Simple Multiple Line Fitting Algorithm

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Simple Multiple Line Fitting Algorithm

• Yan Guo

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Motivation

• To generate better result than EM algorithm, to
  avoid local optimization.

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Line Score

• In order to describe how well certain points fits
  to a line, I developed a score function for the
  line model.
• Higher score indicates more linear line, lower
  score indicates less likely to be a line.
• Higher score can be achieved by either adding a
  point, or removing a point.

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Simple Linear Regression
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is the square of the Pearson correlation.
is the proportion of variation explained by
regression Model. It indicates how well the
prediction line fits the data. In general,
higher value means better fits.

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• Leverage Used to measure the impact of a point
  in a line.
• Student Residual
• Jackknife residual

Jackknife residual follows a t distribution with
(n-3) degree of freedom.
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Line Score

• Two factors are considered into the Line Score
  Function R-Square and Proportion of the points
  in a line.
• Line Score is defined as
• N is the total number of points in the input, n
• is the number of points in the current line.

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Experimenting the Line Score
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The algorithm

• 1. Divide the area into certain finite area of
  stripes.
• 2. Calculate the line score for each stripe.
• 3. Pick the stripe with highest score, filter out
  outliers. Recalculate the stripe area with the
  fitted line in the middle.
• 5. Recalculate the line score with the new points
  inside the stripe.
• 6. If new line score is higher, continue to next
  step, otherwise go back to 3, and pick the next
  highest score stripe.
• 7. Recalculate the stripe with the newly fitted
  line in the middle. Go to step 5. Repeat until no
  more points are getting added into the stripe.
• 8. Remove the points from the final stripe from
  the input, and repeat from step1.
• 9. Finalize the results, detecting noise etc.

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Simple Example
Line Score1.4
Line Score2
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Complicated Example
Line Score 0.578
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Filter out outliers
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Extreme Cases
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Future Improvement

• A Better Scoring Function. Are they more factor
  to be considered in this function. (Press
  Statistic, Cp Statistic, P-value, CVSS, etc)
• Adjusted R square VS R square, VS correlation

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Conclusion

• This Algorithm works on some cases
• It doesnt require initialization
• It works best when line is perfectly straight
• It can detect noise
• It will not work on all case, since it is
  probability based