Forecasting

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ForecastingForecasting Quantitative Approaches to ForecastingQuantitative Approaches to Forecasting The Components of a Time SeriesThe Components of a Time Series Measures of Forecast AccuracyMeasures of Forecast Accuracy Using Smoothing Methods in Forecasting Using Smoothing Methods in Forecasting Using Trend Projection in Forecasting Using Trend Projection in Forecasting Using Regression Analysis in ForecastingUsing Regression Analysis in Forecasting Qualitative Approaches to ForecastingQualitative Approaches to Forecasting

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Quantitative Quantitative Approaches to Approaches to

ForecastingForecasting Quantitative methodsQuantitative methods are based on an analysis of are based on an analysis of historical data concerning one or more time series.historical data concerning one or more time series.

A A time seriestime series is a set of observations measured at is a set of observations measured at successive points in time or over successive periods of successive points in time or over successive periods of time.time.

If the historical data used are restricted to past values of If the historical data used are restricted to past values of the series that we are trying to forecast, the procedure the series that we are trying to forecast, the procedure is called a is called a time series methodtime series method..

If the historical data used involve other time series that If the historical data used involve other time series that are believed to be related to the time series that we are are believed to be related to the time series that we are trying to forecast, the procedure is called a trying to forecast, the procedure is called a causal causal methodmethod. .

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Time Series MethodsTime Series Methods Three time series methods are: Three time series methods are:

smoothingsmoothing trend projectiontrend projection

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Components of a Time Components of a Time SeriesSeries

The The trend componenttrend component accounts for the gradual accounts for the gradual shifting of the time series over a long period of time.shifting of the time series over a long period of time.

Any regular pattern of sequences of values above Any regular pattern of sequences of values above and below the trend line is attributable to the and below the trend line is attributable to the cyclical componentcyclical component of the series. of the series.

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Components of a Time Components of a Time SeriesSeries

The The seasonal componentseasonal component of the series accounts of the series accounts for regular patterns of variability within certain for regular patterns of variability within certain time periods, such as over a year.time periods, such as over a year.

The The irregular componentirregular component of the series is caused of the series is caused by short-term, unanticipated and non-recurring by short-term, unanticipated and non-recurring factors that affect the values of the time series. factors that affect the values of the time series. One cannot attempt to predict its impact on the One cannot attempt to predict its impact on the time series in advance.time series in advance.

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Measures of Forecast Measures of Forecast AccuracyAccuracy Mean Squared ErrorMean Squared Error

The average of the squared forecast errors for The average of the squared forecast errors for the historical data is calculated. The forecasting the historical data is calculated. The forecasting method or parameter(s) which minimize this mean method or parameter(s) which minimize this mean squared error is then selected.squared error is then selected.

Mean Absolute DeviationMean Absolute Deviation

The mean of the absolute values of all forecast The mean of the absolute values of all forecast errors is calculated, and the forecasting method or errors is calculated, and the forecasting method or parameter(s) which minimize this measure is parameter(s) which minimize this measure is selected. The mean absolute deviation measure is selected. The mean absolute deviation measure is less sensitive to individual large forecast errors less sensitive to individual large forecast errors than the mean squared error measure.than the mean squared error measure.

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Smoothing MethodsSmoothing Methods In cases in which the time series is fairly stable In cases in which the time series is fairly stable

and has no significant trend, seasonal, or cyclical and has no significant trend, seasonal, or cyclical effects, one can use effects, one can use smoothing methodssmoothing methods to average to average out the irregular components of the time series. out the irregular components of the time series.

Four common smoothing methods are:Four common smoothing methods are: Moving averagesMoving averages Centered moving averagesCentered moving averages Weighted moving averagesWeighted moving averages Exponential smoothingExponential smoothing

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Smoothing MethodsSmoothing Methods Moving Average MethodMoving Average Method

The The moving average methodmoving average method consists of computing consists of computing an average of the most recent an average of the most recent nn data values for the data values for the series and using this average for forecasting the series and using this average for forecasting the value of the time series for the next period.value of the time series for the next period.

