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Can I solve [x] in the relation [C]*[x]=[d] using x= lsqnonneg(C,d);? Vector dimensions are as follows C(5x2), d(5x1). If not please suggest me better way.
Thank you,
Accepted Answer
Andrew Newell
on 10 Jan 2012
x = C\d;
EDIT: The least squares estimate for x is
xhat = (C'*C)\(C'*d);
You can estimate the error in the fit as follows:
n = length(d);
s2 = norm(x-xhat)^2/(n-2); %estimated variance of the data
covar = s2*inv(C'*C); %covariance matrix for the components of xhat
xerr = sqrt(diag(covar)) %standard errors for xhat components
8 Comments
Abex
on 10 Jan 2012
Titus Edelhofer
on 10 Jan 2012
The backslash is solving the linear system of equations you wrote, even for overdetermined system (e.g. C has more rows then columns). It does this in the least squares sense (because there (usually) is no exact solution).
Andrew Newell
on 10 Jan 2012
MATLAB does not take the inverse of C to solve this equation. In effect, you have two equations for the components of x, and it solves them using Gaussian elimination.
Titus Edelhofer
on 10 Jan 2012
If the matrix C is 5x2 you have 5 equations for 2 unknowns, that is an overdetermined system ...
Abex
on 10 Jan 2012
Andrew Newell
on 10 Jan 2012
Can you give an example?
Abex
on 11 Jan 2012
Andrew Newell
on 11 Jan 2012
Your matrix C is very ill-conditioned because the two rows are so nearly equal. The estimated errors for xhat, using the equations I have added above, are
1.0e+05 *
3.4387
3.4391
i.e., much larger than the components themselves. In other words, it is futile to try to fit such a data set.
More Answers (1)
Titus Edelhofer
on 11 Jan 2012
Hi,
if you take a closer look at your data you observe:
1. the two columns of C are very similar, therefore it's no surprise that using lsqnonneg you get one positive value and a zero.
2. If you do the following:
x = C(:,1)./d
you get values 2.4, 2.3, 2.2, 2.1, ..., 0.62, 0.61. The quadratic mean will be about the value you get from lsqnonneg and (from the data) it's not clear why the value should be less then one.
Titus
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