Note: This page has been translated by MathWorks. Click here to see

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

This example shows how to fit a function to data using `lsqcurvefit`

together with `MultiStart`

.

Many fitting problems have multiple local solutions. `MultiStart`

can help find the global solution, meaning the best fit. While you can use `lsqnonlin`

as the local solver, this example uses `lsqcurvefit`

simply because it has a convenient syntax.

The model is

where the input data is , and the parameters , , , and are the unknown model coefficients.

Write an anonymous function that takes a data matrix `xdata`

with `N`

rows and two columns, and returns a response vector with `N`

rows. It also takes a coefficient matrix `p`

, corresponding to the coefficient vector .

fitfcn = @(p,xdata)p(1) + p(2)*xdata(:,1).*sin(p(3)*xdata(:,2)+p(4));

Create 200 data points and responses. Use the values . Include random noise in the response.

rng default % for reproducibility N = 200; % number of data points preal = [-3,1/4,1/2,1]; % real coefficients xdata = 5*rand(N,2); % data points ydata = fitfcn(preal,xdata) + 0.1*randn(N,1); % response data with noise

Set bounds for `lsqcurvefit`

. There is no reason for to exceed in absolute value, because the sine function takes values in its full range over any interval of width . Assume that the coefficient must be smaller than 20 in absolute value, because allowing a very high frequency can cause unstable responses or spurious convergence.

lb = [-Inf,-Inf,-20,-pi]; ub = [Inf,Inf,20,pi];

Set the initial point arbitrarily to (5,5,5,0).

p0 = 5*ones(1,4); % Arbitrary initial point p0(4) = 0; % so the initial point satisfies the bounds

Fit the parameters to the data, starting at `p0`

.

[xfitted,errorfitted] = lsqcurvefit(fitfcn,p0,xdata,ydata,lb,ub)

Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the default value of the function tolerance.

`xfitted = `*1×4*
-2.6149 -0.0238 6.0191 -1.6998

errorfitted = 28.2524

`lsqcurvefit`

found a local solution that is not particularly close to the model parameter values (-3,1/4,1/2,1).

Create a problem structure so `MultiStart`

can solve the same problem.

problem = createOptimProblem('lsqcurvefit','x0',p0,'objective',fitfcn,... 'lb',lb,'ub',ub,'xdata',xdata,'ydata',ydata);

Solve the fitting problem using `MultiStart`

with 50 iterations. Plot the smallest error as the number of `MultiStart`

iterations.

```
ms = MultiStart('PlotFcns',@gsplotbestf);
[xmulti,errormulti] = run(ms,problem,50)
```

MultiStart completed the runs from all start points. All 50 local solver runs converged with a positive local solver exit flag.

`xmulti = `*1×4*
-2.9852 -0.2472 -0.4968 -1.0438

errormulti = 1.6464

`MultiStart`

found a global solution near the parameter values . (This is equivalent to a solution near `preal`

= , because changing the sign of all the coefficients except the first gives the same numerical values of `fitfcn`

.) The norm of the residual error decreased from about 28 to about 1.6, a decrease of more than a factor of 10.

Was this topic helpful?