Time Series Decomposition

Time Series Decomposition separates a time series into its structural components — trend, seasonal pattern, and residual — and is used in two ways: as a diagnostic before method selection and as a forecasting method in its own right (forecast each component separately, then recombine).

For workforce management, decomposition is the most underused tool in the forecasting toolkit. Plotting the decomposed components of a contact volume series tells you immediately whether the series has a trend (and what shape), whether the seasonality is stable or evolving, and whether the residual contains forecastable structure or random noise. Most method-selection mistakes come from skipping this step.

The components

A time series $y_t$ is decomposed into:

• Trend-cycle $T_t$ — long-run direction plus non-fixed-period swings
• Seasonal $S_t$ — repeating pattern at a fixed period $m$
• Residual $R_t$ — what remains after extracting trend and seasonal

Two combination forms:

Additive decomposition

$y_t=T_t+S_t+R_t$

Use when the seasonal magnitude is roughly constant regardless of the level of the series.

Multiplicative decomposition

$y_t=T_t\times S_t\times R_t$

Use when the seasonal swing scales with the level — the typical pattern in contact volume, where peak-hour volume scales with the day's total.

A diagnostic: plot the series and look at whether the seasonal swings get bigger as the level rises. If yes, multiplicative; if not, additive.

A practitioner shortcut: log-transforming a multiplicative series produces an additive series, since $\log (T_t\times S_t\times R_t)=\log T_t+\log S_t+\log R_t$. Decompose the log, then exponentiate.

Methods of decomposition

Classical decomposition

The original method, dating to the 1920s. Compute trend by moving average; remove trend to get the de-trended series; compute the seasonal pattern by averaging across periods; the residual is what remains.

Trend extraction — for seasonal period $m$:

${\hat{T}}_t=\frac{1}{m}\sum _{j=-k}^k{y}_{t+j}$

with $k=m/2$ (centered moving average for even $m$ requires a slightly different formula).

Seasonal extraction — average the de-trended values within each seasonal slot.

Limitations: classical decomposition is rigid. The trend is smoothed by a fixed-window moving average; it cannot adapt to series with abrupt structural changes. The seasonal pattern is held constant across the whole series; it cannot evolve. The first and last $k$ observations have no trend estimate (the moving average needs both sides).

For most modern WFM use, classical decomposition is a teaching example, not a production tool. STL is the modern replacement.

STL decomposition

Seasonal and Trend decomposition using Loess (Cleveland, Cleveland, McRae, & Terpenning, 1990). Robust, flexible, the modern default.

STL extracts trend and seasonal using locally-weighted regression (Loess) iteratively:

1. Estimate the seasonal component by Loess on each subseries (e.g., all Mondays for a daily series with weekly seasonality)
2. Subtract seasonal; estimate trend on the de-seasonalized series via Loess
3. Subtract trend; re-estimate seasonal
4. Iterate until convergence

Advantages over classical:

• The seasonal component is allowed to evolve over time (controlled by a smoothness parameter)
• The trend is estimated by Loess, which adapts to changes in the rate of change
• Robust to outliers via downweighting in the iteration
• Provides estimates for every observation including the endpoints

STL is the right diagnostic tool for any WFM volume series. It is also a competitive forecasting method when paired with simple forecasting of each component — STL+ETS and STL+ARIMA hybrids are well-established.

X-11, X-12-ARIMA, X-13ARIMA-SEATS

Family of methods developed at the U.S. Census Bureau for adjusting economic time series. More sophisticated than classical decomposition; handles trading-day adjustments, calendar effects, holiday effects directly.

When to use: macroeconomic time series adjustment; not commonly needed for WFM. Mentioned for completeness because some WFM software platforms cite X-12 / X-13 in their methods lists.

