OPTI Toolbox

This page introduces a new model fitting / parameter estimation object in OPTI v2.10, optifit. The purpose of this object is to create a common routine for solving model fitting problems, while also providing a library of common structures suitable for model identification.

The following examples introduce how to use optifit for both curve and surface fitting problems.

Note the optifit function does not currently solve dynamic system parameter regression problems. See the DNLS page for ideas on solving these problems with the opti object.

The optifit Object

optifit is designed to be as simple as possible; just supply the predictor (or independent/experimental) variables as xdata, the outcome (or dependent) variables as ydata and the model as a string or function handle to fit, and optifit will attempt to solve the optimum parameters:

Returned from the above example is an optifit object, which contains the optimum parameters as the class property theta:

If you are fitting a surface, simply concatenate ydata and zdata into a cell array as the second argument:

The full calling form of optifit is:

where theta0 is an initial parameter guess (required for fitting function handles), lb and ub are parameter bounds (optional), wts are data fitting weights (optional, the default is 1), and opts allows optiset options to be supplied to the object.

Example 1: Curve Fitting A

To jump straight into it, here is an example solving a curve fitting problem:

When inspecting an optifit object, it will automatically print the solution statistics:

Noting one of the most important details printed above is the model structure identified, which in this case is a three parameter exponential function: t1*exp(xd*t2) + t3, where t standards for theta and xd stands for xdata.

The overloaded optifit plot command will plot the data, the fitted solution and the confidence bounds (functional, simultaneous):

Example 2: Curve Fitting B

Here is another example where optifit's automatic structure identification is used:

This time a three parameter rational is chosen as the best structure. The resulting fit is shown below.

optifit Model Library

So far we have seen that optifit appears to work, both at solving reasonable parameters, but also in identifying reasonable model structures. The algorithm used in identifying the structure is a collection of heuristics, thus is certainly not robust. Therefore it may be that specifying a particular structure either provides better results, or based on the underlying physics, you know what the model structure should look like. The following table lists the current model structures built into optifit:

Curve Fitting

Surface Fitting

In all models presented, optifit will automatically select a good initial parameter guess (using internal heuristics), as well as having the gradient of the model in its database. This enables it to have a fair chance of finding good parameters, if of course the model is a sensible choice for the data.

Example 3: Fitting A Custom Nonlinear Function

If you do not want to use one of the built in models above, then you are free to pass a function handle of your own model. The function handle must be of the form model(theta,xdata), or for surface fitting model(theta,xdata,ydata).

The example below is one of D. Himmelblau's data fitting examples:

The above code will automatically attempt to derive an analytical expression for the model gradient, build an OPTI object, select the best solver, and then solve the problem and return the solution as an optifit object. The plot below shows the model and 95% confidence bounds.