Abstract

Existing models for spatial data analysis typically rely on mean or quan

tile regression to model the association between a dependent variable and covariates. We in this paper propose a novel spatial modal regression by assuming that

the conditional mode of the response Y given covariates X follows a nonparametric regression structure, defined as m : X 7→m(X) := Mode(Yi | Xi), Xi ∈Rd

and i ∈ZN. The suggested spatial modal regression can be utilized to capture the

“most likely” effect and may reveal new interesting data structures that are possibly missed by the conditional mean or quantiles, especially in cases of asymmetric

data distributions. We derive the asymptotic distributions for the resulting modal

estimators with appropriate choices of bandwidths. To numerically estimate the

developed model, we recommend a modified modal expectation-maximization

(MEM) algorithm with the assistance of a Gaussian kernel. Numerical examples

are presented to demonstrate the favorable finite sample performance of the estimators. We also generalize the propounded spatial modal regression to an addit-

ive sum form to offer a versatile solution to handle high-dimensional datasets.

Information

Preprint No.SS-2024-0339
Manuscript IDSS-2024-0339
Complete AuthorsTao Wang, Weixin Yao
Corresponding AuthorsTao Wang
Emailstaow@uvic.ca

References

  1. Brady, N. C. and Weil, R. R. (2016). The Nature and Properties of Soils. 15th ed. Pearson.
  2. Chen, Y. C. (2018). Modal Regression using Kernel Density Estimation: A Review. Wiley Interdisciplinary Reviewers: Computational Statistics, 10:e1431.
  3. Chen, Y. C., Genovese, C. R., Tibshirani, R. J., and Wasserman, L. (2016). Nonparametric Modal Regression. The Annals of Statistics, 44, 489-514.
  4. Dabo-Niang, S., Ould-Abdi, S., Ould-Abdi, A., and Diop, A. (2014). Consistency of A Nonparametric Conditional Mode Estimator for Random Fields. Statistical Methods & Applications, 23, 1-39.
  5. Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Application. London: Chapman and Hall.
  6. Feng, Y., Fan, J., and Suykens, J. A. K. (2020). A Statistical Learning Approach to Modal Regression. Journal of Machine Learning Research, 21, 1-35.
  7. Gao, J., Lu, Z., and Tjøstheim, D. (2006). Estimation in Semiparametric Spatial Regression. The Annals of Statistics, 34, 1395-1435.
  8. Hallin, M., Lu, Z., and Tran, L. T. (2004). Local Linear Spatial Regression. The Annals of Statistics, 32, 2469-2500.
  9. Hallin, M., Lu, Z., and Yu, K. (2009). Local Linear Spatial Quantile Regression. Bernoulli, 15, 659-686.
  10. Hastie, T. and Tibshirani, R. (1990). Generalised Additive Models. Chapman &
  11. Hall, London.
  12. Kemp, G. C. R., Parente, P. M. D. C., and Santos Silva, J. M. C. (2020). Dynamic Vector Mode Regression. Journal of Business & Economic Statistics, 38, 647661.
  13. Lu, Z. and Chen, X. (2004). Spatial Kernel Regression Estimation: Weak Consistency. Statistics & Probability Letters, 68, 125-136.
  14. Lu, Z., Lundervold, A., Tjøstheim, D., and Yao, Q. (2007). Exploring Spatial Nonlinearity using Additive Approximation. Bernoulli, 13, 447-472.
  15. Lu, Z., Tang, Q., and Cheng, L. (2014). Estimating Spatial Quantile Regression with Functional Coefficients: A Robust Semiparametric Framework. Bernoulli, 20, 164-189.
  16. Nandy, S., Lim, C., and Maiti, T. (2017). Additive Model Building for Spatial Regression. Journal of the Royal Statistical Society Series B, 79, 779-800.
  17. Ortega, J. M. and Rheinboldt, W. C. (1970). Iterative Solution of Nonlinear Equations in Several Variables. Academic Press.
  18. Rio, E. (2017). Asymptotic Theory of Weakly Dependent Random Processes. Springer Berlin, Heidelberg.
  19. Robinson, P. M. (2008). Developments in the Analysis of Spatial Data. Journal of the Japan Statistical Society, 38, 87-96.
  20. Robinson, P. M. (2011). Asymptotic Theory for Nonparametric Regression with Spatial Data. Journal of Econometrics, 165, 5-19.
  21. Ullah, A., Wang, T., and Yao, W. (2021). Modal Regression for Fixed Effects Panel Data. Empirical Economics, 60, 261-308.
  22. Ullah, A., Wang, T., and Yao, W. (2022). Nonlinear Modal Regression for Dependent Data with Application for Predicting COVID-19. Journal of the Royal Statistical Society Series A, 185, 1424-1453.
  23. Ullah, A., Wang, T., and Yao, W. (2023). Semiparametric Partially Linear Varying Coefficient Modal Regression. Journal of Econometrics, 235, 10011026.
  24. Wang, T. (2025). Parametric Modal Regression with Autocorrelated Error Process. Statistica Sinica, 35, 457-478.
  25. White, R. E. (2005). Principles and Practice of Soil Science: The Soil as a Natural Resource. 4th ed. Wiley-Blackwell.
  26. Xiang, S. and Yao, W. (2022). Nonparametric Statistical Learning Based on Modal Regression. Journal of Computational and Applied Mathematics, 409, 114-130.
  27. Yao, W. (2013). A Note on EM Algorithm for Mixture Models. Statistics & Probability Letters, 83, 519-526.
  28. Yao, W. and Li, L. (2014). A New Regression Model: Modal Linear Regression. Scandinavian Journal of Statistics, 41, 656-671.
  29. Yao, W., Lindsay, B. G., and Li, R. (2012). Local Modal Regression. Journal of Nonparametric Statistics, 24, 647-663. a. Department of Economics and Department of Mathematics and Statistics (by

Acknowledgments

We are deeply grateful to the Co-Editor Yi-Hau Chen, Associate Editor, and

two anonymous referees for their constructive comments, leading to the substantial improvement of the paper. This research is supported by SSHRC-

IDG (430-2023-00149), NSERC Discovery Grant (RGPIN-2025-04185 and

DGECR-2025-00343), and NSF Grant (DMS-2210272).

Supplementary Materials

The Supplementary Material contains comments for MEM algorithm and

theoretical conditions, boundary analysis, additional simulations, generalizations to additive and extended spatial modal regression models, as well

as technical proofs of the main theorems and supporting lemmas.

Supplementary materials are available for download.