Formula

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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`

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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`

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