stl

0th

Percentile

Seasonal Decomposition of Time Series by Loess

Decompose a time series into seasonal, trend and irregular components using loess, acronym STL.

Keywords
ts
Usage
stl(x, s.window, s.degree = 0,
t.window = NULL, t.degree = 1,
l.window = nextodd(period), l.degree = t.degree,
s.jump = ceiling(s.window/10),
t.jump = ceiling(t.window/10),
l.jump = ceiling(l.window/10),
robust = FALSE,
inner = if(robust)  1 else 2,
outer = if(robust) 15 else 0,
na.action = na.fail)
Arguments
x
univariate time series to be decomposed. This should be an object of class "ts" with a frequency greater than one.
s.window
either the character string "periodic" or the span (in lags) of the loess window for seasonal extraction, which should be odd and at least 7, according to Cleveland et al. This has no default.
s.degree
degree of locally-fitted polynomial in seasonal extraction. Should be zero or one.
t.window
the span (in lags) of the loess window for trend extraction, which should be odd. If NULL, the default, nextodd(ceiling((1.5*period) / (1-(1.5/s.window)))), is taken.
t.degree
degree of locally-fitted polynomial in trend extraction. Should be zero or one.
l.window
the span (in lags) of the loess window of the low-pass filter used for each subseries. Defaults to the smallest odd integer greater than or equal to frequency(x) which is recommended since it prevents competition between the trend and seasonal components. If not an odd integer its given value is increased to the next odd one.
l.degree
degree of locally-fitted polynomial for the subseries low-pass filter. Must be 0 or 1.
s.jump, t.jump, l.jump
integers at least one to increase speed of the respective smoother. Linear interpolation happens between every *.jumpth value.
robust
logical indicating if robust fitting be used in the loess procedure.
inner
integer; the number of ‘inner’ (backfitting) iterations; usually very few (2) iterations suffice.
outer
integer; the number of ‘outer’ robustness iterations.
na.action
action on missing values.
Details

The seasonal component is found by loess smoothing the seasonal sub-series (the series of all January values, …); if s.window = "periodic" smoothing is effectively replaced by taking the mean. The seasonal values are removed, and the remainder smoothed to find the trend. The overall level is removed from the seasonal component and added to the trend component. This process is iterated a few times. The remainder component is the residuals from the seasonal plus trend fit. Several methods for the resulting class "stl" objects, see, plot.stl.

Value

stl returns an object of class "stl" with components

time.series
a multiple time series with columns seasonal, trend and remainder.
weights
the final robust weights (all one if fitting is not done robustly).
call
the matched call.
win
integer (length 3 vector) with the spans used for the "s", "t", and "l" smoothers.
deg
integer (length 3) vector with the polynomial degrees for these smoothers.
jump
integer (length 3) vector with the ‘jumps’ (skips) used for these smoothers.
ni
number of inner iterations
no
number of outer robustness iterations

Note

This is similar to but not identical to the stl function in S-PLUS. The remainder component given by S-PLUS is the sum of the trend and remainder series from this function.

References

R. B. Cleveland, W. S. Cleveland, J.E. McRae, and I. Terpenning (1990) STL: A Seasonal-Trend Decomposition Procedure Based on Loess. Journal of Official Statistics, 6, 3--73.

plot.stl for stl methods; loess in package stats (which is not actually used in stl). StructTS for different kind of decomposition.
library(stats) require(graphics) plot(stl(nottem, "per")) plot(stl(nottem, s.window = 7, t.window = 50, t.jump = 1)) plot(stllc <- stl(log(co2), s.window = 21)) summary(stllc) ## linear trend, strict period. plot(stl(log(co2), s.window = "per", t.window = 1000)) ## Two STL plotted side by side : stmd <- stl(mdeaths, s.window = "per") # non-robust summary(stmR <- stl(mdeaths, s.window = "per", robust = TRUE)) op <- par(mar = c(0, 4, 0, 3), oma = c(5, 0, 4, 0), mfcol = c(4, 2)) plot(stmd, set.pars = NULL, labels = NULL, main = "stl(mdeaths, s.w = \"per\", robust = FALSE / TRUE )") plot(stmR, set.pars = NULL) # mark the 'outliers' : (iO <- which(stmR $weights < 1e-8)) # 10 were considered outliers sts <- stmR$time.series points(time(sts)[iO], 0.8* sts[,"remainder"][iO], pch = 4, col = "red") par(op) # reset