Stabilizing distribution-free probabilistic forecasts
_{Jente Van Belle, Honglin Wen, Wouter Verbeke, and Pierre Pinson [Preprint 2026]}

This repository provides the code for training (Stable)SQF models for distribution-free probabilistic time series forecasting.

Paper abstract

Multi-step-ahead forecasts are often updated as new observations become available, since shorter forecast horizons typically improve forecast quality. However, such improvements come at the cost of forecast instability, i.e., variability in forecasts for the same target period. This instability can trigger costly changes to plans formulated based on the forecasts and may erode trust in the forecasting system. In this work, we integrate forecast stability alongside forecast quality into the training of distribution-free probabilistic time-series forecasting models, allowing us to control this trade-off. We propose a method for generating stabilized forecasted conditional quantile functions using regression splines parameterized by a neural network. This approach enables joint optimization of stability and quality, as it allows us to directly penalize dissimilarities arising from forecast updates. Furthermore, it allows assigning varying importance to stabilizing different parts of the forecast distributions (e.g., central parts vs. tails) to focus on the parts most relevant for the intended downstream use (e.g., the upper tail for inventory management). We empirically evaluate the proposed method on two datasets with different statistical properties and show that it can effectively reduce forecast instability without a substantial loss in forecast quality, and that it can target stabilization effort toward specific parts of the forecast distributions.

Repository structure

This repository is organised as follows:

|- R_scripts/
    |- Baselines.R           # ETS-G/B, mean-G/B, snaive-G/B + dataset summary statistics
    |- SQF-stabilized.R      # SQF-stabilized Partial and Full
|- data/
    |- processed/            # Saved TimeSeriesDatasets (preprocessed, fixed splits and scaling)
    |- raw/
        |- m5_items.txt
|- src/
    |- data/
        |- M4.py
        |- M5.py
        |- utils/
    |- methods/
        |- StableSQF.py      # SQF forecaster model architecture and optimization procedure to stabilize the forecasts (StableSQF)
    |- utils/                # Utility functions and callbacks
|- main_M4_monthly.py        # Script to train a (Stable)SQF model on the M4 monthly dataset
|- main_M5_items.py          # Script to train a (Stable)SQF model on the M5 items dataset

Installing

We have provided a requirements.txt file:

pip install -r requirements.txt

Please use the above in a newly created virtual environment to avoid conflicting dependencies.

For the R scripts, required R package versions are specified at the top of each R script.

Use

To efficiently train (Stable)SQF models, access to a CUDA-enabled GPU is required.

Weights & Biases is used for performance metric logging. Change the project_name in main_*.py to your wandb project in the script you want to run. A run produces .csv files with the forecasts if you set save_forecasts = True in main_*.py (see WriteQuantileForecastsToCSV callback in src/utils/callbacks.py). A wandb output folder is automatically created.

The SQF-stabilized.R script takes two of these generated .csv files as input: quantile_forecasts.csv and quantile_forecasts_lagged.csv.

Citing

Please cite our work as follows:

@article{vanbelle2026StableSQF,
  title     = {Stabilizing distribution-free probabilistic forecasts},
  author    = {Van Belle, Jente and Wen, Honglin and Verbeke, Wouter and Pinson, Pierre},
  year      = {2026},
  journal   = {arXiv preprint arXiv:2605.28531},
  url       = {https://arxiv.org/abs/2605.28531}
}

Contact

Jente Van Belle (jente.vanbelle@kuleuven.be)