© 2008 Prentice Hall, Inc.4 – 1 Outline – Continued Time-Series Forecasting Decomposition of a Time Series Naive Approach Moving Averages Exponential.

© 2008 Prentice Hall, Inc. 4 – 1

Outline – ContinuedOutline – Continued Time-Series ForecastingTime-Series Forecasting

Decomposition of a Time SeriesDecomposition of a Time Series Naive ApproachNaive Approach Moving AveragesMoving Averages Exponential SmoothingExponential Smoothing Exponential Smoothing with Trend Exponential Smoothing with Trend

AdjustmentAdjustment Trend ProjectionsTrend Projections Seasonal Variations in DataSeasonal Variations in Data Cyclical Variations in DataCyclical Variations in Data


Outline – ContinuedOutline – Continued

Associative Forecasting Methods: Associative Forecasting Methods: Regression and Correlation Regression and Correlation AnalysisAnalysis Using Regression Analysis for Using Regression Analysis for

ForecastingForecasting

Standard Error of the EstimateStandard Error of the Estimate

Correlation Coefficients for Correlation Coefficients for Regression LinesRegression Lines

Multiple-Regression AnalysisMultiple-Regression Analysis


Outline – ContinuedOutline – Continued

Monitoring and Controlling Monitoring and Controlling ForecastsForecasts Adaptive SmoothingAdaptive Smoothing

Focus ForecastingFocus Forecasting

Forecasting In The Service SectorForecasting In The Service Sector


Forecasting at Disney WorldForecasting at Disney World

Global portfolio includes parks in Hong Global portfolio includes parks in Hong Kong, Paris, Tokyo, Orlando, and Kong, Paris, Tokyo, Orlando, and AnaheimAnaheim

Revenues are derived from people – how Revenues are derived from people – how many visitors and how they spend their many visitors and how they spend their moneymoney

Daily management report contains only Daily management report contains only the forecast and actual attendance at the forecast and actual attendance at each parkeach park



Disney generates daily, weekly, monthly, Disney generates daily, weekly, monthly, annual, and 5-year forecastsannual, and 5-year forecasts

Forecast used by labor management, Forecast used by labor management, maintenance, operations, finance, and maintenance, operations, finance, and park schedulingpark scheduling

Forecast used to adjust opening times, Forecast used to adjust opening times, rides, shows, staffing levels, and guests rides, shows, staffing levels, and guests admittedadmitted



20% of customers come from outside the 20% of customers come from outside the USAUSA

Economic model includes gross Economic model includes gross domestic product, cross-exchange rates, domestic product, cross-exchange rates, arrivals into the USAarrivals into the USA

A staff of 35 analysts and 70 field people A staff of 35 analysts and 70 field people survey 1 million park guests, employees, survey 1 million park guests, employees, and travel professionals each yearand travel professionals each year



Inputs to the forecasting model include Inputs to the forecasting model include airline specials, Federal Reserve airline specials, Federal Reserve policies, Wall Street trends, policies, Wall Street trends, vacation/holiday schedules for 3,000 vacation/holiday schedules for 3,000 school districts around the worldschool districts around the world

Average forecast error for the 5-year Average forecast error for the 5-year forecast is 5%forecast is 5%

Average forecast error for annual Average forecast error for annual forecasts is between 0% and 3%forecasts is between 0% and 3%


What is Forecasting?What is Forecasting?

Process of Process of predicting a future predicting a future eventevent

Underlying basis of Underlying basis of

all business all business decisionsdecisions ProductionProduction

InventoryInventory

PersonnelPersonnel

FacilitiesFacilities

??


Short-range forecastShort-range forecast Up to 1 year, generally less than 3 monthsUp to 1 year, generally less than 3 months Purchasing, job scheduling, workforce Purchasing, job scheduling, workforce

levels, job assignments, production levelslevels, job assignments, production levels

Medium-range forecastMedium-range forecast 3 months to 3 years3 months to 3 years Sales and production planning, budgetingSales and production planning, budgeting

Long-range forecastLong-range forecast 33++ years years New product planning, facility location, New product planning, facility location,

research and developmentresearch and development

Forecasting Time HorizonsForecasting Time Horizons


Seven Steps in ForecastingSeven Steps in Forecasting

Determine the use of the forecastDetermine the use of the forecast

Select the items to be forecastedSelect the items to be forecasted

Determine the time horizon of the Determine the time horizon of the forecastforecast

Select the forecasting model(s)Select the forecasting model(s)

Gather the dataGather the data

Make the forecastMake the forecast

Validate and implement resultsValidate and implement results


The Realities!The Realities!

