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1
Constant mean model
Constant mean model: introduction
Suppose demand for a product follows the (very) simple model
x
t
=
μ
+
±
t
Here
•
x
t
= demand for time period
t
•
μ
is the constant expected demand
•
±
1
, ±
2
, . . .
are independent with mean 0
•
the
best forecast of a future value of
x
t
is
μ
•
we want to estimate
μ
and update the estimate as each new
x
t
is observed
Constant mean model: forecasts
•
Let
b
x
n
(
`
)
be the
`
step ahead forecast at time period
n
•
Stated diﬀerently,
b
x
n
(
`
)
is the forecast at time
n
of demand at
time
n
+
`
•
Let
b
μ
n
=
x
1
+
· · ·
+
x
n
n
•
Then
b
x
n
(
`
) =
b
μ
n
,
for all
`
Constant mean model: updating
b
μ
n
•
In this simple model,
μ
does not change, but our estimate of
μ
does
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Here is a simple way to update
b
μ
n
to
b
μ
n
+1
b
μ
n
+1
=
(
x
1
+
· · ·
+
x
n
) +
x
n
+1
n
+ 1
=
n
n
+ 1
b
μ
n
+
1
n
+ 1
x
n
+1
=
b
μ
n
+
1
n
+ 1
(
x
n
+1

b
μ
n
)
2
Simple exponential smoothing
Slowly changing mean model: introduction
•
Now suppose that
x
t
=
μ
n
+
±
t
where
μ
n
is
slowly changing
•
The forecast is the same as for the constant mean model:
b
x
n
(
`
) =
b
μ
n
,
for all
`
•
What changes is the way
b
μ
n
is updated
–
We need for
b
μ
n
to track
μ
n
Slowly changing mean: updating
•
For a constant mean, the update is
b
μ
n
+1
=
b
μ
n
+
1
n
+ 1
(
x
n
+1

b
μ
n
)
•
For a slowly changing mean, the update is
b
μ
n
+1
=
b
μ
n
+
α
(
x
n
+1

b
μ
n
) = (1

α
)
b
μ
n
+
αx
n
+1
for a
constant
α
•
α
is adjusted depending on how fast
μ
n
is changing
2
–
0
< α <
1
–
faster changes in
μ
necessitate larger
α
Exponential weighting
b
μ
n
+1
= (1

α
)
b
μ
n
+
αx
n
+1
= (1

α
)
{
b
μ
n

1
(1

α
) +
αx
n
}
+
αx
n
+1
= (1

α
)
2
b
μ
n

1
+ (1

α
)
αx
n
+
αx
n
+1
= (1

α
)
3
b
μ
n

2
+ (1

α
)
2
αx
n

1
+ (1

α
)
αx
n
+
αx
n
+1
≈
α
±
x
n
+1
+ (1

α
)
x
n
+ (1

α
)
2
x
n

1
+(1

α
)
3
b
μ
n

2
+
· · ·
+ (1

α
)
n
x
1
²
Exponential weighted moving average
•
For previous page:
b
μ
n
+1
≈
α
x
n
+1
+ (1

α
)
x
n
+ (1

α
)
2
x
n

1
+
···
+ (1

α
)
n
x
1
≈
x
n
+1
+ (1

α
)
x
n
+ (1

α
)
2
x
n

1
+
···
+ (1

α
)
n
x
1
1 + (1

α
) +
···
+ (1

α
)
n
Exponential weights: examples
3
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5
10
15
20
25
0.0
0.2
0.4
0.6
0.8
1.0
n
weight =
(
1
α
29
n
α
=0.4
α
=0.2
α
=0.4
3
HoltWinters
Forecasting with trends and seasonality: example
Time
air passengers
1950
1952
1954
1956
1958
1960
100
200
300
400
500
600
4
3.1
Holt’s nonseasonal model
Holt method: forecasting with trend
Holt’s forecasting method uses a linear trend
b
x
n
(
`
) =
b
μ
n
+
b
β
n
(
`

n
)
•
b
μ
n
is called the
level
•
b
β
n
is called the
slope
Both
b
μ
n
and
b
β
n
must be updated
Holt method: Updating the level
In the Holt model, the level
b
μ
n
is updated by the equation:
b
μ
n
+1
= (1

α
)(
b
μ
n
+
b
β
n
) +
αx
n
+1
or, equivalently,
b
μ
n
+1
=
b
μ
n
+ (1

α
)
b
β
n
+
α
(
x
n
+1

b
μ
n
)
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This note was uploaded on 01/09/2009 for the course ORIE 312 taught by Professor D.ruppert,p.jacks during the Spring '08 term at Cornell University (Engineering School).
 Spring '08
 D.RUPPERT,P.JACKS

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