Decomposition of time series

Decomposition based on rates of change

This is an important technique for all types of time series analysis, especially for seasonal adjustment. It seeks to construct, from an observed time series, a number of component series (that could be used to reconstruct the original by additions or multiplications) where each of these has a certain characteristic or type of behavior. For example, time series are usually decomposed into:

• ${\displaystyle T_t}$, the trend component at time t, which reflects the long-term progression of the series (secular variation). A trend exists when there is a persistent increasing or decreasing direction in the data. The trend component does not have to be linear.
• ${\displaystyle C_t}$, the cyclical component at time t, which reflects repeated but non-periodic fluctuations. The duration of these fluctuations depend on the nature of the time series.
• ${\displaystyle S_t}$, the seasonal component at time t, reflecting seasonality (seasonal variation). A seasonal pattern exists when a time series is influenced by seasonal factors. Seasonality occurs over a fixed and known period (e.g., the quarter of the year, the month, or day of the week).
• ${\displaystyle I_t}$, the irregular component (or "noise") at time t, which describes random, irregular influences. It represents the residuals or remainder of the time series after the other components have been removed.

Hence a time series using an additive model can be thought of as

${\displaystyle y_t=T_t+C_t+S_t+I_t,}$

whereas a multiplicative model would be

${\displaystyle y_t=T_t\times C_t\times S_t\times I_t.}$

An additive model would be used when the variations around the trend do not vary with the level of the time series whereas a multiplicative model would be appropriate if the trend is proportional to the level of the time series.

Sometimes the trend and cyclical components are grouped into one, called the trend-cycle component. The trend-cycle component can just be referred to as the "trend" component, even though it may contain cyclical behavior. For example, a seasonal decomposition of time series by Loess (STL) plot decomposes a time series into seasonal, trend and irregular components using loess and plots the components separately, whereby the cyclical component (if present in the data) is included in the "trend" component plot.