Suppose you have data, and want to fit a model of the form

${\mathit{y}}_{\mathit{i}}={\mathit{a}}_1+{\mathit{a}}_2\exp \left(-{\mathit{a}}_3{\mathit{x}}_{\mathit{i}}\right)+{ϵ}_{\mathit{i}}$.

$$a_i$$are the parameters you want to estimate, $$x_i$$are the data points, $$y_i$$are the responses, and $${\epsilon }_i$$are noise terms.

Write a function handle that represents the model:

mdl = @(a,x)(a(1) + a(2)*exp(-a(3)*x));

Generate synthetic data with parametersa = [1;3;2], with thexdata points distributed exponentially with parameter2, and normally distributed noise with standard deviation0.1:

rng(9845,'twister')% for reproducibilitya = [1;3;2]; x = exprnd(2,100,1); epsn = normrnd(0,0.1,100,1); y = mdl(a,x) + epsn;

Fit the model to data starting from the arbitrary guessa0 = [2;2;2]:

a0 = [2;2;2]; [ahat,r,J,cov,mse] = nlinfit(x,y,mdl,a0); ahat

ahat =3×11.0153 3.0229 2.1070

Check whether[1;3;2]is in a 95% confidence interval using the Jacobian argument innlparci:

ci = nlparci(ahat,r,'Jacobian',J)

ci =3×20.9869 1.0438 2.9401 3.1058 1.9963 2.2177

You can obtain the same result using the covariance argument:

ci = nlparci(ahat,r,'covar',cov)

ci =3×20.9869 1.0438 2.9401 3.1058 1.9963 2.2177