Question: The following table shows the past two years of quarterly sales information. Assume that there are both trend and seasonal factors and that the seasonal cycle is one year. Use time-series decomposition to forecast quarterly sales for the next year.

Quarter	Sales
1	160
2	195
3	150
4	140

Quarter	Sales
5	215
6	240
7	205
8	190

The decomposition of your time series could be a statistical task that deconstructs a statistic into several components, each representing one in all the underlying categories of patterns. The theory of your time series analysis makes use of the thought of decomposing a times series into predictable and unpredictable components.

The table is given below:

Column 3 is the seasonal average. For example, the first quarter average is:

$\frac{160+215}{2}=187.5$

Column 4 is the quarter average (column 3) divided by the overall average (186.875)

Column 5 is the actual data divided by the seasonal index. Columns 6 and 7 show $t^2$and $t{y}_d$.

Now we calculate regression results for deseasonalized data

$$\begin{gathered} \overline{\text{y}}=186.875 \\ \overline{\text{t}}=4.5 \\ \text{b}=\frac{{\sum {\text{ty}}_{\text{t}}-\text{n}\overline{\text{ty}}}_{\text{t}}}{\sum {\text{t}}^2-\text{n}{\overline{\text{t}}}^2}=9.91 \\ \text{a}={\overline{\text{y}}}_{\text{t}}-\text{b}\overline{\text{t}}=142.30 \end{gathered}$$

Therefore, the deseasonalized regression results are

$Y=142.30+9.91t$