Confidence and Prediction Intervals

Overview

Confidence and prediction intervals

Adapter support: Batch and Streaming support confidence and prediction intervals; Online supports them in update_mode = "full" only.

Type	Represents	Width	Use
Confidence	Uncertainty in mean curve	Narrow	Where is the trend?
Prediction	Uncertainty for new points	Wide	Where will data fall?

Confidence Intervals

Estimate uncertainty in the smoothed curve itself.

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, intervals = intervals_opts(confidence = 0.95))
result <- fit(model, x, y)

# Plot with bands
plot(x, y, pch = 16, col = "gray",
    main = "LOESS with 95% Confidence Intervals")
lines(result$x, result$y, col = "blue", lwd = 2)
lines(result$x, result$confidence_lower, col = "blue", lty = 2)
lines(result$x, result$confidence_upper, col = "blue", lty = 2)


cat("95% CI at midpoint: [", result$confidence_lower[50], ", ",
    result$confidence_upper[50], "]\n")
#> 95% CI at midpoint: [ -0.1007995 ,  0.1749161 ]

Prediction Intervals

Estimate where new observations might fall.

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, intervals = intervals_opts(prediction = 0.95))
result <- fit(model, x, y)

plot(x, y, pch = 16, col = "gray",
    main = "LOESS with 95% Prediction Intervals")
lines(result$x, result$y, col = "blue", lwd = 2)
lines(result$x, result$prediction_lower, col = "red", lty = 2)
lines(result$x, result$prediction_upper, col = "red", lty = 2)


cat("Prediction bounds: [", result$prediction_lower[1], ", ",
    result$prediction_upper[1], "]\n")
#> Prediction bounds: [ -0.1436505 ,  1.095328 ]

Both Intervals

Request both types simultaneously:

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,
    intervals = intervals_opts(confidence = 0.95, prediction = 0.95)
)
result <- fit(model, x, y)

plot(x, y, pch = 16, col = "gray",
    main = "Confidence vs Prediction Intervals")
lines(result$x, result$y, col = "blue", lwd = 2)

# Confidence interval (narrow, blue)
lines(result$x, result$confidence_lower, col = "blue", lty = 2)
lines(result$x, result$confidence_upper, col = "blue", lty = 2)

# Prediction interval (wide, red)
lines(result$x, result$prediction_lower, col = "red", lty = 3)
lines(result$x, result$prediction_upper, col = "red", lty = 3)

legend("topright",
        c("Data", "Smoothed", "95% CI", "95% PI"),
        pch = c(16, NA, NA, NA), lty = c(NA, 1, 2, 3),
        col = c("gray", "blue", "blue", "red"))


cat("95% CI at midpoint: [", result$confidence_lower[50], ", ",
    result$confidence_upper[50], "]\n")
#> 95% CI at midpoint: [ -0.1007995 ,  0.1749161 ]

Confidence Levels

Common levels and their z-values:

Level	z-value	Interpretation
0.90	1.645	90% of intervals contain true value
0.95	1.960	95% of intervals contain true value
0.99	2.576	99% of intervals contain true value

# 99% confidence interval
model <- Loess(fraction = 0.5, intervals = intervals_opts(confidence = 0.99))
result <- fit(model, x, y)
cat("First lower CI bound (99%):", result$confidence_lower[1], "\n")
#> First lower CI bound (99%): 0.296526

Standard Errors

Access standard errors directly (available when intervals are computed):

model <- Loess(fraction = 0.5, outputs = "se")
result <- fit(model, x, y)
cat("Standard errors (first 5):", head(result$standard_errors, 5), "\n")
#> Standard errors (first 5): 0.06960904 0.07028408 0.07095493 0.07156272 0.07204299

Availability

Adapter support: Confidence and prediction intervals are available in Batch and Streaming; Online supports them in update_mode = "full" only. intervals.

Feature	Batch	Streaming	Online
Confidence intervals	✓	✗	✗
Prediction intervals	✓	✗	✗
Standard errors	✓	✗	✗

sessionInfo()
#> R version 4.6.1 (2026-06-24)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 26.04.1 LTS
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#> [1] rfastloess_3.0.0 rmarkdown_2.32  
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