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jacobian.m
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function J_u_x = jacobian(x,u,probdata)
transf_type = probdata.transf_type;
marg = probdata.marg;
nrv = size(marg,1);
%! NOTE!
%! the same procedure is done in every cases, this whole case stuff could be replaced
% exp(log(pdf1) - log(pdf2)) might be numerically better, log pdf could be
% an option in the functions
switch transf_type
case 1
case 2
%J_u_x = zeros(nrv);
%
%for i = 1 : nrv
% switch marg(i,1)
% case 1 % Normal distribution
% J_u_x(i,i) = 1/marg(i,3);
% case 2 % Lognormal distribution
% ksi = sqrt( log( 1 + ( marg(i,3) / marg(i,2) )^2 ) );
% J_u_x(i,i) = 1 / ( ksi * x(i) );
% case 4 % Shifted exponential marginal distribution
% pdf1 = ferum_pdf(4,x(i),marg(i,2),marg(i,3));
% pdf2 = ferum_pdf(1,u(i),0,1);
% J_u_x(i,i) = pdf1/pdf2;
% case 5 % Shifted Rayleigh marginal distribution
% pdf1 = ferum_pdf(5,x(i),marg(i,2),marg(i,3));
% pdf2 = ferum_pdf(1,u(i),0,1);
% J_u_x(i,i) = pdf1/pdf2;
% case 6 % Uniform marginal distribution
% pdf1 = ferum_pdf(6,x(i),marg(i,2),marg(i,3));
% pdf2 = ferum_pdf(1,u(i),0,1);
% J_u_x(i,i) = pdf1/pdf2;
% case 11 % Type I Largest Value or Gumbel marginal distribution
% pdf1 = ferum_pdf(11,x(i),marg(i,2),marg(i,3));
% pdf2 = ferum_pdf(1,u(i),0,1);
% J_u_x(i,i) = pdf1/pdf2;
% otherwise
% end
%end
case 3
Lo = probdata.Lo;
iLo = probdata.iLo;
z = Lo * u;
J_z_x = zeros(nrv);
for i = 1 : nrv
switch marg(i,1)
case 1 % Normal distribution
pdf1 = ferum_pdf(1,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
% pdf2 = normpdf_mp(z(i));
J_z_x(i,i) = pdf1/pdf2;
%J_z_x(i,i) = 1/marg(i,3);
case 2 % Lognormal distribution
pdf1 = ferum_pdf(2,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
% ksi = sqrt( log( 1 + ( marg(i,3) / marg(i,2) )^2 ) );
% J_z_x(i,i) = 1 / ( ksi * x(i) );
case 3 % Gamma distribution
%lambda = marg(i,5);
%k = marg(i,6);
%pdf1 = lambda * (lambda*x(i))^(k-1) / gamma(k) * exp(-lambda*x(i));
pdf1 = ferum_pdf(3,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 4 % Shifted exponential distribution
%lambda = marg(i,5);
%x_zero = marg(i,6);
%pdf1 = lambda * exp(-lambda*(x(i)-x_zero));
pdf1 = ferum_pdf(4,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 5 % Shifted Rayleigh distribution
%a = marg(i,5);
%x_zero = marg(i,6);
%pdf1 = (x(i)-x_zero)/a^2 * exp(-0.5*((x(i)-x_zero)/a)^2);
pdf1 = ferum_pdf(5,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 6 % Uniform distribution
%a = marg(i,5);
%b = marg(i,6);
%pdf1 = 1 / (b-a);
pdf1 = ferum_pdf(6,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 7 % Beta distribution
%q = marg(i,5);
%r = marg(i,6);
%a = marg(i,7);
%b = marg(i,8);
%pdf1 = (x(i)-a)^(q-1) * (b-x(i))^(r-1) / ( (gamma(q)*gamma(r)/gamma(q+r)) * (b-a)^(q+r-1) );
pdf1 = ferum_pdf(7,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 8 % Chi-square distribution
%lambda = 0.5;