Decomposition as diagnostic

The most valuable use of decomposition for WFM practitioners is before choosing a forecasting method:

1. Decompose the historical series (STL is the recommended method)
2. Inspect the trend component:
   • Is it flat? → method needs no trend term (Simple ES; ARIMA without I)
   • Is it linear and consistent? → Holt's linear or ARIMA with d=1
   • Is it changing direction? → consider damped Holt; ARIMA with d=1; or look for structural break
3. Inspect the seasonal component:
   • Is it stable? → fixed seasonal in any seasonal method (Holt-Winters; SARIMA)
   • Is it evolving? → STL-based method or ETS with high $\gamma$
   • Is there more than one period? → MSTL or specialized multi-seasonal methods (TBATS, dynamic harmonic regression)
4. Inspect the residual:
   • Is it white noise? → the trend + seasonal model captures the structure; the simplest method that fits this shape will work
   • Is there autocorrelation? → ARIMA on the residuals; consider ARIMAX with explanatory variables
   • Are there outliers / spikes? → identify, attribute to events, consider dummy variables

This diagnostic step takes minutes and prevents most "we tried Holt-Winters and it didn't work" misadventures. The decomposition tells you what the series is; method selection follows from that.

Multi-seasonal decomposition

Many WFM series have multiple seasonal periods simultaneously. Half-hour interval call volume has:

• Intraday seasonality (48 intervals per day)
• Weekly seasonality (7 × 48 = 336 intervals per week)
• Possibly annual seasonality (52 weeks)

Standard STL handles one seasonal period. For multi-seasonal data:

• MSTL (Multiple STL) — fits each seasonal pattern iteratively. Available in modern forecasting libraries.
• Dynamic Harmonic Regression — uses Fourier terms for each seasonal frequency, fit via regression (often with ARIMA errors). Effective when the number of seasonal periods is large or non-integer.
• TBATS — exponential smoothing extension supporting multiple seasonal periods explicitly.

For half-hour interval WFM data, MSTL or TBATS is typically the right diagnostic; for weekly or daily data, standard STL is enough.

Decomposition as forecasting method

To forecast using decomposition:

1. Decompose historical data into $T_t$, $S_t$, $R_t$
2. Forecast the seasonal component (typically by repeating the most recent seasonal cycle — equivalent to seasonal naive on the seasonal component)
3. Forecast the trend (e.g., random walk with drift, or Holt's linear method on just the trend)
4. Forecast the residual (typically zero for additive, one for multiplicative)
5. Combine: ${\hat{y}}_{T+h}={\hat{T}}_{T+h}+{\hat{S}}_{T+h}+{\hat{R}}_{T+h}$ (additive) or product (multiplicative)

In practice, STL+ETS is the canonical decomposition-based forecasting method: STL extracts the components, ETS forecasts the de-seasonalized series, the seasonal pattern is added back. The result is competitive with native ETS on series where the seasonal pattern is evolving in ways the native ETS cannot represent.

Common WFM pitfalls

• Skipping the diagnostic step. Reaching for ETS or ARIMA without first looking at the decomposition is the most common cause of bad WFM forecasts.
• Using additive when multiplicative is appropriate. Contact volume is typically multiplicative; using additive decomposition on multiplicative data produces a seasonal pattern that's a function of the level rather than a clean seasonal pattern.
• Single-period decomposition on multi-period data. Don't decompose half-hour data with a single weekly period and expect a clean trend; the daily seasonality contaminates the result.
• Treating the residual as noise without checking. If the residual has structure, your method choice is wrong. Run an ACF on the residual to check.

Connection to method selection

Decomposition feeds the Forecasting Methods decision tree directly. The questions at each branch — does the series have trend? Seasonality? Multiple seasonal periods? Significant residual structure? — are answered by inspecting the decomposed components.

A workflow:

1. Decompose with STL (and MSTL if multi-seasonal data)
2. Inspect each component visually
3. Read off the answers to the decision tree questions
4. Select method
5. Validate against seasonal naive baseline using Forecast Accuracy Metrics

References

• Hyndman, R. J., & Athanasopoulos, G. "Time series decomposition." Forecasting: Principles and Practice (Python edition). otexts.com/fpppy.
• Cleveland, R. B., Cleveland, W. S., McRae, J. E., & Terpenning, I. "STL: A seasonal-trend decomposition procedure based on loess." Journal of Official Statistics 6(1), 1990.

See Also

• Forecasting Methods — overview; decomposition feeds the decision tree
• Naive and Seasonal Naive Forecasting — the baseline that decomposition-based methods must still beat
• Exponential Smoothing — STL+ETS hybrid is a competitive method
• ARIMA Models — STL+ARIMA hybrid is another
• Regression for Forecasting — dynamic harmonic regression with Fourier seasonal terms
• Forecast Accuracy Metrics — how to evaluate decomposition-based forecasts