Chapter 3
Lecture 4
Forecasting
Time Series Forecasts
Time Series is a sequence of measurements
over time, usually obtained at equally
spaced intervals
– Daily
– Monthly
– Quarterly
– Yearly
Time Series Forecasts
Time ordered sequence of observations taken at
regular observations taken at regular intervals.
Statistical techniques that make use of historical
data collected over a long period of time.
Methods assume that what has occurred in the past
will continue to occur in the future.
Forecasts based on only one factor - time.
Time Series Patterns
Time Series Forecasts
Naive
Forecasts
Techniques
for
Averaging
Techniques
for
Trend
Moving
Average
Weighted
Moving
Average
Exponential
Smoothing
Techniques
for
Seasonality
Techniques
for
Cycles
Naive Forecasts
Uh, give me a minute....
We sold 250 wheels last
week.... Now, next week we should sell....
The recent periods are the best predictors of the future
The forecast for any period equals
the previous period’s actual value.
Naïve Forecasts
Simple to use
Virtually no cost
Quick and easy to prepare
Data analysis is nonexistent
Easily understandable
Cannot provide high accuracy
Uses for Naïve Forecasts at
different data patterns
Stable time
series data
(stationer)
• F(t) = A(t-1)
Data with
trends
• F(t) = A(t-1) +
(A(t-1) – A(t-2))
Seasonal
variations
• F(t) = A(t-n)
Naïve Forecasts
Ex.
Year
Demand
2006
390
2007
420
2009
380
2010
400
Stationer
F(t) = A(t-1)
2011
400
Naïve Forecasts
Ex.
Year
Demand
2006
390
2007
420
2009
440
2010
480
Trend
F(t) = A(t-1) + (A(t-1) – A(t-2))
2011
520
Naïve Forecasts
Ex.
Year
Demand
1/2008
10
7/2008
90
1/2009
15
7/2009
100
1/2010
12
7/2010
95
Seasonal
F(t) = A(t-1) + (A(t-1) – A(t-2))
1/2011
12
Naïve Forecast Graph
Wallace Garden - Naive Forecast
25
20
Sheds
15
Actual Value
Naïve Forecast
10
5
0
February
March
April
May
June
July
Period
August
September
October
November
December
Moving Averages, no pattern
( random variation )
A technique that averages a number of recent
actual values, updated as new values become
available.
At-n + … At-2 + At-1
Ft = MAn=
n
Ft = forecast for time period t
MAn = n period moving average
At-1 = actual value in period t-1
n = number of periods ( data points )
Moving Averages
Ex.
Compute a 3-period moving average forecast
given demand for shopping carts for the last
five periods as shown:
2001
2002
2003 2004
2005
period
2006
demand
42
40
43
40
41
At-n + … At-2 + At-1
Ft = MAn=
n
t=6
A 3 + A4 + A5
F6 = MA3 =
3
= 41.33
???
Moving Averages
Ex. (cont.)
forecast
period
demand
2001
2002
2003
2004
2005
42
40
43
40
41
2006 2007
41.33
t=7
???
actual
period
demand
2001
2002
2003
2004
2005
2006
2007
42
40
43
40
41
38
???
F7 = MA3=
period
demand
A4 + A5 + A6
= 39.67
3
forecast
2001
2002
2003
2004
2005
2006
2007
42
40
43
40
41
38
39.67
Ex.
Moving Averages
period
data
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
42
40
43
40
41
39
38
42
44
40
42
41
41
43
39
MA3
MA4
MA5
41.66667
41
41.33333
40
39.33333
39.66667
41.33333
42
42
41
41.33333
41.66667
41
41.25
41
40.75
39.5
40
40.75
41
42
41.75
41
41.75
41
41.2
40.6
40.2
40
40.8
40.6
41.2
41.8
41.6
41.4
41.2
Moving Averages
Ex. (cont.)
45
44
43
42
data 41
MA3 40
MA4
39
MA5
38
37
36
35
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
Moving Averages
Ex.
Period
1
2
3
4
5
6
7
8
9
10
Demand
53
62
84
78
95
75
66
82
71
83
3-Period
MA
Forecast
5-Period
MA
Forecast
66.3
74.7
85.7
82.7
78.7
74.3
73.0
78.7
74.4
78.8
79.6
79.2
77.8
75.4
Moving Averages
Ex. (cont.)
Ex.
Moving Averages
Paper company
Stability vs. Responsiveness
•Should I use a 2-period moving average or a 3period moving average?
•The larger the “n” the more stable the forecast.
•A 2-period model will be more responsive to
change.
•We must balance stability with responsiveness
•If responsiveness is required, average with few
data points should be used,
Moving Averages
Decreasing the number of data points in an moving average
technique, increase the weight of more recent values
As data points in an moving average technique increased, the
sensitivity
( responsiveness ) of the average to new values
decreased.