Forecasts are seldom perfectForecasts are seldom perfect

Most techniques assume an Most techniques assume an underlying stability in the systemunderlying stability in the system

Product family and aggregated Product family and aggregated forecasts are more accurate than forecasts are more accurate than individual product forecastsindividual product forecasts


Forecasting ApproachesForecasting Approaches

Used when situation is ‘stable’ and Used when situation is ‘stable’ and historical data existhistorical data exist Existing productsExisting products

Current technologyCurrent technology

Involves mathematical techniquesInvolves mathematical techniques e.g., forecasting sales of color e.g., forecasting sales of color

televisionstelevisions

Quantitative MethodsQuantitative Methods


Overview of Quantitative Overview of Quantitative ApproachesApproaches

1.1. Naive approachNaive approach

2.2. Moving averagesMoving averages

3.3. Exponential Exponential smoothingsmoothing

4.4. Trend projectionTrend projection

5.5. Linear regressionLinear regression

Time-Series Time-Series ModelsModels

Associative Associative ModelModel


Set of evenly spaced numerical dataSet of evenly spaced numerical data Obtained by observing response Obtained by observing response

variable at regular time periodsvariable at regular time periods

Forecast based only on past values, Forecast based only on past values, no other variables importantno other variables important Assumes that factors influencing Assumes that factors influencing

past and present will continue past and present will continue influence in futureinfluence in future

Time Series ForecastingTime Series Forecasting


Trend

Seasonal

Cyclical

Random

Time Series ComponentsTime Series Components


Components of DemandComponents of DemandD

eman

d f

or

pro

du

ct o

r se

rvic

e

| | | |1 2 3 4

Year

Average demand over four years

Seasonal peaks

Trend component

Actual demand

Random variation

Figure 4.1Figure 4.1


Persistent, overall upward or Persistent, overall upward or downward patterndownward pattern

Changes due to population, Changes due to population, technology, age, culture, etc.technology, age, culture, etc.

Typically several years Typically several years duration duration

Trend ComponentTrend Component


Regular pattern of up and Regular pattern of up and down fluctuationsdown fluctuations

Due to weather, customs, etc.Due to weather, customs, etc.

Occurs within a single year Occurs within a single year

Seasonal ComponentSeasonal Component

Number ofPeriod Length Seasons

Week Day 7Month Week 4-4.5Month Day 28-31Year Quarter 4Year Month 12Year Week 52


Repeating up and down movementsRepeating up and down movements

Affected by business cycle, political, Affected by business cycle, political, and economic factorsand economic factors

Multiple years durationMultiple years duration

Often causal or Often causal or associative associative relationshipsrelationships

Cyclical ComponentCyclical Component

00 55 1010 1515 2020


Erratic, unsystematic, ‘residual’ Erratic, unsystematic, ‘residual’ fluctuationsfluctuations

Due to random variation or Due to random variation or unforeseen eventsunforeseen events

Short duration and Short duration and nonrepeating nonrepeating

Random ComponentRandom Component

MM TT WW TT FF


Naive ApproachNaive Approach

Assumes demand in next Assumes demand in next period is the same as period is the same as demand in most recent perioddemand in most recent period e.g., If January sales were 68, then e.g., If January sales were 68, then

February sales will be 68February sales will be 68

Sometimes cost effective and Sometimes cost effective and efficientefficient

Can be good starting pointCan be good starting point


MA is a series of arithmetic means MA is a series of arithmetic means

Used if little or no trendUsed if little or no trend

Used often for smoothingUsed often for smoothingProvides overall impression of data Provides overall impression of data

over timeover time

Moving Average MethodMoving Average Method

Moving average =Moving average =∑∑ demand in previous n periodsdemand in previous n periods

nn


JanuaryJanuary 1010FebruaryFebruary 1212MarchMarch 1313AprilApril 1616MayMay 1919JuneJune 2323JulyJuly 2626

ActualActual 3-Month3-MonthMonthMonth Shed SalesShed Sales Moving AverageMoving Average

