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Based on your Module topics, what did you find new and interesting?  And what appeared to be a review?  Also, identify at least one discussion post you found interesting, helpful, or beneficial (and why). Business Analytics © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Time Series Analysis and Forecasting Chapter 8 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Introduction (Slide 1 of 2) Forecasting methods can be classified as qualitative or quantitative. Qualitative methods generally involve the use of expert judgment to develop forecasts. Quantitative forecasting methods can be used when: Past information about the variable being forecast is available. The information can be quantified. It is reasonable to assume that past is prologue. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Introduction (Slide 2 of 2) The objective of time series analysis is to uncover a pattern in the time series and then extrapolate the pattern into the future. The forecast is based solely on past values of the variable and/or on past forecast errors. Modern data-collection technologies have enabled individuals, businesses, and government agencies to collect vast amounts of data that may be used for causal forecasting. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 4 Time Series Patterns Horizontal Pattern Trend Pattern Seasonal Pattern Trend and Seasonal Pattern Cyclical Pattern Identifying Time Series Patterns © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Time Series Patterns (Slide 1 of 20) Time series: A sequence of observations on a variable measured at successive points in time or over successive periods of time. The measurements may be taken every hour, day, week, month, year, or any other regular interval. The pattern of the data is important in understanding the series’ past behavior. If the behavior of the times series data of the past is expected to continue in the future, it can be used as a guide in selecting an appropriate forecasting method. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. To identify the underlying pattern in the data, a useful first step is to construct a time series plot, which is a graphical presentation of the relationship between time and the time series variable; time is represented on the horizontal axis and values of the time series variable are shown on the vertical axis. 6 Time Series Patterns (Slide 2 of 20) Horizontal Pattern: Exists when the data fluctuate randomly around a constant mean over time. Stationary time series: It denotes a time series whose statistical properties are independent of time: The process generating the data has a constant mean. The variability of the time series is constant over time. A time series plot for a stationary time series will always exhibit a horizontal pattern with random fluctuations. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Time Series Patterns (Slide 3 of 20) Table 8.1: Gasoline Sales Time Series Week Sales (1,000s of gallons) 1 17 2 21 3 19 4 23 5 18 6 16 7 20 8 18 9 22 10 20 11 15 12 22 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Data in Table 8.1 show the number of gallons of gasoline (in 1000s) sold by a gasoline distributor in Bennington, Vermont, over the past 12 weeks. The average value, or mean, for this time series is 19.25, or 19,250 gallons per week. 8 Time Series Patterns (Slide 4 of 20) Figure 8.1: Gasoline Sales Time Series Plot © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Figure 8.1 shows a time series plot for the data in Table 8.1. Note how the data fluctuate around the sample mean of 19,250 gallons. Although random variability is present, we would say that these data follow a horizontal pattern. 9 Time Series Patterns (Slide 5 of 20) Table 8.2: Gasoline Sales Time Series after Obtaining the Contract with the Vermont State Police Week Sales (1,000s of gallons) Week Sales (1,000s of gallons) 1 17 12 22 2 21 13 31 3 19 14 34 4 23 15 31 5 18 16 33 6 16 17 28 7 20 18 32 8 18 19 30 9 22 20 29 10 20 21 34 11 15 22 33 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Table 8.2 shows the number of gallons of gasoline sold for the original time series and the 10 weeks after signing the new contract with the Vermont State Police to provide gasoline for state police cars located in southern Vermont. 10 Time Series Patterns (Slide 6 of 20) Figure 8.2: Gasoline Sales Time Series Plot after Obtaining the Contract with the Vermont State Police © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Figure 8.2 shows the corresponding time series plot. Note the increased level of the time series beginning in week 13. This change in the level of the time series makes it more difficult to choose an appropriate forecasting method. 