Continuous-Time Multivariate Analysis
Bibliographic record
Abstract
The starting point for much of multivariate analysis (MVA) is an <i>n</i> × <i>p</i> data matrix whose <i>n</i> rows represent observations and whose <i>p</i> columns represent variables. Some multivariate datasets, however, may be best conceptualized not as <i>n</i> discrete <i>p</i>-variate observations, but as <i>p</i> curves or functions defined on a common time interval. Here we introduce a framework for extending techniques of multivariate analysis to such settings. The proposed continuous-time multivariate analysis (CTMVA) framework rests on the assumption that the curves can be represented as linear combinations of basis functions such as <i>B</i>-splines, as in the Ramsay-Silverman representation of functional data; but whereas functional data analysis extends MVA to the case of observations that are curves rather than vectors—heuristically, <i>n</i> × <i>p</i> data with <i>p</i> infinite—we are instead concerned with what happens when <i>n</i> is infinite. We present continuous-time extensions of the classical MVA methods of covariance and correlation estimation, principal component analysis, Fisher’s linear discriminant analysis, and <i>k</i>-means clustering. We show that CTMVA can improve on the performance of classical MVA, in particular for correlation estimation and clustering, and can be applied in some settings where classical MVA cannot, including variables observed at disparate time points. CTMVA is illustrated with a novel perspective on a well-known Canadian weather dataset, and with applications to datasets involving international development, brain signals, and air quality. The proposed methods are implemented in the publicly available R package ctmva. Supplementary materials for this article are available online.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.146 | 0.010 |
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; both teacher heads agree on what is shown here.
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".