Forecasting Techniques

Dr. C. Lightner Fayetteville State University

1

FORECASTING TECHNIQUES

Chapter 16

Qualitative Approaches to Forecasting

Quantitative Approaches to Forecasting

The Components of a Time Series

Using Smoothing Methods in Forecasting

Measures of Forecast Accuracy

Using Trend Projection in Forecasting

Using Regression Analysis in Forecasting


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Forecasting Introduction

An essential aspect of managing any organization is planning for the future.

Organizations employ forecasting techniques to determine future inventory, costs, capacities, and interest rate changes.

There are two basic approaches to forecasting:

-Qualitative

-Quantitative


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Qualitative Approaches to Forecasting

Delphi Approach– A panel of experts, each of whom is physically separated from

the others and is anonymous, is asked to respond to a sequential series of questionnaires.

– After each questionnaire, the responses are tabulated and the information and opinions of the entire group are made known to each of the other panel members so that they may revise their previous forecast response.

– The process continues until some degree of consensus is achieved.


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Qualitative Approaches (continued)

Scenario Writing– Scenario writing consists of developing a conceptual scenario

of the future based on a well defined set of assumptions. – After several different scenarios have been developed, the

decision maker determines which is most likely to occur in the future and makes decisions accordingly.


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Qualitative Approaches (continued)

Subjective or Interactive Approaches– These techniques are often used by committees or panels

seeking to develop new ideas or solve complex problems.– They often involve "brainstorming sessions". – It is important in such sessions that any ideas or opinions be

permitted to be presented without regard to its relevancy and without fear of criticism.


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Quantitative Approaches to Forecasting

Quantitative methods are based on an analysis of historical data concerning one or more time series.A time series is a set of observations measured at successive points in time or over successive periods of time.If the historical data used are restricted to past values of the series that we are trying to forecast, the procedure is called a time series method.If the historical data used involve other time series that are believed to be related to the time series that we are trying to forecast, the procedure is called a causal method. Quantitative approaches are generally preferred. In this chapter we will focus on quantitative approaches to forecasting.


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Time Series Data

Time Series Data is usually plotted on a graph to determine the various characteristics or components of the time series data.

There are 4 Major Components: Trend, Cyclical, Seasonal, and Irregular Components.


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Components of a Time Series

The trend component accounts for the gradual shifting of the time series over a long period of time.Any regular pattern of sequences of values above and below the trend line is attributable to the cyclical component of the series.The seasonal component of the series accounts for regular patterns of variability within certain time periods, such as over a year.The irregular component of the series is caused by short-term, unanticipated and non-recurring factors that affect the values of the time series. One cannot attempt to predict its impact on the time series in advance.


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Time Series Data

We will learn the following Forecasting Approaches:

Smoothing

Trend Projections


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Excel Instructions for Drawing a Scatter Plot

1. Enter data in the Excel spreadsheet.2. Click on Insert on the toolbar and then click on the Chart tab. The

Chart Wizard will appear. In step 1 on select the XY (scatter) chart type and then click next.

3. In step 2 specify the cells where your data is located in the data range box.

4. In step 3 you can give your chart a title and label your axes. In step 4 specify where you want the chart to be placed.


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During the past ten weeks, sales of cases of Comfort brand headache medicine at Robert's Drugs have been as follows:

Week Sales Week Sales 1 110 6 120 2 115 7 130 3 125 8 115 4 120 9 110 5 125 10 130

Plot this data.

Example: Robert’s Drugs


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Plot Robert’s Drugs Example

Excel Spreadsheet Showing Input Data. Specify cells A4:B13 as the Data Range. A B

1 Robert's Drugs2

3 Week (t ) Salest

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

10 7 13011 8 11512 9 11013 10 13014 11


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Plot Robert’s Drugs Example

Robert's Drug Example

105

110

115

120

125

130

135

0 5 10 15

Week, t

Sa

les

I labeled Robert’s DrugExample as The Chart title

I labeled Week, t as My Value (x)axis

I labeled Sales as My Value (y)axis


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Smoothing Methods

In cases in which the time series is fairly stable and has no significant trend, seasonal, or cyclical effects, one can use smoothing methods to average out the irregular components of the time series.

Three common smoothing methods are:– Moving average– Weighted moving average– Exponential smoothing


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Smoothing Methods: Moving Average

Moving Average Method

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


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Robert Drug’s Example: Moving Average

Our scatter plot for Robert’s Drug Sales has no significant trend, seasonal, or cyclical effects. Thus we should employ a smoothing technique for forecasting sales.

Forecast the sales for period 11 using a three period moving average (MA3).


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Example: Robert’s Drugs: Moving Average

Steps to Moving Average Using Excel

Step 1: Select the Tools pull-down menu.

Step 2: Select the Data Analysis option.

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

Step 4: When the Moving Average dialog box appears:

Enter B4:B13 in the Input Range box.

Enter 3 in the Interval box.

Enter C5 in the Output Range box.

Select OK.

This specifies the value of n

This is the column following our data,and one row below whereour data begins.


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Robert’s Drugs: Moving Average

MA3 (Three period Moving average) for Robert’s Drug Example

Ft is the forecast for week t.

