Multivariate LOESS

Overview

Standard LOESS operates on a single predictor x. Setting dimensions > 1 extends the neighbourhood search and local polynomial fit into an n-dimensional predictor space, enabling surface smoothing over spatial grids, time–altitude combinations, and similar multi-predictor datasets.

Multivariate LOESS

Dimensions	Use Case	`x` Input Shape
`1`	Time series, 1D signal (default)	Numeric vector
`2`	Spatial surface, 2-predictor model	`n × 2` matrix
`3+`	High-dimensional regression	`n × d` matrix

Note: Neighbourhood search scales with the number of dimensions. For d ≥ 3 keep fraction small and consider the "interpolation" surface mode (the default) to reduce computation.

1D — Standard (Default)

Single predictor. No configuration required.

library(rfastloess)
set.seed(42)
x <- seq(0, 10, length.out = 200)
y <- sin(x) + rnorm(200, sd = 0.2)

model <- Loess(fraction = 0.3)
result <- fit(model, x, y)
cat("First 6 smoothed values (1D LOESS):\n")
#> First 6 smoothed values (1D LOESS):
print(head(result$y))
#> [1] 0.3808654 0.3912167 0.4028709 0.4154835 0.4286045 0.4422857

2D — Spatial Surface

Two predictors (e.g., latitude/longitude, time/altitude). Pass an n × 2 matrix as x.

set.seed(42)
n <- 100
lat <- runif(n, -pi, pi)
lon <- runif(n, -pi, pi)
z   <- sin(lat) + cos(lon) + rnorm(n, sd = 0.1)

x2d <- cbind(lat, lon)   # n × 2 matrix

model <- Loess(fraction = 0.3, dimensions = 2L)
result <- fit(model, x2d, z)
cat("First 6 smoothed values (2D LOESS, lat/lon):\n")
#> First 6 smoothed values (2D LOESS, lat/lon):
print(head(result$y))
#> [1]  0.8862701  0.1927108 -0.9944883  1.2333466 -0.3672725 -0.8455709

3D and Higher

Pass an n × d matrix as x. Higher dimensions require a larger fraction to keep the local neighbourhood populated.

set.seed(42)
n <- 200
x1 <- runif(n)
x2 <- runif(n)
x3 <- runif(n)
y  <- x1 + sin(2 * pi * x2) + x3^2 + rnorm(n, sd = 0.1)

x3d <- cbind(x1, x2, x3)   # n × 3 matrix

model <- Loess(fraction = 0.5, dimensions = 3L)
result <- fit(model, x3d, y)
cat("First 6 smoothed values (3D LOESS):\n")
#> First 6 smoothed values (3D LOESS):
print(head(result$y))
#> [1] 0.1068976 1.2894880 0.3929455 1.2633451 1.9611165 0.6332138

Distance Metrics for Multivariate Data

When dimensions > 1 you can also control how inter-point distances are computed.

Metric	Description	When to Use
`"normalized"`	Scaled by dimension range (default)	Mixed-scale
`"euclidean"`	Raw Euclidean distance	Same-scale predictors
`"minkowski:p"`	Minkowski (\(L_p\)) norm	Custom geometry
`"weighted"`	Per-dimension via `weighted_metric_weights`	Anisotropic

# Reuse the 2D spatial data from above; upweight the lat dimension
model <- Loess(
    fraction = 0.3,
    dimensions = 2L,
    distance_metric = "weighted",
    weighted_metric_weights = c(2.0, 1.0)  # lat counts twice
)
result <- fit(model, x2d, z)
cat("First 6 smoothed values (2D LOESS, weighted distance):\n")
#> First 6 smoothed values (2D LOESS, weighted distance):
print(head(result$y))
#> [1]  0.7660138  0.1220163 -1.0388792  1.1475979 -0.3815295 -0.9092784

Surface Mode for Multivariate Data

Mode	Behaviour	Speed
`"interpolation"`	Grid-fit, interpolate elsewhere (default)	Fast
`"direct"`	Fit directly at every query point	Exact, slower

# Exact fit at every point — slower but no interpolation artefacts
model <- Loess(fraction = 0.3, dimensions = 2L, surface_mode = "direct")
result <- fit(model, x2d, z)
cat("First 6 smoothed values (2D LOESS, direct surface):\n")
#> First 6 smoothed values (2D LOESS, direct surface):
print(head(result$y))
#> [1]  1.0042372  0.1831346 -0.9723946  1.0257415 -0.3229626 -0.9038626

For large 2D or 3D datasets use "interpolation" (default) and tune cell and interpolation_vertices to control grid resolution.

model <- Loess(
    fraction = 0.3,
    dimensions = 2L,
    surface_mode = "interpolation",
    cell = 0.2
)
result <- fit(model, x2d, z)
cat("First 6 smoothed values (2D LOESS, interpolation surface):\n")
#> First 6 smoothed values (2D LOESS, interpolation surface):
print(head(result$y))
#> [1]  0.8862701  0.1927108 -0.9944883  1.2333466 -0.3672725 -0.8455709

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