# stl

##### 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

`*.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

a multiple time series with columns
`seasonal`

, `trend`

and `remainder`

.

the final robust weights (all one if fitting is not done robustly).

the matched call.

integer (length 3 vector) with the spans used for the `"s"`

,
`"t"`

, and `"l"`

smoothers.

integer (length 3) vector with the polynomial degrees for these smoothers.

integer (length 3) vector with the ‘jumps’ (skips) used for these smoothers.

number of **i**nner iterations

number of **o**uter 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.

##### 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

`library(stats)`

```
# NOT RUN {
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
# }
```

*Documentation reproduced from package stats, version 3.5.0, License: Part of R 3.5.0*