Least Squares for Time Series Forecasting

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.

Time series forecasting aims to predict future values from historical observations and auxiliary features. We propose SteerCast, a retrieval-based latent steering method that improves decoder-only forecaster at inference time, without updating its parameters. SteerCast constructs a database from the training set by storing a representation of each history window together with a steering vector computed in the forecaster's latent space, defined as the difference between representations induced by the ground-truth continuation and by the model's own prediction. At test time, SteerCast retrieves nearest neighbors for a query history, aggregates their steering vectors, and injects the resulting signal into the forecaster's hidden states at every step of autoregressive generation, guiding predictions toward trajectories consistent with similar training cases. Experiments across diverse multivariate benchmarks and multiple horizons show that SteerCast consistently improves forecasting accuracy over the fine-tuned backbone and retrieval-based baselines, while requiring no additional training beyond the original fine-tuning and using only the training set as a retrieval corpus.

Recent advances in instruction-driven time series forecasting highlight a fundamental tension: systems must simultaneously master natural-language instruction following and precise numerical prediction. Current approaches often force a compromise. Large language models (LLMs) can interpret unstructured requests but may be inaccurate when asked to produce forecast values directly, whereas time series foundation models (TSFMs) provide numerical forecasts but typically require structured inputs specifying the target series, context window, and horizon. We present ChronoScribe, which applies semantic-numerical specialization to this workflow. A LoRA-adapted LLM proposes an execution path and context length; deterministic rules resolve the executable configuration. On the specialist path, a frozen TSFM produces quantile trajectories for the task-specified horizon. Deterministic trajectory summaries condition the response, while the LLM neither generates nor revises predicted values. We evaluate a Qwen3.5-9B--Chronos-2 implementation across 23 evaluation dimensions and 30 public datasets against eleven proprietary direct LLMs and three open TSFMs. ChronoScribe has the lowest error on sMAPE, MASE, WAPE, RMSE, and MAE among all evaluated systems, including fixed-context controls. Across these metrics, even the lowest direct-LLM error remains 2.0--2.7 times the corresponding ChronoScribe error. At the system level, ChronoScribe achieves a Task-Level Score (TLS) of 65.5, 10.6 points above the strongest direct-LLM baseline. The results support ChronoScribe's division of labor, in which the LLM handles interpretation, configuration, and communication while the TSFM handles numerical forecasting.