Naïve Forecast
Naïve, & Moving Average
Simple Slope
Ted Mitchell
Year
• Simplest possible forecast
• Tomorrow will be like today
• Naïve is the basis for comparison of all
methods.
• Ignores any historical data previous to today
Actual
Naïve Forecast
error
A
F
E=A-F
0
10
1
?
?
2
3
4
5
6
7
8
Use today’s result to forecast tomorrow
Year
Actual
Naïve Forecast
error
A
F
E=A-F
0
10
1
?
Year
Actual
Naïve Forecast
error
A
F
E=A-F
0
10
1
12
Actual
Naïve Forecast
error
A
F
E=A-F
0
10
1
12
10
2
2
2
2
14
12
2
3
3
3
15
14
1
4
4
4
16
15
1
5
5
5
17
16
1
6
6
6
19
17
2
7
7
7
21
19
2
8
8
8
23
21
2
N=8
N=8
?
23
SE =13
10
Use today’s result to forecast tomorrow
10
Year
Use today’s result to forecast tomorrow
2
Use today’s result to forecast tomorrow
ME = 1.63
1
period actual
Momentum
Total Historical Average
• Fi+1 = (1/n)(ΣAi)
where
• Fi+1 = forecast for next period
• n = number of historical periods
• ΣAi = sum of the actual results for each of
the n historical periods
• If things have momentum they are easier to
predict.
• Averages are a measure of momentum
• Various averages are used for prediction
– Total historical average
– Moving averages
– Weighted averages
period actual
1
2
3
4
5
6
7
8
9
10
12
?
forecast
Error
Get the forecast for the next period
10/1 = 10
(10+12)/2 = 11
2
?
period actual
1
2
3
4
5
6
7
8
9
10
12
14
15
16
17
19
21
23
forecast
1
2
3
4
5
6
7
8
9
10
?
Forecast using average of the total history
Error
Get the forecast for the next period
10/1 = 10
?
Error
Get the forecast for the next period
10/1 = 10
(10+12)/2 = 11
(10+12+14)/3 =12
(10+12+14+15)/4 =12.8
(10+12+14+15+16)/5 =13.4
(10+12+14+15+16+17)/6 =14
(10+12+14+15+16+17+19)/7 =14.7
(10+12+14+15+16+17+19+21)/8 =15.5
2
3
2.2
3.2
3.6
5
6.3
7.5
Using Total Historical Average
• Disadvantage
• Really lags behind a trend! because
– Uses all historical data
– Puts equal weight on every piece of historical
information
2
period actual
Moving Average
• Pick the last n periods that are most relevant
• Fi+1 = (1/n) (ΣAi)
where
• Fi+1 = the forecast for next period
• n = the number of periods in the moving
average
• (ΣAi) = the sum of the last n periods
period actual
1
2
3
4
5
6
7
8
9
10
12
14
15
16
?
Forecast using a moving average on last 3 periods
(10+12+14)/3 =12
(12+14+15)/3 =13.7
(14+15+16)/3 =15
Error
3
2.3
1
2
3
4
5
6
7
8
9
10
12
14
?
period actual
1
2
3
4
5
6
7
8
9
10
12
14
15
16
17
19
21
23
Forecast using a moving average on last 3 periods
Error
(10+12+14)/3 =12
?
Forecast using a moving average on last 3 periods
Error
period actual
1
2
3
4
5
6
7
8
9
10
12
14
15
?
Forecast using a moving average on last 3 periods
Error
(10+12+14)/3 =12
(12+14+15)/3 =13.7
3
?
Moving Average
(10+12+14)/3 =12
(12+14+15)/3 =13.7
(14+15+16)/3 =15
(15+16+17)/3 =16
(16+17+19)/3 =17.3
(17+19+21)/3 =19
3
2.3
2
3
3.7
4
• Still lags behind a trend
• Puts equal weight on each of the historical
results being used
• Gives bias when seasonal data is involved
• If you want more weight on the most recent
data you need a weighted average
3
period actual
Weighted Moving Average
• Three period average with equal weight
• Fjun = (Amar +Aapr + Amay ) / 3
or
• Fjun = (3A mar +3A apr + 3Amay ) / 9
• Weighted average with more on May
• Fjun = (2A mar +3A apr + 4Amay ) / 9
• Naïve Again
• Fjun = (0A mar +0A apr + 9Amay ) / 9
Simple Growth and Slope For
Trends
Ted Mitchell
1
2
3
4
5
6
7
8
9
10
12
14
15
16
17
19
21
23
Forecast using a WEIGHTED moving average on
last 3 periods
(2(10)+3(12)+4(14))/9 = 12.4
(2(12)+3(14)+4(15))/9 = 14
(2(14)+3(15)+4(16))/9 = 15.2
(2(15)+3(16)+4(17))/9 = 16.2
(2(16)+3(17)+4(19))/9 = 17.7
(2(17)+3(19)+4(21))/9 = 19.4
Error
Weighted Moving Average
2.6
2
1.8
2.8
3.3
3.6
• Weighted Moving Average is better at
responding to a trend because it puts more
weight on recent data and less weight on old
data
• They get the appropriate weights by doing a
statistical fit to the data
Trends
Simple Trend Projection
• Trends in the data are not handled well by
moving averages or exponential smoothing
methods.
