1 1 Slides by John Loucks St. Edward’s University

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Slides by

JohnLoucks

St. Edward’sUniversity

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Chapter 6Forecasting

Quantitative Approaches to Forecasting Components of a Time Series Measures of Forecast Accuracy Smoothing Methods Trend Projection Trend and Seasonal Components Regression Analysis Qualitative Approaches

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Quantitative Approaches to Forecasting

A forecast is a prediction of what will happen in the future. A time series is a set of observations measured at successive

points in, or over successive periods of, time. Quantitative forecasting approaches are based on an analysis

of historical data concerning one or more time series. If the data used are limited to past values of the series we

are trying to forecast, the procedure is called a time series method.

If the data used involve other time series that are believed to be related to the time series being forecasted, the procedure is called a causal method.

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Time Series Methods

The objective of time series methods is to discover a pattern in the historical data and then extrapolate this pattern into the future.

The forecast is based solely on past values of the variable that we are trying to forecast and/or on past forecast errors.

Three time series methods are: • smoothing• trend projection• trend projection adjusted for seasonal influence

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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.

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Components of a Time 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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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 averages• Weighted moving averages• Exponential smoothing

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Smoothing Methods

Moving Average MethodThe moving average method consists of computing

an average of the most recent n data values in the series and using this average for forecasting the value of the time series for the next period.

The term moving indicates that, as a new observation becomes available for the time series, it replaces the oldest observation in the equation, and a new average is computed.

(most recent n data points)Moving Average

n

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The data below show the number of gallons of gasoline (in 1,000s) sold by a gasoline distributor in Bennington, Vermont, over the past 12 weeks. The time series appears to be stable over time, so smoothing methods are applicable.

Example: Gasoline Sales

Week Sales Week Sales 1 17 7 20 2 21 8 18 3 19 9 22 4 23 10 20 5 18 11 15 6 16 12 22

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Moving AverageTo forecast gasoline sales for week 13 (in 1,000s)

using a 3-week moving average, we need to compute the average of sales for weeks 10, 11, and 12. The week 13 forecast (F13) is:


1320 15 22

F Moving Average (weeks 10 12) 193

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Smoothing Methods

Weighted Moving Average MethodIn the weighted moving average method for

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

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Three-Period Weighted Moving Average To forecast gasoline sales (in 1,000s) for week 13

using a 3-week weighted moving average, we will compute the weighted average of sales for weeks 10, 11, and 12 using the weights of 1/6, 2/6, and 3/6. The week 13 forecast (F13) is:

F13 = 1/6(20) + 2/6(15) + 3/6(22) = 3.333 + 5.000 + 11.000

= 19.333


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Smoothing Methods

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.

• The forecast is calculated by: [the actual value for the current period] + (1- )[the forecasted value for the current period],

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

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Exponential Smoothing

Ft+1 = aYt + (1 – a)Ft

where: Ft+1 = forecast of the time series for period t+1

Yt = actual value of the time series in period t

Ft = forecast of the time series for period t

a = smoothing constant (0 < a < 1)

(To start the calculations, we let F1 equal the

actual value of the time series in period 1.)

Smoothing Methods

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

F1 = 17

F2 = .2Y1 + .8F1 = .2(17) + .8(17) = 17.00

F3 = .2Y2 + .8F2 = .2(21) + .8(17) = 17.80

F4 = .2Y3 + .8F3 = .2(19) + .8(17.80) = 18.04

F5 = .2Y4 + .8F4 = .2(23) + .8(18.04) = 19.03

F6 = .2Y5 + .8F5 = .2(18) + .8(19.03) = 18.83

F7 = .2Y6 + .8F6 = .2(16) + .8(18.83) = 18.26


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

F8 = .2Y7 + .8F7 = .2(20) + .8(18.26) = 18.61

F9 = .2Y8 + .8F8 = .2(18) + .8(18.61) = 18.49

F10 = .2Y9 + .8F9 = .2(22) + .8(18.49) = 19.19

F11 = .2Y10 + .8F10 = .2(20) + .8(19.19) = 19.35

F12 = .2Y11 + .8F11 = .2(15) + .8(19.35) = 18.48

F13 = .2Y12 + .8F12 = .2(22) + .8(18.48) = 19.18

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Mean Squared Error (MSE)

An important consideration in selecting a forecasting method is the accuracy of the forecast. Clearly, we want forecast errors to be small.

The mean squared error (MSE) is an often-used measure of the accuracy of a forecasting method.

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.

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Mean Squared Error

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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: Tt = b0 + b1t.

where: Tt = trend value in period t

b0 = intercept of the trend line

b1 = slope of the trend line

where: Yt = actual value of the time series in period t

= average value of the time series = average value of t

n = number of periods

0 1b Y b t

Yt

1 2 2( )t tn tY t Y

bn t t

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Consider the time series for bicycle sales (in 1,000s) of a particular manufacturer over the past 10 years, as shown below. Although the data show some up-and-down movement over the past 10 years, the time series for the number of bicycles sold seems to have an overall increasing or upward trend.

