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I am writing a program that inputs a sequence of points $(x_i,y_i)$ based on the user clicking on certain pixels in an image shown. The program should then find the "best -fitting" line in the least squares sense to the set of points received. However, I must account for error in both the $x$ and $y$ variables based on the fact that the user is not exact in his/her clicking, and I must weight the points with predetermined weights $w_i$ based on how important they are in determining the line.

What is the most accurate way of computing the line in this case?

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  • $\begingroup$ Weighted linear least-square fitting is a well kwown topic, eg en.wikipedia.org/wiki/Least_squares#Weighted_least_squares $\endgroup$ – leonbloy Jul 8 '11 at 16:35
  • $\begingroup$ @leonbloy, Thanks for the link, but does that method account for error in both variables? And what will be my variance-covariance matrix in this case? $\endgroup$ – jake Jul 8 '11 at 16:39
  • $\begingroup$ Traditional least squares fixed a "indepentent" variable and seeks to minimize the squared error in the "dependent" variable. That's often reasonable. You can also try (a little more complex) "Total least squares" en.wikipedia.org/wiki/Total_least_squares $\endgroup$ – leonbloy Jul 8 '11 at 16:47
  • $\begingroup$ Here's a different take on it in response to a previous question: math.stackexchange.com/questions/28130/… $\endgroup$ – joriki Jul 8 '11 at 16:51

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