F4 (forecast for week 4)=116.7

F11 (forecast for week 11)=118.3

Thus we would forecast the sales for Week 11 to be 118.3

Robert's Drugn=3

Week (t ) Yt Ft

1 1102 115 #N/A3 125 #N/A4 120 116.66675 125 1206 120 123.33337 130 121.66678 115 1259 110 121.666710 130 118.333311 118.3333


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Smoothing Methods: Weighted Moving Average

Weighted Moving Average Method

The weighted moving average method consists of computing a weighted average of the most recent n data values for the series and using this weighted average for forecasting the value of the time series for the next period. The more recent observations are typically given more weight than older observations. For convenience, the weights usually sum to 1.

The regular moving average gives equal weight to past data values when computing a forecast for the next period. The weighted moving average allows different weights to be allocated to past data values.

There is no Excel command for computing this so you must do this manually. You can either manually enter the formulas into excel and apply to all periods or compute value by hand.


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Smoothing Methods: Weighted Moving Average

Use a 3 period weighted moving average to forecast the sales for week 11 giving a weight of 0.6 to the most recent period, 0.3 to the second most recent period, and 0.1 to the third most recent period.

F11 = (0.6)*130 + (0.3)*110 + (0.1)* 115= 122.5

Thus we would forecast the sales for week 11 to be 122.5.

Sales for themost recentperiod

Sales for 2nd most recentperiod

Sales for 3rd most recentperiod


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Smoothing Methods: Exponential Smoothing

Exponential Smoothing– Using exponential smoothing, the forecast for the next

period is equal to the forecast for the current period plus a

proportion () of the forecast error in the current period.

– Using exponential smoothing, the forecast is calculated by:

Ft+1=Yt + (1- )Ft

where: is the smoothing constant (a number between 0 and

1)Ft is the forecast for period t

Ft +1 is the forecast for period t+1

Yt is the actual data value for period t

This is the same as Ft+1 = Ft + α (Yt – Ft)


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Robert’s Drugs: Exponential Smoothing

Forecast the sales for period 11 using Exponential Smoothing α= 0.1.


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Steps to Exponential Smoothing Using ExcelStep 1: Select the Tools pull-down menu.

Step 2: Select the Data Analysis option.

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

Step 4: When the Exponential Smoothing dialog box appears:

Enter B4:B13 in the Input Range box.

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

Enter C4 in the Output Range box.

Select OK.

Damping factoris always 1-α


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F11 = 0.1 * Y10 + .9 F10

= .1 *130 + .9 * 115.4099 = 116.87

Robert's Drugsα=0.1

Week (t ) Salest Ft

1 110 #N/A2 115 1103 125 110.54 120 111.955 125 112.7556 120 113.97957 130 114.58168 115 116.12349 110 116.0111

10 130 115.409911

Thus we would forecast sales for week 11 to be 116.87


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Questions That You Should Be Asking

For the Moving Average technique, how do I determine the best value of n to use for forecasting?

For Exponential Smoothing, how do I determine the best value of α to use?

If I realize that a smoothing technique should be employed, how do you know which smoothing technique is best?

In order to answer the above questions, we need criteria for judging the accuracy of a forecasting technique. Once we select a criterion, the method (or parameter) which provides the best value for our criterion is the best method (or parameter) to use for forecasting our scenario.


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Measures of Forecast Accuracy

Mean Squared Error (MSE)

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

Mean Absolute Deviation (MAD)

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

You may choose either of the above criteria for evaluating the accuracy of a method (or parameter).


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Selecting the best Smoothing Technique for Robert’s Drugs

Determine the smoothing technique that is best for forecasting Robert’s Drug sales: A two period moving average, a three period moving average, exponential smoothing (α=0.1), or exponential smoothing (α=0.2)

Realistically we should have experimented with more values of n for the moving average, and α for exponential smoothing to determine the absolute best parameters to use for our technique.

On the next slide we randomly chose to use the MSE criterion to judge the best technique.


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Robert’s Drugs :Comparing Smoothing Techniques

Double click on the Excel sheet below to enter actual Excel spreadsheet that I created. Clicking on individual cells will provide the formulas that were entered to compute the observed values.

MSE for MA2

Robert's DrugSales n=2 Error

Week (t ) Yt Ft (Yt - Ft) (Yt - Ft)2

1 1102 115 #N/A3 125 112.5 12.5 156.254 120 120 0 05 125 122.5 2.5 6.256 120 122.5 -2.5 6.257 130 122.5 7.5 56.258 115 125 -10 1009 110 122.5 -12.5 156.2510 130 112.5 17.5 306.2511 120

MSE 98.4375


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MSE for MA3

Robert's DrugSales n=3 Error

Week (t ) Yt Ft (Yt - Ft) (Yt - Ft)2

1 1102 115 #N/A3 125 #N/A4 120 116.6667 3.333333 11.111115 125 120 5 256 120 123.3333 -3.33333 11.111117 130 121.6667 8.333333 69.444448 115 125 -10 1009 110 121.6667 -11.6667 136.111110 130 118.3333 11.66667 136.111111 118.3333

