The Effect of Ensemble Size on the Mean Squared Error and Spread–Error Relationship
Bibliographic record
Abstract
Abstract Most ensemble verification diagnostics are sensitive to ensemble size, complicating the evaluation of a system’s underlying quality and the comparison of different ensemble systems. This study examines how the mean squared error (MSE) of the ensemble mean, and the spread–error relationship used to evaluate ensemble consistency, vary as a function of ensemble size. As the MSE of the ensemble mean (“error” in spread–error) is affected by ensemble size, but the average sample ensemble variance (“spread” in spread–error) is not, these effects must be removed from the MSE for a robust assessment of ensemble consistency. Although the dependence of these diagnostics on ensemble size has been sparsely addressed over several decades, gaps remain concerning the assumptions necessary for quantification. Evidence also suggests these effects are not widely known or fully understood. The impact of ensemble size is examined by assuming exchangeability between ensemble members, allowing us to derive the MSE and spread–error relationship (expressed as a difference) that would be obtained with an infinite-sized ensemble. Ensemble-size effects are removed from both scores by subtracting the average ensemble variance divided by the ensemble size. The unbiased MSE can be used to estimate the error reduction achievable by increasing ensemble size and allows for an “apples-to-apples” comparison of forecast error across systems. The unbiased spread–error relationship eliminates the effects of ensemble size on the original diagnostic, which, when ignored, make ensembles appear too underdispersive.
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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 | 0.018 | 0.080 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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".