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Greetings,

I'm performing research that will help determine the size of observed space and the time elapsed since the big bang. Hopefully you can help!

I have data conforming to a piecewise linear function on which I want to perform two linear regressions. There is a point at which the slope and intercept change, and I need to (write a program to) find this point.

Thoughts?

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see cross posted question: stats.stackexchange.com/questions/5700/… –  Jeromy Anglim Dec 23 '10 at 12:22

3 Answers 3

A simple answer would be to consider the break point a variable, call it x. Collect the points below x and perform a linear fit, returning the error (say sum of squared errors over the points). Collect the points above x and do the same. Then you can define f(x) as the total error over the two fits. Consider this a function in one variable and minimize it.

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I am assuming that you want a continuous piecewise linear fit of the form $$ a(x-\xi)+c\hbox{ if }x\le\xi,\quad b(x-\xi)+c\hbox{ if }x>\xi. $$ There are 4 variables to determine: the slopes $a$, $b$, the point $\xi$ and $c$, the ordinate at $\xi$. Let $\{(x_i,y_i)\}$, $1\le i\le N$, be your data. For simplicity I will assume that $x_i<x_{i+1}$. For each $i=1,\dots,N-1$ minimize the expression $$ O_i=\sum_{j=1}^i(y_j-a(x_j-\xi)-c)^2+\sum_{j=i+1}^N(y_j-b(x_j-\xi)-c)^2. $$ This can be done by solving the system of 4 equations with 4 unknowns $$ \frac{\partial O_i}{\partial a}=\frac{\partial O_i}{\partial b}=\frac{\partial O_i}{\partial c}=\frac{\partial O_i}{\partial \xi}=0. $$ (This will be nonlinear, and will have more than one solution)

Finally, choose the index $i$ that gives the minimum $O_i$.

I tried this in a mock example, and it came out quite right.

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(This was supposed to be a comment, but it got too long.)

The problem of piecewise linear regression has been looked into many times before; I do not currently have access to these papers (and thus cannot say more about them), but you might want to look into these:

This paper (published in a physiology journal, of all places) discusses how to fit a piecewise linear function to certain data sets encountered in neurological research. A FORTRAN routine is included.

This paper also includes a FORTRAN routine for fitting piecewise linear functions to data.

This paper relies on maximum likelihood to find the best fit piecewise linear function.

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