Lehr et al. (2025) Technical note: An illustrative introduction to the domain dependence of spatial principal component patterns

Identification

• Journal: Hydrology and earth system sciences
• Year: 2025
• Date: 2025-11-26
• Authors: Christian Lehr, Tobias L. Hohenbrink
• DOI: 10.5194/hess-29-6735-2025

Research Groups

• Leibniz Centre for Agricultural Landscape Research (ZALF), Müncheberg, Germany
• University of Potsdam, Institute for Environmental Sciences and Geography, Potsdam, Germany
• German Weather Service (DWD), Agrometeorological Research Centre, Braunschweig, Germany

Short Summary

This technical note introduces the concept of Domain Dependence (DD) in S-mode Principal Component Analysis (PCA) to the hydrological community, demonstrating how spatial PC patterns can be determined by the spatial extent and arrangement of data rather than hydrological functioning, and provides methods to detect and diminish this effect.

Objective

• To introduce the Domain Dependence (DD) effect and the application of DD reference patterns to the hydrological community, emphasizing that DD should be considered for any application where PCs are used to draw conclusions about spatially distinct properties of an analyzed system.

Study Configuration

• Spatial Scale: Synthetic data on 20x20 grid cells (e.g., square, rectangular, triangular domains); observed precipitation data over a 200 km x 200 km square in northeast Germany, using 1 km x 1 km grid cells.
• Temporal Scale: Synthetic data time series up to 10,000 time steps; observed precipitation data as monthly sums from 1991 to 2020 (360 months).

Methodology and Data

• Models used: Principal Component Analysis (PCA) (S-mode PCA, correlation matrix-based PCA), Stochastic DD reference method (ensemble of simulated data sets), Analytic DD reference method (eigendecomposition of analytic covariance matrix), Varimax rotation (with Kaiser normalization). Spatial covariance models included isotropic exponential and isotropic spherical.
• Data sources: Synthetic data generated from random fields with defined spatial correlation properties; observed monthly precipitation sums from the HYRAS-DE-PR v6.0 grid provided by the German Weather Service.

Main Results

• Domain Dependence (DD) significantly influences spatial PC patterns, leading to distinct patterns with strong gradients and contrasts that are primarily determined by the spatial domain's size and shape, not necessarily hydrological processes.
• DD can cause substantial accumulation of explained variance in the leading PCs (e.g., 79.96% to 83.22% for PC 1 in precipitation data), which, combined with distinct patterns, can be misleadingly interpreted as dominant hydrological signals.
• Effectively degenerate multiplets (PCs with noticeably similar eigenvalues and ambiguous spatial pattern orientation) are induced by DD, particularly in symmetric domains, and can be misinterpreted as complex spatio-temporal features or mask the presence of DD.
• DD patterns are unique for each combination of spatial domain and spatial correlation properties; subsampling or irregular location distribution can alter these patterns significantly from classical archetypes.
• The ratio of domain size to spatial correlation length critically affects variance distribution and the magnitude of contrasts in PC patterns, with maximum contrasts occurring when correlation length is comparable to domain size.
• While varimax rotation can redistribute variance and simplify patterns, it did not fully resolve DD in the precipitation example, still showing domain-dependent gradients, indicating that rotation alone may not be sufficient without further optimization.

Contributions

• Introduces and thoroughly illustrates the critical concept of Domain Dependence (DD) in S-mode PCA to the hydrological community, highlighting its potential for misinterpretation of hydrological data.
• Develops and presents two practical methods (stochastic and analytic) for generating DD reference patterns, enabling researchers to quantitatively assess the presence and strength of DD in their PCA results.
• Systematically explores the effects of various factors (domain shape, size, spatial correlation length, and effective multiplets) on DD patterns and explained variance using synthetic examples.
• Provides clear guidance and R-scripts for detecting DD through visual comparison of subdomains and statistical comparison with reference patterns, as well as discussing strategies to diminish its influence (e.g., subsampling, rotation).
• Emphasizes the importance of considering PCA constraints (variance maximization, orthogonality, uncorrelatedness) when interpreting PC patterns as distinct physical features.

Funding

• Institutional resources provided by the Leibniz Centre for Agricultural Landscape Research (ZALF) and the University of Potsdam.

Citation

@article{Lehr2025Technical,
  author = {Lehr, Christian and Hohenbrink, Tobias L.},
  title = {Technical note: An illustrative introduction to the domain dependence of spatial principal component patterns},
  journal = {Hydrology and earth system sciences},
  year = {2025},
  doi = {10.5194/hess-29-6735-2025},
  url = {https://doi.org/10.5194/hess-29-6735-2025}
}