Marquardt’s algorithm undertakes a derivative-based search for the maximum-likelihood parameter estimates.

Such an algorithm or an alterative is necessary for non-linear models.

Marquardt’s algorithm requires the values of the first and second derivatives of the likelihood function with respect to the current parameters.

This algorithm was implemented in SmartStats © with the ability to automatically calculate these derivatives numerically.

However, some models have had the equivalent analytical derivatives coded into the model.

In general, the function minimization behaviour of some, but not all, models will be noticeably more efficient when using analytical derivatives.

However, for complex models, calculation of analytical derivatives is neither a practical, nor sufficiently more computationally efficient, option to justify the work involved in calculating them without error.

This is particularly true when you consider that most of us would be reluctant to not verify the analytical derivatives numerically.

The reference for the implementation of Marquardt’s algorithm in SmartStats is:

Press, W.H., S.A. Teukolsky, W.T. Vetterling and B.P. Flannery. 1993. Numerical Recipes in Fortran: The Art of Scientific Computing. Cambridge Univeristy Press. [Numerical recipes]

It is worth noting here that no search algorithm guarantees that the final fit is the global minimum that is usually desired.

Sometimes the so-called ‘best-fit’ is a local minimum, or at worst a singularity, which are useless for providing meaningful estimates for the parameter values.

As such, practitioners may wish to choose different sets of starting parameter values and judge whether the final estimates are consistent among several fitting attempts.

Therefore it is always best to choose parameter values for which you have an a priori suspicion of the final estimate.

It is with this concern in mind that, where possible, models are parameterized using biologically meaningful and interpretable parameters - for example, the values of polynomial coefficients usually have no practical biological interpretation value.

Another approach to assuring the final fit is the desired fit is to perform a systematic grid search for parameter values prior to undertaking a simplex search.

One model offered on this website [Species richness] is coded with this possibility.

The multidimensional grid will methodically select sets of parameter values, usually by changing their values in proportional increments within a reasonable (user chosen) parameter space.

The best set of parameters discovered can then be used as starting values in a simplex search or derivative-based search.

You can appreciate that such grid searches can be time consuming, but they can mitigate the frustration of having to begin an estimation with naïve parameter values.

Once SmartStats © converges on a putative ‘best-fit’, it attempts to confirm that fit by conducting a so-called axial search.

If this axial search (which systematically moves parameter values away from their value at the putative minimum) discovers an improvement in fit greater than the tolerance for convergence, the putative fit is rejected and the search for the ‘best-fit’ continues.