Formula
Examples
1) 3 year Single Exponential Smoothing forecast | year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Sales | 30 | 25 | 35 | 25 | 20 | 30 | 35 | 40 | 30 | 45 |
Calculate 3 year Single Exponential Smoothing forecast
Solution:
(1)
year | (2)
Sales | (3)
Exponential Smoothing
`(alpha=0.3)` |
| 1 | 30 | 30 |
| 2 | 25 | `0.3*30+0.7*30=30` |
| 3 | 35 | `0.3*25+0.7*30=28.5` |
| 4 | 25 | `0.3*35+0.7*28.5=30.45` |
| 5 | 20 | `0.3*25+0.7*30.45=28.815` |
| 6 | 30 | `0.3*20+0.7*28.815=26.1705` |
| 7 | 35 | `0.3*30+0.7*26.1705=27.3193` |
| 8 | 40 | `0.3*35+0.7*27.3193=29.6235` |
| 9 | 30 | `0.3*40+0.7*29.6235=32.7365` |
| 10 | 45 | `0.3*30+0.7*32.7365=31.9155` |
| 11 | | `0.3*45+0.7*31.9155=35.8409` |
(1)
year | (2)
Sales | (3)
Exponential Smoothing | (4)
Error | (5)
|Error| | (6)
`"Error"^2` | (7)
`|%"Error"|` |
| 1 | 30 | 30 | | | | |
| 2 | 25 | 30 | | | | |
| 3 | 35 | 28.5 | | | | |
| 4 | 25 | 30.45 | `25-30.45=-5.45` | `5.45` | `29.7025` | `21.8%` |
| 5 | 20 | 28.815 | `20-28.815=-8.815` | `8.815` | `77.7042` | `44.07%` |
| 6 | 30 | 26.1705 | `30-26.1705=3.8295` | `3.8295` | `14.6651` | `12.77%` |
| 7 | 35 | 27.3193 | `35-27.3193=7.6807` | `7.6807` | `58.9924` | `21.94%` |
| 8 | 40 | 29.6235 | `40-29.6235=10.3765` | `10.3765` | `107.6708` | `25.94%` |
| 9 | 30 | 32.7365 | `30-32.7365=-2.7365` | `2.7365` | `7.4883` | `9.12%` |
| 10 | 45 | 31.9155 | `45-31.9155=13.0845` | `13.0845` | `171.2032` | `29.08%` |
| 11 | | 35.8409 | Total | `51.9725` | `467.4265` | `164.72%` |
Forecasting errors
1. Mean absolute error (MAE), also called mean absolute deviation (MAD)
MAE`=1/n sum |e_i|=51.9725/7=7.4246`
2. Mean squared error (MSE)
MSE`=1/n sum |e_i^2|=467.4265/7=66.7752`
3. Root mean squared error (RMSE)
RMSE`=sqrt(MSE)=sqrt(66.7752)=8.1716`
4. Mean absolute percentage error (MAPE)
MAPE`=1/n sum |e_i/y_i|=164.72/7=23.53`
2) 3 year Single Exponential Smoothing forecast | year | 1 | 2 | 3 | 4 | 5 | 6 |
| Sales | 650 | 700 | 810 | 800 | 900 | 700 |
Calculate 3 year Single Exponential Smoothing forecast
Solution:
(1)
year | (2)
Sales | (3)
Exponential Smoothing
`(alpha=0.1)` |
| 1 | 650 | 650 |
| 2 | 700 | `0.1*650+0.9*650=650` |
| 3 | 810 | `0.1*700+0.9*650=655` |
| 4 | 800 | `0.1*810+0.9*655=670.5` |
| 5 | 900 | `0.1*800+0.9*670.5=683.45` |
| 6 | 700 | `0.1*900+0.9*683.45=705.105` |
| 7 | | `0.1*700+0.9*705.105=704.5945` |
(1)
year | (2)
Sales | (3)
Exponential Smoothing | (4)
Error | (5)
|Error| | (6)
`"Error"^2` | (7)
`|%"Error"|` |
| 1 | 650 | 650 | | | | |
| 2 | 700 | 650 | | | | |
| 3 | 810 | 655 | | | | |
| 4 | 800 | 670.5 | `800-670.5=129.5` | `129.5` | `16770.25` | `16.19%` |
| 5 | 900 | 683.45 | `900-683.45=216.55` | `216.55` | `46893.9025` | `24.06%` |
| 6 | 700 | 705.105 | `700-705.105=-5.105` | `5.105` | `26.061` | `0.73%` |
| 7 | | 704.5945 | Total | `351.155` | `63690.2135` | `40.98%` |
Forecasting errors
1. Mean absolute error (MAE), also called mean absolute deviation (MAD)
MAE`=1/n sum |e_i|=351.155/3=117.0517`
2. Mean squared error (MSE)
MSE`=1/n sum |e_i^2|=63690.2135/3=21230.0712`
3. Root mean squared error (RMSE)
RMSE`=sqrt(MSE)=sqrt(21230.0712)=145.7054`
4. Mean absolute percentage error (MAPE)
MAPE`=1/n sum |e_i/y_i|=40.98/3=13.66`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
