X-Git-Url: https://git.creatis.insa-lyon.fr/pubgit/?p=CreaPhase.git;a=blobdiff_plain;f=octave_packages%2Foptim-1.2.0%2Fwsolve.m;fp=octave_packages%2Foptim-1.2.0%2Fwsolve.m;h=86abe3c908f7e7a8506671efc0c5e5b517743eda;hp=0000000000000000000000000000000000000000;hb=f5f7a74bd8a4900f0b797da6783be80e11a68d86;hpb=1705066eceaaea976f010f669ce8e972f3734b05

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+## Author: Paul Kienzle <pkienzle@gmail.com>
+## This program is granted to the public domain.
+
+## [x,s] = wsolve(A,y,dy)
+##
+## Solve a potentially over-determined system with uncertainty in
+## the values. 
+##
+##     A x = y +/- dy
+##
+## Use QR decomposition for increased accuracy.  Estimate the 
+## uncertainty for the solution from the scatter in the data.
+##
+## The returned structure s contains
+##
+##    normr = sqrt( A x - y ), weighted by dy
+##    R such that R'R = A'A
+##    df = n-p, n = rows of A, p = columns of A
+##
+## See polyconf for details on how to use s to compute dy.
+## The covariance matrix is inv(R'*R).  If you know that the
+## parameters are independent, then uncertainty is given by
+## the diagonal of the covariance matrix, or 
+##
+##    dx = sqrt(N*sumsq(inv(s.R'))')
+##
+## where N = normr^2/df, or N = 1 if df = 0.
+##
+## Example 1: weighted system
+##
+##    A=[1,2,3;2,1,3;1,1,1]; xin=[1;2;3]; 
+##    dy=[0.2;0.01;0.1]; y=A*xin+randn(size(dy)).*dy;
+##    [x,s] = wsolve(A,y,dy);
+##    dx = sqrt(sumsq(inv(s.R'))');
+##    res = [xin, x, dx]
+##
+## Example 2: weighted overdetermined system  y = x1 + 2*x2 + 3*x3 + e
+##
+##    A = fullfact([3,3,3]); xin=[1;2;3];
+##    y = A*xin; dy = rand(size(y))/50; y+=dy.*randn(size(y));
+##    [x,s] = wsolve(A,y,dy);
+##    dx = s.normr*sqrt(sumsq(inv(s.R'))'/s.df);
+##    res = [xin, x, dx]
+##
+## Note there is a counter-intuitive result that scaling the
+## uncertainty in the data does not affect the uncertainty in
+## the fit.  Indeed, if you perform a monte carlo simulation
+## with x,y datasets selected from a normal distribution centered
+## on y with width 10*dy instead of dy you will see that the
+## variance in the parameters indeed increases by a factor of 100.
+## However, if the error bars really do increase by a factor of 10
+## you should expect a corresponding increase in the scatter of 
+## the data, which will increase the variance computed by the fit.
+
+function [x_out,s]=wsolve(A,y,dy)
+  if nargin < 2, usage("[x dx] = wsolve(A,y[,dy])"); end
+  if nargin < 3, dy = []; end
+
+  [nr,nc] = size(A);
+  if nc > nr, error("underdetermined system"); end
+
+  ## apply weighting term, if it was given
+  if prod(size(dy))==1
+    A = A ./ dy;
+    y = y ./ dy;
+  elseif ~isempty(dy)
+    A = A ./ (dy * ones (1, columns(A)));
+    y = y ./ dy;
+  endif
+
+  ## system solution: A x = y => x = inv(A) y
+  ## QR decomposition has good numerical properties:
+  ##   AP = QR, with P'P = Q'Q = I, and R upper triangular
+  ## so
+  ##   inv(A) y = P inv(R) inv(Q) y = P inv(R) Q' y = P (R \ (Q' y))
+  ## Note that b is usually a vector and Q is matrix, so it will
+  ## be faster to compute (y' Q)' than (Q' y).
+  [Q,R,p] = qr(A,0);
+  x = R\(y'*Q)'; 
+  x(p) = x;
+
+  s.R = R;
+  s.R(:,p) = R;
+  s.df = nr-nc;
+  s.normr = norm(y - A*x);
+
+  if nargout == 0,
+    cov = s.R'*s.R
+    if s.df, normalized_chisq = s.normr^2/s.df, end
+    x = x'
+  else
+    x_out = x;
+  endif
+
+## We can show that uncertainty dx = sumsq(inv(R'))' = sqrt(diag(inv(A'A))).
+##
+## Rather than calculate inv(A'A) directly, we are going to use the QR
+## decomposition we have already computed:
+##
+##    AP = QR, with P'P = Q'Q = I, and R upper triangular
+##
+## so 
+##
+##    A'A = PR'Q'QRP' = PR'RP'
+##
+## and
+##
+##    inv(A'A) = inv(PR'RP') = inv(P')inv(R'R)inv(P) = P inv(R'R) P'
+##
+## For a permutation matrix P,
+##
+##    diag(PXP') = P diag(X)
+##
+## so
+##    diag(inv(A'A)) = diag(P inv(R'R) P') = P diag(inv(R'R))
+##
+## For R upper triangular, inv(R') = inv(R)' so inv(R'R) = inv(R)inv(R)'.
+## Conveniently, for X upper triangular, diag(XX') = sumsq(X')', so
+##
+##    diag(inv(A'A)) = P sumsq(inv(R)')'
+## 
+## This is both faster and more accurate than computing inv(A'A)
+## directly.
+##
+## One small problem:  if R is not square then inv(R) does not exist.
+## This happens when the system is underdetermined, but in that case
+## you shouldn't be using wsolve.
+