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Sales of Comfort brand headache Sales of Comfort brand headache medicine formedicine for

the past ten weeks at Rosco Drugsthe past ten weeks at Rosco Drugs

are shown on the next slide. If are shown on the next slide. If

Rosco Drugs uses a 3-periodRosco Drugs uses a 3-period

moving average to forecast sales,moving average to forecast sales,

what is the forecast for Week 11?what is the forecast for Week 11?

Example: Rosco DrugsExample: Rosco Drugs

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Past SalesPast Sales

WeekWeek SalesSales WeekWeek SalesSales 1 110 6 1201 110 6 120 2 115 7 1302 115 7 130 3 125 8 1153 125 8 115 4 120 9 1104 120 9 110 5 125 10 1305 125 10 130

Example: Rosco DrugsExample: Rosco Drugs

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Example: Rosco DrugsExample: Rosco Drugs Excel Spreadsheet Showing Input DataExcel Spreadsheet Showing Input Data

A B C1 Robert's Drugs2

3 Week (t ) Salest Forect+1

4 1 1105 2 1156 3 1257 4 1208 5 1259 6 120

10 7 13011 8 11512 9 11013 10 130

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Example: Rosco DrugsExample: Rosco Drugs Steps to Moving Average Using ExcelSteps to Moving Average Using Excel

Step 1:Step 1: Select the Select the ToolsTools pull-down menu. pull-down menu.

Step 2:Step 2: Select the Select the Data AnalysisData Analysis option. option.

Step 3:Step 3: When the Data Analysis Tools dialog When the Data Analysis Tools dialog appears, choose Mappears, choose Moving Averageoving Average..

Step 4:Step 4: When the Moving Average dialog box When the Moving Average dialog box appears:appears:

Enter B4:B13 in the Enter B4:B13 in the Input RangeInput Range box. box.

Enter 3 in the Enter 3 in the IntervalInterval box. box.

Enter C4 in the Enter C4 in the Output RangeOutput Range box. box.

Select Select OKOK..

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Example: Rosco DrugsExample: Rosco Drugs Spreadsheet Showing Results Using Spreadsheet Showing Results Using nn = 3 = 3

A B C1 Robert's Drugs2

3 Week (t ) Salest Forect+1

4 1 110 #N/A5 2 115 #N/A6 3 125 116.77 4 120 120.08 5 125 123.39 6 120 121.7

10 7 130 125.011 8 115 121.712 9 110 118.313 10 130 118.3

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Smoothing MethodsSmoothing Methods Centered Moving Average MethodCentered Moving Average Method

The The centered moving average methodcentered moving average method consists consists of computing an average of of computing an average of n n periods' data and periods' data and associating it with the midpoint of the periods. For associating it with the midpoint of the periods. For example, the average for periods 5, 6, and 7 is example, the average for periods 5, 6, and 7 is associated with period 6. This methodology is associated with period 6. This methodology is useful in the process of computing season indexes.useful in the process of computing season indexes.

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Smoothing MethodsSmoothing Methods

Weighted Moving Average MethodWeighted Moving Average Method

In the In the weighted moving average methodweighted moving average method for for computing the average of the most recent computing the average of the most recent n n periods, periods, the more recent observations are typically given more the more recent observations are typically given more weight than older observations. For convenience, the weight than older observations. For convenience, the weights usually sum to 1.weights usually sum to 1.

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Smoothing MethodsSmoothing Methods Exponential SmoothingExponential Smoothing

Using Using exponential smoothingexponential smoothing, the forecast for , the forecast for the next period is equal to the forecast for the the next period is equal to the forecast for the current period plus a proportion (current period plus a proportion () of the ) of the forecast error in the current period.forecast error in the current period.

Using exponential smoothing, the forecast is Using exponential smoothing, the forecast is calculated by: calculated by:

[the actual value for the current period] [the actual value for the current period] ++

(1- (1- )[the forecasted value for the current )[the forecasted value for the current period], period],

where the smoothing constant, where the smoothing constant, , is a number , is a number between 0 and 1.between 0 and 1.

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Trend ProjectionTrend Projection If a time series exhibits a linear trend, the If a time series exhibits a linear trend, the

method of method of least squaresleast squares may be used to may be used to determine a trend line (projection) for future determine a trend line (projection) for future forecasts. forecasts.

