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| Meta Title | 6.4.3.5. Triple 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?
Triple Exponential Smoothing
What happens if the data show trend
and
seasonality?
To handle seasonality, we have to add a third parameter
In this case double smoothing will not work. We now introduce a
third equation to take care of seasonality (sometimes called
periodicity). The resulting set of equations is called the
"Holt-Winters" (HW) method after the names of the inventors.
The basic equations for their method are given by:
S
t
=
α
y
t
I
t
−
L
+
(
1
−
α
)
(
S
t
−
1
+
b
t
−
1
)
OVERALL SMOOTHING
b
t
=
γ
(
S
t
−
S
t
−
1
)
+
(
1
−
γ
)
b
t
−
1
TREND SMOOTHING
I
t
=
β
y
t
S
t
+
(
1
−
β
)
I
t
−
L
SEASONAL SMOOTHING
F
t
+
m
=
(
S
t
+
m
b
t
)
I
t
−
L
+
m
FORECAST
,
where
y
is the observation
S
is the smoothed observation
b
is the trend factor
I
is the seasonal index
F
is the forecast at
m
periods ahead
t
is an index denoting a time period
and
α
,
β
, and
γ
are constants that must be estimated in such a way that the MSE of the
error is minimized. This is best left to a good software package.
Complete season needed
To initialize the HW method we need
at least
one complete
season's data to determine initial estimates of the seasonal indices
I
t
−
L
.
L
periods in a season
A complete season's data consists of
L
periods. And we need to
estimate the trend factor from one period to the next. To
accomplish this, it is advisable to use two complete seasons; that
is,
2
L
periods.
Initial values for the trend factor
How to get initial estimates for trend and seasonality
parameters
The general formula to estimate the initial trend is given by
b
=
1
L
(
y
L
+
1
−
y
1
L
+
y
L
+
2
−
y
2
L
+
⋯
+
y
L
+
L
−
y
L
L
)
.
Initial values for the Seasonal Indices
As we will see in the example, we work with data that consist of 6
years with 4 periods (that is, 4 quarters) per year.
Step 1: compute yearly averages
Step 1:
Compute the averages of each of the 6 years.
A
p
=
∑
i
=
1
4
y
i
4
,
p
=
1
,
2
,
…
,
6
.
Step 2: divide by yearly averages
Step 2:
Divide the observations by the appropriate yearly mean.
1
2
3
4
5
6
y
1
/
A
1
y
5
/
A
2
y
9
/
A
3
y
13
/
A
4
y
17
/
A
5
y
21
/
A
6
y
2
/
A
1
y
6
/
A
2
y
10
/
A
3
y
14
/
A
4
y
18
/
A
5
y
22
/
A
6
y
3
/
A
1
y
7
/
A
2
y
11
/
A
3
y
15
/
A
4
y
19
/
A
5
y
23
/
A
6
y
4
/
A
1
y
8
/
A
2
y
12
/
A
3
y
16
/
A
4
y
20
/
A
5
y
24
/
A
6
Step 3: form seasonal indices
Step 3:
Now the seasonal indices are formed by computing the average
of each row. Thus the initial seasonal indices (symbolically) are:
I
1
=
(
y
1
/
A
1
+
y
5
/
A
2
+
y
9
/
A
3
+
y
13
/
A
4
+
y
17
/
A
5
+
y
21
/
A
6
)
/
6
I
2
=
(
y
2
/
A
1
+
y
6
/
A
2
+
y
10
/
A
3
+
y
14
/
A
4
+
y
18
/
A
5
+
y
22
/
A
6
)
/
6
I
3
=
(
y
3
/
A
1
+
y
6
/
A
2
+
y
11
/
A
3
+
y
15
/
A
4
+
y
19
/
A
5
+
y
23
/
A
6
)
/
6
I
4
=
(
y
4
/
A
1
+
y
6
/
A
2
+
y
12
/
A
3
+
y
16
/
A
4
+
y
20
/
A
5
+
y
24
/
A
6
)
/
6
.
We now know the algebra behind the computation of the initial
estimates.
The next page contains an
example
of triple
exponential smoothing.
The case of the Zero Coefficients
Zero coefficients for trend and seasonality parameters
Sometimes it happens that a computer program for triple exponential
smoothing outputs a final coefficient for trend (
γ
)
or for seasonality (
β
)
of zero. Or worse, both are outputted as zero!
Does this indicate that there is no trend and/or no seasonality?
