Parameter uncertainty

[Hofsmo]

Hi,
In Matlab it is possible to get the uncertainty of the parameters in the estimated parameter vector. I was wondering if it is possible to do the same for state space models obtained using newpem? I see that the PredictionStateSpace structure has fields for dynamic covariance, measurement covariance, and cross covariance, so I guess that is a good starting point.

[baggepinnen]

This is not currently possible, but it would be relatively easy to add when abs2 is used as metric (nonlinear_pem supports this)

[Hofsmo]

Thanks for the quick answer :) . Does this mean that nonlinear_pem, already supports parameter uncertainty, and that it is relatively easy to add to newpem?

[baggepinnen]

Correct. I'm traveling at the moment but can have a look at it in a few days

[baggepinnen]

https://baggepinnen.github.io/ControlSystemIdentification.jl/dev/nonlinear/#Covariance-of-the-estimated-parameters

[Hofsmo]

Many thanks for the clarification and the quick answer :-) .

[baggepinnen]

I had a look at it. It's not very hard to compute the Hessian, but since newpem uses a non-minimal parameterization the Hessian will be indefinite and interpreting it as an inverse covariance matrix is not possible. One could possible transform the model to a minimal canonical form and compute the covariance of the parameters in that formulation, but it seems a bit messy. One could also add regularization to the parameter vector, but doing this in a nice way is non-trivial. structured_pem would not suffer from this problem as long as the user-chosen parameterization is minimal

[baggepinnen]

Here's an experimental implementation of this for structured_pem
#188

[Hofsmo]

Wow thanks, that looks very good. I have been using structured_pem a bit and I will check it out :) .