Least squares, also used in regression analysis, Least squares, also used in regression analysis, determines the unique determines the unique trend line forecasttrend line forecast which which minimizes the mean square error between the minimizes the mean square error between the trend line forecasts and the actual observed trend line forecasts and the actual observed values for the time series.values for the time series.

The independent variable is the time period and The independent variable is the time period and the dependent variable is the actual observed the dependent variable is the actual observed value in the time series.value in the time series.

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Trend ProjectionTrend Projection Using the method of least squares, the formula for the Using the method of least squares, the formula for the

trend projection is: trend projection is: TTtt = = bb00 + + bb11tt. .

where: where: TTtt = trend forecast for time period = trend forecast for time period tt

bb1 1 = slope of the trend line= slope of the trend line

bb00 = trend line projection for time 0 = trend line projection for time 0

bb11 = = nntYtYtt - - t t YYtt

nnt t 22 - ( - (t t ))22

where: where: YYtt = observed value of the time series at time = observed value of the time series at time

period period tt

= average of the observed values for = average of the observed values for YYtt

= average time period for the = average time period for the nn observationsobservations

0 1b Y b t 0 1b Y b t

YYtt

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If Rosco Drugs uses exponential smoothing toIf Rosco Drugs uses exponential smoothing to

forecast sales, which value for the smoothing forecast sales, which value for the smoothing constantconstant

, .1 or .8, gives better forecasts?, .1 or .8, gives better forecasts?

WeekWeek SalesSales WeekWeek SalesSales 1 110 6 1201 110 6 120 2 115 7 1302 115 7 130 3 125 8 1153 125 8 115 4 120 9 1104 120 9 110 5 125 10 1305 125 10 130

Example: Rosco Drugs Example: Rosco Drugs (B)(B)

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Example: Rosco Drugs Example: Rosco Drugs (B)(B)

Exponential SmoothingExponential Smoothing

To evaluate the two smoothing constants, determine To evaluate the two smoothing constants, determine how the forecasted values would compare with the actual how the forecasted values would compare with the actual historical values in each case. historical values in each case.

Let: Let: YYtt = actual sales in week = actual sales in week tt

FFt t = forecasted sales in week = forecasted sales in week tt

FF11 = = YY11 = 110 = 110

For other weeks, For other weeks, FFtt+1+1 = .1 = .1YYtt + .9 + .9FFtt

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Example: Rosco Drugs Example: Rosco Drugs (B)(B) Exponential Smoothing (Exponential Smoothing ( = .1, 1 - = .1, 1 - = .9) = .9)

FF11 = 110 = 110

FF2 2 = .1= .1YY11 + .9 + .9FF11 = .1(110) + .9(110) = 110 = .1(110) + .9(110) = 110

FF33 = .1 = .1YY22 + .9 + .9FF22 = .1(115) + .9(110) = 110.5 = .1(115) + .9(110) = 110.5

FF44 = .1 = .1YY33 + .9 + .9FF33 = .1(125) + .9(110.5) = 111.95 = .1(125) + .9(110.5) = 111.95

FF55 = .1 = .1YY44 + .9 + .9FF44 = .1(120) + .9(111.95) = 112.76 = .1(120) + .9(111.95) = 112.76

FF66 = .1 = .1YY55 + .9 + .9FF55 = .1(125) + .9(112.76) = 113.98 = .1(125) + .9(112.76) = 113.98

FF77 = .1 = .1YY66 + .9 + .9FF66 = .1(120) + .9(113.98) = 114.58 = .1(120) + .9(113.98) = 114.58

FF88 = .1 = .1YY77 + .9 + .9FF77 = .1(130) + .9(114.58) = 116.12 = .1(130) + .9(114.58) = 116.12

FF99 = .1 = .1YY88 + .9 + .9FF88 = .1(115) + .9(116.12) = 116.01 = .1(115) + .9(116.12) = 116.01

FF1010= .1= .1YY99 + .9 + .9FF99 = .1(110) + .9(116.01) = 115.41 = .1(110) + .9(116.01) = 115.41

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Example: Rosco Drugs Example: Rosco Drugs (B)(B)

