Measuring Linearity of a Finite Set of Points
Bibliographic record
Abstract
It is often useful to measure how linear a certain set of points is. Our goal is to design algorithms that give a linearity measurement in the interval [0, 1]. There is no explicit discussion on linearity in literature, although some existing shape measures may be adapted. We are interested in linearity measures which are invariant to rotation, scaling, and translation. These linearity measures should also be calculated very quickly and be resistant to protrusions in the data set. The measures of eccentricity and contour smoothness were adapted from literature, the other five being triangle heights, triangle perimeters, rotation correlation, average orientations, and ellipse axis ratio. The algorithms are tested on 30 sample curves and the results are compared against the linear classifications of these curves by human subjects. It is found that humans and computers typically easily identify sets of points that are clearly linear, and sets of points that are clearly not linear. They have trouble measuring sets of points which are in the gray area in between. Although they appear to be conceptually very different approaches, we prove, theoretically and experimentally, that eccentricity and rotation correlations yield exactly the same linearity measurements. They however provide results which are furthest from human measurements. The average orientations method provides the closest results to human perception, while the other algorithms proved themselves to be very competitive
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.038 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.005 | 0.003 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".