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_weightsapplies 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$y[1], "\n")
#> Smoothed y[0]: 2.572621Emphasize Important Points
Assign high weights to measurements you trust most — calibration standards, reference instruments, or low-noise observations.
library(rfastlowess)
x <- seq(0, 2 * pi, length.out = 100)
y <- sin(x) + 0.1
calibration_indices <- c(6, 21, 41, 61, 81)
weights <- rep(1.0, length(x))
weights[calibration_indices] <- 10.0
model <- Lowess(fraction = 0.5)
result <- fit(model, x, y, custom_weights = weights)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 0.3229811Propagate 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)
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]: 0.8612265Validation 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").
See Also
- Robustness — adaptive outlier downweighting via IRLS
?Lowess— full parameter reference
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