The roughness exponent and its model-free estimation
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Bibliographic record
Abstract
Motivated by pathwise stochastic calculus, we say that continuous real-valued function x admits the roughness exponent R if the pth variation of x converges to zero for p>1/R and to infinity for p<1/R. In our main result, we provide a mild condition on the Faber–Schauder coefficients of x under which the roughness exponent exists and is given as the limit of the classical Gladyshev estimates Rˆn(x). This result can be viewed as a strong consistency result for the Gladyshev estimators in an entirely model-free setting, because it works strictly trajectory-wise and requires no probabilistic assumptions. Nonetheless, our proof is probabilistic and relies on a martingale hidden in the Faber–Schauder expansion of x. We show that the condition of our main result is satisfied for the typical sample paths of fractional Brownian motion with drift, and we provide almost sure convergence rates for the corresponding Gladyshev estimates. We also discuss the connections between the roughness exponent and the related concepts of Besov regularity and weighted quadratic variation. Since the Gladyshev estimators are not scale-invariant, we construct several scale-invariant estimators. Finally, we extend our results to the case in which the pth variation of x is defined over a sequence of unequally spaced partitions.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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) | 0.000 | 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