(12 + 13 + 16)/3 = 13 (12 + 13 + 16)/3 = 13 22//33

(13 + 16 + 19)/3 = 16(13 + 16 + 19)/3 = 16(16 + 19 + 23)/3 = 19 (16 + 19 + 23)/3 = 19 11//33

Moving Average ExampleMoving Average Example

101012121313

((1010 + + 1212 + + 1313)/3 = 11 )/3 = 11 22//33


Graph of Moving AverageGraph of Moving Average

| | | | | | | | | | | |

JJ FF MM AA MM JJ JJ AA SS OO NN DD

Sh

ed S

ales

Sh

ed S

ales

30 30 –28 28 –26 26 –24 24 –22 22 –20 20 –18 18 –16 16 –14 14 –12 12 –10 10 –

Actual Actual SalesSales

Moving Moving Average Average ForecastForecast


Increasing n smooths the forecast Increasing n smooths the forecast but makes it less sensitive to but makes it less sensitive to changeschanges

Do not forecast trends wellDo not forecast trends well

Require extensive historical dataRequire extensive historical data

Potential Problems WithPotential Problems With Moving Average Moving Average


Form of weighted moving averageForm of weighted moving average Weights decline exponentiallyWeights decline exponentially

Most recent data weighted mostMost recent data weighted most

Requires smoothing constant Requires smoothing constant (()) Ranges from 0 to 1Ranges from 0 to 1

Subjectively chosenSubjectively chosen

Involves little record keeping of past Involves little record keeping of past datadata

Exponential SmoothingExponential Smoothing


Exponential SmoothingExponential Smoothing

New forecast =New forecast = Last period’s forecastLast period’s forecast+ + ((Last period’s actual demand Last period’s actual demand

– – Last period’s forecastLast period’s forecast))

FFtt = F = Ft t – 1– 1 + + ((AAt t – 1– 1 - - F Ft t – 1– 1))

wherewhere FFtt == new forecastnew forecast

FFt t – 1– 1 == previous forecastprevious forecast

== smoothing (or weighting) smoothing (or weighting) constant constant (0 (0 ≤≤ ≤≤ 1) 1)


Exponential Smoothing Exponential Smoothing ExampleExample

Predicted demand Predicted demand = 142= 142 Ford Mustangs Ford MustangsActual demand Actual demand = 153= 153Smoothing constant Smoothing constant = .20 = .20




New forecastNew forecast = 142 + .2(153 – 142)= 142 + .2(153 – 142)




New forecastNew forecast = 142 + .2(153 – 142)= 142 + .2(153 – 142)

= 142 + 2.2= 142 + 2.2

= 144.2 ≈ 144 cars= 144.2 ≈ 144 cars


Effect ofEffect of Smoothing Constants Smoothing Constants

Weight Assigned toWeight Assigned to

MostMost 2nd Most2nd Most 3rd Most3rd Most 4th Most4th Most 5th Most5th MostRecentRecent RecentRecent RecentRecent RecentRecent RecentRecent

SmoothingSmoothing PeriodPeriod PeriodPeriod PeriodPeriod PeriodPeriod PeriodPeriodConstantConstant (()) (1 - (1 - )) (1 - (1 - ))22 (1 - (1 - ))33 (1 - (1 - ))44

= .1= .1 .1.1 .09.09 .081.081 .073.073 .066.066

= .5= .5 .5.5 .25.25 .125.125 .063.063 .031.031


Impact of Different Impact of Different

225 225 –

200 200 –

175 175 –

150 150 –| | | | | | | | |

11 22 33 44 55 66 77 88 99

QuarterQuarter

De

ma

nd

De

ma

nd

= .1= .1

Actual Actual demanddemand

= .5= .5


Impact of Different Impact of Different

225 225 –

200 200 –

175 175 –

150 150 –| | | | | | | | |

11 22 33 44 55 66 77 88 99

QuarterQuarter

De

ma

nd

De

ma

nd

= .1= .1

Actual Actual demanddemand

= .5= .5Chose high values of Chose high values of when underlying average when underlying average is likely to changeis likely to change

Choose low values of Choose low values of when underlying average when underlying average is stableis stable


Choosing Choosing

The objective is to obtain the most The objective is to obtain the most accurate forecast no matter the accurate forecast no matter the techniquetechnique

