On the computation of the Hessian matrix.

Question

I'm trying to compute the Hessian matrix of a data fit of an ODE model to some data. Below is a cut out of the instructions I'm following (which can also be found at Gutenkunst, R.N., Waterfall, J.J., Casey, F.P., Brown, K.S., Myers, C.R. and Sethna, J.P. (2007) 'Universally Sloppy Parameter Sensitivities in Systems Biology Models', PLoS Comput Biol, 3(10), p. e189).

The function I'm trying to compute the hessian matrix for is:

with:

$ ϴ = $to a parameter vector for ODE model

$ N_c=$Number of experimental conditions

$ N_s=$Number of model variables/species

$ T_c=$measurement time

$ Y_sc(ϴ,t)= $timecourse for species $s$ in condition $c$ given parameters ϴ

$ sigma^2= $ Normalization - maximum value of species $s$ over all $c$

$ $ $ $

And in the paper they define the Hessian Matrix as:

My question is: how do I use this information to compute the hessian matrix for a simple set of ODEs?

Answer 1

Given your description, the Hessian will have entries $(j,k)$ equal to $$ \frac1{2N_cN_s}\sum_{s,c}\frac1{T_c}\int_0^{T_c}\frac{y_{s,c}(\theta,t)-y_{s,c}(\theta^*,t)}{\sigma_s^2}\cdot\frac{\partial^2y_{s,c}(\theta,t)}{\partial(\log\theta_j)\partial(\log\theta_k)}\,dt. $$