dace_代理模型_代理模型工具箱_matlab_Kriging；_

function [dmodel, perf] = dacefit(S, Y, regr, corr, theta0, lob, upb) %DACEFIT Constrained non-linear least-squares fit of a given correlation % model to the provided data set and regression model % % Call % [dmodel, perf] = dacefit(S, Y, regr, corr, theta0) % [dmodel, perf] = dacefit(S, Y, regr, corr, theta0, lob, upb) % % Input % S, Y : Data points (S(i,:), Y(i,:)), i = 1,...,m % regr : Function handle to a regression model % corr : Function handle to a correlation function % theta0 : Initial guess on theta, the correlation function parameters % lob,upb : If present, then lower and upper bounds on theta % Otherwise, theta0 is used for theta % % Output % dmodel : DACE model: a struct with the elements % regr : function handle to the regression model % corr : function handle to the correlation function % theta : correlation function parameters % beta : generalized least squares estimate % gamma : correlation factors % sigma2 : maximum likelihood estimate of the process variance % S : scaled design sites % Ssc : scaling factors for design arguments % Ysc : scaling factors for design ordinates % C : Cholesky factor of correlation matrix % Ft : Decorrelated regression matrix % G : From QR factorization: Ft = Q*G' . % perf : struct with performance information. Elements % nv : Number of evaluations of objective function % perf : (q+2)*nv array, where q is the number of elements % in theta, and the columns hold current values of % [theta; psi(theta); type] % |type| = 1, 2 or 3, indicate 'start', 'explore' or 'move' % A negative value for type indicates an uphill step % hbn@imm.dtu.dk % Last update September 3, 2002 % Check design points [m n] = size(S); % number of design sites and their dimension sY = size(Y); if min(sY) == 1, Y = Y(:); lY = max(sY); sY = size(Y); else, lY = sY(1); end if m ~= lY error('S and Y must have the same number of rows'), end % Check correlation parameters lth = length(theta0); if nargin > 5 % optimization case if length(lob) ~= lth | length(upb) ~= lth error('theta0, lob and upb must have the same length'), end if any(lob <= 0) | any(upb < lob) error('The bounds must satisfy 0 < lob <= upb'), end else % given theta if any(theta0 <= 0) error('theta0 must be strictly positive'), end end % Normalize data mS = mean(S); sS = std(S); mY = mean(Y); sY = std(Y); % 02.08.27: Check for 'missing dimension' j = find(sS == 0); if ~isempty(j), sS(j) = 1; end j = find(sY == 0); if ~isempty(j), sY(j) = 1; end S = (S - repmat(mS,m,1)) ./ repmat(sS,m,1); Y = (Y - repmat(mY,m,1)) ./ repmat(sY,m,1); % Calculate distances D between points mzmax = m*(m-1) / 2; % number of non-zero distances ij = zeros(mzmax, 2); % initialize matrix with indices D = zeros(mzmax, n); % initialize matrix with distances ll = 0; for k = 1 : m-1 ll = ll(end) + (1 : m-k); ij(ll,:) = [repmat(k, m-k, 1) (k+1 : m)']; % indices for sparse matrix D(ll,:) = repmat(S(k,:), m-k, 1) - S(k+1:m,:); % differences between points end if min(sum(abs(D),2) ) == 0 error('Multiple design sites are not allowed'), end % Regression matrix F = feval(regr, S); [mF p] = size(F); if mF ~= m, error('number of rows in F and S do not match'), end if p > mF, error('least squares problem is underdetermined'), end % parameters for objective function par = struct('corr',corr, 'regr',regr, 'y',Y, 'F',F, ... 'D', D, 'ij',ij, 'scS',sS); % Determine theta if nargin > 5 % Bound constrained non-linear optimization [theta f fit perf] = boxmin(theta0, lob, upb, par); if isinf(f) error('Bad parameter region. Try increasing upb'), end else % Given theta theta = theta0(:); [f fit] = objfunc(theta, par); perf = struct('perf',[theta; f; 1], 'nv',1); if isinf(f) error('Bad point. Try increasing theta0'), end end % Return values dmodel = struct('regr',regr, 'corr',corr, 'theta',theta.', ... 