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A linear model estimates the best fit line from a set of points, often through minimization of the sum of squared residuals.

By analogy, is there any established method (possibly implemented in R) for fitting a plane to a set of lines (or segments) ? For example, minimizing sum of area (or squared area ?) between residual-lines and the best-fitting plane ?

Thanks.

Rodolphe
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  • Fascinating question. These lines can be oriented any which-way, or are they maybe parallel to some axis? (e.g. you might have a set of lines where each line has y constant but x-varying, and at a given y the line or lines relate z to x). Just idly supposing for a moment, I wonder if there might be any traction in working in the dual-space. I think that we'd probably look at finding a point (dual to the plane) closest to a set of lines (I think duals of lines would be lines). – Glen_b Jan 21 '16 at 02:31
  • When asking, I was thinking specifically about lines that can be oriented in any direction. I just realized that there might be problems with infinite lines, so I edited my question, talking about segments of lines, too, that should represent an easier problem (IMHO) to tackle. – Rodolphe Jan 21 '16 at 02:39
  • If your data were segments (and you only considered fit where the segment was), you might consider replacing the segment by some set of points (you might have constant density of points per unit distance or per segment, depending on what you were trying to achieve). But I don't think there's necessarily a problem with infinite lines. – Glen_b Jan 21 '16 at 02:39
  • Gary King did some work on tomography that may be what you're after, or perhaps get you started. I'm looking for a reference now... – Sycorax Jan 21 '16 at 02:43
  • @Glen_b You are right the data I am working consists of segments. Because they are not same length, replacing these segments by sets of evenly spaced distance could do the job. However, are'nt all points "from" a given segment correlated (some kind of hierarchy ?) – Rodolphe Jan 21 '16 at 02:59
  • If you wanted to regard them as correlated in the hierarchical sense your certainly could, but what I was suggesting would just treat them as uncorrelated in order to get an appropriate kind of "fit" (that is, I wasn't suggesting a probability model for these point sets, it's just a device for getting a fitted plane to pass close to line segments). An alternative might be to replace each line segment by endpoints or center (and then weight by segment length if needed to get something that would minimize the required integral). Oh, actually, you could probably prove an equivalence. – Glen_b Jan 21 '16 at 04:13

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