TDS_MGWR: Top-Down Scale (TDS) and Adaptive Top-Down Scale (ATDS) Estimation for MGWR

Description

This function implements the "Top-Down Scale" (TDS) methodology for estimating Multiscale Geographically Weighted Regression (MGWR) models. Unlike classical backfitting approaches that fully optimize bandwidths at each iteration, TDS uses a pre-defined sequence of decreasing bandwidths to efficiently identify the optimal spatial scale for each covariate.

The function supports two main algorithms:

'tds_mgwr': A backfitting algorithm that selects a unique optimal bandwidth for each covariate from a decreasing sequence.
'atds_mgwr': Extends 'tds_mgwr' with a second "boosting" stage (Adaptive TDS). It refines estimates by allowing bandwidths to vary locally, capturing complex spatial patterns (e.g., simultaneous large-scale trends and local variations).

Usage

TDS_MGWR(formula, data, coords, Model = 'tds_mgwr',
                    kernels = 'gauss', fixed_vars = NULL, Ht = NULL,
                    control_tds = list(nns = 25, get_AIC = FALSE, init_model = "OLS"),
                    control = list(adaptive = TRUE))

Value

An object of class mgwrsar containing:

Betav: Matrix of spatially varying coefficients.
H: Vector of optimal bandwidths found for each covariate.
RMSE: Root Mean Square Error of the final model.
AICc: Corrected Akaike Information Criterion (if requested).
fitted.values: Vector of fitted values.
residuals: Vector of residuals.

Arguments

formula

A formula object specifying the model (e.g., y ~ x1 + x2).

data

A data frame containing the variables in the model.

coords

A matrix or data frame of coordinates (2 columns for spatial).

Model

A character string specifying the algorithm. Options:

'tds_mgwr' (default): Top-Down Scale MGWR (Stage 1 only).
'atds_mgwr': Adaptive Top-Down Scale MGWR (Stage 1 + Stage 2 boosting).
'atds_gwr': Univariate Adaptive Top-Down Scale GWR.

kernels

A character string or vector specifying the kernel type (e.g., 'triangle', 'bisq', 'gauss'). Default is 'triangle'.

fixed_vars

A character vector indicating the names of variables with spatially stationary (fixed) coefficients. Default is NULL.

Ht

Numeric. Optional bandwidth for the second dimension (time) if using spatio-temporal models (Type 'GDT').

control_tds

A named list of control parameters specific to the TDS algorithm:

nns: Integer. Number of bandwidth steps in the decreasing sequence (default 30).

get_AIC

Logical. If TRUE, computes AICc (slower). Default FALSE (except for 'atds_mgwr').

init_model

Character. Initial model type to start backfitting: 'OLS' (default), 'GWR', 'GTWR', or 'known'.

ncore

Integer. Number of cores for parallelization. Default 1.

tol

Numeric. Convergence tolerance. Default 0.001.

nrounds

Integer. Number of boosting rounds for Stage 2 (only for 'atds_mgwr'). Default 3.

control

A named list of standard control arguments passed to the internal MGWRSAR calls:

adaptive: Logical or Vector. TRUE for adaptive bandwidth (nearest neighbors), FALSE for fixed distance. Can be a vector of length 2 for space/time.

Type

Character. Spatial weighting type: 'GD' (Spatial only) or 'GDT' (Space-Time).

NN

Integer. Maximum number of neighbors for distance matrix computation (truncation). Default is nrow(data).

Details

The TDS algorithm works in two stages:

**Stage 1 (Backfitting):** Starts with a global model (OLS) or a simple GWR. It iteratively updates the bandwidth for each covariate by testing values from a decreasing sequence. This avoids the "yo-yo" effect of standard backfitting and converges faster.
**Stage 2 (Boosting - atds_mgwr only):** Uses the residuals from Stage 1 to iteratively refine coefficients. This stage allows the effective bandwidth to adapt locally, improving accuracy for covariates with spatially heterogeneous scales.

References