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| Meta Title | 6.4.3.2. Forecasting with Single Exponential Smoothing |
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| Boilerpipe Text | 6.
Process or Product Monitoring and Control
6.4.
Introduction to Time Series Analysis
6.4.3.
What is Exponential Smoothing?
Forecasting with Single Exponential Smoothing
Forecasting Formula
Forecasting the next point
The forecasting formula is the basic equation
S
t
+
1
=
α
y
t
+
(
1
−
α
)
S
t
,
0
<
α
≤
1
,
t
>
0
.
New forecast is previous forecast plus an error adjustment
This can be written as:
S
t
+
1
=
S
t
+
α
ϵ
t
,
where
ϵ
t
is the forecast error (actual - forecast) for
period
t
.
In other words, the new forecast is the old one plus an adjustment
for the error that occurred in the last forecast.
Bootstrapping of Forecasts
Bootstrapping forecasts
What happens if you wish to forecast from some origin, usually the
last data point, and no actual observations are available? In this
situation we have to modify the formula to become:
S
t
+
1
=
α
y
o
r
g
i
n
+
(
1
−
α
)
S
t
,
where
y
o
r
i
g
i
n
remains constant. This technique is known as
bootstrapping.
Example of Bootstrapping
Example
The last data point in the previous example was 70 and its forecast
(smoothed value
S
)
was 71.7. Since we do have the data point
and
the forecast available, we can calculate the next forecast
using the regular formula with
α
=
0.1
as
S
t
+
1
=
α
y
o
r
g
i
n
+
(
1
−
α
)
S
t
=
0.1
(
70
)
+
0.9
(
71.7
)
=
71.5
.
But for the next forecast we have no data point (observation).
So now we compute:
S
t
+
2
=
0.1
(
70
)
+
0.9
(
71.5
)
=
71.35
.
Comparison between bootstrap and regular forecasting
Table comparing two methods
The following table displays the comparison between the two methods:
Period
Bootstrap
forecast
Data
Single Smoothing
Forecast
13
71.50
75
71.5
14
71.35
75
71.9
15
71.21
74
72.2
16
71.09
78
72.4
17
70.98
86
73.0
Single Exponential Smoothing with Trend
Single Smoothing (short for single exponential smoothing) is not very
good when there is a trend. The single coefficient
α
is not enough.
Sample data set with trend
Let us demonstrate this with the following data set smoothed with an
α
of 0.3:
Data
Fit
6.4
5.6
6.4
7.8
6.2
8.8
6.7
11.0
7.3
11.6
8.4
16.7
9.4
15.3
11.6
21.6
12.7
22.4
15.4
Plot demonstrating inadequacy of single exponential smoothing
when there is trend
The resulting graph looks like: |
| Markdown | 
| | | | |
|---|---|---|---|
| | | | |
| 6\.4.3.2. | Forecasting with Single Exponential Smoothing | | |
| | **Forecasting Formula** | | |
| *Forecasting the next point* | The forecasting formula is the basic equation S t \+ 1 \= α y t \+ ( 1 − α ) S t , 0 \< α ≤ 1 , t \> 0 . | | |
| *New forecast is previous forecast plus an error adjustment* | This can be written as: S t \+ 1 \= S t \+ α ϵ t , where ϵ t is the forecast error (actual - forecast) for period t. In other words, the new forecast is the old one plus an adjustment for the error that occurred in the last forecast. | | |
| | **Bootstrapping of Forecasts** | | |
| *Bootstrapping forecasts* | What happens if you wish to forecast from some origin, usually the last data point, and no actual observations are available? In this situation we have to modify the formula to become: S t \+ 1 \= α y o r g i n \+ ( 1 − α ) S t , where y o r i g i n remains constant. This technique is known as *bootstrapping.* | | |
| | **Example of Bootstrapping** | | |
| *Example* | The last data point in the previous example was 70 and its forecast (smoothed value S) was 71.7. Since we do have the data point **and** the forecast available, we can calculate the next forecast using the regular formula with α \= 0\.1 as S t \+ 1 \= α y o r g i n \+ ( 1 − α ) S t \= 0\.1 ( 70 ) \+ 0\.9 ( 71\.7 ) \= 71\.5 . But for the next forecast we have no data point (observation). So now we compute: S t \+ 2 \= 0\.1 ( 70 ) \+ 0\.9 ( 71\.5 ) \= 71\.35 . | | |
| | **Comparison between bootstrap and regular forecasting** | | |
| *Table comparing two methods* | | | |
| Period | Bootstrap forecast | Data | Single Smoothing Forecast |
| 13 | 71\.50 | 75 | 71\.5 |
| 14 | 71\.35 | 75 | 71\.9 |
| 15 | 71\.21 | 74 | 72\.2 |
| 16 | 71\.09 | 78 | 72\.4 |
| 17 | 70\.98 | 86 | 73\.0 |
| | **Single Exponential Smoothing with Trend** | | |
| | Single Smoothing (short for single exponential smoothing) is not very good when there is a trend. The single coefficient α is not enough. | | |
| *Sample data set with trend* | | | |
| Data | Fit | | |
| 6\.4 | | | |
| 5\.6 | 6\.4 | | |
| 7\.8 | 6\.2 | | |
| 8\.8 | 6\.7 | | |
| 11\.0 | 7\.3 | | |
| 11\.6 | 8\.4 | | |
| 16\.7 | 9\.4 | | |
| 15\.3 | 11\.6 | | |
| 21\.6 | 12\.7 | | |
| 22\.4 | 15\.4 | | |
| *Plot demonstrating inadequacy of single exponential smoothing when there is trend* | The resulting graph looks like:  | | |
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