## 13 Curve Fitting

### 13.1 Overview

The fitting package deals with curve fitting for univariate real functions. When a univariate real function y = f(x) does depend on some unknown parameters p0, p1 ... pn-1, curve fitting can be used to find these parameters. It does this by fitting the curve so it remains very close to a set of observed points (x0, y0), (x1, y1) ... (xk-1, yk-1). This fitting is done by finding the parameters values that minimizes the objective function Σ(yi - f(xi))2. This is actually a least-squares problem.

For all provided curve fitters, the operating principle is the same. Users must

• create an instance of the fitter using the create factory method of the appropriate class,
• call the fit with a Collection of observed data points as argument, which will return an array with the parameters that best fit the given data points.
The list of observed data points to be passed to fit can be built by incrementally adding instances to an instance of WeightedObservedPoints, and then retrieve the list of WeightedObservedPoint by calling the toList method on that container. A weight can be associated with each observed point; it allows to take into account uncertainty on some points when they come from noisy measurements for example. If no such information exist and all points should be treated the same, it is safe to put 1.0 as the weight for all points (and this is the default when no weight is passed to the add.

Some fitters require that initial values for the parameters are provided by the user, through the withStartPoint method, before attempting to perform the fit. When that's the case the fitter class usually defines a guessing procedure within a ParameterGuesser inner class, that attempts to provide appropriate initial values based on the user-supplied data. When initial values are required but are not provided, the fit method will internally call the guessing procedure.

### 13.2 Implemented Functions

Fitting of specific functions are provided through the following classes:

The following example shows how to fit data with a polynomial function.

// Collect data.
final WeightedObservedPoints obs = new WeightedObservedPoints();