Boundary Handling

Overview

Boundary policy comparison

Standard LOESS neighbourhoods become asymmetric at the boundaries: fewer points exist on one side, pulling the local fit toward the data interior. The boundary_policy parameter controls how the data is padded to mitigate this effect.

Policy	Padding Strategy	Best For
`"extend"`	Repeat first / last value	Most datasets (default)
`"reflect"`	Mirror data at boundaries	Periodic or symmetric data
`"zero"`	Pad with zeros	Data known to approach zero
`"noboundary"`	No padding (Cleveland original)	Reference behaviour

Extend (Default)

Pads beyond both endpoints by replicating the first and last observed values. Prevents the fit from curling toward zero and is a safe default for nearly all use cases.

Use when: No strong prior on boundary behaviour; general-purpose smoothing.

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

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

Reflect

Mirrors the data about both endpoints before fitting, then discards the reflected region from the output. Preserves continuity of derivatives at the endpoints.

Use when: Circular data (e.g., angle, day-of-year), symmetric physical quantities, or when the derivative at the boundary is expected to be zero.

model <- Loess(boundary_policy = "reflect")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.6080511

Zero

Pads beyond both endpoints with zeros before fitting. Appropriate when the underlying process is known to be zero outside the observed range.

Use when: Signal decays to zero at both ends; zero is a meaningful boundary value.

model <- Loess(boundary_policy = "zero")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.253663

No Boundary

Applies no padding. Each local fit uses only the points that are actually available, which may be fewer than the requested neighbourhood at the endpoints. This reproduces the original Cleveland (1979) algorithm exactly.

Use when: Reproducing reference results; you prefer the raw LOESS boundary behaviour.

Note: Without padding, boundary fits can have higher variance and visible edge artefacts, particularly with small fraction values.

model <- Loess(boundary_policy = "noboundary")
result <- fit(model, x, y)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.6893051

Comparing Policies

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

policies <- c("extend", "reflect", "zero", "noboundary")
colors   <- c("blue", "red", "green", "purple")

plot(x, y, pch = 16, col = "gray",
    main = "Boundary Policy Comparison")

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

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

Choosing a Policy

Situation	Recommended Policy
General purpose	`"extend"` (default)
Periodic signal (angle, day-of-year)	`"reflect"`
Signal known to be zero at boundaries	`"zero"`
Replicating original Cleveland behaviour	`"noboundary"`

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              
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#>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
#>  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
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#> 
#> time zone: Etc/UTC
#> tzcode source: system (glibc)
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#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
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#> other attached packages:
#> [1] rfastloess_3.0.0 rmarkdown_2.32  
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#> 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