vanish
2010-05-18, 16:09
我最近正在做毕业论文，是关于不适定反问题的，最后要用多种正则化方法（如Tikhonov正则化方法）用MATLAB编程数值实现得到结果，问题是在一计量经济模型中有三个因自变量x1 x2 x3对结果y有贡献，要通过一些历史数据估计每个变量对结果的贡献系数，即估计y=a1*x1+a2*x2+a3*x3+u中的参数a1 a2 a3，u为随机干扰项。
现在只有一些算法，不会自己修改，求高手不胜感激啊！
function [x_lambda,rho,eta] = tikhonov(U,s,V,b,lambda,x_0)
%TIKHONOV Tikhonov regularization.
%
% [x_lambda,rho,eta] = tikhonov(U,s,V,b,lambda,x_0)
% [x_lambda,rho,eta] = tikhonov(U,sm,X,b,lambda,x_0) , sm = [sigma,mu]
%
% Computes the Tikhonov regularized solution x_lambda. If the SVD
% is used, i.e. if U, s, and V are specified, then standard-form
% regularization is applied:
% min { || A x - b ||^2 + lambda^2 || x - x_0 ||^2 } .
% If, on the other hand, the GSVD is used, i.e. if U, sm, and X are
% specified, then general-form regularization is applied:
% min { || A x - b ||^2 + lambda^2 || L (x - x_0) ||^2 } .
%
% If x_0 is not specified, then x_0 = 0 is used
%
% Note that x_0 cannot be used if A is underdetermined and L ~= I.
%
% If lambda is a vector, then x_lambda is a matrix such that
% x_lambda = [ x_lambda(1), x_lambda(2), ... ] .
%
% The solution norm (standard-form case) or seminorm (general-form
% case) and the residual norm are returned in eta and rho.
% Per Christian Hansen, IMM, April 14, 2003.
% Reference: A. N. Tikhonov & V. Y. Arsenin, "Solutions of
% Ill-Posed Problems", Wiley, 1977.
% Initialization.
if (min(lambda)<0)
error('Illegal regularization parameter lambda')
end
m = size(U,1);
n = size(V,1);
[p,ps] = size(s);
beta = U(:,1:p)'*b;
zeta = s(:,1).*beta;
ll = length(lambda); x_lambda = zeros(n,ll);
rho = zeros(ll,1); eta = zeros(ll,1);
% Treat each lambda separately.
if (ps==1)
% The standard-form case.
if (nargin==6), omega = V'*x_0; end
for i=1:ll
if (nargin==5)
x_lambda(:,i) = V(:,1:p)*(zeta./(s.^2 + lambda(i)^2));
rho(i) = lambda(i)^2*norm(beta./(s.^2 + lambda(i)^2));
else
x_lambda(:,i) = V(:,1:p)*...
((zeta + lambda(i)^2*omega)./(s.^2 + lambda(i)^2));
rho(i) = lambda(i)^2*norm((beta - s.*omega)./(s.^2 + lambda(i)^2));
end
eta(i) = norm(x_lambda(:,i));
end
if (nargout > 1 & size(U,1) > p)
rho = sqrt(rho.^2 + norm(b - U(:,1:n)*[beta;U(:,p+1:n)'*b])^2);
end
elseif (m>=n)
% The overdetermined or square general-form case.
gamma2 = (s(:,1)./s(:,2)).^2;
if (nargin==6), omega = V\x_0; omega = omega(1:p); end
if (p==n)
x0 = zeros(n,1);
else
x0 = V(:,p+1:n)*U(:,p+1:n)'*b;
end
for i=1:ll
if (nargin==5)
xi = zeta./(s(:,1).^2 + lambda(i)^2*s(:,2).^2);
x_lambda(:,i) = V(:,1:p)*xi + x0;
rho(i) = lambda(i)^2*norm(beta./(gamma2 + lambda(i)^2));
else
xi = (zeta + lambda(i)^2*(s(:,2).^2).*omega)./...
(s(:,1).^2 + lambda(i)^2*s(:,2).^2);
x_lambda(:,i) = V(:,1:p)*xi + x0;
rho(i) = lambda(i)^2*norm((beta - s(:,1).*omega)./...
