Abstract
Accurate risk estimation under distribution shifts is critical for deploying machine learning models in real-world spatial applications, from ecological forecasting to medical image analysis. Conventional methods such as No Weighting (NW) and Importance Weighting (IW) fail in spatially structured data due to two challenges: (1) density ratio estimation in high-dimensional clustered distributions and (2) non-stationarity from environmental gradients or sampling biases. Classifier-based approaches offer partial improvements but often yield miscalibrated risk estimates by prioritizing discriminative accuracy over distribution alignment. We conduct a systematic evaluation of four risk estimation methods —NW, IW, Kernel Mean Matching (KMM), and classifier-based reweighting—across synthetic benchmarks (with controlled spatial clustering) and real-world datasets (species distributions and immune cell layouts). Results show that KMM achieves superior robustness, reducing Mean Absolute Percentage Error (MAPE) by 12.3–86.5% compared to alternatives in high-dimensional settings. This advantage stems from KMM’s direct minimization of distributional divergence via kernel embeddings, bypassing error-prone density ratio estimation. Our findings demonstrate that KMM is a principled solution for spatial risk estimation, particularly when source and target distributions exhibit complex clustering or sampling artifacts. Its consistency across ecological and biomedical domains suggests broad applicability for reliable model deployment in spatially heterogeneous environments.
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For the data, preprocessing and modeling details to reproduce the calculations, we refer the reader to the repository of the project https://github.com/awesomeslayer/Importance-reweighting.
References
Shimodaira, H. Improving predictive inference under covariate shift by weighting the log-likelihood function. J. Stat. Plan. Inference 90, 227–244 (2000).
James, F. Monte Carlo theory and practice. Rep. Prog. Phys. 43, 1145 (1980).
Bickel, S., Brückner, M. & Scheffer, T. Discriminative learning under covariate shift. J. Mach. Learn. Res. 10, 2137–2155 (2009).
Zadrozny, B. Learning and evaluating classifiers under sample selection bias. In Proceedings of the 21st International Conference on Machine Learning 114 (ACM, Banff, Alberta, Canada, 2004) https://doi.org/10.1145/1015330.1015425.
Tokdar, S. T. & Kass, R. E. Importance sampling: A review. Wiley Interdiscip. Rev.: Comput. Stat. 2, 54–60 (2010).
Wills, R. C., Dong, Y., Proistosecu, C., Armour, K. C. & Battisti, D. S. Systematic climate model biases in the large-scale patterns of recent sea-surface temperature and sea-level pressure change. Geophys. Res. Lett. 49, e2022GL100011 (2022).
Denissen, J. M. et al. Widespread shift from ecosystem energy to water limitation with climate change. Nat. Clim. Chang. 12, 677–684 (2022).
Ben-Said, M. Spatial point-pattern analysis as a powerful tool in identifying pattern-process relationships in plant ecology: An updated review. Ecol. Process. 10, 1–23 (2021).
Gatrell, A. C., Bailey, T. C., Diggle, P. J. & Rowlingson, B. S. Spatial point pattern analysis and its application in geographical epidemiology. Trans. Inst. Br. Geogr. 256–274 (1996).
Vokinger, K. N., Feuerriegel, S. & Kesselheim, A. S. Mitigating bias in machine learning for medicine. Commun. Med. 1, 25 (2021).
Zhao, Z. et al. Identification of lung cancer gene markers through kernel maximum mean discrepancy and information entropy. BMC Med. Genom. 12, 1–10 (2019).
Vegas, E., Oller, J. M. & Reverter, F. Inferring differentially expressed pathways using kernel maximum mean discrepancy-based test. BMC Bioinform. 17, 399–405 (2016).
Maley, C. C., Koelble, K., Natrajan, R., Aktipis, A. & Yuan, Y. An ecological measure of immune-cancer colocalization as a prognostic factor for breast cancer. Breast Cancer Res. 17, 1–13 (2015).
Roberts, D. et al. Cross-validation strategies for data with temporal, spatial, hierarchical, or phylogenetic structure. Ecography https://doi.org/10.1111/ecog.02881 (2016).
Meyer, H. & Pebesma, E. Predicting into unknown space? Estimating the area of applicability of spatial prediction models. Methods Ecol. Evol. 12, 1620–1633. https://doi.org/10.1111/2041-210x.13650 (2021).
Tuia, D., Persello, C. & Bruzzone, L. Domain adaptation for the classification of remote sensing data: An overview of recent advances. IEEE Geosci. Remote Sensing Mag. 4, 41–57. https://doi.org/10.1109/MGRS.2016.2548504 (2016).
