## Time Series Decomposition

Time series decomposition involves separating a time series into several distinct components. There are three components that are typically of interest:

*T*, a deterministic, nonseasonal secular trend component. This component is sometimes restricted to being a linear trend, though higher-degree polynomials are also used._{t}*S*, a deterministic seasonal component with known periodicity. This component captures level shifts that repeat systematically within the same period (e.g., month or quarter) between successive years. It is often considered to be a nuisance component, and seasonal adjustment is a process for eliminating it._{t}*I*, a stochastic irregular component. This component is not necessarily a white noise process. It can exhibit autocorrelation and cycles of unpredictable duration. For this reason, it is often thought to contain information about the business cycle, and is usually the most interesting component._{t}

There are three functional forms that are most often used for representing a time series *y _{t}* as a function of its trend, seasonal, and irregular components:

*Additive decomposition*, where$${y}_{t}={T}_{t}+{S}_{t}+{I}_{t}.$$

This is the classical decomposition. It is appropriate when there is no exponential growth in the series, and the amplitude of the seasonal component remains constant over time. For identifiability from the trend component, the seasonal and irregular components are assumed to fluctuate around zero.

*Multiplicative decomposition*, where$${y}_{t}={T}_{t}{S}_{t}{I}_{t}.$$

This decomposition is appropriate when there is exponential growth in the series, and the amplitude of the seasonal component grows with the level of the series. For identifiability from the trend component, the seasonal and irregular components are assumed to fluctuate around one.

*Log-additive decomposition*, where$$\mathrm{log}{y}_{t}={T}_{t}+{S}_{t}+{I}_{t}.$$

This is an alternative to the multiplicative decomposition. If the original series has a multiplicative decomposition, then the logged series has an additive decomposition. Using the logs can be preferable when the time series contains many small observations. For identifiability from the trend component, the seasonal and irregular components are assumed to fluctuate around zero.

You can estimate the trend and seasonal components by using filters (moving averages) or parametric regression models. Given estimates $${\widehat{T}}_{t}$$ and $${\widehat{S}}_{t}$$, the irregular component is estimated as

$${\widehat{I}}_{t}={y}_{t}-{\widehat{T}}_{t}-{\widehat{S}}_{t}$$

using the additive decomposition, and

$${\widehat{I}}_{t}=\frac{{y}_{t}}{\left({\widehat{T}}_{t}{\widehat{S}}_{t}\right)}$$

using the multiplicative decomposition.

The series

$${y}_{t}-{\widehat{T}}_{t}$$

(or $${y}_{t}/{\widehat{T}}_{t}$$ using the multiplicative decomposition) is called a *detrended* series.

Similarly, the series $${y}_{t}-{\widehat{S}}_{t}$$ (or $${y}_{t}/{\widehat{S}}_{t}$$) is called a *deseasonalized* series.

## Related Examples

- Moving Average Trend Estimation
- Seasonal Adjustment Using a Stable Seasonal Filter
- Seasonal Adjustment Using S(n,m) Seasonal Filters
- Parametric Trend Estimation