11 Time Series Patterns (Slide 7 of 20) Trend Pattern: A trend pattern shows gradual shifts or movements to relatively higher or lower values over a longer period of time. A trend is usually the result of long-term factors such as: Population increases or decreases. Shifting demographic characteristics of the population. Improving technology. Changes in the competitive landscape. Changes in consumer preferences. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Time Series Patterns (Slide 8 of 20) Table 8.3: Bicycle Sales Time Series Year Sales (1,000s) 1 21.6 2 22.9 3 25.5 4 21.9 5 23.9 6 27.5 7 31.5 8 29.7 9 28.6 10 31.4 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Table 8.3 shows the time series of bicycle sales for a particular manufacturer over the past 10 years. 13 Time Series Patterns (Slide 9 of 20) Figure 8.3: Bicycle Sales Time Series Plot © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Figure 8.3 shows the time series of bicycle sales for a particular manufacturer over the past 10 years. Note that 21,600 bicycles were sold in year 1, 22,900 were sold in year 2, and so on. In year 10, the most recent year, 31,400 bicycles were sold. Visual inspection of the time series plot shows some up-and-down movement over the past 10 years. 14 Time Series Patterns (Slide 10 of 20) Table 8.4: Cholesterol Drug Revenue Times Year Revenue ($ millions) 1 23.1 2 21.3 3 27.4 4 34.6 5 33.8 6 43.2 7 59.5 8 64.4 9 74.2 10 99.3 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The data in Table 8.4 and the corresponding time series plot in Figure 8.4 show the sales revenue for a cholesterol drug since the company won FDA approval for the drug 10 years ago. The time series increases in a nonlinear fashion; that is, the rate of change of revenue does not increase by a constant amount from one year to the next. In fact, the revenue appears to be growing in an exponential fashion. 15 Time Series Patterns (Slide 11 of 20) Figure 8.4: Cholesterol Drug Revenue Times Series Plot ($ millions) © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The data in Table 8.4 and the corresponding time series plot in Figure 8.4 show the sales revenue for a cholesterol drug since the company won FDA approval for the drug 10 years ago. The time series increases in a nonlinear fashion; that is, the rate of change of revenue does not increase by a constant amount from one year to the next. In fact, the revenue appears to be growing in an exponential fashion. 16 Time Series Patterns (Slide 12 of 20) Seasonal Pattern: Seasonal patterns are recurring patterns over successive periods of time. Example: A retailer that sells bathing suits expects low sales activity in the fall and winter months, with peak sales in the spring and summer months to occur every year. The time series plot not only exhibits a seasonal pattern over a one-year period but also for less than one year in duration. Example: daily traffic volume shows within-the-day “seasonal” behavior, with peak levels occurring during rush hour, moderate flow during the rest of the day, and light flow from midnight to early morning. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Time Series Patterns (Slide 13 of 20) Table 8.5: Umbrella Sales Time Series Year Quarter Sales 1 1 125   2 153   3 106   4 88 2 1 118   2 161   3 133   4 102 3 1 138   2 144   3 113   4 80 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 18 Time Series Patterns (Slide 14 of 20) Table 8.5: Umbrella Sales Time Series (cont.) Year Quarter Sales 4 1 109   2 137   3 125   4 109 5 1 130   2 165   3 128   4 96 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 19 Time Series Patterns (Slide 15 of 20) Figure 8.5: Umbrella Sales Time Series Plot © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 20 Time Series Patterns (Slide 16 of 20) Trend and Seasonal Pattern: Some time series include both a trend and a seasonal pattern. Table 8.6: Quarterly Smartphone Sales Time Series Year Quarter Sales ($1,000s) 1 1 4.8   2 4.1   3 6.0   4 6.5 2 1 5.8   2 5.2   3 6.8   4 7.4 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Table 8.6 and Figure 8.6 show quarterly smartphone sales for a particular manufacturer over the past four years. Clearly an increasing trend is present. However, Figure 8.6 also indicates that sales are lowest in the second quarter of each year and highest in quarters 3 and 4. Thus, we conclude that a seasonal pattern also exists for smartphone sales. 21 Time Series Patterns (Slide 17 of 20) Table 8.6: Quarterly Smartphone Sales Time Series (cont.) Year Quarter Sales ($1,000s) 3 1 6.0   2 5.6   3 7.5   4 7.8 4 1 6.3   2 5.9   3 8.0   4 8.4 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Figure 8.6 shows quarterly smartphone sales for a particular manufacturer over the past four years. Clearly an increasing trend is present. However, Figure 8.6 also indicates that sales are lowest in the second quarter of each year and highest in quarters 3 and 4. Thus, we conclude that a seasonal pattern also exists for smartphone sales. 