Scaled Fisher Information Matrix

[Ruslan Neshev]

Hello, Copasi profis!

I have a question regarding the results of a parameter estimation that I am running. Since it is an enzyme cascade with more than one species, I usually assumed that the scaled Fisher Information Matrix is the one with the weights applied to it as in this formula: J^T*W*J, where J is the Jacobian and W the weights matrix. However, when I run the parameter estimation with only one species, the weights matrix becmes the identity matrix (it disappears), so I would have assumed that FIM and the scaled FIM would become equal, which isn't the case! So I am now confused as to how exactly the scaled Fisher Information Matrix is calculated and how is it different from the unscaled one with regards to interpreting it (especially in the sense of its conditioning number: max eigenvalue/min eigenvalue)?

Thank you for any insights you could provide in advance!

Best regards,
Ruslan Neshev

[sven....@gmail.com]

Hello Ruslan, 

I guess this is a case for us to update the documentation...

The "scaled" FIM does not refer to the scaling by weights. Indeed, even the "unscaled" FIM includes the weights. The scaling refers to the parameter values. That means that the scaled FIM is about relative changes of parameter values (or parameter values in log space), while the unscaled FIM is about absolut changes in parameter values. In my experience the Eigenvalues of the "scaled" FIM are more useful when analyzing the information content of the data, while the unscaled FIM seems to be more useful when actually trying to seperate unidentifiable parameters. 

best, 

Sven

[Ruslan Neshev]

Hello Sven,

Thank you for your reply and sorry for my rather late one, I was quite busy the past couple of weeks.. May I ask, what do you mean by "absolute" and "relative" changes? Relative to what - the starting values of the parameters? If that is the case, then a starting value of 0 or 1 should give me the same results for the FIM and scaled FIM, which I tested for 1 and turned out to not be the case (I cannto test for 0 due to the specifications of the model). Furthermore, do I understand it correctly that the scaled FIM is in some way the log10-transformed of the FIM, because for the same index of the FIM, let say a value of 10^3, it should have a value of around 3 in the scaled FIM - which is also not really the case.. I think I would have an easier time understanding the meaning, if I could find a formula (like the one for the FIM = J^T*W*J) for the scaled FIM. But Google is sadly not very helpful here...

Thank you again for your answer and I hope I wasn't too late with my follow-up question!

Best regards,
Ruslan

[sven....@gmail.com]

Hello Ruslan,

first of all, the scaled Fischer-Matrix is scaled with the actual estimated values of the parameters, not with the initial values. 

Now for explaining what I meant with abs/rel changes or log-space, I have to explain how the FIM is calculated. 

In COPASI we first calculate the sensitivities of the simulation result (corresponding to each data point) with respect to each parameter, at the parameter value that is the result of the fitting run: 

sensitivity = (change in simulation result)/(change in parameter value) 

Since we have several (N) data points and several (M) parameters, this is a NxM matrix of numbers, the Jacobian of the parameter estimation.  It turns out the FIM is the matrix product of this matrix with its own transposed matrix (as you wrote). 

Now for the scaled FIM we calculate the Jacobian by using the sensitivities scaled by the parameter values: 

scaled sensitivity = (change in simulation result)/(change in parameter value) * (actual parameter value) 

which means the change of the simulation result with respect to a relative change of a parameter. This is the same as 

scaled sensitivity = (change in simulation result)/(change in log of parameter value)

This means that the scaled FIM should be the FIM, where each row and column is multiplied with the corresponding parameter value (so that each number is effectively multiplied by two parameter values, the diagonal elements are multiplied by the square of the corresponding parameter). 

You should (in my understanding, I should probably test this) get the scaled FIM, if you replace each parameter p_i in the model with a e^q_i, so that the new parameters q_i are essentially the same as the p_i, but in log space. 

I hope this helps, otherwise please do not hesitate to ask again...

Sven