Find the best fit 2 piecewise polynomial function for a data set with 1 optimal breakpoint location

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Yutao Wang 2021-5-25

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https://ww2.mathworks.cn/matlabcentral/answers/838705-find-the-best-fit-2-piecewise-polynomial-function-for-a-data-set-with-1-optimal-breakpoint-location

编辑： Torsten 2021-5-26

I have a 2D data set (2 variables x and y). I would like to use 2 piecewise polynomial functions with 1 breakpoint to best fit the dataset.

For example, function 1 = polynomial1 for mini_data < t < a and function 2 = polynomial2 for a<t<max_data. Also, I need to find the optimal breakpoint a value.

What maths tool/algorithm should I use, please?

Thanks in advance!

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Yutao Wang 2021-5-25

编辑：Yutao Wang 2021-5-25

Thanks, @Torsten.

Any degree of polynomial is fine, perhaps degree = 2 to 6 is preferred.

At breapoint, it needs to be continuous + continuity of the first derivative. An example of the data is given here. Thanks in advance!

Torsten 2021-5-26

编辑：Torsten 2021-5-26

min: sum_{i=1}^{i=ndata} (p(xdata(i),a) - ydata(i))^2

subject to

p1(a,a) = p2(a,a)

p1'(a,a) = p2'(a,a)

xdata(1) <= a <= xdata(ndata)

with

p(x,a) = p1(x,a) for x<=a

p(x,a) = p2(x,a) for x>=a

Parameters to be optimized are a and the coefficients of the polynomials p1 and p2.

A typical application for Matlab's "fmincon".

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