• Before the era of simple statistical tools on
every PC managers used simple calculations
of trends based on the naïve forecast.
• The naïve forecast is
sales in next period t = sales in the last
period (t-1) or
• Rt = Rt-1
4
Do a Forecast for Period 2
• The sales in the last period plus the
percentage growth over the last two periods
Sales
Revenue
Two
Periods
Ago
Last period plus X Percent
Do a Forecast for Period 2
Simple Percentage
Projection uses the
same growth as
between the last
two periods
Sales
Revenue
Two
Periods
Ago
Last period
Last period
Naïve Forecast
Naïve Forecast
g
0
1
2
time
Getting the slope
• The percentage growth over the last two
periods = g
• Prediction for the last period would be
R1 = g R 0
• We know R 1 and R0 so we can calculate g
0
Example Calculate historical g
• R1 = g R0
where
• R0 = $150
• R1 = $175
then calculate g
• 175 = g(150)
• g = 175 / 150
• g = 1.17 or growth is 117%
1
2
time
g = Growth rate between 0 and 1
Sales
Revenue
Two
Periods
Ago
Simple last period
plus percent
growth Projection
uses the same
slope as the last
two periods
Naïve Forecast
Last period
g = R1/R0
0
1
2
time
5
Prediction of Revenue in period 2
• (Revenue in period 2) = g (Revenue in period 1)
• R2 = gR1
Prediction for R2
Prediction for R2 in period 2
Sales
Revenue
Where
R0 = 150
• R1 = revenue in 1 = 175
• g = 1.17
Then
R2 = 1.17(175) = 204.17
Sales
Revenue
R2 = 1.17(175) = 204
R1 = 175
R0 = 150
R2 = 204.17
R1 = 175
Naïve Forecast
= 175
Naïve Forecast
= 175
g = 1.17
0
1
g = 1.17
2
time
0
1
2
time
The Problem with
• The “last period result + percent
improvement” method
• Very dependent on the base used in the
percentage. If you use the same percentage
as time passes then the method inflates the
forecasted values
• But it is simple and very popular!
Examples: Naïve Method
& Last Period Plus Rate of
Change Method
• Home Market in this example is
experiencing a long run decline in sales as it
nears the end of the Product Life Cycle
Ted Mitchell
New Shoes Home Market Spring 478
6
Period
3
4
Actual Units
Sold
Naïve Forecast
error
A
F
E=A-F
1,193,000
1,193,000
Actual Units
Sold
Naïve Forecast
error
A
F
E=A-F
3
1,193,000
4
1,023,000
Naïve Forecast
error
A
F
E=A-F
170,000
3
1,193,000
4
1,023,000
1,193,000
5
5
?
1,023,000
6
6
7
7
7
You have two pieces of information
Industry Sales in period 3 = 1,193,000
Industry Sales in period 4 = 1,023,000
And the idea that the market is in decline
phase of the Product Life Cycle (PLC)
• Do you naïve or last period + decline %
170,000
Actual Units
Sold
6
What to do Next?
1,193,000
Period
5
Use today’s result to forecast tomorrow
•
•
•
•
Period
Use today’s result to forecast tomorrow
Last period + change %
• Consider the last period plus the decline rate
from the two previous periods
• What is the decline rate
• Sales in 4 = decline rate (Sales in 3)
1,023 = decline rate (1,193 )
• Decline rate = 1,023 / 1,193 = 85.75%
Use today’s result to forecast tomorrow
Forecasting period 5
• Sales in 5 = decline rate (sales in 4)
• Sales in 5 = 85.75% (1,023,000)
• Sales in 5 = 877,225 units
7
Period
Actual Units
Sold
Naïve Forecast
error
A
F
E=A-F
3
1,193,000
4
1,023,000
1,193,000
5
?
877.225
170,000
Period
Naïve Forecast
error
A
F
E=A-F
3
1,193,000
4
1,023,000
1,193,000
5
1,000,000
877.225 or the
naïve method
1,023,000
6
7
Actual Units
Sold
6
Period
Actual Units
Sold
Naïve Forecast
error
A
F
E=A-F
3
1,193,000
170,000
4
1,023,000
1,193,000
170,000
Smallest error
is naive
5
1,000,000
877.225 or the
naïve method
1,023,000
Smallest error
is naive
6
885,000
977,517 or the
naïve method
1,000,000
Smallest error
is last period +
decline rate
7
7
Use last period and decline rate to
forecast period 5
Use last period and decline rate to
forecast period 5 or naïve method
Use last period and decline rate to
forecast period 5
8