STL

215. STL

215.1. STL Decomposition (Seasonal-Trend using Loess)

Modern alternative to classical decomposition. Robust, handles changing seasonal patterns, and produces complete trend/seasonal estimates with no endpoint gaps.

Developed by Cleveland, Cleveland, McRae, and Terpenning (1990). Standard tool in statsmodels, R::stats, forecast::stl.

215.1.1. What it does

Decompose a series additively:

𝑦_{𝑡} = 𝑇_{𝑡} + 𝑆_{𝑡} + 𝑅_{𝑡}

where $𝑇$ is trend, $𝑆$ is seasonal, $𝑅$ is the remainder (noise). Multiplicative decomposition is handled by first taking $log 𝑦_{𝑡}$ , decomposing additively, then exponentiating.

Same components as classical decomposition, but the estimation method is different:

Loess (Locally Estimated Scatterplot Smoothing): a non-parametric regression that fits local low-order polynomials to nearby points. Produces smooth trend and seasonal curves.
Iterative: STL alternates between estimating trend and seasonal components, refining each pass. After few iterations the components stabilize.
Robustness option: a robust-loess variant downweights outliers, making STL insensitive to anomalies.

215.1.2. Key parameters

Parameter	Typical value	Effect
`np` (period)	seasonal period $𝑚$ (e.g., 12 for monthly, 7 for daily-with-weekly-seasonality)	Defines the seasonal cycle
`ns` (seasonal window)	7-15 (must be odd ≥ 7)	Smaller → seasonal pattern can change rapidly. Larger → seasonal pattern is locked-in over time. Set based on how stable seasonality is.
`nt` (trend window)	auto: $1.5 𝑚 / (1 - 1.5 / 𝑛 𝑠)$	Larger → smoother trend. Smaller → trend tracks short-term moves.
`nl` (low-pass window)	next odd integer $\geq 𝑚$	Internal; usually leave default.
`robust`	True / False	If True, downweights outliers via a second pass. Use when there are known anomalies.

215.1.3. Why STL beats classical decomposition

Issue	Classical	STL
Seasonal pattern	Constant across cycles	Can evolve gradually (controlled by `ns`)
Endpoints	Gaps of $𝑚 / 2$ at start and end	Smooth all the way through
Outliers	Sensitive — one anomaly distorts seasonal index	Robust mode downweights anomalies
Multiplicative decomp	Separate procedure	Take logs, decompose additively, exponentiate

215.1.4. Workflow

Plot the series, look at seasonality and trend.
Choose np (= seasonal period).
Try default STL — usually good. Inspect residuals: should be structureless.
If seasonal pattern is clearly shifting, lower ns. If residuals show seasonal structure (under-fit), lower ns.
If outliers, enable robust mode.
Use the components for: forecasting (forecast each component, recombine), anomaly detection (flag large residuals), seasonal-adjustment (subtract seasonal for trend-only view).

215.1.5. When STL doesn’t work

Multiple seasonalities: STL assumes one seasonal period. For daily data with both weekly and yearly seasonality, use MSTL (Multiple STL) or Fourier-based methods.
Non-additive structure: if seasonality interacts with trend in a way that’s not pure multiplicative, STL on $log 𝑦$ won’t capture it. Use a structural state-space model.
Short series: STL needs at least two full seasonal cycles; ideally more.

Example

Given: 5 years of monthly retail sales with growing seasonal swing.

Setup:

$𝑚 = 12$ (monthly data with yearly seasonality)
$𝑛 𝑠 = 13$ (seasonal pattern can evolve slowly)
Robust mode on (Black-Friday-style anomalies expected)

Output: three series, each of length 60.

Trend $𝑇_{𝑡}$ : smooth upward curve, captures the multi-year growth.
Seasonal $𝑆_{𝑡}$ : 12-month repeating pattern, but the December peak grows from year 1 (+10%) to year 5 (+18%) — that’s the evolving seasonality STL captures.
Remainder $𝑅_{𝑡}$ : small, mean-zero, ideally autocorrelation-free.

Use the output:

Forecast: extend trend (e.g., linear extrapolation or random-walk) + use the most-recent seasonal pattern.
Seasonally-adjusted series: $𝑦_{𝑡} - 𝑆_{𝑡}$ . Useful for tracking underlying performance ignoring seasonal effects (Black Friday, summer, etc.).
Anomaly detection: large residuals $| 𝑅_{𝑡} |$ flag anomalous months.

Code (Python):

from statsmodels.tsa.seasonal import STL
result = STL(y, period=12, seasonal=13, robust=True).fit()
trend = result.trend
seasonal = result.seasonal
remainder = result.resid

Plot all four (result.plot()) for visual diagnosis.