Of course not! It only means that the initial values for trend and/or
seasonality were right on the money. No updating was necessary in order
to arrive at the lowest possible MSE. We should inspect the updating
formulas to verify this. |
| Markdown | 
| | | | | | |
|---|---|---|---|---|---|
| | | | | | |
| 6\.4.3.5. | Triple Exponential Smoothing | | | | |
| | What happens if the data show trend **and** seasonality? | | | | |
| *To handle seasonality, we have to add a third parameter* | In this case double smoothing will not work. We now introduce a third equation to take care of seasonality (sometimes called periodicity). The resulting set of equations is called the "Holt-Winters" (HW) method after the names of the inventors. The basic equations for their method are given by: S t \= α y t I t − L \+ ( 1 − α ) ( S t − 1 \+ b t − 1 ) OVERALL SMOOTHING b t \= γ ( S t − S t − 1 ) \+ ( 1 − γ ) b t − 1 TREND SMOOTHING I t \= β y t S t \+ ( 1 − β ) I t − L SEASONAL SMOOTHING F t \+ m \= ( S t \+ m b t ) I t − L \+ m FORECAST , wherey is the observation S is the smoothed observation b is the trend factor I is the seasonal index F is the forecast at *m* periods ahead t is an index denoting a time period and α, β, and γ are constants that must be estimated in such a way that the MSE of the error is minimized. This is best left to a good software package. | | | | |
| *Complete season needed* | To initialize the HW method we need at least one complete season's data to determine initial estimates of the seasonal indices I t − L. | | | | |
| *L periods in a season* | A complete season's data consists of L periods. And we need to estimate the trend factor from one period to the next. To accomplish this, it is advisable to use two complete seasons; that is, 2 L periods. | | | | |
| | **Initial values for the trend factor** | | | | |
| *How to get initial estimates for trend and seasonality parameters* | The general formula to estimate the initial trend is given by b \= 1 L ( y L \+ 1 − y 1 L \+ y L \+ 2 − y 2 L \+ ⋯ \+ y L \+ L − y L L ) . | | | | |
| | **Initial values for the Seasonal Indices** | | | | |
| | As we will see in the example, we work with data that consist of 6 years with 4 periods (that is, 4 quarters) per year. | | | | |
| *Step 1: compute yearly averages* | **Step 1:** Compute the averages of each of the 6 years. A p \= ∑ i \= 1 4 y i 4 , p \= 1 , 2 , … , 6 . | | | | |
| *Step 2: divide by yearly averages* | | | | | |
| 1 | 2 | 3 | 4 | 5 | 6 |
| y 1 / A 1 | y 5 / A 2 | y 9 / A 3 | y 13 / A 4 | y 17 / A 5 | y 21 / A 6 |
| y 2 / A 1 | y 6 / A 2 | y 10 / A 3 | y 14 / A 4 | y 18 / A 5 | y 22 / A 6 |
| y 3 / A 1 | y 7 / A 2 | y 11 / A 3 | y 15 / A 4 | y 19 / A 5 | y 23 / A 6 |
| y 4 / A 1 | y 8 / A 2 | y 12 / A 3 | y 16 / A 4 | y 20 / A 5 | y 24 / A 6 |
| *Step 3: form seasonal indices* | **Step 3:** Now the seasonal indices are formed by computing the average of each row. Thus the initial seasonal indices (symbolically) are: I 1 \= ( y 1 / A 1 \+ y 5 / A 2 \+ y 9 / A 3 \+ y 13 / A 4 \+ y 17 / A 5 \+ y 21 / A 6 ) / 6 I 2 \= ( y 2 / A 1 \+ y 6 / A 2 \+ y 10 / A 3 \+ y 14 / A 4 \+ y 18 / A 5 \+ y 22 / A 6 ) / 6 I 3 \= ( y 3 / A 1 \+ y 6 / A 2 \+ y 11 / A 3 \+ y 15 / A 4 \+ y 19 / A 5 \+ y 23 / A 6 ) / 6 I 4 \= ( y 4 / A 1 \+ y 6 / A 2 \+ y 12 / A 3 \+ y 16 / A 4 \+ y 20 / A 5 \+ y 24 / A 6 ) / 6 . We now know the algebra behind the computation of the initial estimates. The next page contains an [example](https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc436.htm) of triple exponential smoothing. | | | | |
| | **The case of the Zero Coefficients** | | | | |
| *Zero coefficients for trend and seasonality parameters* | Sometimes it happens that a computer program for triple exponential smoothing outputs a final coefficient for trend (γ) or for seasonality (β) of zero. Or worse, both are outputted as zero! Does this indicate that there is no trend and/or no seasonality?Of course not! It only means that the initial values for trend and/or seasonality were right on the money. No updating was necessary in order to arrive at the lowest possible MSE. We should inspect the updating formulas to verify this. | | | | |
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| Readable Markdown | null |
| Shard | 183 (laksa) |
| Root Hash | 4377278747177273583 |
| Unparsed URL | gov,nist!itl,www,/div898/handbook/pmc/section4/pmc435.htm s443 |