Linear least square approximation error estimation with error in both coordinates
Im stuck with the following problem and would be very grateful for any help:
How large will the estimated error be (95% confidence intervall) for my least square fit to data that are uncertain in both coordinates?
The problem in numbers:
The data is temperature measurements with an accuracy of 0.1K (uncorrelated errors) measured at different positions with an accuracy of 1e-5m (also uncorrelated).
Temperatures [C]__ Position [m]
94.6 +-0.1 _______ 0.01 +-1e-5
85.7 +-0.1 _______ 0.02 +-1e-5
76.3 +-0.1 _______ 0.03 +-1e-5
66.1 +-0.1 _______ 0.04 +-1e-5
T=64.06+-dT _____ 0.05 +-1e-5
I got T=64.06 from the least square fit to the data but I wanna know the estimated error dT of this value.
I'm very grateful for any kind of help or directions.
Update: Just found out that the problem had a pretty simple solution and can be found in two ways:
1) Through Error estimations with pertubations to A and b from eq. Ax = b
The error eq is then dx = A^-1 (db-dAx - dAdx). This expression needs some rewriting through taking the norm and triangle inequality. But this can be found in eg. Heath, Scientific computing
2) Through error progression formula which also can be found in Heath, Scientific computing.
PS. Why can't I thank myself? =)