%nu = marg(i,5);
%k = nu/2;
%pdf1 = lambda * (lambda*x(i))^(k-1) * exp(-lambda*x(i)) / gamma(k) ;
pdf1 = ferum_pdf(8,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i)= pdf1/pdf2;
case 11 % Type I largest value distribution ( same as Gumbel distribution )
%u_n = marg(i,5);
%a_n = marg(i,6);
%pdf1 = a_n * exp( -a_n*(x(i)-u_n) - exp(-a_n*(x(i)-u_n)) );
pdf1 = ferum_pdf(11,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
% pdf2 = normpdf_mp(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 12 % % Type I smallest value distribution
%u_1 = marg(i,5);
%a_1 = marg(i,6);
%pdf1 = a_1 * exp( a_1*(x(i)-u_1) - exp(a_1*(x(i)-u_1)) );
pdf1 = ferum_pdf(12,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 13 % Type II largest value distribution
%u_n = marg(i,5);
%k = marg(i,6);
%pdf1 = k/u_n * (u_n/x(i))^(k+1) * exp(-(u_n/x(i))^k);
pdf1 = ferum_pdf(13,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 14 % Type III smallest value distribution
%u_1 = marg(i,5);
%k = marg(i,6);
%epsilon = marg(i,7);
%pdf1 = k/(u_1-epsilon) * ((x(i)-epsilon)/(u_1-epsilon))^(k-1) ...
% * exp(-((x(i)-epsilon)/(u_1-epsilon))^k);
pdf1 = ferum_pdf(14,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 15 % Gumbel distribution ( same as type I largest value distribution )
%u_n = marg(i,5);
%a_n = marg(i,6);
%pdf1 = a_n * exp( -a_n*(x(i)-u_n) - exp(-a_n*(x(i)-u_n)) );
pdf1 = ferum_pdf(15,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 16 % Weibull distribution ( same as Type III smallest value distribution with epsilon = 0 )
%u_1 = marg(i,5);
%k = marg(i,6);
%pdf1 = k/u_1 * (x(i)/u_1)^(k-1) * exp(-(x(i)/u_1)^k);
pdf1 = ferum_pdf(16,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 20
pdf1 = ferum_pdf(20, x(i), marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 25
pdf1 = ferum_pdf(25, x(i), marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 30
pdf1 = ferum_pdf(30, x(i), marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 31
pdf1 = ferum_pdf(31, x(i), marg(i,5:8));
pdf2 = normpdf(z(i));
% if isnan(pdf1/pdf2)
% keyboard
% end
J_z_x(i,i) = pdf1/pdf2;
case 32
pdf1 = ferum_pdf(32, x(i), marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 33
pdf1 = ferum_pdf(33, x(i), marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
case 51 % Truncated normal marginal distribution
pdf1 = ferum_pdf(51,x(i),marg(i,5:8));
pdf2 = normpdf(z(i));
J_z_x(i,i) = pdf1/pdf2;
otherwise
end
end
J_u_x = iLo * J_z_x;
case 4
%alpha12 = 1;
%
%J_u_x = zeros(2);
%
%p1 = ferum_pdf(marg(1,1),x(1),marg(1,2),marg(1,3));
%p2 = ferum_pdf(marg(2,1),x(2),marg(2,2),marg(2,3));
%
%P1 = ferum_cdf(marg(1,1),x(1),marg(1,2),marg(1,3));
%P2 = ferum_cdf(marg(2,1),x(2),marg(2,2),marg(2,3));
%
%J_u_x(1,1) = pdf_morgenstern(x,marg,1) / ferum_pdf(1,u(1),0,1);
%J_u_x(2,2) = pdf_morgenstern(x,marg,2) / ferum_pdf(1,u(2),0,1);
%
%J_u_x(2,1) = -2 * alpha12 * P2 * (1-P2) * p1 / ferum_pdf(1,u(2),0,1);
otherwise
end