Polynomial Degree

Overview

At each target point, LOESS fits a polynomial to the neighbouring data using weighted least squares. The degree parameter controls the order of that polynomial.

Degree comparison

Degree	Local Fit	Captures	Risk
`0` / `"constant"`	Weighted mean	Level only	Over-smooth; edge bias
`1` / `"linear"`	Weighted line	Trend (default)	Rarely overfits
`2` / `"quadratic"`	Parabola	Curvature	Overfits with small `fraction`
`3` / `"cubic"`	Cubic curve	Inflections	Requires larger `fraction`
`4` / `"quartic"`	Quartic curve	Fine structure	High variance, rare

Degree can be specified as an integer (0L–4L) or as a string ("constant", "linear", "quadratic", "cubic", "quartic").

Degree 0 — Local Constant

The fit at each point is a weighted mean. Produces very smooth results but ignores local slope, introducing bias wherever the true function changes.

Use when: Maximum smoothness matters more than accuracy; computationally cheapest.

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

model <- Loess(fraction = 0.5, degree = "constant")
result <- fit(model, x, y)
cat("First 6 smoothed values (constant/Nadaraya-Watson):\n")
#> First 6 smoothed values (constant/Nadaraya-Watson):
print(head(result$y))
#> [1] 0.4694689 0.4748509 0.4785029 0.4798484 0.4836073 0.4936813

Degree 1 — Local Linear (Default)

Fits a weighted line through the neighbourhood. Removes first-order bias and handles boundary regions correctly. The right choice for the vast majority of applications.

Use when: Default; monotone or gently curved data; boundary accuracy matters.

model <- Loess(fraction = 0.5, degree = "linear")
result <- fit(model, x, y)
cat("First 6 smoothed values (linear local regression):\n")
#> First 6 smoothed values (linear local regression):
print(head(result$y))
#> [1] 0.4758389 0.4846388 0.4944609 0.5054102 0.5169758 0.5286606

Degree 2 — Local Quadratic

Fits a weighted parabola. Captures curvature in the data and reduces bias in regions with strong local curvature, at the cost of higher variance.

Use when: Data has visible curvature or local extrema; use a larger fraction to stabilise the fit.

model <- Loess(fraction = 0.5, degree = "quadratic")
result <- fit(model, x, y)
cat("First 6 smoothed values (quadratic local regression):\n")
#> First 6 smoothed values (quadratic local regression):
print(head(result$y))
#> [1] 0.4043043 0.4131994 0.4244604 0.4384709 0.4565813 0.4791656

Degree 3 — Local Cubic

Fits a weighted cubic polynomial. Captures inflections and fine structure but requires a sufficiently large neighbourhood to remain stable.

Use when: Data has clear S-shaped curves or multiple inflection points; use fraction >= 0.5.

model <- Loess(fraction = 0.6, degree = "cubic")
result <- fit(model, x, y)
cat("First 6 smoothed values (cubic local regression):\n")
#> First 6 smoothed values (cubic local regression):
print(head(result$y))
#> [1] 0.4183075 0.4299695 0.4442197 0.4608883 0.4798054 0.5008015

Degree 4 — Local Quartic

Fits a weighted quartic polynomial. Rarely needed in practice; only useful for capturing highly oscillatory local structure. Very prone to overfitting.

Use when: Fine oscillatory structure is physically meaningful and the dataset is large; always cross-validate.

model <- Loess(fraction = 0.7, degree = "quartic")
result <- fit(model, x, y)
cat("First 6 smoothed values (quartic local regression):\n")
#> First 6 smoothed values (quartic local regression):
print(head(result$y))
#> [1] 0.3964750 0.4064069 0.4182002 0.4319700 0.4478315 0.4658997

Degree and Boundary Fallback

At the dataset boundary, the neighbourhood is one-sided and may not contain enough points to support the requested degree. Setting boundary_degree_fallback = TRUE automatically reduces the degree at boundaries to avoid instability.

model <- Loess(degree = 2L, boundary_degree_fallback = TRUE)
result <- fit(model, x, y)
cat("First 6 smoothed values (quadratic, boundary fallback):\n")
#> First 6 smoothed values (quadratic, boundary fallback):
print(head(result$y))
#> [1] 0.4577829 0.4682789 0.4793020 0.4908341 0.5028573 0.5153538

Choosing the Right Degree

Situation	Recommended Degree
Monotone trend, general purpose	`1` (default)
Maximum smoothness, speed	`0`
Clear peaks / valleys / inflections	`2` (with `fraction` >= 0.4)
S-shaped curves, multiple inflections	`3` (with `fraction` >= 0.5)
Fine oscillatory structure (rare)	`4` (fraction >= 0.6, CV)
Boundary accuracy is critical	`1` or `2` (not `0`)
Very small dataset (n < 50)	`1`

Rule of thumb: Start with degree = 1 (default). Move to degree = 2 only if you see systematic bias in regions of high curvature, and increase fraction at the same time.

Higher Degree Effects

Higher Degree Comparison

Surface Mode

The surface_mode parameter controls whether LOESS evaluates the local polynomial at every query point or at a sparser grid of vertices with Hermite cubic interpolation in between.

Mode	Behaviour	Speed	Accuracy
`"interpolation"` (default)	Anchor + Hermite blend	Faster	Approx
`"direct"`	Evaluate at every query point	Exact	Full precision

Surface Mode Comparison

Degree x Interpolation

See vignette("dimensions", package = "rfastloess") for runnable surface_mode examples.

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] rfastloess_3.0.0 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