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A Top-Down Scale Approach for Multiscale Geographically and Temporally Weighted Regression

Ghislain Geniaux, César Martinez, Samuel Soubeyrand

Abstract

This paper proposes tds mgtwr, a multiscale geographically and temporally weighted regression (MGTWR) model with covariate-specific spatial and temporal scales. The approach combines a separable spatio-temporal kernel with a Top-Down Scale (TDS) calibration scheme, where spatial and temporal bandwidths are selected for each covariate through a coordinate-wise search over ordered grids guided by the corrected Akaike Information Criterion (AICc). By avoiding unconstrained multidimensional optimization, this strategy extends to the spatio-temporal setting the stabilizing properties of TDS calibration scheme Geniaux (2026). The multiscale backfitting procedure combines the Top-Down Scale calibration scheme with an adaptive, importance-driven update schedule that prioritizes covariates according to their current scale-normalized contribution to the fitted signal, thereby limiting the number of local recalibrations required and accelerating convergence while maintaining estimator fidelity. We also introduce a generic prediction method for MGWR and MGTWR based on kernel sharpening. Monte Carlo experiments show that modeling both space and time improves coefficient recovery and predictive accuracy relative to purely spatial multiscale models when temporal variation is present and sufficiently supported by the data. Gains increase with sample size and signal-to-noise ratio. Two empirical applications illustrate the method under contrasting regimes. For Beet Yellows severity, a plant epidemiology and pest management problem, multiscale spatial modeling is essential, while spatio-temporal extensions yield additional gains when temporal information is rich. In modeling house prices, MGTWR consistently outperforms spatial local and STVC models. In both cases, predictive performance rivals flexible machine-learning benchmarks while preserving interpretable spatio-temporal scales.

A Top-Down Scale Approach for Multiscale Geographically and Temporally Weighted Regression

Abstract

This paper proposes tds mgtwr, a multiscale geographically and temporally weighted regression (MGTWR) model with covariate-specific spatial and temporal scales. The approach combines a separable spatio-temporal kernel with a Top-Down Scale (TDS) calibration scheme, where spatial and temporal bandwidths are selected for each covariate through a coordinate-wise search over ordered grids guided by the corrected Akaike Information Criterion (AICc). By avoiding unconstrained multidimensional optimization, this strategy extends to the spatio-temporal setting the stabilizing properties of TDS calibration scheme Geniaux (2026). The multiscale backfitting procedure combines the Top-Down Scale calibration scheme with an adaptive, importance-driven update schedule that prioritizes covariates according to their current scale-normalized contribution to the fitted signal, thereby limiting the number of local recalibrations required and accelerating convergence while maintaining estimator fidelity. We also introduce a generic prediction method for MGWR and MGTWR based on kernel sharpening. Monte Carlo experiments show that modeling both space and time improves coefficient recovery and predictive accuracy relative to purely spatial multiscale models when temporal variation is present and sufficiently supported by the data. Gains increase with sample size and signal-to-noise ratio. Two empirical applications illustrate the method under contrasting regimes. For Beet Yellows severity, a plant epidemiology and pest management problem, multiscale spatial modeling is essential, while spatio-temporal extensions yield additional gains when temporal information is rich. In modeling house prices, MGTWR consistently outperforms spatial local and STVC models. In both cases, predictive performance rivals flexible machine-learning benchmarks while preserving interpretable spatio-temporal scales.
Paper Structure (40 sections, 20 equations, 5 figures, 7 tables)

This paper contains 40 sections, 20 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Kernel Formulation
  4. 1. Linear temporal kernel.
  5. 2. Seasonal (cyclic) temporal kernel.
  6. tds_mgtwr Algorithm
  7. Steepest Discrete Descent Strategy.
  8. Inertial Stability and Robustness.
  9. Boundary Solutions and Nested Model Selection.
  10. Update-order effects in multiscale backfitting.
  11. Computational Efficiency.
  12. Out-of-sample Prediction.
  13. Software.
  14. Monte Carlo simulations
  15. Monte Carlo Design
  16. ...and 25 more sections

Figures (5)

  • Figure 1: Schematic Representation of the tds_mgtwr Algorithm (Fixed Update Ordering)
  • Figure 2: Simulated space-time varying coefficients $\beta_1(u,v,t)$
  • Figure 3: Simulated space-time varying coefficients $\beta_2(u,v,t)$
  • Figure 4: Simulated space varying coefficients $\beta_3(u,v)$ and $\beta_4(u,v)$
  • Figure 5: Marginal effect of a 1% increase in living area (100 m$^2$ reference), MGTWR (tds_mgtwr)