Problem in Curve fitting

Question

maryam amiri 2021-5-20

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/835418-problem-in-curve-fitting

编辑： Matt J 2021-5-20

采纳的回答： Matt J

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Hi,

I have a sets of data [x,y] that I want to fit with a function F(v,x) where v contains six free parameters.

x=[70,75,80,83,90,100]; y=[1,1,0.97,0.95,0.9,0];

I found the best fitted curve by cftool for this data set (polynomial degree 5):

but the result is different when I use lsqcurvefit.

v0=[0,0,0,0,0,0];
fun = @(v,x)v(1)*x.^5 + v(2)*x.^4 + v(3)*x.^3 + v(4)*x.^2 + v(5)*x + v(6);
x=[70,75,80,83,90,100];y=[1,1,0.97,0.95,0.9,0];
v=lsqcurvefit(fun,v0,x,y);
times = linspace(x(1),x(end));
plot(x,y,'ko',times,fun(v,times),'b-')

this is the result:

It seems lsqurvefit did not fitted the curve to the points.

any idea that why it does not work for me?

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Answer 1

Matt J 2021-5-20

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https://ww2.mathworks.cn/matlabcentral/answers/835418-problem-in-curve-fitting#answer_704808

编辑：Matt J 2021-5-20

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Although polyfit is the better tool here, both polyfit and lsqcurvefit will be challenged by the scaling of your xdata, which is making the problem highly ill-conditioned. Rescaling helps considerably, as shown below,

v0=[0,0,0,0,0,0];

fun = @(v,x)v(1)*x.^5 + v(2)*x.^4 + v(3)*x.^3 + v(4)*x.^2 + v(5)*x + v(6);

x=[70,75,80,83,90,100];y=[1,1,0.97,0.95,0.9,0];

x=(x-mean(x))/std(x);

[v,fval,~,exitflag]=lsqcurvefit(fun,v0,x,y)

Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance.

v = 1×6

-0.0381 -0.0852 -0.0060 0.0310 -0.0643 0.9500

fval = 5.8273e-19

exitflag = 1

times = linspace(x(1),x(end));

plot(x,y,'ko',times,fun(v,times),'b-')

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Matt J 2021-5-20

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Another way to see the need for scaling is to look its effect on the condition number of the Vandermonde matrix,

x=[70,75,80,83,90,100];
cond(vander(x)),
ans = 2.2962e+15
cond(vander((x-mean(x))/std(x)))
ans = 230.1084

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Answer 2

Walter Roberson 2021-5-20

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fun = @(v,x)v(1)*x.^5 + v(2)*x.^4 + v(3)*x.^3 + v(4)*x.^2 + v(5)*2 + v(6);
                                                            ^^^^^^

Should be

v(5)*x

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Walter Roberson 2021-5-20

However, it stops when it thinks the residue is good enough, or if it gets too very small step sizes.

It is a convex problem

Your v = -0.0381 -0.0852 -0.0060 0.0310 -0.0643 0.9500 has two sign changes, so the function itself is not convex.

Matt J 2021-5-20

编辑：Matt J 2021-5-20

However, it stops when it thinks the residue is good enough, or if it gets too very small step sizes.

Yes, the ill-conditioning of the problem does cause one of these lsqcurvefit stopping criteria to be triggered prematurely, and where it stops will indeed depend on the initial point.

Your v = -0.0381 -0.0852 -0.0060 0.0310 -0.0643 0.9500 has two sign changes, so the function itself is not convex.

Yes, the polynomial being fitted is surely not convex as a function of x as we can also see from the plots. However, the least squares objective is convex as a function of v, which is why, in theory, lsqcurvefit should be globally convergent for this problem.

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