Seasonal Decomposition of Time Series by Loess
Description
Decompose a time series into seasonal, trend and irregular components
using loess, acronym STL.
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
*.jump-th 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
Author(s)
B.D. Ripley; Fortran code by Cleveland et al. (1990) from ‘netlib’.
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.
See Also
plot.stl for stl methods;
loess in package stats (which is not actually
used in stl).
StructTS for different kind of decomposition.
Examples
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