Figure Properties - SERC

lscov
2-1378
'orth' uses orthogonal decompositions, and is more appropriate when V is
ill-conditioned or singular, but is computationally more expensive.
[x,stdx] = lscov(...) returns the estimated standard errors of x. When A
is rank deficient, stdx contains zeros in the elements corresponding to the
necessarily zero elements of x.
[x,stdx,mse] = lscov(...) returns the mean squared error.
[x,stdx,mse,S] = lscov(...) returns the estimated covariance matrix of x.
When A is rank deficient, S contains zeros in the rows and columns
corresponding to the necessarily zero elements of x. lscov cannot return S if it
is called with multiple right-hand sides, that is, if size(B,2) > 1.
The standard formulas for these quantities, when A and V are full rank, are
x = inv(A'*inv(V)*A)*A'*inv(V)*B
mse = B'*(inv(V) - inv(V)*A*inv(A'*inv(V)*A)*A'*inv(V))*B./(m-n)
S = inv(A'*inv(V)*A)*mse
stdx = sqrt(diag(S))
However, lscov uses methods that are faster and more stable, and are
applicable to rank deficient cases.
lscov assumes that the covariance matrix of B is known only up to a scale
factor. mse is an estimate of that unknown scale factor, and lscov scales the
outputs S and stdx appropriately. However, if V is known to be exactly the
covariance matrix of B, then that scaling is unnecessary. To get the appropriate
estimates in this case, you should rescale S and stdx by 1/mse and
sqrt(1/mse), respectively.
Algorithm The vector x minimizes the quantity (A*x-b)'*inv(V)*(A*x-b). The classical
linear algebra solution to this problem is
See Also lsqnonneg, qr
x = inv(A'*inv(V)*A)*A'*inv(V)*b
but the lscov function instead computes the QR decomposition of A and then
modifies Q by V.
The arithmetic operator \