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

2026-05-08

AI Summary

This work addresses the issue of noise introduced by over-modeling in highly periodic time series forecasting by proposing FLAIR, a method that reshapes the input sequence into an approximately rank-1 matrix and forecasts by combining a fixed average shape from historical periods with dynamic day-level scaling—eliminating the need to learn complex structures. Built upon rank-1 decomposition, singular value decomposition (SVD), and generalized cross-validation (GCV) ridge regression, FLAIR admits a closed-form solution and requires only 28–57 parameters, operating without GPU acceleration or task-specific hyperparameter tuning. Evaluated across all 97 configurations of GIFT-Eval, FLAIR achieves performance comparable to PatchTST (relMASE: 0.838) in just 22 minutes on a single CPU core, demonstrating that in high rank-1, multi-periodic settings, parsimonious modeling can outperform sophisticated learning paradigms.

Abstract

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.

Problem

• periodic time series
• forecasting
• rank-1 decomposition
• shape learning
• low-parameter models

Innovation

• rank-1 decomposition
• periodic time series forecasting
• parameter efficiency
• closed-form solution
• FLAIR