Modeling multi‐way data with linearly dependent loadings
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Bibliographic record
Abstract
Abstract A generalization/specialization of the PARAFAC model is developed that improves its properties when applied to multi‐way problems involving linearly dependent factors. This model is called PARALIND (PARAllel profiles with LINear Dependences). Linear dependences can arise when the empirical sources of variation being modeled by factors are causally or logically linked during data generation, or circumstantially linked during data collection. For example, this can occur in a chemical context when end products are related to the precursor or in a psychological context when a single stimulus generates two incompatible feelings at once. For such cases, the most theoretically appropriate PARAFAC model has loading vectors that are linearly dependent in at least one mode, and when collinear, are nonunique in the others. However, standard PARAFAC analysis of fallible data will have neither of these features. Instead, latent linear dependences become high surface correlations and any latent nonuniqueness is replaced by a meaningless surface‐level ‘unique orientation’ that optimally fits the particular random noise in that sample. To avoid these problems, any set of components that in theory should be rank deficient are re‐expressed in PARALIND as a product of two matrices, one that explicitly represents their dependency relationships and another, with fewer columns, that captures their patterns of variation. To demonstrate the approach, we apply it first to fluorescence spectroscopy (excitation‐emission matrices, EEM) data in which concentration values for two analytes covary exactly, and then to flow injection analysis (FIA) data in which subsets of columns are logically constrained to sum to a constant, but differently in each of two modes. In the PARAFAC solutions of the EEM data, all factors are ‘unique’ but this is only meaningful for two of the factors that are also unique at the latent level. In contrast, the PARALIND solutions directly display the extent and nature of partial nonuniqueness present at the latent level by exhibiting a corresponding partial uniqueness in their recovered loadings. For the FIA data, PARALIND constraints restore latent uniqueness to the concentration estimates. Comparison of the solutions shows that PARALIND more accurately recovers latent structure, presumably because it uses fewer parameters and hence fits less error. Copyright © 2009 John Wiley & Sons, Ltd.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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