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Confidence and Prediction Intervals

Source: vignettes/intervals.Rmd
intervals.Rmd

Overview

Confidence and prediction intervals
Confidence and prediction intervals

Adapter support: Confidence and prediction intervals are available in Batch mode, Streaming mode (computed per chunk and merged across overlap boundaries via merge_strategy, like y/derivative), and Online mode when update_mode = "full" is set (errors if combined with the default "incremental" mode).

Confidence and prediction coverage levels are independent; for example, a 90% confidence interval can be paired with a 99% prediction interval.

Type Represents Width Use
Confidence Uncertainty in mean curve Narrow Trend location
Prediction Uncertainty for new points Wide New data location

Confidence Intervals

Estimate uncertainty in the smoothed curve itself.

library(rfastlowess)
set.seed(42)
x <- seq(0, 2 * pi, length.out = 100)
y <- sin(x) + rnorm(100, sd = 0.3)

model <- Lowess(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 = "LOWESS 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: [ -0.04412499 ,  0.1214997 ]

Prediction Intervals

Estimate where new observations might fall.

library(rfastlowess)
set.seed(42)
x <- seq(0, 2 * pi, length.out = 100)
y <- sin(x) + rnorm(100, sd = 0.3)

model <- Lowess(fraction = 0.5, intervals = intervals_opts(prediction = 0.95))
result <- fit(model, x, y)

plot(x, y, pch = 16, col = "gray",
    main = "LOWESS 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: [ -0.1513472 ,  1.114249 ]

Both Intervals

Request both types simultaneously:

library(rfastlowess)
set.seed(42)
x <- seq(0, 2 * pi, length.out = 100)
y <- sin(x) + rnorm(100, sd = 0.3)

model <- Lowess(
    fraction = 0.5,
    intervals = intervals_opts(confidence = 0.90, prediction = 0.99)
)
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: [ -0.03081586 ,  0.1081906 ]

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 <- Lowess(fraction = 0.5, intervals = intervals_opts(confidence = 0.99))
result <- fit(model, x, y)
cat("99% CI at midpoint: [", result$confidence_lower[50], ", ",
    result$confidence_upper[50], "]\n")
#> 99% CI at midpoint: [ -0.07015174 ,  0.1475265 ]

Bootstrap Intervals

Set bootstrap to compute intervals from resampled fits instead of the analytic formula. Pass seed for reproducible resampling:

model <- Lowess(
    fraction = 0.5,
    intervals = intervals_opts(
        confidence = 0.95,
        prediction = 0.95,
        bootstrap = 200
    ),
    seed = 42
)
result <- fit(model, x, y)
cat("Bootstrap 95% CI at midpoint: [", result$confidence_lower[50], ", ",
    result$confidence_upper[50], "]\n")
#> Bootstrap 95% CI at midpoint: [ -0.07091939 ,  0.1507282 ]

The same intervals and seed arguments are accepted by StreamingLowess(), OnlineLowess(), and predict().

Standard Errors

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

model <- Lowess(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): 0.03763669 0.03931471 0.04100582 0.04268883 0.04433928

Availability

Supported In All Three Adapters: Confidence and prediction intervals are available in Batch, Streaming, and Online mode (update_mode = "full" only).

Feature Batch Streaming Online
Standard errors ✓ ✓ ✓ (update_mode = "full" only)
Confidence intervals ✓ ✓ ✓ (update_mode = "full" only)
Prediction intervals ✓ ✓ ✓ (update_mode = "full" only) 
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