gkrreg

Gaussian Kernel Robust Regression — a convergence-guaranteed robust regression method that down-weights outliers and leverage points via an iteratively re-weighted least squares algorithm driven by the Gaussian kernel.

The method was proposed by De Carvalho, Lima Neto & Ferreira (2017, Neurocomputing, 234, 58–74). This package extends the original work with full statistical inference: an analytic sandwich variance estimator (HC0) and a pairs bootstrap with BCa confidence intervals and centred-t p-values.

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Installation

# Stable release from CRAN
install.packages("gkrreg")

# Development version from GitHub
# install.packages("remotes")
remotes::install_github("marcelorpf/gkrreg")

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Quick start

library(gkrreg)

data(belgium_calls)

# Fit with sandwich inference (default)
fit <- gkrr(calls ~ year, data = belgium_calls, sigma_method = "s3")
summary(fit)

# Fit with bootstrap inference (BCa, B = 999)
fit_b <- gkrr(calls ~ year, data = belgium_calls,
              sigma_method = "s3",
              boot      = TRUE,
              boot_args = list(B = 999, type = "bca", seed = 1))
summary(fit_b)

# Diagnostic plots
plot(fit, which = 1:4)

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Key features

• Robust fitting via IRWLS with Gaussian kernel weights — outliers and leverage points receive near-zero weight automatically
• Four estimators for γ² (the kernel width hyper-parameter):
  • "s1" Caputo sub-sample quantiles — clean data
  • "s2" Pairwise median — mild contamination
  • "s3" Residual variance — heavy contamination / leverage points
  • "s4" AICc bandwidth via sm::h.select() — large samples (n ≥ 200)
  • "auto" Data-driven selection among S1–S4 by out-of-bag bootstrap MSE
• Sandwich inference (HC0) — analytic, computed at fit time, always available via summary() and vcov()
• Bootstrap inference — pairs bootstrap re-estimating γ² on every replicate; percentile, normal and BCa intervals; centred-t p-values
• Automatic guidance — summary() warns when sandwich and bootstrap SEs diverge by more than 25%, and notes when bootstrap is recommended (n < 50 or heavy contamination detected)
• Six bundled datasets covering all outlier scenarios (Y-space, X-space, leverage, bad leverage)
• Six diagnostic panels — point size inversely proportional to kernel weight, with the theoretical kernel curve G(e) overlaid on panel 3

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Inference at a glance

# Sandwich (fast, deterministic — default when boot = FALSE)
fit <- gkrr(y ~ x, sigma_method = "s3")
summary(fit)          # SE, 95% CI, Wald z-test p-values
vcov(fit)             # full sandwich covariance matrix

# Bootstrap (recommended for n < 50 or heavy contamination)
boot <- gkrr_boot(fit, B = 999, type = "bca", seed = 1)
summary(fit, boot = boot)   # BCa CI, centred-t p-values
plot(boot, which = 1)       # histogram + shaded CI per coefficient
plot(boot, which = 2)       # scatter-plot matrix of bootstrap replicates

	Sandwich (HC0)	Bootstrap (BCa)
Cost	O(np²)	O(Bnp²)
Accounts for γ̂² variability	No	Yes
Reliable for small n	Limited	Yes
Corrects for skewness	No	Yes
Recommended	n ≥ 50, mild contamination	n < 50 or heavy contamination

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Bundled datasets

Dataset	n	Outlier type	Source
belgium_calls	24	Y-space	Rousseeuw & Leroy (1987)
cloud_point	19	Leverage	Draper & Smith (1998)
kootenay	13	X-space	Neter et al. (1996)
delivery	25	Bad leverage	Montgomery & Peck (1992)
mammals	62	Leverage (log scale)	Allison & Cicchetti (1976)
stars_cyg	47	Bad leverage	Humphreys (1978)

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Citation

If you use gkrreg in your research, please cite both the original method and the package:

De Carvalho, F.A.T., Lima Neto, E.A., Ferreira, M.R.P. (2017).
A robust regression method based on exponential-type kernel functions.
Neurocomputing, 234, 58–74. https://doi.org/10.1016/j.neucom.2016.12.035

Ferreira, M.R.P. and Lima Neto, E.A. (2025).
gkrreg: Gaussian Kernel Robust Regression. R package version 0.4.0.
https://CRAN.R-project.org/package=gkrreg

Or in BibTeX:

@Article{DeCarvalho+LimaNeto+Ferreira:2017,
  author  = "{De Carvalho}, Francisco de A.T. and {Lima Neto}, Eufrásio de A.
             and Ferreira, Marcelo R.P.",
  title   = "A Robust Regression Method Based on Exponential-Type Kernel Functions",
  journal = "Neurocomputing",
  year    = "2017",
  volume  = "234",
  pages   = "58--74",
  doi     = "10.1016/j.neucom.2016.12.035",
}

@Manual{gkrreg:2025,
  author = "Ferreira, Marcelo Rodrigo Portela and {Lima Neto}, Eufrásio de Andrade",
  title  = "{gkrreg}: {Gaussian} Kernel Robust Regression",
  year   = "2025",
  note   = "R package version 0.4.0",
  url    = "https://CRAN.R-project.org/package=gkrreg",
}

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Authors

• Marcelo Rodrigo Portela Ferreira — Departamento de Estatística, UFPB (marcelo@de.ufpb.br)
• Eufrásio de Andrade Lima Neto — Departamento de Estatística, UFPB (eufrasio@de.ufpb.br)

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License

GPL-3 © Marcelo Rodrigo Portela Ferreira, Eufrásio de Andrade Lima Neto

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