# Error propagation through a nonlinear model with error on constants as well as observations

I'm using the following equation to invert (non-linearly) for three parameters, L Vr and f.

$$T_r=\frac{L}{V_r} - \frac{(L\,\cos( \theta )\,\cos( \alpha - f))}{V}$$

I have observational errors on $T_r$ which can be dealt with using weighting in the LM algorithms in Matlab, but how do I deal with the errors in my three variables V, $\alpha$ and $\theta$?

I'm looking to get a value for my three parameters ($L$, $V_r$ and $f$) and 95% confidence limits.

• Take a look at en.wikipedia.org/wiki/Total_least_squares . – Mark L. Stone Jun 29 '17 at 15:02
• It looks like you have a non-linear errors-in-variables problem. In respect of the remaining symbols, please clarify their role -- are they variables without observational error, constants, parameters to be estimated? I'm guessing $L$ $f$ and $V_r$ are meant to be parameters to be estimated but you should be explicit. (Additional information is likely to be required as well.) – Glen_b Jun 30 '17 at 0:12
• No problem I'll edit above, V, alpha and theta are meant to be variables in the equation with some error associated with them and I want to see how much of a difference it makes to L, Vr and f. – Whitt Jun 30 '17 at 8:41