Multiscale Similarity GWR (M-SGWR) Model

• Multiscale Similarity GWR (M-SGWR) is a spatial regression framework that blends geographic proximity with attribute similarity to capture both contiguous and network-based dependencies.
• It employs dual weighting via geographic and attribute kernels, with each predictor’s spatial scale and mixing parameter optimized through iterative backfitting using criteria like AICc and CV.
• Empirical results, such as in COVID-19 case studies, show that M-SGWR yields improved fit, lower RMSE, and reduced residual spatial autocorrelation compared to traditional GWR models.

Multiscale Similarity Geographically Weighted Regression (M-SGWR) is a spatial regression framework that extends the classical geographically weighted regression (GWR) and its multiscale generalizations. M-SGWR explicitly quantifies local regression relationships using both spatial proximity and attribute (covariate) similarity, allowing different predictors to exhibit varying degrees of dependence on geographic versus contextual, non-contiguous similarity. This dual weighting addresses limitations of conventional GWR/MGWR models in settings where local similarity is not purely spatially contiguous, as in network diffusion or socio-demographic contagion. The model was formally introduced by Lessani et al. (Lessani et al., 27 Jan 2026).

1. Mathematical Foundations and Model Specification

Given a response vector $y\in\mathbb{R}^n$ and predictors $$X=[\mathbf 1, \mathbf x_1, \ldots, \mathbf x_{m-1}]\in\mathbb{R}^{n\times m}$$ at spatial locations $(u_i, v_i)$, M-SGWR fits local weighted least-squares (WLS) models at each site $i$. Each predictor $j$ is associated with two key parameters:

• Bandwidth $b_j$ specifying the spatial scale for geographic proximity.
• Mixing parameter $\alpha_j \in [0,1]$ quantifying the balance between geographic and attribute-based weighting.

For each location $i$ and covariate $j$, two diagonal weight matrices are constructed:

• Geographic kernel: $W^{\rm geo}_{i,j}$ uses a fixed bandwidth $$X=[\mathbf 1, \mathbf x_1, \ldots, \mathbf x_{m-1}]\in\mathbb{R}^{n\times m}$$0:

$$X=[\mathbf 1, \mathbf x_1, \ldots, \mathbf x_{m-1}]\in\mathbb{R}^{n\times m}$$1

with $$X=[\mathbf 1, \mathbf x_1, \ldots, \mathbf x_{m-1}]\in\mathbb{R}^{n\times m}$$2.

• Attribute similarity kernel: $$X=[\mathbf 1, \mathbf x_1, \ldots, \mathbf x_{m-1}]\in\mathbb{R}^{n\times m}$$3, defined over geographic neighbors $$X=[\mathbf 1, \mathbf x_1, \ldots, \mathbf x_{m-1}]\in\mathbb{R}^{n\times m}$$4:

$$X=[\mathbf 1, \mathbf x_1, \ldots, \mathbf x_{m-1}]\in\mathbb{R}^{n\times m}$$5

with $$X=[\mathbf 1, \mathbf x_1, \ldots, \mathbf x_{m-1}]\in\mathbb{R}^{n\times m}$$6.

The combined weight matrix for covariate $$X=[\mathbf 1, \mathbf x_1, \ldots, \mathbf x_{m-1}]\in\mathbb{R}^{n\times m}$$7:

$$X=[\mathbf 1, \mathbf x_1, \ldots, \mathbf x_{m-1}]\in\mathbb{R}^{n\times m}$$8

Local parameter estimates $$X=[\mathbf 1, \mathbf x_1, \ldots, \mathbf x_{m-1}]\in\mathbb{R}^{n\times m}$$9 are computed as:

$(u_i, v_i)$0

with $(u_i, v_i)$1 and $(u_i, v_i)$2.

2. Parameter Selection and Optimization

Efficient estimation in M-SGWR requires the determination of both $(u_i, v_i)$3 and $(u_i, v_i)$4 for each predictor.

2.1 Model Selection Criteria

The two primary objective functions used are:

• Corrected Akaike Information Criterion (AICc):

$(u_i, v_i)$5

where $(u_i, v_i)$6 is the WLS residual variance and $(u_i, v_i)$7 is the hat-matrix.

