Use Case: Time Series Analysis

Overview

LOESS provides flexible trend extraction from time series without parametric assumptions.

Basic Trend Extraction

fraction = 0.1 sizes the neighbourhood as 10% of the data at each evaluation point — narrow enough to follow a slowly varying trend without smearing periodic variation. Three robustness iterations down-weight noise spikes so they cannot bias the fitted curve; this is especially important when the signal-to-noise ratio is low or when occasional outliers are expected.

library(rfastloess)

set.seed(42)
t <- seq(0, 100, length.out = 500)
trend <- 10 + 0.5 * t + 3 * sin(t / 10)
noise <- rnorm(500, sd = 3)
y <- trend + noise

model <- Loess(fraction = 0.1, iterations = 3)
result <- fit(model, t, y)

plot(t, y, col = "gray", pch = ".",
    xlab = "Time", ylab = "Value", main = "Trend Extraction")
lines(result$x, result$y, col = "blue", lwd = 2)
legend("topleft", c("Observed", "Trend (LOESS)"),
        pch = c(1, NA), lty = c(NA, 1), col = c("gray", "blue"))

Seasonal Decomposition

Setting outputs = "residuals" stores observed − smoothed alongside the smooth. A slightly wider fraction = 0.4 produces a smoother baseline trend, so short-duration oscillations end up in the residuals rather than being absorbed into the trend component. The residual series is then ready for spectral analysis, seasonality detection, or change-point methods.

library(rfastloess)
set.seed(42)

t        <- 1:120
trend    <- 100 + 0.5 * t
seasonal <- 15 * sin(2 * pi * t / 12)
noise    <- rnorm(120, sd = 5)
y        <- trend + seasonal + noise

model    <- Loess(fraction = 0.4, iterations = 2, outputs = "residuals")
result   <- fit(model, t, y)
residual <- result$residuals

par(mfrow = c(2, 1), mar = c(4, 4, 2, 0.5))
plot(t, y, type = "l", col = "gray",
    main = "Original + LOESS Trend", xlab = "Month", ylab = "Value")
lines(result$x, result$y, col = "blue", lwd = 2)

plot(t, residual, type = "l", col = "darkgreen",
    main = "Detrended Residual (Seasonal Component)",
    xlab = "Month", ylab = "Residual")
abline(h = 0, lty = 2, col = "red")

par(mfrow = c(1, 1))

Irregular Time Grids

library(rfastloess)
set.seed(42)

t_irregular <- c(sort(runif(200, 0, 50)), sort(runif(50, 50, 100)))
y_irregular <- sin(t_irregular / 10) + rnorm(length(t_irregular), sd = 0.5)

model  <- Loess(fraction = 0.2, iterations = 2)
result <- fit(model, t_irregular, y_irregular)

plot(t_irregular, y_irregular, pch = 16, cex = 0.4, col = "gray",
    xlab = "Time", ylab = "Value", main = "Irregularly Sampled Time Series")
lines(result$x, result$y, col = "blue", lwd = 2)

Forecasting with Prediction Intervals

Prediction intervals widen the uncertainty band to include both the uncertainty in the fitted curve (confidence interval) and the expected scatter of new observations around it. fraction = 0.2 offers a balance between local detail and stable interval width — too small a fraction produces jagged interval edges; too large a fraction underestimates local variance near turning points.

library(rfastloess)
set.seed(42)
t <- seq(0, 100, length.out = 500)
y <- 10 + 0.3 * t + sin(t / 5) + rnorm(500, sd = 2)

model <- Loess(
    fraction = 0.2,
    iterations = 3,
    intervals = intervals_opts(confidence = 0.95, prediction = 0.95)
)
result <- fit(model, t, y)

plot(t, y, pch = ".", col = "gray",
    xlab = "Time", ylab = "Value",
    main = "Trend with 95% Prediction Interval")
polygon(c(result$x, rev(result$x)),
        c(result$prediction_upper, rev(result$prediction_lower)),
        col = rgb(0, 0, 1, 0.15), border = NA)
lines(result$x, result$y, col = "blue", lwd = 2)

Handling Missing Data

LOESS naturally handles irregular time sampling:

library(rfastloess)
set.seed(42)

t_irregular <- sort(runif(200, 0, 100))
y_irregular <- 10 + t_irregular * 0.3 + rnorm(200, sd = 2)

model <- Loess(fraction = 0.2)
result <- fit(model, t_irregular, y_irregular)
cat("Smoothed y[0]:", result$y[1], "\n")
#> Smoothed y[0]: 12.10549

Multi-Scale Analysis

Use different fractions to extract features at different scales:

library(rfastloess)
set.seed(42)
t <- seq(0, 100, length.out = 500)
trend_true <- 10 + 0.5 * t + 3 * sin(t / 10)
y <- trend_true + rnorm(length(t), sd = 3)

fractions <- c(0.05, 0.2, 0.5)

plot(t, y, col = "gray", pch = ".",
    xlab = "Time", ylab = "Value", main = "Multi-Scale LOESS")

for (f in fractions) {
    model <- Loess(fraction = f)
    result <- fit(model, t, y)
    lines(result$x, result$y, lwd = 2)
}
legend("topleft", paste0("fraction=", fractions), lty = 1)

Gene Expression Time Course

Biological application:

library(rfastloess)
set.seed(42)

hours <- seq(0, 24, by = 0.5)
expression <- 100 * (1 + 0.5 * sin(hours * pi / 12)) +
    rnorm(length(hours), sd = 10)

model <- Loess(
    fraction = 0.3,
    iterations = 3,
    outputs = "diagnostics",
    intervals = intervals_opts(confidence = 0.95)
)
result <- fit(model, hours, expression)

cat("R2:", result$diagnostics$r_squared, "\n")
#> R2: 0.9096872

Choosing Fraction for Time Series

Data Type	Recommended Fraction	Rationale
Daily data (years)	0.3–0.5	Capture annual trends
Hourly data (days)	0.1–0.2	Capture daily patterns
Sensor data (minutes)	0.05–0.1	Preserve short-term features
Noisy data	Higher	Reduce noise impact
Clean data	Lower	Preserve detail

vignette("use-case-real-time", package = "rfastloess") — For streaming time series
vignette("cross-validation", package = "rfastloess") — Optimal fraction selection
vignette("degree", package = "rfastloess") — Degree 2 for curved trends
vignette("boundary", package = "rfastloess") — Edge bias in trend extraction
?Loess — Full parameter reference

sessionInfo()
#> R version 4.6.1 (2026-06-24)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 26.04.1 LTS
#> 
#> Matrix products: default
#> BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
#> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.32.so;  LAPACK version 3.12.0
#> 
#> locale:
#>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
#>  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
#>  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
#>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
#>  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
#> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
#> 
#> time zone: Etc/UTC
#> tzcode source: system (glibc)
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] rfastloess_3.0.0 rmarkdown_2.32  
#> 
#> loaded via a namespace (and not attached):
#>  [1] digest_0.6.39    R6_2.6.1         fastmap_1.2.0    xfun_0.61       
#>  [5] maketools_1.3.2  cachem_1.1.0     knitr_1.52       htmltools_0.5.9 
#>  [9] buildtools_1.0.0 lifecycle_1.0.5  cli_3.6.6        sass_0.4.10     
#> [13] jquerylib_0.1.4  compiler_4.6.1   sys_3.4.3        tools_4.6.1     
#> [17] evaluate_1.0.5   bslib_0.12.0     yaml_2.3.12      otel_0.2.0      
#> [21] jsonlite_2.0.0   rlang_1.3.0