Exponential Smoothing (Exponential Smoothing ( = .8, 1 - = .8, 1 - = .2) = .2)

FF11 = 110 = 110

FF22 = .8(110) + .2(110) = 110 = .8(110) + .2(110) = 110

FF33 = .8(115) + .2(110) = 114 = .8(115) + .2(110) = 114

FF44 = .8(125) + .2(114) = 122.80 = .8(125) + .2(114) = 122.80

FF55 = .8(120) + .2(122.80) = 120.56 = .8(120) + .2(122.80) = 120.56

FF66 = .8(125) + .2(120.56) = 124.11 = .8(125) + .2(120.56) = 124.11

FF77 = .8(120) + .2(124.11) = 120.82 = .8(120) + .2(124.11) = 120.82

FF88 = .8(130) + .2(120.82) = 128.16 = .8(130) + .2(120.82) = 128.16

FF99 = .8(115) + .2(128.16) = 117.63 = .8(115) + .2(128.16) = 117.63

FF1010= .8(110) + .2(117.63) = 111.53= .8(110) + .2(117.63) = 111.53

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Example: Rosco Drugs Example: Rosco Drugs (B)(B)

Mean Squared ErrorMean Squared Error

In order to determine which smoothing constant In order to determine which smoothing constant gives the better performance, calculate, for each, the gives the better performance, calculate, for each, the mean squared error for the nine weeks of forecasts, mean squared error for the nine weeks of forecasts, weeks 2 through 10 by:weeks 2 through 10 by:

[([(YY22--FF22))22 + ( + (YY33--FF33))22 + ( + (YY44--FF44))22 + . . . + ( + . . . + (YY1010--FF1010))22]/9]/9

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Example: Rosco Drugs Example: Rosco Drugs (B)(B)

αα = .1 = .1 αα = .8 = .8

Week Week YYtt FFtt ( (YYtt - - FFtt))22 FFt t ((YYtt - - FFtt))22

1 110 1 110 2 115 110.00 25.00 2 115 110.00 25.00 110.00 25.00110.00 25.00 3 125 110.50 210.25 3 125 110.50 210.25 114.00 121.00114.00 121.00 4 120 111.95 64.80 4 120 111.95 64.80 122.80 7.84122.80 7.84 5 125 112.76 149.94 5 125 112.76 149.94 120.56 19.71120.56 19.71 6 120 113.98 36.25 6 120 113.98 36.25 124.11 16.91124.11 16.91 7 130 114.58 237.73 7 130 114.58 237.73 120.82 84.23120.82 84.23 8 115 116.12 1.26 8 115 116.12 1.26 128.16 173.30128.16 173.30 9 110 116.01 36.12 9 110 116.01 36.12 117.63 58.26117.63 58.26 10 130 115.41 212.87 10 130 115.41 212.87 111.53 341.27111.53 341.27

Sum 974.22 Sum 974.22 Sum 847.52Sum 847.52 MSE Sum/9 MSE Sum/9 Sum/9Sum/9

108.25108.25108.25108.25 94.1794.1794.1794.17

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Example: Rosco Drugs Example: Rosco Drugs (B)(B)

Excel Spreadsheet Showing Input DataExcel Spreadsheet Showing Input Data

A B C1 Robert's Drugs23 Week Sales4 1 1105 2 1156 3 1257 4 1208 5 1259 6 120

10 7 13011 8 11512 9 11013 10 130

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Example: Rosco Drugs Example: Rosco Drugs (B)(B) Steps to Exponential Smoothing Using ExcelSteps to Exponential Smoothing Using Excel

Step 1:Step 1: Select the Select the ToolsTools pull-down menu. pull-down menu.

Step 2:Step 2: Select the Select the Data AnalysisData Analysis option. option.

Step 3:Step 3: When the Data Analysis Tools dialog When the Data Analysis Tools dialog appears, choose appears, choose Exponential SmoothingExponential Smoothing..

Step 4:Step 4: When the Exponential Smoothing dialog When the Exponential Smoothing dialog box box appears:appears:

Enter B4:B13 in the Enter B4:B13 in the Input RangeInput Range box. box.