• Cross Validation (CV):

$(u_i, v_i)$8

with $(u_i, v_i)$9.

2.2 Iterative Backfitting

Simultaneous optimization across all predictors is performed via an iterative backfitting routine:

• Initialize $i$0 and $i$1 for all $i$2.
• For each predictor, hold others fixed, optimize $i$3 by minimizing AICc or CV.
• Update WLS estimates using new weights.
• Iterate until convergence (e.g., residual sum-of-squares or coefficient change).

This approach efficiently explores the combinatorial search space by reducing the high-dimensional problem into a series of one-dimensional optimizations, such as via golden-section or greedy search strategies.

3. Theoretical Properties and Model Generalization

M-SGWR encompasses several established local regression models:

• Setting all $i$4 and equal $i$5 recovers standard GWR.
• Allowing each $i$6 but setting all $i$7 yields MGWR.
• Fixed $i$8, shared global $i$9 is equivalent to SGWR.

The parameter $j$0 provides an explicit control: as $j$1, predictor $j$2’s kernel becomes purely geographic; as $j$3, attribute similarity dominates, supporting remote (non-contiguous) similarity. This generalization enables modeling of processes where spatial interaction is defined by contextual or attribute matching, not just physical proximity.

A plausible implication is that M-SGWR is structurally capable of capturing both spatially contiguous and “network-like” contextual dependencies, accommodating complex patterns in spatial social or epidemiological data.

4. Simulation Studies

M-SGWR has been empirically compared to MGWR under simulated scenarios:

• Mixed “Geographic + Contextual” Effects (n=1200):
  • M-SGWR yields lower RMSE and higher Pearson correlation for local $j$6 estimation than MGWR.
  • Estimated $j$7 demonstrates that $j$8 is primarily driven by attribute similarity.
• Pure Geographic Effects:

All $j$9 are smooth spatial fields. M-SGWR with $b_j$0 reproduces MGWR coefficients exactly.

Goodness-of-fit metrics used include: RMSE for local $b_j$1 estimates, adjusted $b_j$2, AICc, residual sum-of-squares (RSS), and residual Moran’s $b_j$3.

Model	RMSE(β₀,β₁,β₂)	Pearson(β₀,β₁,β₂)
MGWR	(0.260, 0.385, 0.544)	(0.971, 0.766, 0.704)
M-SGWR	(0.224, 0.361, 0.511)	(0.972, 0.798, 0.744)

5. Empirical Application: COVID-19 Analysis

An application to COVID-19 confirmed case rates across 616 counties in seven southern U.S. states involved covariates such as %Black, %Hispanic, %Bachelor’s, median income, %65+, %18–29, population density, %foreign born, and %uninsured.

5.1 Comparative Model Performance

Model	Adj. $b_j$4	AICc	RMSE
MGWR	0.701	1093.5	0.513
M-SGWR	0.782	943.2	0.433

M-SGWR achieves the highest adjusted $b_j$5, lowest AICc, and lowest RMSE. Residual spatial autocorrelation (Moran’s $b_j$6) is also minimized under M-SGWR.

5.2 Interpretation of Estimated Parameters

Optimized $b_j$7 values reveal the role of attribute similarity in different predictors:

• $b_j$8
• $b_j$9
• $\alpha_j \in [0,1]$0
• $\alpha_j \in [0,1]$1

Low $\alpha_j \in [0,1]$2 (e.g., population density) indicates significant influence from attribute similarity, enabling the model to identify shared dynamics across non-local counties. High $\alpha_j \in [0,1]$3 values indicate primarily spatially contiguous effects.

6. Implications and Summary

M-SGWR generalizes previous geographically weighted regression models by allowing each covariate to select both its spatial scale $\alpha_j \in [0,1]$4 and its own mix $\alpha_j \in [0,1]$5 of geographic versus attribute-based weighting. The methodology retains the interpretability and computational structure of local WLS regression while expanding the scope of spatial interaction to include both traditional spatial and non-spatial (attribute-based, network) similarity. In both synthetic and real spatial datasets, including epidemiological contexts, M-SGWR demonstrates superior fit and ability to model complex spatial-connectivity structures compared to GWR, SGWR, and MGWR (Lessani et al., 27 Jan 2026).