The 'simplex' algorithm undertakes a geometric search for the maximum-likelihood parameter estimates.

This methodology is also sometimes referred to as a ‘direct search’.

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

The conventional reference to the simplex method is:

Nelder, J. A. and R. Mead, 1965. A simplex method for function minimization. Comput. J. 7: 308-313. 1965.

However, a very pedagogical reference in the biological world for this methodology for model fitting is:

Mittertreiner, A., and J. Schnute. 1985. Simplex: a manual and software package for easy nonlinear parameter estimation and interpretation in fishery research. Canadian Technical Report of Fisheries and Aquatic Science 1384, Ottawa, Ontario, Canada.

Though most of the implementation aspects of this technical report are out-of-date, perhaps even obsolete, the conceptual descriptions of the simplex method, algorithm files, and covariance calculations are informative.

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 Marquardt's 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.