Enter 0.9 (for Enter 0.9 (for = 0.1) in = 0.1) in Damping FactorDamping Factor box.box.

Enter C4 in the Enter C4 in the Output RangeOutput Range box. box.

Select Select OKOK..

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Example: Rosco Drugs Example: Rosco Drugs (B)(B) Spreadsheet Showing Results Using Spreadsheet Showing Results Using = 0.1 = 0.1

A B C1 Robert's Drugs2 = 0.1

3 Week (t ) Salest Forect +1

4 1 110 #N/A5 2 115 110.06 3 125 110.57 4 120 112.08 5 125 112.89 6 120 114.0

10 7 130 114.611 8 115 116.112 9 110 116.013 10 130 115.4

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Example: Rosco Drugs Example: Rosco Drugs (B)(B) Repeating the Process for Repeating the Process for = 0.8 = 0.8

Step 4: When the Exponential Smoothing dialog Step 4: When the Exponential Smoothing dialog box box appears:appears:

Enter B4:B13 in the Enter B4:B13 in the Input RangeInput Range box. box.

Enter 0.2 (for Enter 0.2 (for = 0.8) in = 0.8) in Damping FactorDamping Factor box.box.

Enter D4 in the Enter D4 in the Output RangeOutput Range box. box.

Select Select OKOK..

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Example: Rosco Drugs Example: Rosco Drugs (B)(B) Spreadsheet Results for Spreadsheet Results for = 0.1 and = 0.1 and = 0.8 = 0.8

A B C D1 Robert's Drugs2 = 0.1 = 0.8

3 Week (t ) Salest Forect +1 Forect +1

4 1 110 #N/A #N/A5 2 115 110.0 110.06 3 125 110.5 114.07 4 120 112.0 122.88 5 125 112.8 120.69 6 120 114.0 124.1

10 7 130 114.6 120.811 8 115 116.1 128.212 9 110 116.0 117.613 10 130 115.4 111.5

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The number of plumbing repair jobs performed The number of plumbing repair jobs performed byby

Auger's Plumbing Service in each of the last nineAuger's Plumbing Service in each of the last nine

months is listed on the next slide. Forecastmonths is listed on the next slide. Forecast

the number of repair jobs Auger's willthe number of repair jobs Auger's will

perform in December using the leastperform in December using the least

squares method. squares method.

Example: Auger’s Example: Auger’s Plumbing ServicePlumbing Service

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MonthMonth JobsJobs MonthMonth JobsJobs MonthMonth JobsJobs

March 353 June 374 March 353 June 374 September 399September 399

April 387 July 396 October April 387 July 396 October 412 412

May 342 August 409 May 342 August 409 November 408November 408

Example: Auger’s Example: Auger’s Plumbing ServicePlumbing Service

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Example: Auger’s Example: Auger’s Plumbing ServicePlumbing Service

Trend ProjectionTrend Projection

(month) (month) tt YYtt tYtYtt t t 22

(Mar.) 1 353 353 1 (Mar.) 1 353 353 1 (Apr.) 2 387 774 4(Apr.) 2 387 774 4 (May) 3 342 1026 9(May) 3 342 1026 9 (June) 4 374 1496 16(June) 4 374 1496 16 (July) 5 396 1980 25(July) 5 396 1980 25 (Aug.) 6 409 2454 36(Aug.) 6 409 2454 36 (Sep.) 7 399 2793 49(Sep.) 7 399 2793 49 (Oct.) 8 412 3296 64(Oct.) 8 412 3296 64 (Nov.) 9 408 3672 81(Nov.) 9 408 3672 81

Sum 45 3480 17844 285Sum 45 3480 17844 285

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Example: Auger’s Example: Auger’s Plumbing ServicePlumbing Service

Trend Projection (continued)Trend Projection (continued)

= 45/9 = 5 = 3480/9 = 386.667= 45/9 = 5 = 3480/9 = 386.667

nntYtYtt - - t t YYtt (9)(17844) - (45)(3480) (9)(17844) - (45)(3480) bb11 = = = = = 7.4 = 7.4 nnt t 22 - ( - (tt))22 (9)(285) - (45) (9)(285) - (45)22