'beta',fit.beta, 'gamma',fit.gamma, 'sigma2',sY.^2.*fit.sigma2, ... 'S',S, 'Ssc',[mS; sS], 'Ysc',[mY; sY], ... 'C',fit.C, 'Ft',fit.Ft, 'G',fit.G); % >>>>>>>>>>>>>>>> Auxiliary functions ==================== function [obj, fit] = objfunc(theta, par) % Initialize obj = inf; fit = struct('sigma2',NaN, 'beta',NaN, 'gamma',NaN, ... 'C',NaN, 'Ft',NaN, 'G',NaN); m = size(par.F,1); % Set up R r = feval(par.corr, theta, par.D); idx = find(r > 0); o = (1 : m)'; mu = (10+m)*eps; R = sparse([par.ij(idx,1); o], [par.ij(idx,2); o], ... [r(idx); ones(m,1)+mu]); % Cholesky factorization with check for pos. def. [C rd] = chol(R); if rd, return, end % not positive definite % Get least squares solution C = C'; Ft = C \ par.F; [Q G] = qr(Ft,0); if rcond(G) < 1e-10 % Check F if cond(par.F) > 1e15 T = sprintf('F is too ill conditioned\nPoor combination of regression model and design sites'); error(T) else % Matrix Ft is too ill conditioned return end end Yt = C \ par.y; beta = G \ (Q'*Yt); rho = Yt - Ft*beta; sigma2 = sum(rho.^2)/m; detR = prod( full(diag(C)) .^ (2/m) ); obj = sum(sigma2) * detR; if nargout > 1 fit = struct('sigma2',sigma2, 'beta',beta, 'gamma',rho' / C, ... 'C',C, 'Ft',Ft, 'G',G'); end % -------------------------------------------------------- function [t, f, fit, perf] = boxmin(t0, lo, up, par) %BOXMIN Minimize with positive box constraints % Initialize [t, f, fit, itpar] = start(t0, lo, up, par); if ~isinf(f) % Iterate p = length(t); if p <= 2, kmax = 2; else, kmax = min(p,4); end for k = 1 : kmax th = t; [t, f, fit, itpar] = explore(t, f, fit, itpar, par); [t, f, fit, itpar] = move(th, t, f, fit, itpar, par); end end perf = struct('nv',itpar.nv, 'perf',itpar.perf(:,1:itpar.nv)); % -------------------------------------------------------- function [t, f, fit, itpar] = start(t0, lo, up, par) % Get starting point and iteration parameters % Initialize t = t0(:); lo = lo(:); up = up(:); p = length(t); D = 2 .^ ([1:p]'/(p+2)); ee = find(up == lo); % Equality constraints if ~isempty(ee) D(ee) = ones(length(ee),1); t(ee) = up(ee); end ng = find(t < lo | up < t); % Free starting values if ~isempty(ng) t(ng) = (lo(ng) .* up(ng).^7).^(1/8); % Starting point end ne = find(D ~= 1); % Check starting point and initialize performance info [f fit] = objfunc(t,par); nv = 1; itpar = struct('D',D, 'ne',ne, 'lo',lo, 'up',up, ... 'perf',zeros(p+2,200*p), 'nv',1); itpar.perf(:,1) = [t; f; 1]; if isinf(f) % Bad parameter region return end if length(ng) > 1 % Try to improve starting guess d0 = 16; d1 = 2; q = length(ng); th = t; fh = f; jdom = ng(1); for k = 1 : q j = ng(k); fk = fh; tk = th; DD = ones(p,1); DD(ng) = repmat(1/d1,q,1); DD(j) = 1/d0; alpha = min(log(lo(ng) ./ th(ng)) ./ log(DD(ng))) / 5; v = DD .^ alpha; tk = th; for rept = 1 : 4 tt = tk .* v; [ff fitt] = objfunc(tt,par); nv = nv+1; itpar.perf(:,nv) = [tt; ff; 1]; if ff <= fk tk = tt; fk = ff; if ff <= f t = tt; f = ff; fit = fitt; jdom = j; end else itpar.perf(end,nv) = -1; break end end end % improve % Update Delta if jdom > 1 D([1 jdom]) = D([jdom 1]); itpar.D = D; end end % free variables itpar.nv = nv; % -------------------------------------------------------- function [t, f, fit, itpar] = explore(t, f, fit, itpar, par) % Explore step nv = itpar.nv; ne = itpar.ne; for k = 1 : length(ne) j = ne(k); tt = t; DD = itpar.D(j); if t(j) == itpar.up(j) atbd = 1; tt(j) = t(j) / sqrt(DD); elseif t(j) == itpar.lo(j) atbd = 1; tt(j) = t(j) * sqrt(DD); else atbd = 0; tt(j) = min(itpar.up(j), t(j)*DD); end [ff fitt] = objfunc(tt,par); nv = nv+1; itpar.perf(:,nv)