Wilson, G. & Cook, D. J. A survey of unsupervised deep domain adaptation. ACM Trans. Intell. Syst. Technol. (TIST) 11, 1–46 (2020).
Gretton, A. et al. Covariate shift by kernel mean matching. Dataset Shift Mach. Learn. 3, 5 (2009).
Martynova, E. & Textor, J. A uniformly bounded correlation function for spatial point patterns. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, 2177–2188 (2024).
Scott, D. W. On optimal and data-based histograms. Biometrika 66, 605–610. https://doi.org/10.1093/biomet/66.3.605 (1979).
Huang, J., Gretton, A., Borgwardt, K., Schölkopf, B. & Smola, A. Correcting sample selection bias by unlabeled data. Adv. Neural Inf. Process. Syst. 19, (2006).
Quionero-Candela, J., Sugiyama, M., Schwaighofer, A. & Lawrence, N. D. Dataset Shift in Machine Learning (The MIT Press, 2009).
GBIF.org. Occurrence download: Tussilago farfara l. (2024). https://www.gbif.org/occurrence/download/0031125-240626123714530. Accessed 20 Jul 2024.
GBIF.org. Occurrence download: Anemone nemorosa l. (2024). https://www.gbif.org/occurrence/download/0031144-240626123714530. Accessed 20 Jul 2024.
GBIF.org. Occurrence download: Caltha palustris l. https://www.gbif.org/occurrence/download/0031146-240626123714530 (2024). Accessed 20 Jul 2024.
Hijmans, R. J. et al. Package ‘raster’. R package 734, 473 (2015).
Bivand, R. et al. Package ‘rgdal’. Bindings for the Geospatial Data Abstraction Library. Available online: https://cran.r-project.org/web/packages/rgdal/index.html (Accessed on 15 Oct 2017) 172 (2015).
Pebesma, E. J. et al. Simple features for r: Standardized support for spatial vector data. R J. 10, 439 (2018).
Hijmans, R. J. et al. Package ‘terra’ (Vienna, Austria, Maintainer, 2022).
van der Hoorn, I. A. et al. Detection of dendritic cell subsets in the tumor microenvironment by multiplex immunohistochemistry. Eur. J. Immunol. 54, 2350616 (2024).
van der Woude, L. L., Gorris, M. A., Halilovic, A., Figdor, C. G. & de Vries, I. J. M. Migrating into the tumor: A roadmap for t cells. Trends Cancer 3, 797–808 (2017).
Sultan, S. et al. A segmentation-free machine learning architecture for immune land-scape phenotyping in solid tumors by multichannel imaging. BioRxiv 2021–10 (2021).
Friedman, J. H. Greedy function approximation: A gradient boosting machine. Ann. Stat. 1189–1232 (2001).
Liu, A. & Ziebart, B. D. Robust classification under sample selection bias. In (eds. Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N. & Weinberger, K.) Advances in Neural Information Processing Systems, vol. 27 (Curran Associates, Inc., 2014).
Cauchois, M., Gupta, S., Ali, A. & Duchi, J. C. Robust validation: Confident predictions even when distributions shift. J. Am. Stat. Assoc. 119, 3033–3044 (2024).
Lam, H. & Zhang, H. Doubly robust stein-kernelized monte carlo estimator: Simultaneous bias-variance reduction and supercanonical convergence (2023). arXiv:2110.12131.
Zech, J. R. et al. Variable generalization performance of a deep learning model to detect pneumonia in chest radiographs: A cross-sectional study. PLoS Med. 15, e1002683 (2018).
Funding
The work was supported by the grant for research centers in the field of AI provided by the Ministry of Economic Development of the Russian Federation in accordance with the agreement 000000C313925P4F0002 and the agreement with Skoltech №139-10-2025-033.
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Conceptualization: A.Z., E.S. and D.K.; methodology: E.S., A.Z., D.K.; software: E.S.; validation: E.S., A.Z.; formal analysis: E.S., D.K.; investigation: A.Z., E.S.; data curation: D.K., E.S.; writing—original draft preparation: E.S., D.K.; writing—review and editing: D.K., E.S. and A.Z.; visualization: E.S., D.K.; supervision: A.Z.; project administration: A.Z., D.K. All authors have read and agreed to the published version of the manuscript.
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Serov, E., Koldasbayeva, D. & Zaytsev, A. Kernel mean matching enhances risk estimation under spatial distribution shifts. Sci Rep (2026). https://doi.org/10.1038/s41598-026-36740-7
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DOI: https://doi.org/10.1038/s41598-026-36740-7