= 386.667 - 7.4(5) = 349.667= 386.667 - 7.4(5) = 349.667

TT1010 = 349.667 + (7.4)(10) = = 349.667 + (7.4)(10) =

423.667423.667423.667423.667

0 1b Y b t 0 1b Y b t

YYtt

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Example: Auger’s Example: Auger’s Plumbing ServicePlumbing Service

Excel Spreadsheet Showing Input DataExcel Spreadsheet Showing Input DataA B C

1 Auger's Plumbing Service23 Month Calls4 1 3535 2 3876 3 3427 4 3748 5 3969 6 409

10 7 39911 8 41212 9 40813

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Example: Auger’s Example: Auger’s Plumbing ServicePlumbing Service

Steps to Trend Projection Using ExcelSteps to Trend Projection Using ExcelStep 1:Step 1: Select an empty cell (B13) in the worksheet. Select an empty cell (B13) in the worksheet.

Step 2:Step 2: Select the Select the InsertInsert pull-down menu. pull-down menu.

Step 3:Step 3: Choose the Choose the FunctionFunction option. option.

Step 4:Step 4: When the Paste Function dialog box appears: When the Paste Function dialog box appears:Choose Choose StatisticalStatistical in Function Category box. in Function Category box.

Choose Choose ForecastForecast in the Function Name box. in the Function Name box.

Select Select OKOK..

more . . . . . . .more . . . . . . .

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Example: Auger’s Example: Auger’s Plumbing ServicePlumbing Service

Steps to Trend Projecting Using Excel Steps to Trend Projecting Using Excel (continued)(continued)Step 5:Step 5: When the Forecast dialog box appears: When the Forecast dialog box appears:

Enter 10 in the Enter 10 in the xx box (for month 10). box (for month 10).

Enter B4:B12 in the Enter B4:B12 in the Known y’sKnown y’s box. box.

Enter A4:A12 in the Enter A4:A12 in the Known x’sKnown x’s box. box.

Select Select OKOK..

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Example: Auger’s Example: Auger’s Plumbing ServicePlumbing Service

Spreadsheet with Trend Projection for Month 10Spreadsheet with Trend Projection for Month 10

A B C1 Auger's Plumbing Service23 Month Calls4 1 3535 2 3876 3 3427 4 3748 5 3969 6 409

10 7 39911 8 41212 9 40813 10 423.667 Projected

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Example: Auger’s Example: Auger’s Plumbing Service (B)Plumbing Service (B)

Forecast for December (Month 10) using aForecast for December (Month 10) using a

three-period (three-period (nn = 3) weighted moving average with = 3) weighted moving average with

weights of .6, .3, and .1. weights of .6, .3, and .1.

Then, compare this Month 10 weighted movingThen, compare this Month 10 weighted moving

average forecast with the Month 10 trend projectionaverage forecast with the Month 10 trend projection

forecast.forecast.

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Example: Auger’s Example: Auger’s Plumbing Service (B)Plumbing Service (B)

Three-Month Weighted Moving AverageThree-Month Weighted Moving Average

The forecast for December will be the weighted The forecast for December will be the weighted average of the preceding three months: September, average of the preceding three months: September, October, and November.October, and November.

FF1010 = .1 = .1YYSep.Sep. + .3 + .3YYOct.Oct. + .6 + .6YYNov.Nov.

= .1(399) + .3(412) + .6(408) = .1(399) + .3(412) + .6(408)

= =

Trend ProjectionTrend Projection

FF1010 = 423.7 (from earlier slide) = 423.7 (from earlier slide)

408.3408.3408.3408.3

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Example: Auger’s Example: Auger’s Plumbing Service (B)Plumbing Service (B)

ConclusionConclusion

Due to the positive trend component in the time Due to the positive trend component in the time series, the trend projection produced a forecast that series, the trend projection produced a forecast that is more in tune with the trend that exists. The is more in tune with the trend that exists. The weighted moving average, even with heavy (.6) placed weighted moving average, even with heavy (.6) placed on the current period, produced a forecast that is on the current period, produced a forecast that is lagging behind the changing data. lagging behind the changing data.