We generally do this by selecting the We generally do this by selecting the model that gives us the lowest forecast model that gives us the lowest forecast errorerror

Forecast errorForecast error = Actual demand - Forecast value= Actual demand - Forecast value

= A= Att - F - Ftt


Common Measures of ErrorCommon Measures of Error

Mean Absolute Deviation Mean Absolute Deviation ((MADMAD))

MAD =MAD =∑∑ |Actual - Forecast||Actual - Forecast|

nn

Mean Squared Error Mean Squared Error ((MSEMSE))

MSE =MSE =∑∑ ((Forecast ErrorsForecast Errors))22

nn


Common Measures of ErrorCommon Measures of Error

Mean Absolute Percent Error Mean Absolute Percent Error ((MAPEMAPE))

MAPE =MAPE =∑∑100100|Actual|Actualii - Forecast - Forecastii|/Actual|/Actualii

nn

nn

i i = 1= 1


Comparison of Forecast Comparison of Forecast Error Error

RoundedRounded AbsoluteAbsolute RoundedRounded AbsoluteAbsoluteActualActual ForecastForecast DeviationDeviation ForecastForecast DeviationDeviation

TonnageTonnage withwith forfor withwith forforQuarterQuarter UnloadedUnloaded = .10 = .10 = .10 = .10 = .50 = .50 = .50 = .50

11 180180 175175 5.005.00 175175 5.005.0022 168168 175.5175.5 7.507.50 177.50177.50 9.509.5033 159159 174.75174.75 15.7515.75 172.75172.75 13.7513.7544 175175 173.18173.18 1.821.82 165.88165.88 9.129.1255 190190 173.36173.36 16.6416.64 170.44170.44 19.5619.5666 205205 175.02175.02 29.9829.98 180.22180.22 24.7824.7877 180180 178.02178.02 1.981.98 192.61192.61 12.6112.6188 182182 178.22178.22 3.783.78 186.30186.30 4.304.30

82.4582.45 98.6298.62





11 180180 175175 5.005.00 175175 5.005.0022 168168 175.5175.5 7.507.50 177.50177.50 9.509.5033 159159 174.75174.75 15.7515.75 172.75172.75 13.7513.7544 175175 173.18173.18 1.821.82 165.88165.88 9.129.1255 190190 173.36173.36 16.6416.64 170.44170.44 19.5619.5666 205205 175.02175.02 29.9829.98 180.22180.22 24.7824.7877 180180 178.02178.02 1.981.98 192.61192.61 12.6112.6188 182182 178.22178.22 3.783.78 186.30186.30 4.304.30

82.4582.45 98.6298.62

MAD =∑ |deviations|

n

= 82.45/8 = 10.31

For = .10

= 98.62/8 = 12.33

For = .50





11 180180 175175 5.005.00 175175 5.005.0022 168168 175.5175.5 7.507.50 177.50177.50 9.509.5033 159159 174.75174.75 15.7515.75 172.75172.75 13.7513.7544 175175 173.18173.18 1.821.82 165.88165.88 9.129.1255 190190 173.36173.36 16.6416.64 170.44170.44 19.5619.5666 205205 175.02175.02 29.9829.98 180.22180.22 24.7824.7877 180180 178.02178.02 1.981.98 192.61192.61 12.6112.6188 182182 178.22178.22 3.783.78 186.30186.30 4.304.30

82.4582.45 98.6298.62MADMAD 10.3110.31 12.3312.33

= 1,526.54/8 = 190.82

For = .10

= 1,561.91/8 = 195.24

For = .50

MSE =∑ (forecast errors)2

n





11 180180 175175 5.005.00 175175 5.005.0022 168168 175.5175.5 7.507.50 177.50177.50 9.509.5033 159159 174.75174.75 15.7515.75 172.75172.75 13.7513.7544 175175 173.18173.18 1.821.82 165.88165.88 9.129.1255 190190 173.36173.36 16.6416.64 170.44170.44 19.5619.5666 205205 175.02175.02 29.9829.98 180.22180.22 24.7824.7877 180180 178.02178.02 1.981.98 192.61192.61 12.6112.6188 182182 178.22178.22 3.783.78 186.30186.30 4.304.30

