Actual Naïve Forecast error

(1)

Naïve, & Moving Average

Simple Slope

Ted Mitchell

Naïve Forecast

• Simplest possible forecast • Tomorrow will be like today

• Naïve is the basis for comparison of all methods.

• Ignores any historical data previous to today

8 7 6 5 4 3 2 ? ? 1 10 0 E=A-F F A error Naïve Forecast Actual Year

Use today’s result to forecast tomorrow

8 7 6 5 4 3 2 10 ? 1 10 0 E=A-F F A error Naïve Forecast Actual Year

N = 8 8 7 6 5 4 3 2 2 10 12 1 10 0 E=A-F F A error Naïve Forecast Actual Year

Use today’s result to forecast tomorrow ME = 1.63

SE =13 23 ? N = 8 2 21 23 8 2 19 21 7 2 17 19 6 1 16 17 5 1 15 16 4 1 14 15 3 2 12 14 2 2 10 12 1 10 0 E=A-F F A error Naïve Forecast Actual Year

(2)

Momentum

• 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

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

? Error 9 8 7 6 5 4 3 10/1 = 10 ? 2 10 1

Forecast using average of the total history actual

period

Get the forecast for the next period

? 2 Error 9 8 7 6 5 4 (10+12)/2 = 11 ? 3 10/1 = 10 12 2 10 1 forecast actual period

7.5 6.3 5 3.6 3.2 2.2 3 2 Error (10+12+14+15+16+17+19+21)/8 =15.5 23 9 (10+12+14+15+16+17+19)/7 =14.7 21 8 (10+12+14+15+16+17)/6 =14 19 7 (10+12+14+15+16)/5 =13.4 17 6 (10+12+14+15)/4 =12.8 16 5 (10+12+14)/3 =12 15 4 (10+12)/2 = 11 14 3 10/1 = 10 12 2 10 1 forecast actual period

Using Total Historical Average

• Disadvantage

• Really lags behind a trend! because

– Uses all historical data

– Puts equal weight on every piece of historical information

(3)

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

? Error 9 8 7 6 5 (10+12+14)/3 =12 ? 4 14 3 12 2 10 1

Forecast using a moving average on last 3 periods actual period ? 3 Error 9 8 7 6 (12+14+15)/3 =13.7 ? 5 (10+12+14)/3 =12 15 4 14 3 12 2 10 1

Forecast using a moving average on last 3 periods actual period 2.3 3 Error 9 8 7 (14+15+16)/3 =15 ? 6 (12+14+15)/3 =13.7 16 5 (10+12+14)/3 =12 15 4 14 3 12 2 10 1

Forecast using a moving average on last 3 periods actual period 4 3.7 3 2 2.3 3 Error (17+19+21)/3 =19 23 9 (16+17+19)/3 =17.3 21 8 (15+16+17)/3 =16 19 7 (14+15+16)/3 =15 17 6 (12+14+15)/3 =13.7 16 5 (10+12+14)/3 =12 15 4 14 3 12 2 10 1

Forecast using a moving average on last 3 periods actual

period

Moving Average

• 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

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Weighted Moving Average

• Three period average with equal weight • Fjun = (Amar +Aapr + Amay ) / 3

or

• Fjun = (3Amar +3Aapr + 3Amay ) / 9 • Weighted average with more on May • Fjun = (2Amar +3Aapr + 4Amay ) / 9 • Naïve Again

• Fjun = (0Amar +0Aapr + 9Amay ) / 9 3.6

3.3 2.8 1.8 2 2.6 Error (2(17)+3(19)+4(21))/9 = 19.4 23 9 (2(16)+3(17)+4(19))/9 = 17.7 21 8 (2(15)+3(16)+4(17))/9 = 16.2 19 7 (2(14)+3(15)+4(16))/9 = 15.2 17 6 (2(12)+3(14)+4(15))/9 = 14 16 5 (2(10)+3(12)+4(14))/9 = 12.4 15 4 14 3 12 2 10 1

Forecast using a WEIGHTED moving average on last 3 periods

actual period

Weighted Moving Average

• 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

Simple Growth and Slope For

Trends

Ted Mitchell

Trends

• Trends in the data are not handled well by moving averages or exponential smoothing methods.

Simple Trend Projection

• 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

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Do a Forecast for Period 2

time 0 1 2 Sales Revenue Last period Naïve Forecast Two Periods Ago

Last period plus X Percent

• The sales in the last period plus the percentage growth over the last two periods

Do a Forecast for Period 2

time 0 1 2 Sales Revenue Last period Naïve Forecast Two Periods Ago Simple Percentage Projection uses the same growth as between the last two periods

g

Getting the slope

• The percentage growth over the last two periods = g

• Prediction for the last period would be R1 = g R0

• We know R1 and R0 so we can calculate g

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%

g = Growth rate between 0 and 1

time Sales Revenue Last period Naïve Forecast Two Periods Ago

Simple last period plus percent growth Projection uses the same slope as the last two periods

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Prediction of Revenue in period 2

• (Revenue in period 2) = g (Revenue in period 1) • R2 = gR1 Where • R1 = revenue in 1 = 175 • g = 1.17 Then R2 = 1.17(175) = 204.17

Prediction for R2

time 0 1 2 Sales Revenue R₁ = 175 Naïve Forecast = 175 R₀ = 150 R₂ = 1.17(175) = 204 g = 1.17

Prediction for R2 in period 2

time 0 1 2 Sales Revenue R₁ = 175 Naïve Forecast = 175 R₀ = 150 R2 = 204.17 g = 1.17

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

Ted Mitchell

New Shoes Home Market Spring 478

• Home Market in this example is

experiencing a long run decline in sales as it nears the end of the Product Life Cycle

(7)

7 6 5 1,193,000 4 1,193,000 3 E=A-F F A error Naïve Forecast Actual Units Sold Period

7 6 5 170,000 1,193,000 1,023,000 4 1,193,000 3 E=A-F F A error Naïve Forecast Actual Units Sold Period

7 6 1,023,000 ? 5 170,000 1,193,000 1,023,000 4 1,193,000 3 E=A-F F A error Naïve Forecast Actual Units Sold Period

What to do Next?

• 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 %

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%

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

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7 6 877.225 ? 5 170,000 1,193,000 1,023,000 4 1,193,000 3 E=A-F F A error Naïve Forecast Actual Units Sold Period

Use last period and decline rate to forecast period 5 7 6 Smallest error is naive 877.225 or the naïve method 1,023,000 1,000,000 5 170,000 1,193,000 1,023,000 4 1,193,000 3 E=A-F F A error Naïve Forecast Actual Units Sold Period

Use last period and decline rate to forecast period 5 or naïve method

7 Smallest error is last period + decline rate 977,517 or the naïve method 1,000,000 885,000 6 Smallest error is naive 877.225 or the naïve method 1,023,000 1,000,000 5 170,000 1,193,000 1,023,000 4 1,193,000 3 E=A-F F A error Naïve Forecast Actual Units Sold Period

Use last period and decline rate to forecast period 5