Efficient Correlation Matching for Normal-Copula Dependence when Univariate Marginals Are Discrete
Bibliographic record
Abstract
AbstractA popular approach for modeling dependence in a finite-dimensional random vectorX with given univariate marginals is via a normal copula that fits the rank or linearcorrelations for the bivariate marginals of X. In this approach, known as the NORTAmethod, the normal distribution function is applied to each coordinate of a vector Zof correlated standard normals to produce a vector U of correlated uniforms randomvariables over (0,1); then X is obtained by applying the inverse of the target marginaldistribution function for each coordinate of U. The fitting requires finding the ap-propriate correlation ρ between any two given coordinates of Z that would yield thetarget rank or linear correlation r between the corresponding coordinates of X. Thisroot-finding problem is easy to solve when the marginals are continuous, but not whenthey are discrete. In this paper, we provide a detailed analysis of the NORTA methodfor discrete marginals. We prove key properties of r and of its derivative as a functionof ρ. It turns out that the derivative is easier to evaluate than the function itself.Based on that, we propose and compare alternative methods for finding or approxi-mating the appropriate ρ. The case of discrete distributions with unbounded supportis covered as well. In our numerical experiments, a derivative-supported method isfaster and more accurate than a state-of-the-art, non-derivative-based method. Wealso characterize the asymptotic convergence rate of the function r (as a function of ρ)to the continuous-marginals limiting function, when the discrete marginals convergeto continuous distributions.Key Words: Statistics; distribution; estimation; correlation; mathematics; simula-tion.Acknowledgments: This work has been supported by a grant from Bell Canada viathe Bell University Laboratories, Grants No. CRDPJ-251320 and ODGP0110050 fromNSERC-Canada, and a Canada Research Chair to the third author. We thank RichardSimard for his assistance in programming aspects and some numerical experiments.
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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 | 0.006 | 0.026 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 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".