Don't Learn the Shape: Forecasting Periodic Time Series by Rank-1 Decomposition

How few parameters do we really need to forecast a periodic time series? An hourly electricity series, reshaped as a 24-row matrix with one column per day, is approximately rank-1: a daily shape modulated by a daily level (median centered rank-1 energy 0.82 on GIFT-Eval). Should we learn the shape? Smoothing, shrinkage, and low-rank fits all seem like obvious upgrades over the simple average of the last K=2 cycles. On all 97 GIFT-Eval configurations, we tested 8 such alternatives (e.g., Fourier, EWMA, James-Stein, rank-r SVD): none significantly beats the frozen baseline under Holm correction; two are significantly worse. The resulting method, FLAIR, is (a) Effective: matches PatchTST on aggregate GIFT-Eval (relMASE 0.838 vs 0.849); (b) Compact: 28 scalars for hourly, 57 for weekly; (c) Fast: 22 minutes on one CPU core of a MacBook Pro; (d) Closed-form & Hands-Off: one SVD per period candidate, GCV-averaged Ridge, no GPU, no pre-training, no per-task tuning. In the high-rank-1, many-cycle regime, extra flexibility is estimation noise.