MSE 69.84127


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MSE for ExponentialSmoothing α=0.1

Sales α=0.1 ErrorWeek (t ) Yt Ft (Yt - Ft) (Yt - Ft)

2

1 110 #N/A2 115 110 5 253 125 110.5 14.5 210.254 120 111.95 8.05 64.80255 125 112.755 12.245 149.946 120 113.9795 6.0205 36.246427 130 114.5816 15.41845 237.72868 115 116.1234 -1.1234 1.2620169 110 116.0111 -6.01106 36.1327910 130 115.4099 14.59005 212.869611

MSE 108.248


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MSE for ExponentialSmoothing α=0.2

Sales α=0.2 ErrorWeek (t ) Yt Ft (Yt - Ft) (Yt - Ft)

2

1 110 #N/A2 115 110 5 253 125 111 14 1964 120 113.8 6.2 38.445 125 115.04 9.96 99.20166 120 117.032 2.968 8.8090247 130 117.6256 12.3744 153.12588 115 120.1005 -5.10048 26.01499 110 119.0804 -9.08038 82.4533710 130 117.2643 12.73569 162.197911

MSE 87.91584


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Since the three period moving average technique (MA3) provides to lowest MSE value, this is the best smoothing technique to use for forecasting Robert’s Drug Sales.


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Trend Projection

If a time series exhibits a linear trend, the method of least squares may be used to determine a trend line (projection) for future forecasts.

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

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


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Trend Projection

Using the method of least squares, the formula for the trend projection is: Yt = b0 + b1t.

where: Yt = trend forecast for time period t b1 = slope of the trend line

b0 = trend line projection for time 0

b1 = ntYt - t Yt

nt 2 - (t )2

where: Yt = observed value of the time series at time period t

= average of the observed values for Yt

= average time period for the n observations

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

tt

tt YY

tt


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Example: Auger’s Plumbing Service

The number of plumbing repair jobs performed by Auger's Plumbing Service in each of the last nine months are listed below.

Month Jobs Month Jobs Month Jobs March 353 June 374 September 399 April 387 July 396 October 412 May 342 August 409 November 408

Forecast the number of repair jobs Auger's will perform in December using the least squares method.


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Auger’s Plumbing Service: Trend ProjectionTrend Projection

(month) t Yt tYt t 2

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

Sum 45 3480 17844 285


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Trend Projection (continued)

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

ntYt - t Yt (9)(17844) - (45)(3480)

b1 = = = 7.4 n t 2 - ( t)2 (9)(285) - (45)2 = 386.667 - 7.4(5) = 349.667

Thus our trend line is Yt = 349.667 + 7.4 t.

Y10 = 349.667 + (7.4)(10) = 423.667423.667

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

YYtt t

For December t=10


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Auger’s Plumbing Service: Trend Line in Excel

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

Excel Spreadsheet Showing Input Data


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Steps to Trend Projection Using Excel

Step 1: Select an empty cell (B13) in the worksheet.

Step 2: Select the Insert pull-down menu.

Step 3: Choose the Function option.

Step 4: When the Select Category dialog box appears:

Choose Statistical in Function Category box.Choose Forecast in the Function Name box.Select OK.

Step 5: When the Forecast dialog box appears:

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

Enter B4:B12 in the Known y’s box.

Enter A4:A12 in the Known x’s box.

Select OK.


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Spreadsheet Showing Trend Projection for Month 10 Auger's Plumbing Service

Month Calls1 3532 3873 3424 3745 3966 4097 3998 4129 408

10 423.667 Projected


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Roberts Drug Example

Suppose we neglected to plot Robert’s Drug example, and therefore we do not know that a trend does not exist. Use trend analysis to forecast the sales for month 11.

Week (t ) Yt

1 1102 1153 1254 1205 1256 1207 1308 1159 11010 13011 124 Forecast


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Question????

How could we use the MSE or MAD to verify that the MA3 is a better smoothing technique than trend analysis for Robert’s Drug Sales data?


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Causal Method: Regression Analysis

Regression Analysis is similar to trend analysis, except the independent variable is not restricted to time. Refer to Robert’s Drug example. Instead of letting time represent our independent variable, we could forecast sales based upon the price of the product. Since products often go on sale, we could collect data over several months collecting the weekly price and number of items sold for the week. For this model, we would find the regression equation in the same manner in which we found the trend line except we would call the independent variable x, instead of t.


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Regression Equation

Using the method of least squares, the formula for the regression line is: Y = b0 + b1x.

where: Y= dependent variable which depends on the value of x b1 = slope of the regression line

b0 = regression line projection for x= 0

b1 = nXiYi - Xi Yi

nXi2 - (Xi)2

where: Yt = observed value of the time series at time period t

= average of the observed values for Yt

= average time period for the n observations

tt

tt

YY

tt

b y b x0 1


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Regression Analysis in Excel

The dependent variable Y can predicted using the same forecast function in Excel as used to forecast a trend line. Follow the same steps provided on slide 39.


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THE END

See your textbook for more examples and detailed explanations

of all topics discussed in these notes.

Forecasting Techniques

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