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% Usage: prodmom(V,[i1 i2 ... ir],[nu1 nu2 ... nur])
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% Example: To get E[X_2X_4^3X_7^2], use prodmom(V,[2 4 7],[1 3 2])
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%
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function dy = prodmom_deriv(V,ii,nu,dV,dC);
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if nargin<3
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nu = ones(size(ii));
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end
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s = sum(nu);
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if s==0
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dy = zeros(1,1,size(dV,3));
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return
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end
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if rem(s,2)==1
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dy = zeros(1,1,size(dV,3));
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return
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end
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nuz = nu==0;
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nu(nuz) = [];
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ii(nuz) = [];
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m = length(ii);
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V = V(ii,ii);
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dV = dV(ii,ii,:);
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s2 = s/2;
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%
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% Use univariate normal results
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%
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if m==1
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dy = s2*V^(s2-1)*dV*prod([1:2:s-1]);
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dy = reshape(dy,1,size(dV,3));
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return
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end
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%
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% Use bivariate normal results when there are only two distinct indices
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%
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if m==2
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rho = V(1,2)/sqrt(V(1,1)*V(2,2));
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drho = dC(ii(1),ii(2),:);
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[tmp,dtmp] = bivmom(nu,rho);
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dy = (nu(1)/2)*V(1,1)^(nu(1)/2-1)*dV(1,1,:) * V(2,2)^(nu(2)/2) * tmp...
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+ V(1,1)^(nu(1)/2) * (nu(2)/2)*V(2,2)^(nu(2)/2-1)*dV(2,2,:) * tmp...
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+ V(1,1)^(nu(1)/2) * V(2,2)^(nu(2)/2) * dtmp * drho;
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dy = reshape(dy,1,size(dV,3));
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return
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end
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%
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% Regular case
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%
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[nu,inu] = sort(nu,2,'descend');
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V = V(inu,inu); % Extract only the relevant part of V
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dV = dV(inu,inu,:); % Extract only the relevant part of dV
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x = zeros(1,m);
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V = V./2;
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dV = dV./2;
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nu2 = nu./2;
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p = 2;
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q = nu2*V*nu2';
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%dq = nu2*dV*nu2';
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%dq = multiprod(multiprod(nu2,dV),nu2');
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dq = NaN(size(q,1), size(q,2), size(dV,3));
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for jp = 1:size(dV,3)
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dq(:,:,jp) = nu2*dV(:,:,jp)*nu2';
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end
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dy = 0;
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for i=1:fix(prod(nu+1)/2)
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dy = dy+p*s2*q^(s2-1)*dq;
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for j=1:m
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if x(j)<nu(j)
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x(j) = x(j)+1;
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p = -round(p*(nu(j)+1-x(j))/x(j));
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%dq = dq-2*(nu2-x)*dV(:,j,:)-dV(j,j,:);
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%dq = dq-2*multiprod((nu2-x),dV(:,j,:))-dV(j,j,:);
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for jp=1:size(dV,3)
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dq(:,:,jp) = dq(:,:,jp)-2*(nu2-x)*dV(:,j,jp)-dV(j,j,jp);
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end
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q = q-2*(nu2-x)*V(:,j)-V(j,j);
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break
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else
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x(j) = 0;
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if rem(nu(j),2)==1
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p = -p;
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end
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%dq = dq+2*nu(j)*multiprod((nu2-x),dV(:,j,:))-nu(j)^2*dV(j,j,:);
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for jp=1:size(dV,3)
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dq(:,:,jp) = dq(:,:,jp)+2*nu(j)*(nu2-x)*dV(:,j,jp)-nu(j)^2*dV(j,j,jp);
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end
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q = q+2*nu(j)*(nu2-x)*V(:,j)-nu(j)^2*V(j,j);
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end
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end
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end
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dy = dy/prod([1:s2]);
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dy = reshape(dy,1,size(dV,3)); |