MATLAB CURVE FITTING TOOLBOX - RELEASE NOTES Guide de l'utilisateur télécharger pdf (Page 90)

3 Fitting Data

3-14

Nonlinear Least Squares

The Curve Fitting Toolbox uses the nonlinear least squares formulation to fit

a nonlinear model to data. A nonlinear model is defined as an equation that is

nonlinear in the coefficients, or a combination of linear and nonlinear in the

coefficients. For example, Gaussians, ratios of polynomials, and power

functions are all nonlinear.

In matrix form, nonlinear models are given by the formula

where

• y is an n-by-1 vector of responses.

• f is a function of β and X.

• β is a m-by-1 vector of coefficients.

• X is the n-by-m design matrix for the model.

• ε is an n-by-1 vector of errors.

Nonlinear models are more difficult to fit than linear models because the

coefficients cannot be estimated using simple matrix techniques. Instead, an

iterative approach is required that follows these steps:

1 Start with an initial estimate for each coefficient. For some nonlinear

models, a heuristic approach is provided that produces reasonable starting

values. For other models, random values on the interval [0,1] are provided.

2 Produce the fitted curve for the current set of coefficients. The fitted

response value is given by

and involves the calculation of the Jacobian of f(X,b), which is defined as a

matrix of partial derivatives taken with respect to the coefficients.

yfXβ,()ε+=

fXb,()=

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MATLAB CURVE FITTING TOOLBOX - RELEASE NOTES Guide de l'utilisateur Page 90