Scaling Methods • rfastloess

Overview

When iterations > 0, LOESS computes robustness weights by comparing each residual to the current residual scale estimate $\hat{\sigma}$ . The scaling_method parameter controls how $\hat{\sigma}$ is estimated.

$w_i = B\!\left(\frac{|r_i|}{6 \cdot \hat{\sigma}}\right)$

Method	Formula	Robustness	Speed
`"mad"`	Median absolute deviation from median	Very robust	Moderate
`"mar"`	Median of \|residuals\|	Robust	Fast
`"mean"`	Mean of \|residuals\|	Less robust	Fastest

Scaling method comparison

MAD — Median Absolute Deviation (Default)

$\hat{\sigma} = \text{median}(|r_i - \text{median}(r_i)|)$

First centers residuals at their median, then takes the median of the absolute deviations. Double use of the median makes it highly resistant to extreme outliers.

Use when: Data may contain outliers (default for most applications).

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

model <- Loess(iterations = 3, scaling_method = "mad")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.4897657

MAR — Median Absolute Residual

$\hat{\sigma} = \text{median}(|r_i|)$

Uses the uncentered median — unlike MAD it does not subtract the residual median first. Still robust (median-based), faster (one partial sort instead of two).

Use when: Speed matters and data have minimal systematic bias in residuals.

model <- Loess(iterations = 3, scaling_method = "mar")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.5096452

Mean — Mean Absolute Residual

$\hat{\sigma} = \frac{1}{n}\sum_i |r_i|$

Arithmetic mean of absolute residuals. Non-robust: a single extreme outlier inflates $\hat{\sigma}$ , causing the bisquare weight function to under-downweight other outliers.

Use when: Clean data with no outliers; maximum computation speed required.

model <- Loess(iterations = 3, scaling_method = "mean")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.512746

Choosing a Scaling Method

Situation	Recommended Method
General purpose, possible outliers	`"mad"` (default)
Speed matters; residuals have minimal systematic bias	`"mar"`
Clean data, no outliers	`"mean"`

See vignette("robustness", package = "rfastloess") for a broader discussion of outlier handling.

Comparing Scaling Methods

library(rfastloess)
set.seed(42)
x <- 1:100
y <- sin(x / 10) + rnorm(100, sd = 0.3)
y[c(20, 50, 80)] <- y[c(20, 50, 80)] + 5  # outliers

methods <- c("mad", "mar", "mean")
colors  <- c("blue", "red", "green")

plot(x, y, pch = 16, col = "gray",
    main = "Scaling Method Comparison (with outliers)")

for (i in seq_along(methods)) {
    model  <- Loess(iterations = 3, scaling_method = methods[i])
    result <- fit(model, x, y)
    lines(result$x, result$y, col = colors[i], lwd = 2)
}

legend("topright", methods, col = colors, lwd = 2)

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] rfastloess_3.0.0
#> 
#> 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_0.62         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