# MultiStart Using `lsqcurvefit`

or `lsqnonlin`

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

together with `MultiStart`

. The end of the example shows the same solution using `lsqnonlin`

.

Many fitting problems have multiple local solutions. `MultiStart`

can help find the global solution, meaning the best fit. This example first uses `lsqcurvefit`

because of its convenient syntax.

The model is

$$y=a+b{x}_{1}\mathrm{sin}(c{x}_{2}+d),$$

where the input data is $$x=({x}_{1},{x}_{2})$$, and the parameters $$a$$, $$b$$, $$c$$, and $$d$$ are the unknown model coefficients.

### Step 1. Create the objective function.

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

with `N`

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

rows. The function also takes a coefficient matrix `p`

, corresponding to the coefficient vector $$(a,b,c,d)$$.

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

### Step 2. Create the training data.

Create 200 data points and responses. Use the values $$a=-3,b=1/4,c=1/2,d=1$$. 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

### Step 3. Set bounds and initial point.

Set bounds for `lsqcurvefit`

. There is no reason for $$d$$ to exceed $$\pi $$ in absolute value, because the sine function takes values in its full range over any interval of width $$2\pi $$. Assume that the coefficient $$c$$ must be smaller than 20 in absolute value, because allowing a high frequency can cause unstable responses or inaccurate 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; % Ensure the initial point satisfies the bounds

### Step 4. Find the best local fit.

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 value of the function tolerance.

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

errorfitted = 28.2524

`lsqcurvefit`

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

### Step 5. Set up the problem for `MultiStart`

.

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);

### Step 6. Find a global solution.

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 exitflag.

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

errormulti = 1.6464

`MultiStart`

finds a global solution near the parameter values (–3,–1/4,–1/2,–1). (This is equivalent to a solution near `preal`

= (–3,1/4,1/2,1), because changing the sign of all the coefficients except the first gives the same numerical values of `fitfcn`

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

### Formulate Problem for `lsqnonlin`

For an alternative approach, use `lsqnonlin`

as the fitting function. In this case, use the difference between predicted values and actual data values as the objective function.

fitfcn2 = @(p)fitfcn(p,xdata)-ydata; [xlsqnonlin,errorlsqnonlin] = lsqnonlin(fitfcn2,p0,lb,ub)

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

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

errorlsqnonlin = 28.2524

Starting from the same initial point `p0`

, `lsqnonlin`

finds the same relatively poor solution as `lsqcurvefit`

.

Run `MultiStart`

using `lsqnonlin`

as the local solver.

problem2 = createOptimProblem('lsqnonlin','x0',p0,'objective',fitfcn2,... 'lb',lb,'ub',ub'); [xmultinonlin,errormultinonlin] = run(ms,problem2,50)

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

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

errormultinonlin = 1.6464

Again, `MultiStart`

finds a much better solution than the local solver alone.