Stochastic persistence: why average the clear-sky gap?

Worked example / Exemple chiffré. All input values below are fictitious teaching values, not observations or results from the cited paper.

Result: three fictitious observations give an additive stochastic-persistence forecast of 510 W/m². The multiplicative form gives 497.97 W/m². They are different models.

Why average before persisting?

A single measurement carries useful recent information and fluctuations. The idea is to persist a short-window summary relative to clear sky rather than copy every fluctuation. “Stochastic” here does not mean adding a random draw to the forecast.

Three past observations

Additive form, step by step

Equation 5 of Voyant and Notton (2018) uses the mean deficit Δ = CS − GHI over the last N available observations. With N = 3, Δ̄ = (100+150+320)/3 = 190 W/m². Given CS(t+h) = 700 W/m², forecast GHI = 700−190 = 510 W/m². Equivalently: mean(GHI) + future CS − mean(CS).

Multiplicative form

The other form uses the geometric mean of the clear-sky ratios: (0.80 × 0.75 × 0.60)^(1/3) ≈ 0.711379. Multiplying by future clear sky 700 gives 497.97 W/m². An arithmetic mean of these ratios would implement a different rule.

Compare without claiming a winner

Ordinary persistence predicts 480 W/m². With N = 1, the additive clear-sky correction predicts 700−320 = 380 W/m²; ratio persistence predicts 0.60 × 700 = 420 W/m². A held-out observation is needed to rank these forecasts.

What this study contributes

Voyant and Notton develop additive and multiplicative stochastic-persistence forecasts using short-window summaries relative to clear sky. The source provides the definitions, evaluation conditions and selection of the window length.

Limits and reproducibility

Averaging can suppress useful rapid changes as well as noise. Choose N on a validation period, then keep the test period untouched. Ratios near zero clear sky need filtering; geometric means need an explicit policy for zero or invalid ratios. The example contains only positive daylight values. Any physical clipping must be reported. It proves the arithmetic, not superiority at a real station.

Author manuscript on HAL, equations 5–11 · Author’s public code (no dedicated implementation of this paper is asserted).

À retenir : lisser l’écart au ciel clair avant de le prolonger. Ne pas confondre moyenne arithmétique du déficit et moyenne géométrique du rapport ; la taille de fenêtre reste à valider.

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