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u_to_x.m
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function x = u_to_x(u,probdata)
%U_TO_X Transformation between u and x space
%
% x = u_to_x(u,marg,varargin)
%
% Function 'u_to_x' perform the transformation between standard
% normal space and original space.
%
% Output: - x = values of r.v.'s in original space
% Input: - u = values of r.v.'s in standard normal space
% - probdata
%! NOTE!
%! the same procedure is done in every cases, this whole case stuff could be replaced
%! inverse cdf would be needed
transf_type = probdata.transf_type;
marg = probdata.marg;
nx = size(u,2);
nrv = size(marg,1);
x = zeros(nrv,nx);
switch transf_type
case 1
case 2
%for i = 1 : nrv
% switch marg(i,1)
% case 1 % Normal distribution
% x(i,:) = u(i,:) * marg(i,3) + marg(i,2);
% case 2 % Lognormal distribution
% zeta = ( log( ( 1 + (marg(i,3)/marg(i,2))^2 ) ) )^0.5;
% lambda = log( marg(i,2) ) - 0.5 * zeta^2;
% x(i,:) = exp( u(i,:) * zeta + lambda ) ;
% case 4 % Shifted exponential marginal distribution
% b = marg(i,3);
% a = marg(i,2) - b;
% x(i,:) = a - b * log(1-ferum_cdf(1,u(i,:),0,1));
% case 5 % Shifted Rayleigh marginal distribution
% b = marg(i,3) / (1-pi/4)^0.5;
% a = marg(i,2) - b*pi^0.5/2;
% x(i,:) = a + b * ( -log( 1-ferum_cdf(1,u(i,:),0,1) ) ).^0.5;
% case 6 % Uniform marginal distribution
% a = marg(i,2) - sqrt(3)*marg(i,3);
% b = marg(i,2) + sqrt(3)*marg(i,3);
% x(i,:) = a + (b-a) * ferum_cdf(1,u(i,:),0,1);
% case 11 % Type I Largest Value or Gumbel marginal distribution
% b = 6^0.5/pi*marg(i,3);
% a = marg(i,2) - 0.5772156649*b;
% x(i,:) = a - b * log( -log( ferum_cdf(1,u(i,:),0,1) ) );
% otherwise
% end
%end
case 3
Lo = probdata.Lo;
z = Lo * u;
for i = 1 : nrv
switch marg(i,1)
case 1 % Normal distribution
mean = marg(i,5);
stdv = marg(i,6);
% marg(i,7) not relevant
n = marg(i,8);
if n == 1 %it dies when u~>8 (gives 1 for cdf)
x(i,:) = z(i,:) * stdv + mean;
else
x(i,:) = norminv(normcdf(z(i,:)).^(1/n), mean, stdv);
end
case 2 % Lognormal distribution
lambda = marg(i,5);
zeta = marg(i,6);
% marg(i,7) not relevant
n = marg(i,8);
%x(i,:) = exp( z(i,:) * zeta + lambda );
mu = lambda;
sigma = zeta;
if n == 1
x(i,:) = exp(sigma*z(i,:) + mu);
else
x(i,:) = logninv(normcdf(z(i,:)).^(1/n), mu, sigma);
end
case 3 % Gamma distribution
lambda = marg(i,5);
k = marg(i,6);
% marg(i,7) not relevant
n = marg(i,8);
mean = marg(i,2);
for j = 1 : nx
normal_val = normcdf(z(i,j)).^(1/n);
%x(i,j) = fzero('zero_gamma',mean,optimset('fzero'),k,lambda,normal_val); % Doesn't work
x(i,j) = fminsearch('zero_gamma',mean,optimset('fminsearch'),k,lambda,normal_val);
%A = k;
%B = -1/(log(lambda^k)*lambda);
%x(i,:) = gaminv(normcdf(z(i,:)).^(1/n), A, B);
% should be verified
end
case 4 % Shifted exponential distribution
lambda = marg(i,5);
x_zero = marg(i,6);
% marg(i,7) not relevant
n = marg(i,8);
x(i,:) = x_zero + 1/lambda * log( 1 ./ ( 1 - normcdf(z(i,:)).^(1/n) ) );
case 5 % Shifted Rayleigh distribution
a = marg(i,5);
x_zero = marg(i,6);
% marg(i,7) not relevant
n = marg(i,8);
x(i,:) = x_zero + a * ( 2*log( 1 ./ (1 - normcdf(z(i,:)).^(1/n)) ) ) .^0.5;
case 6 % Uniform distribution
a = marg(i,5);
b = marg(i,6);
% marg(i,7) not relevant
n = marg(i,8);
x(i,:) = a + (b-a) * normcdf(z(i,:)).^(1/n);
case 7 % Beta distribution
q = marg(i,5);
r = marg(i,6);
a = marg(i,7);
b = marg(i,8);
mean = marg(i,2);
for j = 1 : nx
normal_val = normcdf(z(i,:));
%x01 = fzero('zero_beta',(mean-a)/(b-a),optimset('fzero'),q,r,normal_val); % Doesn't work
x01 = fminbnd('zero_beta',0,1,optimset('fminbnd'),q,r,normal_val);
% Transform x01 from [0,1] to [a,b] interval
x(i,j) = a + x01 * ( b - a );
end
case 8 % Chi-square distribution
lambda = 0.5;
nu = marg(i,5);
% marg(i,6) not relevant
% marg(i,7) not relevant
n = marg(i,8);
k = nu/2 ;
mean = marg(i,2);
for j = 1 : nx
normal_val = normcdf(z(i,j)).^(1/n);
%x(i,j) = fzero('zero_gamma',mean,optimset('fzero'),k,lambda,normal_val); % Doesn't work
x(i,j) = fminsearch('zero_gamma',mean,optimset('fminsearch'),k,lambda,normal_val);
end
case 11 % Type I largest value distribution ( same as Gumbel distribution )
u_n = marg(i,5);
a_n = marg(i,6);
% marg(i,7) not relevant
n = marg(i,8);
% % % x(i,:) = u_n - (1/a_n) * log( log( 1 ./ (normcdf(z(i,:))).^(1/n)) ) ;
% WARNING!!!