82.4582.45 98.6298.62MADMAD 10.3110.31 12.3312.33MSEMSE 190.82190.82 195.24195.24

= 44.75/8 = 5.59%

For = .10

= 54.05/8 = 6.76%

For = .50

MAPE =∑100|deviationi|/actuali

n

n

i = 1





11 180180 175175 5.005.00 175175 5.005.0022 168168 175.5175.5 7.507.50 177.50177.50 9.509.5033 159159 174.75174.75 15.7515.75 172.75172.75 13.7513.7544 175175 173.18173.18 1.821.82 165.88165.88 9.129.1255 190190 173.36173.36 16.6416.64 170.44170.44 19.5619.5666 205205 175.02175.02 29.9829.98 180.22180.22 24.7824.7877 180180 178.02178.02 1.981.98 192.61192.61 12.6112.6188 182182 178.22178.22 3.783.78 186.30186.30 4.304.30

82.4582.45 98.6298.62MADMAD 10.3110.31 12.3312.33MSEMSE 190.82190.82 195.24195.24MAPEMAPE 5.59%5.59% 6.76%6.76%


Trend ProjectionsTrend Projections

Fitting a trend line to historical data points Fitting a trend line to historical data points to project into the medium to long-rangeto project into the medium to long-range

Linear trends can be found using the least Linear trends can be found using the least squares techniquesquares technique

y y = = a a + + bxbx^̂

where ywhere y= computed value of the = computed value of the variable to be predicted (dependent variable to be predicted (dependent variable)variable)aa= y-axis intercept= y-axis interceptbb= slope of the regression line= slope of the regression linexx= the independent variable= the independent variable

^̂


Least Squares MethodLeast Squares Method

Time periodTime period

Va

lue

s o

f D

ep

end

en

t V

ari

able


DeviationDeviation11

(error)(error)







Actual observation Actual observation (y value)(y value)

Trend line, y = a + bxTrend line, y = a + bx^̂



Time periodTime period

Va

lue

s o

f D

ep

end

en

t V

ari

able









Actual observation Actual observation (y value)(y value)

Trend line, y = a + bxTrend line, y = a + bx^̂

Least squares method minimizes the sum of the

squared errors (deviations)



Equations to calculate the regression variablesEquations to calculate the regression variables

b =b =xy - nxyxy - nxy

xx22 - nx - nx22

y y = = a a + + bxbx^̂

a = y - bxa = y - bx


Least Squares ExampleLeast Squares Example

b b = = = 10.54= = = 10.54∑∑xy - nxyxy - nxy

∑∑xx22 - nx - nx22

3,063 - (7)(4)(98.86)3,063 - (7)(4)(98.86)

140 - (7)(4140 - (7)(422))

aa = = yy - - bxbx = 98.86 - 10.54(4) = 56.70 = 98.86 - 10.54(4) = 56.70

TimeTime Electrical Power Electrical Power YearYear Period (x)Period (x) DemandDemand xx22 xyxy

20012001 11 7474 11 747420022002 22 7979 44 15815820032003 33 8080 99 24024020042004 44 9090 1616 36036020052005 55 105105 2525 52552520052005 66 142142 3636 85285220072007 77 122122 4949 854854

∑∑xx = 28 = 28 ∑∑yy = 692 = 692 ∑∑xx22 = 140 = 140 ∑∑xyxy = 3,063 = 3,063xx = 4 = 4 yy = 98.86 = 98.86



b b = = = 10.54= = = 10.54xy - nxyxy - nxy

xx22 - nx - nx22

3,063 - (7)(4)(98.86)3,063 - (7)(4)(98.86)

140 - (7)(4140 - (7)(422))

aa = = yy - - bxbx = 98.86 - 10.54(4) = 56.70 = 98.86 - 10.54(4) = 56.70

TimeTime Electrical Power Electrical Power YearYear Period (x)Period (x) DemandDemand xx22 xyxy

19991999 11 7474 11 747420002000 22 7979 44 15815820012001 33 8080 99 24024020022002 44 9090 1616 36036020032003 55 105105 2525 52552520042004 66 142142 3636 85285220052005 77 122122 4949 854854

xx = 28 = 28 yy = 692 = 692 xx22 = 140 = 140 xyxy = 3,063 = 3,063xx = 4 = 4 yy = 98.86 = 98.86

