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Multivariate LOESS

Source: vignettes/dimensions.Rmd
dimensions.Rmd

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
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 (LpL_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()
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