Forecasting irradiance and its uncertainty with complex numbers
A single complex autoregression, six coefficients, forecasts irradiance and its volatility at once. For a target coverage of 80%, the observed coverage stays between 79.5 and 80.7% up to six hours, and its intervals are narrower than those of quantile regression at every horizon.
The idea
Compute the clear-sky index κ = GHI/GHICS, then its volatility στ: the standard deviation of its hour-to-hour changes over the last 30 hours. Instead of forecasting one and then the other, put them in a single complex number, z(t) = κ(t) + j·στ(t), and fit an autoregression of order p to z: ẑ(t+1) = Σ ωi·z(t−i), with complex coefficients ωi. The real part predicts κ; multiplying it by GHICS recovers predicted irradiance. The imaginary part predicts the volatility of κ. The coefficients come from complex least squares, ω = (IHI + λ·1)−1IHo, with p = 6 and λ = 3.74 for Ajaccio.
Setting α
The 1 − α prediction interval is ĜHI ± μ·GHICS·σ̂. Everything lies in the multiplier μ. Under a Gaussian assumption, μ = erf−1(1 − α)/√(1 − β); but irradiance is not Gaussian, and the paper recommends calibrating μ on training data, μ = f1·ef2·α, one pair (f1, f2) per horizon. Move α: the blue curve is the calibrated μ at the chosen horizon, the dashed orange curve the Gaussian μ at 1 h.
The paper's results (target coverage 80%, test year 2018)
| Horizon | nRMSE | PICP Gauss | PICP Quant | PICP Compl | MIL Gauss | MIL Quant | MIL Compl |
|---|---|---|---|---|---|---|---|
| 1 h | 0.196 | 83.8 | 79.7 | 80.0 | 51.2 | 42.2 | 41.2 |
| 2 h | 0.252 | 81.4 | 79.9 | 80.7 | 64.6 | 58.7 | 56.7 |
| 3 h | 0.282 | 80.7 | 80.1 | 80.1 | 71.0 | 67.3 | 63.1 |
| 4 h | 0.303 | 80.4 | 80.6 | 80.0 | 75.4 | 74.5 | 67.1 |
| 5 h | 0.317 | 81.0 | 80.8 | 79.5 | 78.7 | 79.1 | 69.3 |
| 6 h | 0.325 | 81.4 | 81.4 | 79.7 | 81.4 | 82.7 | 70.0 |
PICP (Prediction Interval Coverage Probability): share of measurements inside the interval (target 80%). MIL (Mean Interval Length): mean interval width, normalized. nRMSE of the complex method. Gauss: residual variance; Quant: ridge quantile regression. Source: Table II of the paper.
What to keep in mind
- The slider evaluates μ, not a new observed coverage at each α. Table results remain those at α=0.2.
- One site, Ajaccio, hourly daytime data; training 2008–2017, test 2018.
- Point accuracy is of a similar order of magnitude to the other methods (nRMSE 0.196 to 0.325): the gain is in the interval.
- On CRPS and MSIS, quantile regression remains best (MSIS 0.89 versus 0.95).
- The appendix suggests replacing the μ(α) curve by a lookup table, an idea taken up in the probabilistic animation on the home page.
- To compare point forecasts on public data: the forecast comparison.
The paper
Cyril Voyant, Philippe Lauret, Gilles Notton, Jean-Laurent Duchaud, Luis Garcia-Gutierrez, Ghjuvan Antone Faggianelli. Complex-valued time series based solar irradiance forecast. Journal of Renewable and Sustainable Energy 14, 066502 (2022). HAL · arXiv.