stlDecompose#

Purpose#

Seasonal-Trend decomposition via LOESS (STL).

Format#

stl = stlDecompose(y, period)#
stl = stlDecompose(y, period, s_window=7)
Parameters:

  • y (Nx1 vector) – time series data.

  • period (scalar) – seasonal period (e.g., 12 for monthly, 4 for quarterly, 52 for weekly).

  • s_window (scalar) – Optional keyword, seasonal smoothing window (must be odd, >= 7). Default = auto.

  • quiet (scalar) – Optional keyword, set to 1 to suppress output. Default = 0.

Returns:

stl (struct) –

An instance of a stlResult structure containing:

stl.seasonal

Nx1 vector, seasonal component.

stl.trend

Nx1 vector, trend component.

stl.remainder

Nx1 vector, remainder (y - seasonal - trend).

Examples#

Monthly Data#

new;
library timeseries;

y = loadd(getGAUSSHome("pkgs/timeseries/examples/airline.dat"), "passengers");

stl = stlDecompose(y, 12);

print "Seasonal component (first year):";
print stl.seasonal[1:12];

print "Trend (first 5):";
print stl.trend[1:5];

Deseasonalize then Fit ARIMA#

new;
library timeseries;

y = loadd(getGAUSSHome("pkgs/timeseries/examples/airline.dat"), "passengers");

stl = stlDecompose(y, 12, quiet=1);

// Fit ARIMA on seasonally adjusted series
y_adj = y - stl.seasonal;
result = arimaFit(y_adj);

Weekly Data#

new;
library timeseries;

y = loadd(getGAUSSHome("pkgs/timeseries/examples/weekly.dat"), "sales");

stl = stlDecompose(y, 52, s_window=15);

Remarks#

STL decomposes a time series into three additive components:

\[y_t = S_t + T_t + R_t\]

where \(S_t\) is the seasonal component, \(T_t\) is the trend, and \(R_t\) is the remainder.

The seasonal smoothing window (s_window) controls how rapidly the seasonal pattern can change. Larger values produce a more stable seasonal pattern. Must be odd and >= 7. The default is typically period + 1 (rounded to odd).

Algorithm#

STL uses an inner loop of iterative LOESS (locally weighted regression) smoothing:

  1. Detrend: Subtract the current trend estimate from the series.

  2. Seasonal smoothing: Apply LOESS to each subseries (e.g., all Januaries) with window s_window.

  3. Low-pass filter: Remove high-frequency artifacts from the seasonal estimate.

  4. Deseasonalize: Subtract the seasonal estimate from the original series.

  5. Trend smoothing: Apply LOESS to the deseasonalized series.

  6. Iterate steps 1-5 (typically 2 inner iterations, 1 outer iteration for robustness weights).

See Cleveland et al. (1990) for the complete specification.

Troubleshooting#

Seasonal component is too smooth / not smooth enough: Increase s_window for smoother seasonal patterns; decrease for more adaptive patterns. The default is typically period + 1.

Remainder has visible seasonal pattern: s_window is too large — the seasonal smoother can’t track changes in the seasonal pattern. Reduce s_window or use a multiplicative decomposition (take logs first, decompose, exponentiate).

Verification#

STL decomposition verified against R stats::stl() with s.window=13 on the AirPassengers dataset. Seasonal, trend, and remainder components match at \(10^{-4}\) tolerance.

See crossval/03_arima_crossval.R.

References#

  • Cleveland, R.B., W.S. Cleveland, J.E. McRae, and I. Terpenning (1990). “STL: A seasonal-trend decomposition procedure based on Loess.” Journal of Official Statistics, 6(1), 3-73.

Library#

timeseries

Source#

stl.src

See also

Functions arimaFit()

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