Preliminary Course
Applied Mathematics
Parameter estimation
Identification
-
Identification of dynamic systems
Given a system S described by a model M(X, P, U) and a set of measured experimental data Yobs, the parameter estimation of M consists in finding a set of parameters Þ that minimizes a distance measure between Yobs and Y = g(n, Xn(Un, P, n), P).
A requested preliminary is to check the parameter identifiability.
The parameter vector P is structurally globally identifiable if for (almost*) [Note: except on a negligible subset] any P* in the definition set of the parameter vector,
M(P) = M(P*) ==> P = P*
An alternative and equivalent way of seeing it is to say that it should not be possible to find two different parameter vectors that would give exactly the same model outputs if simulated under the same conditions.
Studying the parameter identifiability of a model amounts to answering the question "will it be possible to uniquely estimate the model unknown parameters under given ideal (noise free and as frequent as needed) experimental conditions (stimuli and observables)? "
If the answer is "no", then the model design should be modified (e.g. some parameters should be removed) before moving to the step of parameter estimation.
Exercise
-
Let us consider the following simple model representing forest demography.
A virtual forest stand (Image Digiplante team, ECOLE CENTRALE PARIS)
-
Assume that there is a yearly constant rate of tree reproduction r and of tree death, d.
Then the number of trees in the forest at year n, Nn, obeys the following recurrent equation:
Nn+1 = r Nn - d Nn
1. Assume that you can ask foresters to count trees every year and thus have access to any values of Nn for n=0, ..., Nmax.
Is the parameter set (r, d) identifiable ?