The trend line is

y = 56.70 + 10.54x^



| | | | | | | | |20012001 20022002 20032003 20042004 20052005 20062006 20072007 20082008 20092009

160 160 –

150 150 –

140 140 –

130 130 –

120 120 –

110 110 –

100 100 –

90 90 –

80 80 –

70 70 –

60 60 –

50 50 –

YearYear

Po

wer

dem

and

Po

wer

dem

and

Trend line,Trend line,y y = 56.70 + 10.54x= 56.70 + 10.54x^̂


Seasonal Variations In DataSeasonal Variations In Data

The multiplicative The multiplicative seasonal model seasonal model can adjust trend can adjust trend data for seasonal data for seasonal variations in variations in demanddemand


Seasonal Variations In DataSeasonal Variations In Data

1.1. Find average historical demand for each Find average historical demand for each season season

2.2. Compute the average demand over all Compute the average demand over all seasons seasons

3.3. Compute a seasonal index for each season Compute a seasonal index for each season

4.4. Estimate next year’s total demandEstimate next year’s total demand

5.5. Divide this estimate of total demand by the Divide this estimate of total demand by the number of seasons, then multiply it by the number of seasons, then multiply it by the seasonal index for that seasonseasonal index for that season

Steps in the process:Steps in the process:


Seasonal Index ExampleSeasonal Index Example

JanJan 8080 8585 105105 9090 9494

FebFeb 7070 8585 8585 8080 9494

MarMar 8080 9393 8282 8585 9494

AprApr 9090 9595 115115 100100 9494

MayMay 113113 125125 131131 123123 9494

JunJun 110110 115115 120120 115115 9494

JulJul 100100 102102 113113 105105 9494

AugAug 8888 102102 110110 100100 9494

SeptSept 8585 9090 9595 9090 9494

OctOct 7777 7878 8585 8080 9494

NovNov 7575 7272 8383 8080 9494

DecDec 8282 7878 8080 8080 9494

DemandDemand AverageAverage AverageAverage Seasonal Seasonal MonthMonth 20052005 20062006 20072007 2005-20072005-2007 MonthlyMonthly IndexIndex



