Pypkowski et al. (2025) The Whittle likelihood for mixed models with application to groundwater level time series

Identification

• Journal: arXiv (Cornell University)
• Year: 2025
• Date: 2025-12-23
• Authors: Jakub J. Pypkowski, Adam M. Sykulski, James S. Martin, Ben P. Marchant
• DOI: 10.48550/arxiv.2512.20810

Research Groups

• Department of Statistical Science, University College London (UCL), United Kingdom.
• British Geological Survey (BGS), Keyworth, Nottingham, United Kingdom.

Short Summary

This paper introduces a frequency-domain approach using the Whittle likelihood to jointly estimate fixed and random effect parameters in mixed models. The method significantly reduces computational complexity for large groundwater level datasets while remaining robust to missing values and non-Gaussian noise.

Objective

• To develop a computationally efficient framework for the joint estimation of all parameters in mixed models (combining fixed mean predictors with autocorrelated random errors) to improve forecasting and hazard response in hydrogeology.

Study Configuration

• Spatial Scale: Local to regional (focused on individual groundwater monitoring boreholes).
• Temporal Scale: Long-term time series; capable of handling high-frequency data and large datasets that are computationally prohibitive for standard maximum likelihood estimation.

Methodology and Data

• Models used: Mixed models (Fixed effects + autocorrelated random errors), Whittle likelihood (frequency-domain quasi-likelihood), and Maximum Likelihood Estimation (MLE) for benchmarking.
• Data sources: Simulated time series data and real-world groundwater level observations.
• Statistical Approach: The method transforms the time series into the frequency domain to approximate the likelihood, allowing for joint parameter estimation instead of the traditional two-stage approach (which estimates fixed and random effects separately).

Main Results

• Computational Efficiency: The Whittle likelihood approach handles much larger datasets than exact maximum likelihood by reducing the complexity of matrix inversions.
• Joint Estimation: Successfully estimates both fixed effect coefficients (e.g., responses to rainfall or pumping) and random effect parameters (autocorrelation structures) simultaneously.
• Robustness: The method maintains accuracy even when data are non-Gaussian or contain significant gaps (missing data), which are common in environmental monitoring.
• Performance: Simulation studies show that the proposed method outperforms the common two-stage estimation approach and provides results comparable to MLE but at a fraction of the computational cost.

Contributions

• Provides a scalable mathematical framework for hydrogeological modeling that avoids the restrictive simplifying assumptions (such as fixing parameters) often required by current software.
• Bridges the gap between frequency-domain statistics and mixed-effects modeling in the context of environmental time series.
• Offers a robust solution for filling missing values and forecasting groundwater droughts using large-scale observational networks.

Funding

• Simons Foundation.
• Support from member institutions and contributors to the arXiv/scientific community.

Citation

@article{Pypkowski2025Whittle,
  author = {Pypkowski, Jakub J. and Sykulski, Adam M. and Martin, James S. and Marchant, Ben P.},
  title = {The Whittle likelihood for mixed models with application to groundwater level time series},
  journal = {arXiv (Cornell University)},
  year = {2025},
  doi = {10.48550/arxiv.2512.20810},
  url = {https://doi.org/10.48550/arxiv.2512.20810}
}

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