Estimation and testing for separable variance–covariance structures
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Bibliographic record
Abstract
The statistical analysis of data for a p ‐variate response observed repeatedly on q occasions or of spatiotemporal data recorded at p locations by q times for n individuals may require that constraints be imposed on the modeling of the variance–covariance structure of the underlying process, not because of the repeated‐measures or spatiotemporal nature of the data but because there is not enough data otherwise to estimate the model parameters. Besides stationarity and isotropy, separability is an interesting option for that purpose because it reduces the number of variance‐covariance parameters to estimate, from pq ( pq + 1)/2 to the Kronecker product of two matrices with p ( p + 1)/2 and q ( q + 1)/2 parameters. Originally, in the late 1980s, separability of the variance–covariance structure was assumed . Under this model, combined with the normality assumption on the underlying distribution, novel theoretical developments were thus made. The question of estimation of the parameters of a separable variance–covariance structure, more particularly by maximum likelihood, was raised from the early 1990s on, the question of testing for this structure being effectively addressed several years later. The existence and uniqueness of maximum likelihood estimators for the matrix normal distribution (i.e., the doubly multivariate normal distribution characterized by a simply separable variance–covariance structure) have been and remain questions of interest, as shown by recent results. Below, the reader is guided throughout the field of study of the separable variance–covariance structures as the author provides a fair treatment of the topic, its components, extensions (e.g., double separability), and future perspectives. This article is categorized under Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 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) | 0.000 | 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 it