# Parameter uncertainty after non-linear least squares estimation

I've fit a system of non-linear ODE to some experimental data using Levemberg-Marquardt. After the algorithm converged, I estimated the Hessian matrix of the system using:

$H = (J^TJ)$

The covariance matrix is then the inverse of H:

$cov = H^{-1}$

To get an unbiased estimate, I rescaled cov like so:

$cov_{scaled} = cov * (RSS / (m - n))$

Where $m$ is the number of measurements, and $n$ is the number of parameters.

The diagonal of $cov_{scaled}$ gives me the uncertainty in the parameters.

However, if I am interested in the uncertainty of a 'meta parameter', such as:

$p_{meta} = p_1 + p_2$

How do I estimate that from $cov_{scaled}$?

What if $p_{meta}$ is a slightly more complex function, such as:

$p_{meta} = p_1/p_2$

Is there a generic approach?

I cannot really re-parametrize the system and fit p_new directly unfortunately.

• I suggest writing the equations in latex. Commented Apr 10, 2014 at 12:47
• Fixed, thanks for the suggestion, didn't realize you could use mathjax. Commented Apr 10, 2014 at 13:10
• Heard of the delta method? Commented Apr 10, 2014 at 13:13
• I've heard of it, but I'm not very familiar how it would apply to this case, while taking into account of the covariance of parameters p1 and p2. Commented Apr 11, 2014 at 7:32
• en.wikipedia.org/wiki/… ... but check the various threads here on the subject for plenty of caveats Commented Jan 13, 2015 at 6:15

The linear one is easy. Suppose $p_1$ and $p_2$ and $p_3$ are all parameters of your model (call the vector of these parameters $P$), then the variance-covariance matrix of the three parameters is the 3x3 matrix H. So to get Var($p_1$ + $p_2$), which is Var($(1,1,0)*P$), just compute $(1,1,0)*H*(1,1,0)'$.