22 Time Series Patterns (Slide 18 of 20) Figure 8.6: Quarterly Smartphone Sales Time Series Plot © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Table 8.6 and Figure 8.6 show quarterly smartphone sales for a particular manufacturer over the past four years. Clearly an increasing trend is present. However, Figure 8.6 also indicates that sales are lowest in the second quarter of each year and highest in quarters 3 and 4. Thus, we conclude that a seasonal pattern also exists for smartphone sales. 23 Time Series Patterns (Slide 19 of 20) Cyclical Pattern: A cyclical pattern exists if the time series plot shows an alternating sequence of points below and above the trendline that lasts for more than one year. Example: Periods of moderate inflation followed by periods of rapid inflation can lead to a time series that alternates below and above a generally increasing trendline. Cyclical effects are often combined with long-term trend effects and referred to as trend-cycle effects. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 24 Time Series Patterns (Slide 20 of 20) Identifying Time Series Patterns: The underlying pattern in the time series is an important factor in selecting a forecasting method. A time series plot should be one of the first analytic tools. We need to use a forecasting method that is capable of handling the pattern exhibited by the time series effectively. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 25 Forecast Accuracy © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Forecast Accuracy (Slide 1 of 10) Table 8.7: Computing Forecasts and Measures of Forecast Accuracy Using the Most Recent Value as the Forecast for the Next Period Week Time Series Value Forecast Forecast Error Absolute Value of Forecast Error Squared Forecast Error Percentage Error Absolute Value of Percentage Error 1 17             2 21 17 4 4 16 19.05 19.05 3 19 21 −2 2 4 −10.53 10.53 4 23 19 4 4 16 17.39 17.39 5 18 23 −5 5 25 −27.78 27.78 6 16 18 −2 2 4 −12.50 12.50 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Table 8.7 shows the forecasts for the gasoline time series shown in Table 8.1 using the simplest of all the forecasting methods. We use the most recent week’s sales volume as the forecast for the next week. For instance, the distributor sold 17 thousand gallons of gasoline in week 1; this value is used as the forecast for week 2. Next, we use 21, the actual value of sales in week 2, as the forecast for week 3, and so on. This method is often referred to as a naïve forecasting method because of its simplicity. 27 Forecast Accuracy (Slide 2 of 10) Table 8.7: Computing Forecasts and Measures of Forecast Accuracy Using the Most Recent Value as the Forecast for the Next Period (cont.) Week Time Series Value Forecast Forecast Error Absolute Value of Forecast Error Squared Forecast Error Percentage Error Absolute Value of Percentage Error 7 20 16 4 4 16 20.00 20.00 8 18 20 −2 2 4 −11.11 11.11 9 22 18 4 4 16 18.18 18.18 10 20 22 −2 2 4 −10.00 10.00 11 15 20 −5 5 25 −33.33 33.33 12 22 15 7 7 49 31.82 31.82 Totals 5 41 179 1.19 211.69 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Table 8.7 shows the forecasts for the gasoline time series shown in Table 8.1 using the simplest of all the forecasting methods. We use the most recent week’s sales volume as the forecast for the next week. For instance, the distributor sold 17 thousand gallons of gasoline in week 1; this value is used as the forecast for week 2. Next, we use 21, the actual value of sales in week 2, as the forecast for week 3, and so on. This method is often referred to as a naïve forecasting method because of its simplicity. 28 Forecast Accuracy (Slide 3 of 10) Naïve forecasting method: Using the most recent data to predict future data. The key concept associated with measuring forecast accuracy is forecast error. Measures to determine how well a particular forecasting method is able to reproduce the time series data that are already available. Forecast error. Mean forecast error (MFE). Mean absolute error (MAE). Mean squared error (MSE). Mean absolute percentage error (MAPE). © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Forecast Accuracy (Slide 4 of 10) Forecast Error: Difference between the actual and the forecasted values for period t. Mean Forecast Error: Mean or average of the forecast errors. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Forecast error: = = actual value = forecasted value Example: Consider Table 8.7. The distributor actually sold 21 thousand gallons of gasoline in week 2, and the forecast, using the sales volume in week 1, was 17 thousand gallons. The forecast error in week 2 is = = 21 17 = 4. Mean Forecast Error (MFE): MFE = n = Number of periods in time series k = Number of periods at the beginning of the time series for which we cannot produce a naïve forecast Example: Table 8.7 shows that the sum of the forecast errors for the gasoline sales time series is 5. Thus, the mean or average error is 5/11 = 0.45. 