z_i = z(i,:);
if z_i < 8
x(i,:) = u_n - (1/a_n) * log( log( 1 ./ (normcdf(z_i)).^(1/n)) ) ;
else
f = lognlognormcdf(z_i);
x(i,:) = u_n - (1/a_n) * f;
warning('Hip hip hip barba trick. Gumbel - double precision.')
end
case 12 % Type I smallest value distribution
u_1 = marg(i,5);
a_1 = marg(i,6);
% marg(i,7) not relevant
n = marg(i,8);
x(i,:) = u_1 + (1/a_1) * log( log( 1 ./ ( 1 - normcdf(z(i,:)).^(1/n)) ) ) ;
case 13 % Type II largest value distribution
u_n = marg(i,5);
k = marg(i,6);
% marg(i,7) not relevant
n = marg(i,8);
x(i,:) = u_n * log( 1 ./ normcdf(z(i,:)).^(1/n)).^ (-1/k);
case 14 % Type III smallest value distribution
u_1 = marg(i,5);
k = marg(i,6);
epsilon = marg(i,7);
n = marg(i,8);
x(i,:) = epsilon + ( u_1 - epsilon ) * log( 1 ./ ( 1 - normcdf(z(i,:)).^(1/n)) ).^(1/k);
case 15 % Gumbel distribution ( same as type I largest value distribution )
u_n = marg(i,5);
a_n = marg(i,6);
% marg(i,7) not relevant
n = marg(i,8);
x(i,:) = u_n - (1/a_n) * log( log( 1 ./ normcdf(z(i,:)).^(1/n)) );
case 16 % Weibull distribution ( same as Type III smallest value distribution with epsilon = 0 )
u_1 = marg(i,5);
k = marg(i,6);
% marg(i,7) not relevant
n = marg(i,8);
x(i,:) = u_1 * log( 1 ./ ( 1 - normcdf(z(i,:)).^(1/n)) ).^(1/k);
case 18 % (Reserved for Laplace distribution)
case 19 % (Reserved for Pareto distribution)
case 20 % Generalized extreme value (GEV) distribution
k = marg(i,5);
sigma = marg(i,6);
mu = marg(i,7);
n = marg(i,8);
x(i,:) = gevinv( normcdf(z(i,:)).^(1/n), k, sigma, mu );
case 25 % Three-parameter lognormal (LN3) distribution
shape = marg(i,5);
scale = marg(i,6);
thres = marg(i,7);
n = marg(i,8);
x(i,:) = lognorm3inv( normcdf(z(i,:)).^(1/n), shape, scale, thres, 'par');
case 30 % sample based custom distribution
ID = marg(i,5);
% x(i,:) = custom_invcdf( normcdf(z(i,:)), ID, 'sample');
error('NYAAAK')
case 31 % vector based custom distribution
ID = marg(i,5);
% x(i,:) = custom_invcdf( normcdf(z(i,:)), ID, 'point');
x(i,:) = custom_invcdf( z(i,:), ID, 'point');
case 32 % vector based custom distribution
ID = marg(i,5);
shift = marg(i,6);
scale = marg(i,7);
x(i,:) = nonparametric_invcdf(z(i,:), ID, shift, scale);
case 33 % hardcoded custom distribution
ID = marg(i,5);
shift = marg(i,6);
scale = marg(i,7);
x(i,:) = hardcoded_invcdf(z(i,:), ID, shift, scale);
case 51 % Truncated normal marginal distribution
mean = marg(i,5);
stdv = marg(i,6);
xmin = marg(i,7);
xmax = marg(i,8);
x(i,:) = mean + stdv * inv_norm_cdf( ...
normcdf((xmin-mean)/stdv) + ...
(normcdf((xmax-mean)/stdv)-normcdf((xmin-mean)/stdv)) * normcdf(z(i,:)) ...
);
Imin = find(x(i,:)<xmin); x(i,Imin) = xmin;
Imax = find(x(i,:)>xmax); x(i,Imax) = xmax;
otherwise
end
end
case 4
%opt = optimset([]);
%
%str = '';
%for i = 1:nrv
% eval(['x(' num2str(i) ') = fzero(''zero_u_to_x'',marg(' num2str(i) ',2),opt,u,marg,' num2str(i) str ');']);
% str = [str ',x(' num2str(i) ')'];
%end
%
%% x(1) = fzero('zero_u_to_x',marg(1,2),opt,u,marg,1);
%% x(2) = fzero('zero_u_to_x',marg(2,2),opt,u,marg,2,x(1));
%% ... opt = optimset([]);
otherwise
end