MultiStart Using`lsqcurvefit`或者`lsqnonlin`

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此示例显示如何使用函数来使用lsqcurvefittogether withMultiStart。The end of the example shows the same solution usinglsqnonlin。

许多拟合问题有多个本地解决方案。金宝搏官方网站MultiStartcan help find the global solution, meaning the best fit. This example first useslsqcurvefitbecause of its convenient syntax.

该模型是

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

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

Step 1. Create the objective function.

编写一个采用数据矩阵的匿名函数xdatawithN行和两列，并返回响应矢量N行。该功能也采用系数矩阵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 $a = - 3, b = 1 / 4, c = 1 / 2, d = 1$ 。在响应中包括随机噪声。

rng.default再现性的百分比n = 200;％数据点数preal = [-3,1/4,1/2,1];% Real coefficientsxdata = 5*rand(N,2);% Data pointsydata = fitfcn（preal，xdata）+ 0.1 * randn（n，1）;% Response data with noise

Step 3. Set bounds and initial point.

Set bounds forlsqcurvefit。There is no reason for $d$ 超过 $π.$ in absolute value, because the sine function takes values in its full range over any interval of width $2 π.$ 。假设系数 $c$ 必须小于20的绝对值，因为允许高频可能导致不稳定的响应或不准确的收敛。

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 pointP0.(4) = 0;％确保初始点满足界限

Step 4. Find the best local fit.

Fit the parameters to the data, starting atP0.。

[Xfited，errorfited] = 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

lsqcurvefitfinds 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`。

创建一个问题结构MultiStartcan solve the same problem.

问题=创造eOptimProblem('lsqcurvefit','x0',p0,'objective',fitfcn,。。。'磅'，磅，'UB'，UB，'xdata'，xdata，'ydata'，ydata）;

Step 6. Find a global solution.

Solve the fitting problem usingMultiStartwith 50 iterations. Plot the smallest error as the number ofMultiStartiterations.

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

Figure MultiStart contains an axes object. The axes object with title Best Function Value: 1.6464 contains 2 objects of type line, text.

MultiStart完成了所有起始点的运行。所有50个本地求解器运行融合，并使用正本地求解器退出标志。

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

errormulti = 1.6464

MultiStartfinds a global solution near the parameter values (–3,–1/4,–1/2,–1). (This is equivalent to a solution nearpreal=（-3,1 / 4,1 / 2,1），因为更改所有系数的符号，除了第一个给出相同的数字值fitfcn.。）残余误差的规范从约28到约1.6减小，降低超过10倍。

制定问题`lsqnonlin`

For an alternative approach, uselsqnonlinas 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 pointP0.,lsqnonlinfinds the same relatively poor solution aslsqcurvefit。

RunMultiStartusinglsqnonlin作为当地的解决者。

问题2 = createOptimproblem（'lsqnonlin','x0',p0,'objective'，fitfcn2，。。。'磅'，磅，'UB',ub'); [xmultinonlin,errormultinonlin] = run(ms,problem2,50)

Figure MultiStart contains an axes object. The axes object with title Best Function Value: 1.6464 contains 2 objects of type line, text.

MultiStart完成了所有起始点的运行。所有50个本地求解器运行融合，并使用正本地求解器退出标志。

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

errormultinonlin = 1.6464

Again,MultiStartfinds a much better solution than the local solver alone.

MultiStart Usinglsqcurvefit或者lsqnonlin