# 6.3 Classical decomposition

The classical decomposition method originated in the 1920s. It is a relatively simple procedure and forms the basis for most other methods of time series decomposition. There are two forms of classical decomposition: an additive decomposition and a multiplicative decomposition. These are described below for a time series with seasonal period $m$ (e.g., $m=4$ for quarterly data, $m=12$ for monthly data, $m=7$ for daily data with a weekly pattern).

In classical decomposition, we assume the seasonal component is constant from year to year. These $m$ values are sometimes called the “seasonal indices”.

Step 1
If $m$ is an even number, compute the trend-cycle component using a $2\times m$-MA to obtain $\hat{T}_t$. If $m$ is an odd number, compute the trend-cycle component using an $m$-MA to obtain $\hat{T}_t$.
Step 2
Calculate the detrended series: $y_t - \hat{T}_t$.
Step 3
To estimate the seasonal component for each month, simply average the detrended values for that month. For example, the seasonal index for March is the average of all the detrended March values in the data. These seasonal indexes are then adjusted to ensure that they add to zero. The seasonal component is obtained by stringing together all the seasonal indices for each year of data. This gives $\hat{S}_t$.
Step 4
The remainder component is calculated by subtracting the estimated seasonal and trend-cycle components: $\hat{E}_t = y_t - \hat{T}_t - \hat{S}_t$.

## Multiplicative decomposition

A classical multiplicative decomposition is very similar except the subtractions are replaced by divisions.

Step 1
If $m$ is an even number, compute the trend-cycle component using a $2\times m$-MA to obtain $\hat{T}_t$. If $m$ is an odd number, compute the trend-cycle component using an $m$-MA to obtain $\hat{T}_t$.
Step 2
Calculate the detrended series: $y_t/ \hat{T}_t$.
Step 3
To estimate the seasonal component for each month, simply average the detrended values for that month. For example, the seasonal index for March is the average of all the detrended March values in the data. These seasonal indexes are then adjusted to ensure that they add to $m$.  The seasonal component is obtained by stringing together all the seasonal indices for each year of data. This gives $\hat{S}_t$.
Step 4
The remainder component is calculated by dividing out the estimated seasonal and trend-cycle components: $\hat{E}_t = y_t /( \hat{T}_t \hat{S}_t)$.
R code
# x is the time series
fit <- decompose(x, type="multiplicative")
plot(fit)

• The estimate of the trend is unavailable for the first few and last few observations. For example, if $m=12$, there is no trend estimate for the first six and last six observations. Consequently, there is also no estimate of the remainder component for the same time periods.