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[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.