Custom Weights

How Custom Weights Work

Standard LOWESS assigns equal prior trust to all observations. Custom weights let you override this assumption point by point — before any distance or robustness weighting is applied.

The effective weight of observation $j$ in a local fit centred at $x_i$ is:

$w_{ij} = \text{custom\_weights}[j] \times K\!\left(\frac{d_{ij}}{h_i}\right) \times r_j$

where $K$ is the distance kernel, $h_i$ is the local bandwidth, and $r_j$ is the robustness weight from the current iteration.

Batch adapter only: custom_weights applies in Batch mode. It is silently ignored in Streaming and Online adapters.

When to Use Custom Weights

Situation	Recommended weight
Point known to be erroneous	`0.0` — fully excluded
Unreliable sensor / low precision	`0.1 – 0.5`
Standard observation	`1.0` (default)
Carefully calibrated measurement	`> 1.0`
Measurement uncertainty $\sigma_i$	$1 / \sigma_i^2$

Custom Weights vs. Robustness Iterations

Both mechanisms handle unreliable data, but they serve different purposes:

	Custom Weights	Robustness Iterations
When known	Before fitting	Computed from residuals
Knowledge required	Prior knowledge of quality	None — data-driven
Effect	Fixed throughout fit	Adapts each iteration
Use case	Known bad sensors	Unknown outlier contamination

They compose: you can use both simultaneously. Custom weights suppress a priori bad points; robustness iterations then handle any residual outliers that remain.

Basic Usage

Suppress a Known Outlier

Set the weight to 0 at the bad point — it is excluded from every local fit that would otherwise include it.

library(rfastlowess)

x <- 1:10
y <- x * 2.0
y[6] <- 100.0              # spike at index 6

weights <- rep(1.0, 10)
weights[6] <- 0.0          # exclude the spike

model <- Lowess(fraction = 0.5, iterations = 0L)
result <- fit(model, x, y, custom_weights = weights)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 2.572621

Emphasize Important Points

Assign high weights to measurements you trust most — calibration standards, reference instruments, or low-noise observations.

library(rfastlowess)
x <- seq(0, 2 * pi, length.out = 100)
y <- sin(x) + 0.1
calibration_indices <- c(6, 21, 41, 61, 81)

weights <- rep(1.0, length(x))
weights[calibration_indices] <- 10.0

model <- Lowess(fraction = 0.5)
result <- fit(model, x, y, custom_weights = weights)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.3229811

Propagate Measurement Uncertainty

If each observation has a known standard deviation $\sigma_i$ , set $w_i = 1 / \sigma_i^2$ to give the fit information-theoretically optimal weighting.

library(rfastlowess)
x <- seq(0, 2 * pi, length.out = 100)
y <- sin(x) + 0.1
sigma <- 0.1 + (seq_along(x) %% 4) * 0.1

weights <- 1 / sigma^2
model <- Lowess(fraction = 0.5)
result <- fit(model, x, y, custom_weights = weights)

plot(x, y, pch = 16, cex = 0.5 + weights / max(weights),
    col = "gray", main = "Inverse-Variance Weighted LOWESS")
lines(result$x, result$y, col = "blue", lwd = 2)


cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.1737171

Combined with Robustness Iterations

Custom weights and robustness iterations compose naturally: use custom weights for known bad points and robustness for unknown contamination.

library(rfastlowess)
x <- 0:19
y <- x * 1.5
y[4]  <- -50.0   # known bad
y[13] <- 80.0    # unknown outlier

weights <- rep(1.0, 20)
weights[4] <- 0.0

model <- Lowess(fraction = 0.4, iterations = 3)
result <- fit(model, x, y, custom_weights = weights)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.8612265

Validation Rules

Rule	Effect
Length must equal `n`	Error at fit time if mismatched
All values must be ≥ 0	Negative weights are rejected
All-zero weight vector	Error: no points remain for any local fit
Uniform weights (`1.0` everywhere)	Identical result to omitting weights

Zero-weight windows: If a local neighbourhood contains only zero-weight points, the fit at that centre point falls back to the behaviour specified by zero_weight_fallback (default: "use_local_mean").

sessionInfo()
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