Robustness • rfastlowess

How Robustness Works

Standard LOWESS can be biased by outliers. Robustness iterations downweight points with large residuals:

Fit initial LOWESS
Compute residuals
Assign robustness weights (large residuals → low weight)
Refit using combined distance × robustness weights
Repeat steps 2–4

Robustness Methods

Robustness Iterations

Robustness Methods

Bisquare (Default)

Smooth downweighting. Points transition gradually from full weight to zero.

$w(u) = \begin{cases} (1 - u^2)^2 & |u| < 1 \\ 0 & |u| \geq 1 \end{cases}$

Use when: General purpose, balanced approach.

library(rfastlowess)
set.seed(42)
x <- seq(0, 2 * pi, length.out = 100)
y <- sin(x) + rnorm(100, sd = 0.3)

model <- Lowess(iterations = 3, robustness_method = "bisquare")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.5023423

Huber

Linear penalty beyond threshold. Less aggressive than Bisquare.

$w(u) = \begin{cases} 1 & |u| \leq k \\ k/|u| & |u| > k \end{cases}$

Use when: Moderate outliers, want to retain some influence.

model <- Lowess(iterations = 3, robustness_method = "huber")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.4995519

Talwar

Hard threshold. Points are either fully weighted or completely excluded.

$w(u) = \begin{cases} 1 & |u| \leq k \\ 0 & |u| > k \end{cases}$

Use when: Extreme outliers, want binary exclusion.

model <- Lowess(iterations = 3, robustness_method = "talwar")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.4919233

Comparison

Method	Transition	Aggressiveness	Use Case
Bisquare	Smooth	Moderate	General purpose
Huber	Gradual	Mild	Preserve influence
Talwar	Hard	Strong	Extreme contamination

Detecting Outliers

Use robustness weights to identify potential outliers:

library(rfastlowess)
set.seed(42)
x <- seq(0, 2 * pi, length.out = 100)
y <- sin(x) + rnorm(100, sd = 0.3)
y[c(20, 50, 80)] <- y[c(20, 50, 80)] + 5  # inject outliers

model <- Lowess(iterations = 5, outputs = "weights")
result <- fit(model, x, y)

shown <- 0
for (i in seq_along(result$robustness_weights)) {
    if (result$robustness_weights[i] < 0.5 && shown < 5) {
        cat(sprintf("Potential outlier at index %d: weight = %.3f\n",
                    i, result$robustness_weights[i]))
        shown <- shown + 1
    }
}
#> Potential outlier at index 12: weight = 0.460
#> Potential outlier at index 20: weight = 0.000
#> Potential outlier at index 24: weight = 0.465
#> Potential outlier at index 25: weight = 0.218
#> Potential outlier at index 50: weight = 0.000

Scale Estimation

Residuals are scaled before computing robustness weights. Two methods:

Method	Formula	Robustness
MAD	`median(\\|r - median(r)\\|)`	Very robust (default)
MAR	`median(\\|r\\|)`	Robust, uncentered
Mean	`mean(\\|r\\|)`	Less robust, fastest

Scaling Methods Comparison

library(rfastlowess)
set.seed(42)
x <- seq(0, 2 * pi, length.out = 100)
y <- sin(x) + rnorm(100, sd = 0.3)

model <- Lowess(iterations = 3, scaling_method = "mad")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.5023423

Auto-Convergence

Stop iterations early when weights stabilize:

Performance: Auto-convergence can significantly reduce computation when weights stabilize before reaching max iterations.

library(rfastlowess)
set.seed(42)
x <- seq(0, 2 * pi, length.out = 100)
y <- sin(x) + rnorm(100, sd = 0.3)

model <- Lowess(iterations = 10, auto_converge = 1e-6)
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.5015125

sessionInfo()
#> R version 4.6.1 (2026-06-24)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 24.04.5 LTS
#> 
#> Matrix products: default
#> BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
#> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0
#> 
#> locale:
#>  [1] LC_CTYPE=C.UTF-8       LC_NUMERIC=C           LC_TIME=C.UTF-8       
#>  [4] LC_COLLATE=C.UTF-8     LC_MONETARY=C.UTF-8    LC_MESSAGES=C.UTF-8   
#>  [7] LC_PAPER=C.UTF-8       LC_NAME=C              LC_ADDRESS=C          
#> [10] LC_TELEPHONE=C         LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C   
#> 
#> time zone: UTC
#> tzcode source: system (glibc)
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] rfastlowess_5.0.0
#> 
#> loaded via a namespace (and not attached):
#>  [1] digest_0.6.39     desc_1.4.3        R6_2.6.1          fastmap_1.2.0    
#>  [5] xfun_0.61         cachem_1.1.0      knitr_1.52        htmltools_0.5.9  
#>  [9] rmarkdown_2.32    lifecycle_1.0.5   cli_3.6.6         sass_0.4.10      
#> [13] pkgdown_2.2.1     textshaping_1.0.5 jquerylib_0.1.4   systemfonts_1.3.2
#> [17] compiler_4.6.1    tools_4.6.1       ragg_1.5.2        bslib_0.12.0     
#> [21] evaluate_1.0.5    yaml_2.3.12       otel_0.2.0        jsonlite_2.0.0   
#> [25] rlang_1.3.0       fs_2.1.0          htmlwidgets_1.6.4