Practical Identifiability
A new method for a priori practical identifiability
Peter Thompson, Benjamin Andersson, and Gunnar Cedersund
Mathematical modeling uses differential equations to study biological processes. One often does not know the numerical values of the parameters before measurements are taken and these measurements are used to estimate the parameter values. However, a parameter may be impossible to estimate this way due to the nature of the equations (structurally unidentifiable) or due to the level of noise in the signal (practically unidentifiable). Much progress has been made in recent decades on determining structural identifiability, but the problem of practical identifiability remains a challenge. However, it is often the case that a parameter is structurally identifiable but is not practically identifiable for any realistic level of noise. This could be because the structural identifiability relies on high order derivatives of the output signal that are hard to accurately estimate.
We present a new algorithm that can help determine whether a parameter is practically identifiable a priori, that is, before measurements are made, by determining whether it can be estimated from derivatives of the outputs specified by the experimenter. Additionally, we show this method can determine if a user-specified combination of states and parameters is practically identifiable. We present a fast version of this algorithm using random integers that succeeds with high probability, and provide rigorous justification not shown in similar algorithms in the literature.
The graph below shows, for a model with 5 states and 17 parameters, the number of identifiable states and parameters as a function of the highest derivative of the output(s) that can be estimated. The blue points are for an experiment with one output and the red for two outputs.
