Dealing with uncertainty in the principal directions of tensors
Bibliographic record
Abstract
The variability of tensor fields is usually analyzed with multivariate statistical distributions. Multivariate distributions model every component of the tensor, which are not invariant under rotation. They therefore tell very little information about the true shape of the tensor. A statistical analysis on the eigenvalues of the tensor would be more revealing. The eigenvalues determine if a tensor is uniaxial, i.e.: only one eigenvalue is different from zero, isotropic, volume preserving or purely anisotropic. However, the eigenvalues of a normally distributed tensor are not, in general, normally distributed. In this paper, we solve this problem directly for small sizes of samples by determining the probability that the maximum error is within a reasonable bound. When the error is likely to be within a reasonable bound, we consider the eigenvalues of a tensor to be normally distributed along a mean eigendirection. Monte Carlo simulation shows that the computed bound is tight and becomes tighter when the number of sample increases. An application of the method, analysis of deformations on the cortical surface, is presented in this paper. On this data, we found that 80% of the anisotropic deformations could be analyzed by modeling the eigenvalues directly. Thus, the proposed method allows formulating statistical hypothesis directly on eigenvalues in many cases of measured deformations. Although the method was used in only one application, the method could be extended to application involving diffusion MRI or other imaging technique involving tensors.
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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 | 0.027 | 0.135 |
| Meta-epidemiology (narrow) | 0.001 | 0.002 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.005 | 0.003 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.005 | 0.007 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.003 | 0.004 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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".