Weight Functions (Kernels) • rfastlowess

Overview

Weight functions (kernels) determine how neighboring points contribute to each local fit. Points closer to the target receive higher weights.

Weight function comparison

Available Kernels

Kernel	Efficiency	Smoothness	Support
Tricube	0.998	Very smooth	Compact
Epanechnikov	1.000	Smooth	Compact
Gaussian	0.961	Infinite	Unbounded
Biweight	0.995	Very smooth	Compact
Cosine	0.999	Smooth	Compact
Triangle	0.989	Moderate	Compact
Uniform	0.943	None	Compact

Efficiency = AMISE relative to Epanechnikov (1.0 = optimal)

Tricube (Default)

Cleveland’s original choice. Best all-around performance.

$w(u) = (1 - |u|^3)^3$

Use when: Default choice for most applications.

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

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

Epanechnikov

Theoretically optimal for kernel density estimation.

$w(u) = \frac{3}{4}(1 - u^2)$

Use when: Optimal MSE properties desired.

model <- Lowess(weight_function = "epanechnikov")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.5158928

Gaussian

Infinitely smooth. No boundary effects.

$w(u) = \exp(-u^2/2)$

Use when: Maximum smoothness needed, computational cost acceptable.

model <- Lowess(weight_function = "gaussian")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.4790245

Biweight

Good balance of efficiency and smoothness.

$w(u) = (1 - u^2)^2$

Use when: Alternative to Tricube with slightly different properties.

model <- Lowess(weight_function = "biweight")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.4989114

Cosine

Smooth and computationally efficient.

$w(u) = \cos(\pi u / 2)$

Use when: Want smooth kernel with simple form.

model <- Lowess(weight_function = "cosine")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.5131147

Triangle

Simple linear taper.

$w(u) = 1 - |u|$

Use when: Simple, interpretable weights.

model <- Lowess(weight_function = "triangle")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.5013121

Uniform

Equal weights within window. Fastest but least smooth.

$w(u) = 1$

Use when: Speed is critical, smoothness less important.

model <- Lowess(weight_function = "uniform")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.5600545

Choosing a Kernel

Choose the first row below whose condition applies:

Condition	Kernel
Need maximum smoothness	`"gaussian"`
Default is acceptable	`"tricube"`
Need optimal asymptotic MSE	`"epanechnikov"`
Speed is critical	`"uniform"`
None of the above	`"biweight"`

Recommendation: Stick with Tricube (default) unless you have specific requirements. The differences between kernels are usually small in practice.

sessionInfo()
#> R version 4.6.1 (2026-06-24)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 24.04.5 LTS
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#> BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
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#> [1] rfastlowess_5.0.0
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#>  [9] rmarkdown_2.32    lifecycle_1.0.5   cli_3.6.6         sass_0.4.10      
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