(gamma2 + lambda(i)^2));
end
eta(i) = norm(s(:,2).*xi);
end
if (nargout > 1 & size(U,1) > p)
rho = sqrt(rho.^2 + norm(b - U(:,1:n)*[beta;U(:,p+1:n)'*b])^2);
end
else
% The underdetermined general-form case.
gamma2 = (s(:,1)./s(:,2)).^2;
if (nargin==6), error('x_0 not allowed'), end
if (p==m)
x0 = zeros(n,1);
else
x0 = V(:,p+1:m)*U(:,p+1:m)'*b;
end
for i=1:ll
xi = zeta./(s(:,1).^2 + lambda(i)^2*s(:,2).^2);
x_lambda(:,i) = V(:,1:p)*xi + x0;
rho(i) = lambda(i)^2*norm(beta./(gamma2 + lambda(i)^2));
eta(i) = norm(s(:,2).*xi);
end
end
相关数据见下：
表二 1978-2008我国宏观经济数据增长率
年度 国内生产总值增长率 最终消费增长率 资本形成总额增长率 货物和服务净出口增长率
1978 - - - -
1979 11% 17% 7% 72%
1980 12% 14% 8% -24%
1981 8% 11% -1% -176%
1982 9% 10% 11% 706%
1983 12% 11% 14% -44%
1984 21% 17% 23% -97%
1985 25% 23% 37% -28323%
1986 14% 13% 14% -30%
1987 17% 14% 12% -105%
1988 25% 26% 27% -1414%
1989 13% 13% 11% 23%
1990 10% 8% 6% -375%
1991 17% 16% 17% 21%
1992 23% 21% 28% -55%
1993 30% 27% 56% -347%
1994 35% 33% 28% -193%
1995 25% 26% 24% 57%
1996 18% 19% 13% 46%
1997 9% 9% 6% 96%
1998 6% 6% 4% 7%
1999 3% 7% 4% -26%
2000 9% 10% 6% 6%
2001 7% 8% 22% -3%
2002 9% 7% 15% 33%
2003 12% 7% 23% -3%
2004 17% 29% 24% 37%
2005 33% 11% 17% 151%
2006 16% 14% 17% 63%
2007 21% 17% 18% 40%
2008 17% 16% 20% 3%
现在只有一些算法，不会自己修改，求高手不胜感激啊！
function [x_lambda,rho,eta] = tikhonov(U,s,V,b,lambda,x_0)
%TIKHONOV Tikhonov regularization.
%
% [x_lambda,rho,eta] = tikhonov(U,s,V,b,lambda,x_0)
% [x_lambda,rho,eta] = tikhonov(U,sm,X,b,lambda,x_0) , sm = [sigma,mu]
%
% Computes the Tikhonov regularized solution x_lambda. If the SVD
% is used, i.e. if U, s, and V are specified, then standard-form
% regularization is applied:
% min { || A x - b ||^2 + lambda^2 || x - x_0 ||^2 } .
% If, on the other hand, the GSVD is used, i.e. if U, sm, and X are
% specified, then general-form regularization is applied:
% min { || A x - b ||^2 + lambda^2 || L (x - x_0) ||^2 } .
%
% If x_0 is not specified, then x_0 = 0 is used
%
% Note that x_0 cannot be used if A is underdetermined and L ~= I.
%
% If lambda is a vector, then x_lambda is a matrix such that
% x_lambda = [ x_lambda(1), x_lambda(2), ... ] .
%
% The solution norm (standard-form case) or seminorm (general-form
% case) and the residual norm are returned in eta and rho.
% Per Christian Hansen, IMM, April 14, 2003.
% Reference: A. N. Tikhonov & V. Y. Arsenin, "Solutions of
% Ill-Posed Problems", Wiley, 1977.