JanJan 8080 8585 105105 9090 9494

FebFeb 7070 8585 8585 8080 9494

MarMar 8080 9393 8282 8585 9494

AprApr 9090 9595 115115 100100 9494

MayMay 113113 125125 131131 123123 9494

JunJun 110110 115115 120120 115115 9494

JulJul 100100 102102 113113 105105 9494

AugAug 8888 102102 110110 100100 9494

SeptSept 8585 9090 9595 9090 9494

OctOct 7777 7878 8585 8080 9494

NovNov 7575 7272 8383 8080 9494

DecDec 8282 7878 8080 8080 9494


0.9570.957

Seasonal index = average 2005-2007 monthly demand

average monthly demand

= 90/94 = .957



JanJan 8080 8585 105105 9090 9494 0.9570.957

FebFeb 7070 8585 8585 8080 9494 0.8510.851

MarMar 8080 9393 8282 8585 9494 0.9040.904

AprApr 9090 9595 115115 100100 9494 1.0641.064

MayMay 113113 125125 131131 123123 9494 1.3091.309

JunJun 110110 115115 120120 115115 9494 1.2231.223

JulJul 100100 102102 113113 105105 9494 1.1171.117

AugAug 8888 102102 110110 100100 9494 1.0641.064

SeptSept 8585 9090 9595 9090 9494 0.9570.957

OctOct 7777 7878 8585 8080 9494 0.8510.851

NovNov 7575 7272 8383 8080 9494 0.8510.851

DecDec 8282 7878 8080 8080 9494 0.8510.851




JanJan 8080 8585 105105 9090 9494 0.9570.957

FebFeb 7070 8585 8585 8080 9494 0.8510.851

MarMar 8080 9393 8282 8585 9494 0.9040.904

AprApr 9090 9595 115115 100100 9494 1.0641.064

MayMay 113113 125125 131131 123123 9494 1.3091.309

JunJun 110110 115115 120120 115115 9494 1.2231.223

JulJul 100100 102102 113113 105105 9494 1.1171.117

AugAug 8888 102102 110110 100100 9494 1.0641.064

SeptSept 8585 9090 9595 9090 9494 0.9570.957

OctOct 7777 7878 8585 8080 9494 0.8510.851

NovNov 7575 7272 8383 8080 9494 0.8510.851

DecDec 8282 7878 8080 8080 9494 0.8510.851


Expected annual demand = 1,200

Jan x .957 = 961,200

12

Feb x .851 = 851,200

12

Forecast for 2008



140 140 –

130 130 –

120 120 –

110 110 –

100 100 –

90 90 –

80 80 –

70 70 –| | | | | | | | | | | |

JJ FF MM AA MM JJ JJ AA SS OO NN DD

TimeTime

Dem

and

Dem

and

2008 Forecast2008 Forecast

2007 Demand 2007 Demand

2006 Demand2006 Demand

2005 Demand2005 Demand


Associative ForecastingAssociative Forecasting

Used when changes in one or more Used when changes in one or more independent variables can be used to predict independent variables can be used to predict

the changes in the dependent variablethe changes in the dependent variable

Most common technique is linear Most common technique is linear regression analysisregression analysis

We apply this technique just as we did We apply this technique just as we did in the time series examplein the time series example


Associative ForecastingAssociative Forecasting

Forecasting an outcome based on predictor Forecasting an outcome based on predictor variables using the least squares techniquevariables using the least squares technique

y y = = a a + + bxbx^̂

where ywhere y= computed value of the = computed value of the variable to be predicted (dependent variable to be predicted (dependent variable)variable)aa= y-axis intercept= y-axis interceptbb= slope of the regression line= slope of the regression linexx= the independent variable though = the independent variable though to predict the value of the to predict the value of the dependent variabledependent variable

^̂


Associative Forecasting Associative Forecasting ExampleExample

SalesSales Local PayrollLocal Payroll($ millions), y($ millions), y ($ billions), x($ billions), x

2.02.0 113.03.0 332.52.5 442.02.0 222.02.0 113.53.5 77

4.0 –

3.0 –

2.0 –

1.0 –

| | | | | | |0 1 2 3 4 5 6 7

Sal

es

Area payroll



Sales, y Payroll, x x2 xy

2.0 1 1 2.03.0 3 9 9.02.5 4 16 10.02.0 2 4 4.02.0 1 1 2.03.5 7 49 24.5

∑y = 15.0 ∑x = 18 ∑x2 = 80 ∑xy = 51.5

xx = = ∑∑xx/6 = 18/6 = 3/6 = 18/6 = 3

yy = = ∑∑yy/6 = 15/6 = 2.5/6 = 15/6 = 2.5

bb = = = .25 = = = .25∑∑xy - nxyxy - nxy

∑∑xx22 - nx - nx22

51.5 - (6)(3)(2.5)51.5 - (6)(3)(2.5)

80 - (6)(380 - (6)(322))

aa = = yy - - bbx = 2.5 - (.25)(3) = 1.75x = 2.5 - (.25)(3) = 1.75



4.0 –

3.0 –

2.0 –

1.0 –

| | | | | | |0 1 2 3 4 5 6 7

Sal

es

Area payroll

y y = 1.75 + .25= 1.75 + .25xx^̂ Sales Sales = 1.75 + .25(= 1.75 + .25(payrollpayroll))

If payroll next year If payroll next year is estimated to be is estimated to be $6$6 billion, then: billion, then:

Sales Sales = 1.75 + .25(6)= 1.75 + .25(6)SalesSales = $3,250,000 = $3,250,000

3.25


Standard Error of the Standard Error of the EstimateEstimate

A forecast is just a point estimate of a A forecast is just a point estimate of a future valuefuture value

This point is This point is actually the actually the mean of a mean of a probability probability distributiondistribution


4.0 –

3.0 –

2.0 –

1.0 –

| | | | | | |0 1 2 3 4 5 6 7

Sal

es

Area payroll

3.25



wherewhere yy == y-value of each data y-value of each data pointpoint

yycc == computed value of the computed value of the dependent variable, from the dependent variable, from the regression equationregression equation

nn == number of data pointsnumber of data points

SSy,xy,x = =∑∑((y - yy - ycc))22

n n - 2- 2



Computationally, this equation is Computationally, this equation is considerably easier to useconsiderably easier to use

