Module decomposition 

Seasonal-trend decomposition of a univariate series by Loess (STL).

A series y with a known period is split additively into a trend, a seasonal component and a remainder, y = trend + seasonal + residual, by the procedure of Cleveland, Cleveland, McRae & Terpenning (1990). The period is an input — crate::seasonality::estimate_period estimates it when it is not known.

Algorithm

An inner loop refines the two components, starting from a zero trend:

1. Detrend: y − trend.
2. Cycle-subseries smoothing: the values at each position of the cycle (every period-th value) are smoothed by Loess with span seasonal, and each subseries is extended one cycle before and after by Loess extrapolation.
3. Low-pass filter of that extended series: moving averages of length period, period and 3, then Loess with span low_pass.
4. Seasonal = step 2 − step 3 (the low-pass removes what of the trend leaked into the subseries).
5. Trend = Loess with span trend of y − seasonal.

The outer loop (only when robust) gives each point a robustness weight from its remainder — the bisquare (1 − (r/h)²)² with h = 6 · median|r| — and runs the inner loop again with every Loess fit weighted by it, so that outliers go to the remainder instead of bending the trend and seasonal components.

Each Loess fit is a local polynomial of degree 0 or 1 with tricube weights over the span nearest points. This is the original Fortran implementation (netlib stl), which statsmodels’ STL also follows, with its defaults: degree 1 everywhere, seasonal = 7, trend the smallest odd integer at least 1.5 · period / (1 − 1.5 / seasonal), low_pass the smallest odd integer above period, and 5 inner / 0 outer iterations (2 / 15 when robust).

Strength of trend and seasonality

The result also reports F_T = max(0, 1 − Var(R)/Var(T + R)) and F_S = max(0, 1 − Var(R)/Var(S + R)) (Wang, Smith & Hyndman 2006): near 1 when the component explains most of what the remainder does not, near 0 when it is indistinguishable from noise.

Complexity

O(iterations · n · span) for spans seasonal, trend and low_pass.

References

• Cleveland, R. B., Cleveland, W. S., McRae, J. E. & Terpenning, I. (1990). “STL: A seasonal-trend decomposition procedure based on Loess.” Journal of Official Statistics, 6(1), 3–73.
• Wang, X., Smith, K. & Hyndman, R. J. (2006). “Characteristic-based clustering for time series data.” Data Mining and Knowledge Discovery, 13(3), 335–364.

Structs

Stl

STL decomposition with a builder for its options.

StlResult

An STL decomposition: data[i] = trend[i] + seasonal[i] + residual[i].

Enums

StlError

Why a decomposition was refused.

Constants

DEFAULT_SEASONAL

Default span of the seasonal smoother (statsmodels’ seasonal=7).