Weight Functions (Kernels)
Overview
Weight functions (kernels) determine how neighboring points contribute to each local fit. Points closer to the target receive higher weights.
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]: 0.5023423Epanechnikov
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]: 0.5158928Gaussian
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]: 0.4790245Biweight
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]: 0.4989114Cosine
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]: 0.5131147Triangle
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]: 0.5013121Uniform
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]: 0.5600545Choosing 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 26.04.1 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-r0.3.32.so; LAPACK version 3.12.0
#>
#> locale:
#> [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
#> [3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
#> [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
#> [7] LC_PAPER=en_US.UTF-8 LC_NAME=C
#> [9] LC_ADDRESS=C LC_TELEPHONE=C
#> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
#>
#> time zone: Etc/UTC
#> tzcode source: system (glibc)
#>
#> attached base packages:
#> [1] stats graphics grDevices utils datasets methods base
#>
#> other attached packages:
#> [1] rfastlowess_5.0.0 rmarkdown_2.32
#>
#> loaded via a namespace (and not attached):
#> [1] digest_0.6.39 R6_2.6.1 fastmap_1.2.0 xfun_0.61
#> [5] maketools_1.3.2 cachem_1.1.0 knitr_1.52 htmltools_0.5.9
#> [9] buildtools_1.0.0 lifecycle_1.0.5 cli_3.6.6 sass_0.4.10
#> [13] jquerylib_0.1.4 compiler_4.6.1 sys_3.4.3 tools_4.6.1
#> [17] evaluate_1.0.5 bslib_0.12.0 yaml_2.3.12 otel_0.2.0
#> [21] jsonlite_2.0.0 rlang_1.3.0