30 Forecast Accuracy (Slide 5 of 10) Mean Absolute Error (MAE): Measure of forecast accuracy that avoids the problem of positive and negative forecast errors offsetting one another. Mean Squared Error (MSE): Measure that avoids the problem of positive and negative errors offsetting each other is obtained by computing the average of the squared forecast errors. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Mean Absolute Error (MAE) It is also referred to as the mean absolute deviation (MAD). Example: Table 8.7 shows that the sum of the absolute values of the forecast errors is 41. Thus MAE = average of the absolute value of the forecast errors = 41/11 = 3.73. MEAN SQUARED ERROR (MSE) Example: From Table 8.7, the sum of the squared errors is 179. Hence, MSE = average of the square of the forecast errors = 179/11 = 16.27. 31 Forecast Accuracy (Slide 6 of 10) Mean Absolute Percentage Error (MAPE): Average of the absolute value of percentage forecast errors. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Mean Absolute Percentage Error (MAPE): The size of MAE or MSE depends upon the scale of the data. As a result, it is difficult to make comparisons for different time intervals (such as comparing a method of forecasting monthly gasoline sales to a method of forecasting weekly sales) or to make comparisons across different time series (such as monthly sales of gasoline and monthly sales of oil filters). To make comparisons such as these, we need to work with relative or percentage error measures. The mean absolute percentage error (MAPE) is such a measure. Example: Table 8.7 shows that the sum of the absolute values of the percentage errors is Thus the MAPE, which is the average of the absolute value of percentage forecast errors, is 211.69/11 = 19.24\% 32 Forecast Accuracy (Slide 7 of 10) Table 8.8: Computing Forecasts and Measures of Forecast Accuracy Using the Average of All the Historical Data as the Forecast for the Next Period Week Time Series Value Forecast Forecast Error Absolute Value of Forecast Error Squared Forecast Error Percentage Error Absolute Value of Percentage Error 1 17             2 21 17.00 4.00 4.00 16.00 19.05 19.05 3 19 19.00 0.00 0.00 0.00 0.00 0.00 4 23 19.00 4.00 4.00 16.00 17.39 17.39 5 18 20.00 −2.00 2.00 4.00 −11.11 11.11 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. We begin by developing a forecast for week 2. Because there is only one historical value available prior to week 2, the forecast for week 2 is just the time series value in week 1; thus, the forecast for week 2 is 17 thousand gallons of gasoline. To compute the forecast for week 3, we take the average of the sales values in weeks 1 and 2. Thus, = (17 + 21)/2 = 19 Similarly, the forecast for week 4 is = (17 + 21 + 19)/3 = 19. Using the results shown in Table 8.8, we obtain the following values of MAE, MSE, and MAPE: MAE = 26.81/11 = 2.44 MSE = 89.07/11 = 8.10 MAPE = 141.34/11 = 12.85\% 33 Forecast Accuracy (Slide 8 of 10) Table 8.8: Computing Forecasts and Measures of Forecast Accuracy Using the Average of All the Historical Data as the Forecast for the Next Period (cont.) Week Time Series Value Forecast Forecast Error Absolute Value of Forecast Error Squared Forecast Error Percentage Error Absolute Value of Percentage Error 6 16 19.60 −3.60 3.60 12.96 −22.50 22.50 7 20 19.00 1.00 1.00 1.00 5.00 5.00 8 18 19.14 −1.14 1.14 1.31 −6.35 6.35 9 22 19.00 3.00 3.00 9.00 13.64 13.64 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. We begin by developing a forecast for week 2. Because there is only one historical value available prior to week 2, the forecast for week 2 is just the time series value in week 1; thus, the forecast for week 2 is 17 thousand gallons of gasoline. To compute the forecast for week 3, we take the average of the sales values in weeks 1 and 2. Thus, = (17 + 21)/2 = 19 Similarly, the forecast for week 4 is = (17 + 21 + 19)/3 = 19. Using the results shown in Table 8.8, we obtain the following values of MAE, MSE, and MAPE: MAE = 26.81/11 = 2.44 MSE = 89.07/11 = 8.10 MAPE = 141.34/11 = 12.85\% 34 Forecast Accuracy (Slide 9 of 10) Table 8.8: Computing Forecasts and Measures of Forecast Accuracy Using the Average of All the Historical Data as the Forecast for the Next Period (cont.) Week Time Series Value Forecast Forecast Error Absolute Value of Forecast Error Squared Forecast Error Percentage Error Absolute Value of Percentage Error 10 20 19.33 0.67 0.67 0.44 3.33 3.33 11 15 19.40 −4.40 4.40 19.36 −29.33 29.33 12 22 19.00 3.00 3.00 9.00 13.64 13.64 Totals 4.52 26.81 89.07 2.75 141.34 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. We begin by developing a forecast for week 2. Because there is only one