6. Process or Product Monitoring and Control
6.4. Introduction to Time Series Analysis
6.4.3. What is Exponential Smoothing?

## Example of Triple Exponential Smoothing

Example comparing single, double, triple exponential smoothing This example shows comparison of single, double and triple exponential smoothing for a data set.

The following data set represents 24 observations. These are six years of quarterly data (each year has four quarters).

Table showing the data for the example
Quarter Period Sales     Quarter Period Sales

90 1 1 362   93 1 13 544
2 2 385     2 14 582
3 3 432     3 15 681
4 4 341     4 16 557
91 1 5 382   94 1 17 628
2 6 409     2 18 707
3 7 498     3 19 773
4 8 387     4 20 592
92 1 9 473   95 1 21 627
2 10 513     2 22 725
3 11 582     3 23 854
4 12 474     4 24 661
Plot of raw data with single, double, and triple exponential forecasts
Plot of raw data with triple exponential forecasts
Actual Time Series with forecasts

Comparison of MSEs
Comparison of MSEs
MSE $$\alpha$$
demand
$$\gamma$$
trend
$$\beta$$
seasonality

6906 0.4694
5054 0.1086 1.0000
936 1.0000   1.0000
520 0.7556 0.0000 0.9837

The updating coefficients were chosen by a computer program such that the MSE for each of the methods was minimized.

Example of the computation of the Initial Trend
Computation of initial trend The data set consists of quarterly sales data. The season is 1 year and since there are 4 quarters per year, $$L=4$$. Using the formula we obtain: $$\begin{eqnarray} b_1 & = & \frac{1}{4} \left[ \left( \frac{y5-y1}{4} \right) + \left( \frac{y6-y2}{4} \right) + \left( \frac{y7-y3}{4} \right) + \left( \frac{y8-y4}{4} \right) \right] \\ & & \\ & & \\ & = & \frac{1}{4} \left[ \left( \frac{382-362}{4} \right) + \left( \frac{409-385}{4} \right) + \left( \frac{498-432}{4} \right) + \left( \frac{387-341}{4} \right) \right] \\ & & \\ & & \\ & = & \frac{5 + 6 + 16.5 + 11.5}{4} = 9.75 \, . \end{eqnarray}$$
Example of the computation of the Initial Seasonal Indices
Table of initial seasonal indices
1 2 3 4 5 6

1 362 382 473 544 628 627
2 385 409 513 582 707 725
3 432 498 582 681 773 854
4 341 387 474 557 592 661

$$\bar{X}$$ 380 419 510.5 591 675 716.75

In this example we used the full 6 years of data. Other schemes may use only 3, or some other number of years. There are also a number of ways to compute initial estimates.