Bibliographic record
Abstract
This paper investigates the definition and the estimation of the Fréchet mean of a random rigid body motion in &Ropf;<it>p</it>. The sample space <it>SE</it>(<it>p</it>) contains objects <it>M</it>=(<it>R</it>,<it>t</it>) where <it>R</it> is a <it>p</it>×<it>p</it> rotation matrix and <it>t</it> is a <it>p</it>×1 translation vector. This work is motivated by applications in biomechanics where the posture of a joint at a given time is expressed as <it>M</it>∈<it>SE</it>(3), the rigid body displacement needed to map a system of axes on one segment of the joint to a similar system on the other segment. This posture can also be reported as <it>M</it>−1=(<it>R</it><it>T</it>,−<it>R</it><it>T</it><it>t</it>) by interchanging the role of the two segments. Several definitions of a Fréchet mean for a random motion are proposed using weighted least squares distances. A special emphasis is given to a Fréchet mean that is equivariant with respect to the inverse transform; this means that if <it>P</it> is the Fréchet mean for <it>M</it> then <it>P</it>−1 is the Fréchet mean for <it>M</it>−1, where <it>M</it> is a random <it>SE</it>(<it>p</it>) object. The sampling properties of moment estimators of the Fréchet means are studied in a large concentration setting, where the scatter of the random <it>M</it>s around their mean value <it>P</it> is small, and as the sample size goes to ∞. Some simple exponential family models for <it>SE</it>(<it>p</it>) data that generalize Downs’ (1972) Fisher–von Mises matrix distribution for rotation matrices are introduced and the least squares mean values for these distributions are calculated. Asymptotic comparisons between the estimators presented in this work are carried out for a particular model when <it>p</it>=2. A numerical example involving the motion of the ankle is presented to illustrate the methodology.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 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 teacher head, 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".