Fractional relaxation noises, motions and the fractionalenergy balance equation
Bibliographic record
Abstract
We consider the statistical properties of solutions of the stochastic fractional relaxation equation that has been proposed as a model for the earth’s energy balance. In thisequation, the (scaling) fractional derivative term modelsenergy storage processes that occur over a wide range of space and time scales. Up until now, stochastic fractionalrelaxation processes have only been considered withRiemann-Liouville fractional derivatives in the context of random walk processes where it yields highlynonstationary behaviour. For our purposes we require the stationary processes that are the solutions of the Weyl fractional relaxation equations whose domain is −∞ to t rather than 0 to t. We develop a framework for handling fractional equationsdriven by white noise forcings. To avoid divergences, wefollow the approach used in fractional Brownian motion(fBm). The resulting fractional relaxation motions (fRm) and fractional relaxation noises (fRn) generalize the more familiar fBm and fGn (fractional Gaussian noise). Weanalytically determine both the small and large scale limitsand show extensive analytic and numerical results on the autocorrelation functions, Haar fluctuations and spectra. We display sample realizations. Finally, we discuss the prediction of fRn, fRm which – due to long memories - is a past value problem, not an initial value problem. We develop an analytic formula for the fRnforecast skill and compare it to fGn. Although the large scale limit is an (unpredictable) white noise that is attainedin a slow power law manner, when the temporal resolutionof the series is small compared to the relaxation time, fRncan mimick a long memory process with a wide range of exponents ranging from fGn to fBm and beyond. Wediscuss the implications for monthly, seasonal, annualforecasts of the earth’s temperature.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), 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".