How Good Can Linear Models Be for Time-Series Forecasting?

📄 Paper (arXiv)  |  🌐 Project Page  |  🕹️ Interactive Demo

Optimal forecasting context is heterogeneous and non-monotonic in horizon: as the forecast horizon H sweeps 24→720, the searched optimal lookback L*(H) grows, plateaus, or shrinks per dataset.

Automated hyperparameter search for Ridge regression forecasting using Optuna. Finds optimal combinations of scaling strategies, lookback windows, regularization parameters, and data augmentation for multivariate time series datasets.

Installation

pip install -r requirements.txt

Quick Start

Single Run

# Run optimization on weather dataset (default)
python optuna_ridge.py

# Specify dataset and output directory
python optuna_ridge.py --input_csv data/ETTh1.csv --output_dir exps/my_experiment

# Faster search with horizon grouping (must divide 96, 192, 336, 720)
python optuna_ridge.py --local_horizon_group_size 48

# Use k-fold cross-validation
python optuna_ridge.py --n_folds 3 --n_trials 20

Reproducing the paper

The paper sweeps --local_series_group_size (sgs) over the divisors of each dataset's series count and selects the value with the lowest mean Local MSE. The scripts/reproduce.sh helper runs this sweep with the canonical configuration for every shipped dataset:

scripts/reproduce.sh                 # all datasets
scripts/reproduce.sh etth1 weather   # a subset

Each run uses the canonical setting below (sgs is the swept value):

python optuna_ridge.py \
  --input_csv data/ETTh1.csv \
  --output_dir exps/exp8_etth1_sgs7_pooled \
  --scaler_scope local \
  --scaler_method mean \
  --local_horizon_group_size 24 \
  --local_series_group_size 7 \
  --n_folds 3 \
  --n_trials 20 \
  --pool_series \
  --instance_norm   # only affects the global baseline, not the tuned Ridge model

Command Line Arguments

Basic Arguments

Argument	Default	Description
--input_csv	data/weather.csv	Input CSV file path
--output_dir	results	Output directory for results
--local_horizon_group_size	24	Group horizons for HP search (valid: 1,2,3,4,6,8,12,16,24,48)
--local_series_group_size	1	Group series for HP search (1=per-series, -1=all share HPs)
--pool_series	False	When set, pool training windows across all series in each series group and train a single forecast model per group; otherwise, series in the group only share selected hyperparameters, but each series is refit with its own model
--instance_norm	False	Use instance normalization for global baseline
--n_folds	1	Number of folds for expanding window CV
--fold_reg_lambda	0.0	Regularization weight for fold variance
--n_trials	30	Number of Optuna trials per search

Ablation Controls

Argument	Default	Description
--scaler_scope	search	Fix scaler scope: global, local, or search
--scaler_method	search	Fix scaler method: mean, robust, or search
--fixed_local_ratio	None	Fix local_ratio value (disables search)
--fixed_noise_type	None	Fix augmentation: none, time, freq, or None (search)
--fixed_aug_sigma	None	Fix augmentation intensity (disables search)

Architecture

Data Pipeline

Raw CSV → Sliding Windows (lookback, horizon) → Train/Val/Test Split → Scaling → Optional Augmentation → Ridge Solver

Search Space

Parameter	Range	Scale
lookback	32-2048	log
local_ratio	0.001-1.0	log
scaler_method	mean, robust	categorical
noise_type	none, time, freq	categorical
aug_sigma	0.001-0.5	log

Scaling Strategies

• StandardStrategy: Mean/std normalization
• RobustStrategy: Median/IQR normalization
• LocalNormScaler: Per-sample normalization with configurable last_k (supports NowNormalization)
• GlobalScaler: Dataset-level normalization

Augmentation

• TimeDomainNoise: Gaussian noise with random per-sample intensity
• FreqDomainNoise: FFT-based amplitude/phase perturbation

Solver

Closed-form Ridge regression via normal equations with batched alpha search and Cholesky decomposition.

Output Files

After running, the output directory contains:

File	Description
local_results.csv	Per-series/horizon best hyperparameters and metrics
benchmark_comparison.csv	Local vs global MSE/MAE at cutoffs 96, 192, 336, 720
predictions.npy	Raw predictions for visualization
forecast_comparison_*.png	Visualization plots

Supported Datasets

Place datasets in data/ directory:

• ETT: ETTh1.csv, ETTh2.csv, ETTm1.csv, ETTm2.csv
• Others: weather.csv, exchange_rate.csv

ETT datasets use fixed 12/4/4 month train/val/test splits. Other datasets use 70/10/20 splits.

Citation

If you find this work useful, please cite:

@article{huang2026linear,
  title   = {How Good Can Linear Models Be for Time-Series Forecasting?},
  author  = {Huang, Lang and Xu, Jinglue and Darlow, Luke},
  journal = {arXiv preprint arXiv:2606.27282},
  year    = {2026}
}