Smoothing Spline Matlab Code (This sparse assembly could be simplified)
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function [mu,m]=SmoothSpline(X,f,p) | |
% [mu,m]=SmoothSpline(X,f,p) | |
% | |
% This code compute the parameter of "the smoothing natural spline" | |
% for the data (x_i,f_i) with weights {p_i} i.e. the solution of | |
% | |
% \min\Biggl\{ \sum_{i=0}^N p_i\left(\hat{f}(x_i)-f_i \right)^2 | |
% +\int_{a}^{b} \bigl|\hat{f}''(x)\bigr|^2dx \Biggr\} | |
% Input: | |
% X := {x_1,...,x_N} | |
% f := {f_1,...,f_N} Set of obserbations | |
% p := weigths of the method | |
% | |
% Output: | |
% mu := {mu_0,...,mu_N} the values of the Smothing Spline at X | |
% m := Second derivative of the Smothing Spline at X. | |
% Last modified: March 21, 2018. | |
N=length(X); % Number of observations | |
f=f(:); | |
p=1./2*p; % | |
P=sparse(1:N,1:N,p,N,N); % P^{-1} Sparse matrix. | |
h=X(2:end)-X(1:end-1); % vector with h_{j+1}-h_j. | |
h2=(1/6)*h(2:end-1); | |
h3=(1/3)*(X(3:1:end)-X(1:1:end-2)); % H(j) =(1/3)*(h_j+h_{j+1}) | |
CoefR=zeros(3*N-8,1); | |
CoefR(1:N-2)=1:N-2; CoefR(N-1:2*N-5)=2:N-2; CoefR(2*N-4:end)=1:N-3; | |
CoefC=zeros(3*N-8,1); | |
CoefC(1:N-2)=1:N-2; CoefC(N-1:2*N-5)=1:N-3; CoefC(2*N-4:end)=2:N-2; | |
A=sparse(CoefR,CoefC,[h3 h2 h2],N-2,N-2); | |
CoefR=repmat((1:N-2)',3,1); | |
CoefC=zeros(3*N-6,1); | |
CoefC(1:N-2)=1:N-2; CoefC(N-1:2*N-4)=2:N-1; CoefC(2*N-3:3*N-6)=3:N; | |
b=[1./h(1:end-1) -(1./h(1:end-1)+1./h(2:end)) 1./h(2:end)]; | |
H=sparse(CoefR,CoefC,b,N-2,N); | |
h=h(:); | |
f2=(f(3:end)-f(2:end-1))./(h(2:end))-(f(2:end-1)-f(1:end-2))./h(1:end-1); | |
f2=f2(:); | |
m=zeros(N,1); m(2:end-1)=(H*P*H'+A)\f2; | |
mu=f-P*(H')*m(2:end-1); % this could be simplified. | |
end | |
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