lsqnonneg
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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,
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Andrew Newell
2012-1-10
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 个评论
Abex
2012-1-10
Titus Edelhofer
2012-1-10
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
2012-1-10
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
2012-1-10
If the matrix C is 5x2 you have 5 equations for 2 unknowns, that is an overdetermined system ...
Abex
2012-1-10
Andrew Newell
2012-1-10
Can you give an example?
Abex
2012-1-11
Andrew Newell
2012-1-11
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.
更多回答（1 个）
Titus Edelhofer
2012-1-11
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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