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Math Help - best fit, minimizing point-to-line error, not y error

  1. #1
    Mar 2011

    best fit, minimizing point-to-line error, not y error

    I have a set of [x, y] points that trend in a path, and I would like to find piece-wise equations to model the entire path.

    What I am thinking of doing is:

    1) Order the points, based on proximity to one another, so that I have an ordered set [x0, y0] ... [xn, yn]
    2) Find contiguous subsets of points that can be modeled by the equation ax + by + c = 0. I would start a new piecewise set when the average deviation from the line becomes too high.

    I might end up with:
    For [x0, y0] through [x5, y5]: ax +  by + c = 0
    For [x6, y6] through [x9, y9]:  a'x + b'y + c' = 0
    For [x10, y10] through [x12, y12]: a''x + b''y + c'' = 0

    ... More piecewise functions until I reach [xn, yn]

    Anyway, I'm not sure how to use least squares or something like it to solve for a, b, and c. As I've seen it done, least squares just solves for  f(x) = y = ax + b, but I want to be able to have lines that are potentially vertical and never intersect the y axis. I don't want to minimize the y-distance error. I want to minimize the point-to-line distance.

    Thanks for any ideas
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  2. #2
    A Plied Mathematician
    Jun 2010
    CT, USA
    This link will help you with the perpendicular regression part. The rest of your problem appears to me to be clever programming.
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