Ridge, Lasso, Elastic Net, SCAD and MCP with gwrs

Bahadir Yuzbasi

Scope and interfaces

This guide covers geographically weighted ridge, lasso, elastic-net, SCAD and MCP regression. All five estimators fit a local intercept and local slopes. Ridge shrinks coefficients; Lasso, Elastic Net, SCAD and MCP can also set local slopes to zero. SCAD and MCP use nonconvex penalties, so convergence does not by itself establish a global optimum or recovery of the true active set.

Estimator	Formula	Matrix
GWR-Ridge	gwr_ridge()	gwr_ridge_fit()
GWR-Lasso	gwr_lasso()	gwr_lasso_fit()
GWR-ENET	gwr_en()	gwr_en_fit()
GWR-SCAD	gwr_scad()	gwr_scad_fit()
GWR-MCP	gwr_mcp()	gwr_mcp_fit()

Run a small example

The installed example constructs a deterministic grid on the unit square, uses the first 48 locations for training and the last 16 for prediction, and fits all five estimators. Coordinates and distances are dimensionless. The example makes no statistical comparison or performance claim.

The demonstration fixes the adaptive bisquare neighborhood at 18 training neighbors. It uses penalty 0.08 for the initial fits, Elastic Net mixing 0.5, and the default SCAD/MCP concavity constants 3.7 and 3. Spatial CV then uses three common folds (seed 42) and the explicit penalty grid 0.2, 0.08, 0.02. These are example settings, not the settings used in the manuscript.

example_file <- system.file("examples", "five-methods.R", package = "gwrs")
source(example_file, local = knitr::knit_global())
#> FIVE_METHOD_USAGE_PASS: fits, shared-fold CV, coefficients and new-location predictions

Inspect the selected penalty for each method:

lapply(cv, function(z) z$best)
#> $ridge
#>   lambda d      rmse       mse       mae convergence_rate
#> 3   0.02 0 0.3094922 0.0957854 0.2395486                1
#> 
#> $lasso
#>   lambda d      rmse       mse       mae convergence_rate
#> 3   0.02 0 0.3676823 0.1351902 0.2905299                1
#> 
#> $elastic_net
#>   lambda d     rmse       mse       mae convergence_rate
#> 3   0.02 0 0.358257 0.1283481 0.2776355                1
#> 
#> $scad
#>   lambda      rmse       mse       mae convergence_rate
#> 3   0.02 0.3362073 0.1130353 0.2535229                1
#> 
#> $mcp
#>   lambda      rmse       mse       mae convergence_rate
#> 3   0.02 0.3362073 0.1130353 0.2535229                1

Inspect SCAD coefficients and new-location predictions:

head(coef(cv$scad$fit))
#>      (Intercept)        x1         x2
#> [1,]   1.0116238 0.8978659 -0.6488478
#> [2,]   1.0082939 0.9520375 -0.6512968
#> [3,]   1.0044745 1.0496238 -0.6647299
#> [4,]   0.9990337 1.2171170 -0.7095883
#> [5,]   0.9680645 1.4490780 -0.7876932
#> [6,]   0.9215754 1.7015540 -0.8839481
head(as.data.frame(prediction))
#>         ridge      lasso elastic_net        scad         mcp
#> 1  0.43369460  0.7426695  0.56788760  0.22530960  0.22530958
#> 2  0.36477466  0.6552055  0.49266148  0.16179572  0.16179572
#> 3  0.20863631  0.3947105  0.29357838  0.04253704  0.04253704
#> 4  0.01394306  0.1098925  0.05861234 -0.13117765 -0.13117765
#> 5 -0.31935765 -0.3125693 -0.31622899 -0.42455670 -0.42455670
#> 6 -0.55231892 -0.5637020 -0.55797957 -0.62960456 -0.62960456

Paths, tuning and interpretation

Use gwr_ridge_path(), gwr_lasso_path(), gwr_en_path(), gwr_scad_path() or gwr_mcp_path() when the complete penalty path is needed. The predictors passed to matrix interfaces exclude the intercept; the package handles it separately. Predictor standardization is enabled by default and coefficients are transformed back to the original predictor units.

path <- gwr_scad_path(x, train$y, coords = coords, k = 18,
  lambda = c(0.2, 0.08, 0.02), control = control)
path$lambda
#> [1] 0.20 0.08 0.02

cv_gwr_penalized() accepts an explicit penalty and returns CV diagnostics, the selected tuning row and, with refit = TRUE, a fitted model. Dedicated cv_gwr_scad() and cv_gwr_mcp() interfaces are also available. Training-fold standardization and training-only neighbors prevent preprocessing and neighborhood leakage. Hold folds fixed for a paired method comparison.

The fixed penalty grid in this example is specified before CV. An adaptive bandwidth or mixing-parameter search requires an explicitly designed outer search; this guide does not perform such a search. Examine convergence, candidate eligibility and numerical diagnostics before interpreting results. The default SCAD/MCP solver remains coordinate.

Joint ordinary-GWR selection

Use gwr_select() to compare unpenalized GWR predictor subsets, kernels and bandwidths on a common sample. The joint-selection guide provides an executable example, diagnostic limits and verified resume. This search selects a predictor subset for each candidate model; local penalized fits can instead have different active sets at different locations.

F reference probabilities are exploratory and have no established calibration, including for unpenalized fits. F1 and F2 share one residual-sum-of-squares ratio; they do not provide independent confirmations. These diagnostics do not rank candidate models or validate a selected coefficient map.

Paper-specific source versions

This guide describes the current package. Reproducing the paper requires its recorded simulation (0.4.0) and county-analysis (0.4.0.9006) source versions, settings, inputs and software records. Use the accompanying reproduction archive for those studies. This demonstration is intentionally small and is not a substitute for either numerical experiment.