Hi everyone,

Here's the shortest version of this optimization problem:

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I need to find out the value of one growth rate and one mortality rate by simulating a population between two assessment campaigns. I initiate the model from the first one's data and try to fit the second one's with the virtual population.

The input of my model is then:

- one exponential growth rate (a number tau)
- one constant rate of natural mortality (a number mu)

The ouput of my model is:

- the final weight distribution of my simulated population (a density function d : R→R+)
- the final abundance of my simulated population (a number A)

.. that I have to compare with the in-situ measured final weight distribution dobs and abundance Aobs.

What is the most natural way to compute a stress value from this output, that will allow me to optimize the function stress = f(tau,mu)?


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Here's some more information if needed:

- the in-situ measurements are habitat-stratified. Therefore, both the initial/final "observed" abundances/weight distributions are actually rebuilt from several locally measured abundances and weight distributions knowing the respective areas of the various kind of habitats covering the whole site. This is the reason why:

I have to manipulate a "pure" density function with no straightforward "underlying sample"
I don't have any "individual point" or "scatter plot" to compare at the final state

- Several fishing events occur between the two campaigns, and I have enough data about'em to make them occur in the simulation. As the fishing occurs only above a given weigth, these events do link the growth phenomenon and the mortality phenomemon to each other. This is the reason why I can't work with, on one side, the couple (mortality, abundance) and, on the other side, the couple (growth, weight distribution).


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Some naive things I've tried :

- compute a "distribution stress" by integrating the function delta(x) = (d(x)-dobs(x))^2 on R, but the result is kinda "nervous" because the fishing events sometimes sharpen d a lot.

- use instead the repartition functions : delta(x) = (F(x)-Fobs(x))^2), this seems more stable and may make more sense.

- compute an "abundance stress" with the chi-square-like formula : ((A-Aobs)/A)^2.

- use instead ((A-Aobs)/mean(A,Aobs))^2 not to make the stress-value depending on the model surestimating or underestimating the final abundance.

- multiply the two, but as one comes close to zero, the other one becomes optimization-meaningless

- sum up the two, but they are barely likely to be straighforwardly compared, and I don't want to make an arbitrary choice in weighting them.

- come back to a "straightforward stress" integrating the function delta(x) = (A d(x) - Aobs dobs(x))^2 on R, but I feel like this makes the final abundance far too much important compared to the final weight distribution.


And this is it, thank you for your help if you knew any simple and natural way to deal with this kind of ouput.

Cheers,
:)


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Iago-lito