% Initialization.
if (min(lambda)<0)
error('Illegal regularization parameter lambda')
end
m = size(U,1);
n = size(V,1);
[p,ps] = size(s);
beta = U(:,1:p)'*b;
zeta = s(:,1).*beta;
ll = length(lambda); x_lambda = zeros(n,ll);
rho = zeros(ll,1); eta = zeros(ll,1);
% Treat each lambda separately.
if (ps==1)
% The standard-form case.
if (nargin==6), omega = V'*x_0; end
for i=1:ll
if (nargin==5)
x_lambda(:,i) = V(:,1:p)*(zeta./(s.^2 + lambda(i)^2));
rho(i) = lambda(i)^2*norm(beta./(s.^2 + lambda(i)^2));
else
x_lambda(:,i) = V(:,1:p)*...
((zeta + lambda(i)^2*omega)./(s.^2 + lambda(i)^2));
rho(i) = lambda(i)^2*norm((beta - s.*omega)./(s.^2 + lambda(i)^2));
end
eta(i) = norm(x_lambda(:,i));
end
if (nargout > 1 & size(U,1) > p)
rho = sqrt(rho.^2 + norm(b - U(:,1:n)*[beta;U(:,p+1:n)'*b])^2);
end
elseif (m>=n)
% The overdetermined or square general-form case.
gamma2 = (s(:,1)./s(:,2)).^2;
if (nargin==6), omega = V\x_0; omega = omega(1:p); end
if (p==n)
x0 = zeros(n,1);
else
x0 = V(:,p+1:n)*U(:,p+1:n)'*b;
end
for i=1:ll
if (nargin==5)
xi = zeta./(s(:,1).^2 + lambda(i)^2*s(:,2).^2);
x_lambda(:,i) = V(:,1:p)*xi + x0;
rho(i) = lambda(i)^2*norm(beta./(gamma2 + lambda(i)^2));
else
xi = (zeta + lambda(i)^2*(s(:,2).^2).*omega)./...
(s(:,1).^2 + lambda(i)^2*s(:,2).^2);
x_lambda(:,i) = V(:,1:p)*xi + x0;
rho(i) = lambda(i)^2*norm((beta - s(:,1).*omega)./...
(gamma2 + lambda(i)^2));
end
eta(i) = norm(s(:,2).*xi);
end
if (nargout > 1 & size(U,1) > p)
rho = sqrt(rho.^2 + norm(b - U(:,1:n)*[beta;U(:,p+1:n)'*b])^2);
end
else
% The underdetermined general-form case.
gamma2 = (s(:,1)./s(:,2)).^2;
if (nargin==6), error('x_0 not allowed'), end
if (p==m)
x0 = zeros(n,1);
else
x0 = V(:,p+1:m)*U(:,p+1:m)'*b;
end
for i=1:ll
xi = zeta./(s(:,1).^2 + lambda(i)^2*s(:,2).^2);
x_lambda(:,i) = V(:,1:p)*xi + x0;
rho(i) = lambda(i)^2*norm(beta./(gamma2 + lambda(i)^2));
eta(i) = norm(s(:,2).*xi);
end
end
相关数据见下：
表二 1978-2008我国宏观经济数据增长率
年度 国内生产总值增长率 最终消费增长率 资本形成总额增长率 货物和服务净出口增长率
1978 - - - -
1979 11% 17% 7% 72%
1980 12% 14% 8% -24%
1981 8% 11% -1% -176%
1982 9% 10% 11% 706%
1983 12% 11% 14% -44%
1984 21% 17% 23% -97%
1985 25% 23% 37% -28323%
1986 14% 13% 14% -30%
1987 17% 14% 12% -105%
1988 25% 26% 27% -1414%
1989 13% 13% 11% 23%
1990 10% 8% 6% -375%
1991 17% 16% 17% 21%
1992 23% 21% 28% -55%
1993 30% 27% 56% -347%
1994 35% 33% 28% -193%
1995 25% 26% 24% 57%
1996 18% 19% 13% 46%
1997 9% 9% 6% 96%
1998 6% 6% 4% 7%
1999 3% 7% 4% -26%
2000 9% 10% 6% 6%
2001 7% 8% 22% -3%
2002 9% 7% 15% 33%
2003 12% 7% 23% -3%
2004 17% 29% 24% 37%
2005 33% 11% 17% 151%
2006 16% 14% 17% 63%
2007 21% 17% 18% 40%
2008 17% 16% 20% 3%