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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 = -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 = -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.

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