Timeline for Least-square fit of line with rational slope to points on a square with periodic boundary conditions
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Oct 20, 2010 at 1:49 | vote | accept | Tamas Kalmar-Nagy | ||
| Oct 18, 2010 at 5:19 | comment | added | Gerry Myerson | I think the answer will display sensitive dependence on $\alpha$. E.g., if $N=3$ and $\alpha=1,000,000$, then no doubt the minimizing line will be of the form $y=b$. | |
| Oct 18, 2010 at 5:07 | comment | added | Tamas Kalmar-Nagy | Thanks for the comments. I edited the entry to clarify the question. | |
| Oct 18, 2010 at 5:06 | history | edited | Tamas Kalmar-Nagy | CC BY-SA 2.5 |
Clarified question
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| Oct 18, 2010 at 4:40 | comment | added | Peter Shor | Let me explain more carefully what I think you might want (based on your motivation). Join opposite sides of the unit square to make a torus. A line with slope $r=p/q$ is going to pass through the square (torus) $q$ times horizontally and $p$ times vertically before finally meeting up with itself. Question: given $t$, is there a good way to find (or approximately find) the line with $p,q \leq t$ which minimizes the least-squares distance to all the points. By working on a torus, I expect you've given up convexity, which means the best line is probably going to be really difficult to find. | |
| Oct 18, 2010 at 3:53 | comment | added | Gerry Myerson | I echo Peter Shor's confusion. What do you want if $N=2$ and the line through the two points has irrational slope? There are rational lines arbitrarily close to that irrational line. It may be that what you are looking at is the diophantine approximation problem called simultaneous approximation, in which case there is a considerable literature. But we shouldn't have to read your mind. | |
| Oct 18, 2010 at 3:47 | answer | added | Aaron Meyerowitz | timeline score: 0 | |
| Oct 18, 2010 at 3:15 | comment | added | Peter Shor | I'm confused by the question. Do you want the line to go through the square $k$ times, where $k$ is the denominator of $r$, and have the line pass near all the points? If that's what you want, how do you control for $k$ (as increasing it will presumably give you better and better solutions). Maybe you want the best fit with $r$ having numerator and denominator less than $n$. | |
| Oct 18, 2010 at 1:21 | history | asked | Tamas Kalmar-Nagy | CC BY-SA 2.5 |