We use the standard error to set up We use the standard error to set up prediction intervals around the prediction intervals around the

point estimatepoint estimate

SSy,xy,x = =∑∑yy22 - a - a∑∑y - by - b∑∑xyxy

n n - 2- 2



4.0 –

3.0 –

2.0 –

1.0 –

| | | | | | |0 1 2 3 4 5 6 7

Sal

es

Area payroll

3.25

SSy,xy,x = = = =∑∑yy22 - a - a∑∑y - by - b∑∑xyxyn n - 2- 2

39.5 - 1.75(15) - .25(51.5)39.5 - 1.75(15) - .25(51.5)6 - 26 - 2

SSy,xy,x = = .306.306

The standard error The standard error of the estimate is of the estimate is $306,000$306,000 in sales in sales


How strong is the linear How strong is the linear relationship between the relationship between the variables?variables?

Correlation does not necessarily Correlation does not necessarily imply causality!imply causality!

Coefficient of correlation, r, Coefficient of correlation, r, measures degree of associationmeasures degree of association Values range from Values range from -1-1 to to +1+1

CorrelationCorrelation


Correlation CoefficientCorrelation Coefficient

r = r = nnxyxy - - xxy y

[[nnxx22 - ( - (xx))22][][nnyy22 - ( - (yy))22]]


Correlation CoefficientCorrelation Coefficient

r = r = nnxyxy - - xxy y

[[nnxx22 - ( - (xx))22][][nnyy22 - ( - (yy))22]]

y

x(a) Perfect positive correlation: r = +1

y

x(b) Positive correlation: 0 < r < 1

y

x(c) No correlation: r = 0

y

x(d) Perfect negative correlation: r = -1


Coefficient of Determination, rCoefficient of Determination, r22, , measures the percent of change in measures the percent of change in y predicted by the change in xy predicted by the change in x Values range from Values range from 00 to to 11

Easy to interpretEasy to interpret

CorrelationCorrelation

For the Nodel Construction example:For the Nodel Construction example:

r r = .901= .901

rr22 = .81 = .81


Multiple Regression Multiple Regression AnalysisAnalysis

If more than one independent variable is to be If more than one independent variable is to be used in the model, linear regression can be used in the model, linear regression can be

extended to multiple regression to extended to multiple regression to accommodate several independent variablesaccommodate several independent variables

y y = = a a + + bb11xx11 + b + b22xx22 … …^̂

Computationally, this is quite Computationally, this is quite complex and generally done on the complex and generally done on the

computercomputer


Multiple Regression Multiple Regression AnalysisAnalysis

y y = 1.80 + .30= 1.80 + .30xx11 - 5.0 - 5.0xx22^̂

In the Nodel example, including interest rates in In the Nodel example, including interest rates in the model gives the new equation:the model gives the new equation:

An improved correlation coefficient of r An improved correlation coefficient of r = .96= .96 means this model does a better job of predicting means this model does a better job of predicting the change in construction salesthe change in construction sales

Sales Sales = 1.80 + .30(6) - 5.0(.12) = 3.00= 1.80 + .30(6) - 5.0(.12) = 3.00Sales Sales = $3,000,000= $3,000,000


Measures how well the forecast is Measures how well the forecast is predicting actual valuespredicting actual values

Ratio of running sum of forecast errors Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)(RSFE) to mean absolute deviation (MAD) Good tracking signal has low valuesGood tracking signal has low values

If forecasts are continually high or low, the If forecasts are continually high or low, the forecast has a bias errorforecast has a bias error

Monitoring and Controlling Monitoring and Controlling ForecastsForecasts

Tracking SignalTracking Signal


Monitoring and Controlling Monitoring and Controlling ForecastsForecasts

Tracking Tracking signalsignal

RSFERSFEMADMAD==

Tracking Tracking signalsignal ==

∑∑(Actual demand in (Actual demand in period i - period i -

Forecast demand Forecast demand in period i)in period i)

∑∑|Actual - Forecast|/n|Actual - Forecast|/n))


Tracking SignalTracking Signal

Tracking signalTracking signal

++

00 MADs MADs

––

Upper control limitUpper control limit

Lower control limitLower control limit

TimeTime

Signal exceeding limitSignal exceeding limit

Acceptable Acceptable rangerange

© 2008 Prentice Hall, Inc.4 – 1 Outline – Continued Time-Series Forecasting Decomposition of a Time Series Naive Approach Moving Averages Exponential.

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