If responsiveness is required, average with few data points
should be used,
Moving Averages
It is easy to
compute
It is easy to
understand
All values in the
average are weighted
equally, the oldest value
has the same weight as
the most recent value
But
Idea
most recent observations must be better indicators of the
future than older observations
Weighted Moving Averages
Historical values of the
time series are assigned
different weights when
performing the forecast
Weighted Moving Averages
More recent values in a series are
given more weight in computing the
forecast.
Trial and error used to find the
suitable weighting scheme
The sum of all weights must be = 1
Weighted average technique is more
reflective of the most recent
occurrences.
Weighted Moving Averages
In a weighted moving average, weights are assigned to the
most recent data.
Formula:
n
WMAn   W A
i1 i i
where W  the weight for period i, between 0% and 100%
i
Wi  1.00
Example : Paper company we ights 50% for October, 33%
for September, 17% for August
forecast for November ?:
3
WMA   W A  (.50)(90)  (.33)(110)  (.17)(130)  103.4 orders
3 i1 i i
Weighted Moving Averages
Ex.
Compute a 4-period weighted moving average forecast
given demand for shopping carts for the last periods
values and weights as shown:
period
1
2
3
4
5
6
demand
weight
42
40
43
40
41
???
0
0.1
0.2
0.3
0.4
F6 = W5*A5 + W4*A4 + W3*A3 + W2*A2
F6 = 0.4(41) + 0.3(40) + 0.2(43) + 0.1(40) = 41
If the actual value of F6 is 39 then
F7 = 0.4(39) + 0.3(41) + 0.2(40) + 0.1(43) = 40.2
Weighted Moving Averages
Market Mixer, Inc. sells can openers. Monthly sales for an eight-month period were as follows:
Month
1
2
3
4
Sales
450
425
445
435
Month
5
6
7
8
Sales
460
455
430
420
Forecast next month’s sales using a 3-month weighted moving average, where the weight for the
most recent data value is 0.60; the next most recent, 0.30; and the earliest, 0.10.
Solution:
Period
1
2
3
4
5
6
7
8
9
Sales
450
425
445
435
460
455
430
420
Weighted Moving Average Forecast
(450*.10) + (425*.30) + (445*.60) = 440
(425*.10) + (445*.30) + (435*.60) = 437
(445*.10) + (435*.30) + (460*.60) = 451
(435*.10) + (460*.30) + (445*.60) = 455
(460*.10) + (455*.30) + (430*.60) = 441
(455*.10) + (430*.30) + (420*.60) = 427
Comments:
1. Any forecasts beyond Period
9 will have the same value as
the Period 9 forecast, i.e.,
427.
3. WMA gives greater weight
to more recent values in the
moving average and is more
responsive to recent changes
in the data.
Exponential Smoothing
•The most recent observations might
have the highest predictive value.
•Therefore,
we should give more
weight to the more recent time
periods when forecasting.
Exponential Smoothing
Weighted averaging method based on
previous forecast plus a percentage of
the forecast error
Ft = Ft-1 + (At-1 - Ft-1)
Determination of  is usually judgmental and subjective and
often based on trial-and -error experimentation.
The most commonly used values of  are between
.10 and .50.
Exponential Smoothing
Ex.
Period
1
2
3
4
5
6
7
8
9
10
11
12
Actual
42
40
43
40
41
39
46
44
45
38
40
Alpha = 0.1
Error
Alpha = 0.4
Error
42
41.8
41.92
41.73
41.66
41.39
41.85
42.07
42.36
41.92
41.73
-2.00
1.20
-1.92
-0.73
-2.66
4.61
2.15
2.93
-4.36
-1.92
42
41.2
41.92
41.15
41.09
40.25
42.55
43.13
43.88
41.53
40.92
-2
1.8
-1.92
-0.15
-2.09
5.75
1.45
1.87
-5.88
-1.53
Picking a Smoothing Constant
Actual
Demand
50
.4
 .1
45
40
35
1
2
3
4
5
6
7
8
9
10 11 12
Period
Selecting a smoothing constant α is a matter
of judgment or trial and error
Techniques for Trend
Nonlinear Trends
Parabolic
Exponential
Growth
Techniques for Trend
Linear Trends
Chart Title
8
7
7
6
6
5
5
4
4
3
3
2
2
1
0
1
0
Techniques for Trend
Ft
Ft = a + bt
0 1 2 3 4 5
•
•
•
•
Ft = Forecast for period t
t = Specified number of time periods
a = Value of Ft at t = 0
b = Slope of the line
t
Calculating a and b
b
a
=
=
n  (ty)
2
n t
 y
 t y
2
- (  t)
-
- b  t
n
Linear Trend Equation
Ex.
t
Week
1
2
3
4
5
2
t
1
4
9
16
25
 t = 15
t = 55
2
(t) = 225
2
y
Sales
150
157
162
166
177
ty
150
314
486
664
885
 y = 812  ty = 2499
Linear Trend Equation
b =
5 (2499) - 15(812)
5(55) - 225
=
12495 -12180
275 -225
812 - 6.3(15)
a =
= 143.5
5
y = 143.5 + 6.3t
= 6.3
Techniques for Averaging
Moving
average
Weighted
moving
average
Exponential
smoothing