Matérn's hard core models of types I and II with arbitrary compact grains
Bibliographic record
Abstract
Matérn's classical hard core models can be interpreted as models obtained from a stationary marked Poisson process by dependent thinning. The marks are balls of fixed radius, and a point is retained when its associated ball does not hit any other balls (type I) or when its random birth time is strictly smaller than the birth times of all balls hitting it (type II). Extending ideas of [M. Månsson and M. Rudemo. Random patterns of nonoverlapping convex grains. Adv. in Appl. Probab., 34:718--738, 2002.], who considered grains that are isotropic rotations or random scalings of a fixed convex set, we discuss these two models in d-dimensional space when the marks are arbitrary random compact grains. We determine the intensity and the mark distribution after thinning, and find the second order factorial moment density of the ground process for model II under weak additional assumptions. By Brunn-Minkowski's inequality, the volume density associated to model II turns out to be bounded by 2−d. This bound is sharp. It is attained asymptotically (when the proposal intensity tends to infinity) only when all grains coincide with one deterministic origin-symmetric convex set. We also discuss how known connections of this model with the process of intact grains of the dead leaves model and the Stienen model leads to analogous results for the latter.
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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 | 0.003 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".