How can I fit data to a piecewise function, where the breakpoint of the function is also a parameter to be optimised?

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Rahul
Rahul 2025-9-29，20:45
评论： Image Analyst 2025-10-3，14:58
I have data with x and y values. This data should conform to a function: an assymmetric parabola. Here, the parameters that define the shape of the parabola should be different on either side of the maximum point of the parabola i.e. the breakpoint is where the maximum value of y occurs.
I was hoping to use 'fit' and to define an anonymous function for my data. But I'm not able to work out how to define an anonymous, piecewise function, especially where the breakpoint is one of the parameters to be determined by the fitting procedure, as it is not immediately clear from the data itself where the maximum value of y should occur.
Any help would be appreciated.

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采纳的回答

Matt J
Matt J 2025-9-30，4:09
编辑：Matt J 2025-9-30，11:20
Ran in:
Once you've chosen the coefficients of the first parabola [a1,b1,c1], the breakpoint is determined from,
d=-b1/(2*a1)
Only the leading coefficient of the second parabola is a free parameter:
F=@(x) asymParabola(-2,1,0,-0.6,x);
fplot(F,[-10,10]);axis padded %example plot
ft = fittype(@(a1,b1,c1,a2, x) asymParabola(a1,b1,c1,a2, x) )
ft =
General model: ft(a1,b1,c1,a2,x) = asymParabola(a1,b1,c1,a2,x)
function y=asymParabola(a1,b1,c1,a2, x)
d=-b1/(2*a1);
b2=-d*2*a2;
c2=polyval([a1,b1,c1],d)-polyval([a2,b2,0],d);
left=(x<=d);
y=x;
y(left)=polyval([a1,b1,c1],x(left));
y(~left)=polyval([a2,b2,c2],x(~left));
end
  6 个评论
显示 4更早的评论
Rahul
Rahul 2025-10-2，15:09
You're right Torsten that things could be interpreted in this way, and I could have been clearer in my original question.
Image Analyst
Image Analyst 2025-10-3，14:58
@Rahul what would have been best is if you had shown a screenshot of your noisy data plotted, and attached the data so that people would have something to work with.
I think the fit values will change because of noise. With one set of noise, you'd have one set of parabola coefficients but if you had a different set of noise, then you have a different set of coefficients. If there is no noise, there is no need for a fit because you have the analytical formula already.
If you have any more questions, then attach your data and code to read it in with the paperclip icon after you read this:
TUTORIAL: How to ask a question (on Answers) and get a fast answer

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更多回答（3 个）

Walter Roberson
Walter Roberson 2025-9-29，21:20
(a1*x.^2 + b1*x + c1) .* (x <= d) + (a2*x.^2 + b2*x + c2) .* (x > d)
Note that for this to work, the coefficients must be constrained to be finite
  2 个评论
Paul
Paul 2025-9-30，0:37
Sounds like both sides of the function should have the same value at x = d, at least that's how interpret the question. If so, then I think the function would look something like
(a1*(x-d).^2 + b1*(x-d) + c) .* (x <= d) + (a2*(x-d).^2 + b2*(x-d) + c) .* (x > d)
Rahul
Rahul 2025-9-30，13:57
Thanks Paul, I think this would also work, but as Matt pointed out I think some of extra parameters are not needed (as they are actually constrained by the others). I am not sure how Matlab deals with this in the curve fitting/optimisation algorithms.

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Catalytic
Catalytic 2025-9-30，14:13
编辑：Catalytic 2025-9-30，14:18
Ran in:
You can also parametrize the model function directly in terms of the break point coordinates (xbreak, ybreak) and two curvature parameters -
F= @(a1,a2,xbreak,ybreak, x) modelFun(a1,a2,xbreak,ybreak, x);
xbreak=3; ybreak=5;
fplot( @(x) F(-2,-0.6,xbreak,ybreak,x), [1,5]);
xline(xbreak,'--')
fType = fittype(F);
function y=modelFun(a1,a2,xbreak,ybreak, x)
X=x-xbreak;
LHS=(X<=0);
RHS=~LHS;
y=X.^2;
y(LHS)=a1.*y(LHS) + ybreak;
y(RHS)=a2.*y(RHS) + ybreak;
end
Matt J
Matt J 2025-9-30，21:15
编辑：Matt J 2025-10-3，13:47
Ran in:
Why ? Both parabola can intersect below their respective maxima, and nonetheless the point of intersection can be the maximum y-value of the piecewise function.
If I understand @Torsten, that would be a 6-parameter function,
F=@(x) asymParabola(-2,1,0,-6,5,-20 ,x);
fplot(F,[-10,-1]);axis padded %example plot
function y=asymParabola(a1,b1,c1, a2, s, rightSlope, x)
%Requirements: a1<0, a2<0, s>=0, rightSlope<=0
d=-b1/(2*a1)-s;
c2=polyval([a1,b1,c1],d);
left=(x<=d);
right=~left;
xright=x(right);
y=x;
y(left)=polyval([a1,b1,c1],x(left));
y(right)=a2*(xright-d).^2 + rightSlope*(xright-d) +c2;
end

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