Example: Bicycle Sales

Year Sales Year Sales

1 21.6 6 27.5

2 22.9 7 31.5

3 25.5 8 29.7

4 21.9 9 28.6

5 23.9 10 31.4

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Trend Projection

t Yt tYt t 2

1 21.6 21.6 1 2 22.9 45.8 4 3 25.5 76.5 9 4 21.9 87.6 16 5 23.9 119.5 25 6 27.5 165.0 36 7 31.5 220.5 49 8 29.7 237.6 64 9 28.6 257.4 81 10 31.4 314.0 100

55 264.5 1545.5 385

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Trend Projection (continued)

= 55/10 = 5.5 = 264.5/10 = 26.45

ntYt - t Yt (10)(1545.5) - (55)(264.5) b1 = = = 1.1 nt 2 - (t)2 (10)(385) - (55)2

= 26.45 - 1.1(5.5) = 20.4

Tt = 20.4 + 1.1(t)

T11 = 20.4 + 1.1(11) = 32.5 (or 32,500 bicycles)

0 1b Y b t

Yt


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Forecasting with Trendand Seasonal Components

Steps of Multiplicative Time Series Model1. Calculate the centered moving averages (CMAs).2. Center the CMAs on integer-valued periods.3. Determine the seasonal and irregular factors (StIt ).4. Determine the average seasonal factors.5. Scale the seasonal factors (St ).6. Determine the deseasonalized data.7. Determine a trend line of the deseasonalized data.8. Determine the deseasonalized predictions.9. Take into account the seasonality.

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Example: Television Sales

The data below show quarterly television sales (in thousands of units) for a particular manufacturer over the past four years.

Quarter Year 1 2 3 4 1 4.8 4.1 6.0

6.5 2 5.8 5.2 6.8

7.4 3 6.0 5.6 7.5

7.8 4 6.3 5.9 8.0

8.4

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1. Calculate the centered moving averages.There are four distinct seasons in each year. Hence,

take a four-period centered moving average to eliminate seasonal and irregular factors. For example:

1st CMA = (4.8 + 4.1 + 6.0 + 6.5)/4 = 5.35 2nd CMA = (4.1 + 6.0 + 6.5 + 5.8)/4 = 5.60

etc.


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2. Center the CMAs on integer-valued periods.The first centered moving average computed in step

1 (5.35) will be centered on quarter 2.5 of year 1. The second centered moving average computed (5.60) will be centered on quarter 3.5 of year 1.

Note that the moving averages from step 1 do not center themselves on integer-valued periods because n (the number of seasons) is an even number.

In this case, 2-period CMAs are computed for the original 4-period CMAs. These 2-CMAs will center themselves on integer-valued periods.

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2. Center the CMAs on integer-valued periods.

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3. Determine the seasonal & irregular factors (St It ). Isolate the trend and cyclical components. For each period t, this is given by:

St It = Yt /(Moving Average for period t )

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3. Determine the seasonal & irregular factors (St It ).

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4. Determine the average seasonal factors. Averaging all St It values corresponding to that season:

Season 1: (0.971 + 0.918 + 0.908) /3 = 0.9323 Season 2: (0.840 + 0.839 + 0.834) /3 = 0.8377 Season 3: (1.096 + 1.075 + 1.109) /3 = 1.0933 Season 3: (1.133 + 1.156 + 1.141) /3 = 1.1433

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5. Scale the seasonal factors (St ).

Average the seasonal factors = (0.9323 + 0.8377 + 1.0933 + 1.1433)/4 = 1.00165. Then, divide each

seasonal factor by the average of the seasonal factors.

Season 1: 0.9323/1.00165 = 0.9308 Season 2: 0.8377/1.00165 = 0.8363

Season 3: 1.0933/1.00165 = 1.0915 Season 4: 1.1433/1.00165 = 1.1414

Total = 4.0000

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6. Determine the deseasonalized data.Divide the data point values, Yt , by St .

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6. Determine the deseasonalized data.

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7. Determine a trend line of the deseasonalized data. Using the least squares method for t = 1, 2, ..., 16,

gives:

Tt = 5.101 + 0.148(t)

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8. Determine the deseasonalized (trend) predictions.Substitute t = 17, 18, 19, and 20 into the least

squares equation:

Year 5, Quarter 1: T17 = 5.101 + 0.148(17) = 7.617

Year 5, Quarter 2: T18 = 5.101 + 0.148(18) = 7.765

Year 5, Quarter 3: T19 = 5.101 + 0.148(19) = 7.913

Year 5, Quarter 4: T20 = 5.101 + 0.148(20) = 8.061

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9. Take into account the seasonality.Multiply each deseasonalized (trend) prediction by

its seasonal factor to give the following forecasts for year 5:

Year 5, Quarter 1: F17 = 7.617(0.9308) = 7.090

Year 5, Quarter 2: F18 = 7.765(0.8363) = 6.494

Year 5, Quarter 3: F19 = 7.913(1.0915) = 8.637

Year 5, Quarter 4: F20 = 8.061(1.1414) = 9.201

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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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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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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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End of Chapter 6

Documents

1 1 Slides by John Loucks St. Edward’s University