historical value available prior to week 2, the forecast for week 2 is just the time series value in week 1; thus, the forecast for week 2 is 17 thousand gallons of gasoline. To compute the forecast for week 3, we take the average of the sales values in weeks 1 and 2. Thus, = (17 + 21)/2 = 19 Similarly, the forecast for week 4 is = (17 + 21 + 19)/3 = 19. Using the results shown in Table 8.8, we obtain the following values of MAE, MSE, and MAPE: MAE = 26.81/11 = 2.44 MSE = 89.07/11 = 8.10 MAPE = 141.34/11 = 12.85\% 35 Forecast Accuracy (Slide 10 of 10) Compare the accuracy of the two forecasting methods by comparing the values of MAE, MSE, and MAPE for each method. Naïve Method Average of Past Values MAE 3.73 2.44 MSE 16.27 8.10 MAPE 19.24\% 12.85\% The average of past values provides more accurate forecasts for the next period than using the most recent observation. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 36 Moving Averages and Exponential Smoothing Moving Averages Exponential Smoothing © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Moving Averages and Exponential Smoothing (Slide 1 of 16) Moving Averages: Moving averages method: Uses the average of the most recent k data values in the time series as the forecast for the next period. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. In the formula for moving average forecast, = forecast of the time series for period t + 1 = actual value of the time series in period t k = number of periods of time series data used to generate the forecast To use moving averages to forecast a time series, we must first select the order k. If only the most recent values of the time series are considered relevant, a small value of k is preferred. If a greater number of past values are considered relevant, then we generally opt for a larger value of k. A moving average will adapt to the new level of the series and continue to provide good forecasts in k periods. Thus, a smaller value of k will track shifts in a time series more quickly. On the other hand, larger values of k will be more effective in smoothing out random fluctuations. 38 Moving Averages and Exponential Smoothing (Slide 2 of 16) Table 8.9: Summary of Three-Week Moving Average Calculations Week Time Series Value Forecast Forecast Error Absolute Value of Forecast Error Squared Forecast Error Percentage Error Absolute Value of Percentage Error 1 17             2 21             3 19             4 23 19 4 4 16 17.39 17.39 5 18 21 −3 3 9 −16.67 16.67 6 16 20 −4 4 16 −25.00 25.00 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. To illustrate the moving averages method, let us return to the original 12 weeks of gasoline sales data in Table 8.1 and Figure 8.1. We will use a three-week moving average (k = 3). We begin by computing the forecast of sales in week 4 using the average of the time series values in weeks 1 to 3. = average for weeks 1 to 3 = (17 + 21 + 19)/3 = 19 The moving average forecast of sales in week 4 is 19, or 19,000 gallons of gasoline. Because the actual value observed in week 4 is 23, the forecast error in week 4 is e4 = 23 - 19 = 4. We next compute the forecast of sales in week 5 by averaging the time series values in weeks 2 to 4. = average for weeks 2 to 4 = (21 + 19 + 23)/3 = 21 Hence, the forecast of sales in week 5 is 21 and the error associated with this forecast is e5 = 18 - 21 = -3. A complete summary of the three-week moving average forecasts for the gasoline sales time series is provided in Table 8.9. 39 Moving Averages and Exponential Smoothing (Slide 3 of 16) Table 8.9: Summary of Three-Week Moving Average Calculations (cont.) Week Time Series Value Forecast Forecast Error Absolute Value of Forecast Error Squared Forecast Error Percentage Error Absolute Value of Percentage Error 7 20 19 1 1 1 5.00 5.00 8 18 18 0 0 0 0.00 0.00 9 22 18 4 4 16 18.18 18.18 10 20 20 0 0 0 0.00 0.00 11 15 20 −5 5 25 −33.33 33.33 12 22 19 3 3 9 13.64 13.64 Totals 0 24 92 −20.79 129.21 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. To illustrate the moving averages method, let us return to the original 12 weeks of gasoline sales data in Table 8.1 and Figure 8.1. We will use a three-week moving average (k = 3). We begin by computing the forecast of sales in week 4 using the average of the time series values in weeks 1 to 3. = average for weeks 1 to 3 = (17 + 21 + 19)/3 = 19 The moving average forecast of sales in week 4 is 19 or 19,000 gallons of gasoline. Because the actual value observed in week 4 is 23, the forecast error in week 4 is e4 = 23 - 19 = 4. We next compute the forecast of sales in week 5 by averaging the time series values in weeks 2 to 4. = average for weeks 2 to 4 = (21 + 19 + 23)/3 = 21 Hence, the forecast of sales in week 5 is 21 and the error associated with this forecast is e5 = 18 - 21 = -3. A complete summary of the three-week moving average forecasts …