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![Weighted Moving Average
MONTH ACTUAL SHED SALES 3-MONTH WEIGHTED MOVING AVERAGE
January 10
February 12
March 13
April 16
May 19
June 23
July 26
August 30
September 28
October 18
November 16
December 14
WEIGHTS APPLIED PERIOD
3 Last month
2 Two months ago
1 Three months ago
6 Sum of the weights
Forecast for this month =
3 x Sales last mo. + 2 x Sales 2 mos. ago + 1 x Sales 3 mos. ago
Sum of the weights
[(3 x 13) + (2 x 12) + (10)]/6 = 12 1/6
10
12
13](https://image.slidesharecdn.com/forecasting-180721091808/75/Forecasting-29-2048.jpg)
![Weighted Moving Average
MONTH ACTUAL SHED SALES 3-MONTH WEIGHTED MOVING AVERAGE
January 10
February 12
March 13
April 16
May 19
June 23
July 26
August 30
September 28
October 18
November 16
December 14
[(3 x 13) + (2 x 12) + (10)]/6 = 12 1/6
10
12
13
[(3 x 16) + (2 x 13) + (12)]/6 = 14 1/3
[(3 x 19) + (2 x 16) + (13)]/6 = 17
[(3 x 23) + (2 x 19) + (16)]/6 = 20 1/2
[(3 x 26) + (2 x 23) + (19)]/6 = 23 5/6
[(3 x 30) + (2 x 26) + (23)]/6 = 27 1/2
[(3 x 28) + (2 x 30) + (26)]/6 = 28 1/3
[(3 x 18) + (2 x 28) + (30)]/6 = 23 1/3
[(3 x 16) + (2 x 18) + (28)]/6 = 18 2/3](https://image.slidesharecdn.com/forecasting-180721091808/75/Forecasting-30-2048.jpg)
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Forecasting
- 1. Forecasting Dr. K. M.Salah Uddin Associate Professor Department of Management Information Systems University of Dhaka © 2014 Pearson Education, Inc.
- 2. Outline ▶ What IsForecasting? ▶ The Strategic Importance of Forecasting ▶ Seven Steps in the Forecasting System ▶ Forecasting Approaches
- 3. What is Forecasting? ►Process of predicting a future event ► Underlying basis of all business decisions ► Production ► Inventory ► Personnel ► Facilities ??
- 4. 1. Short-range forecast ►Up to 1 year, generally less than 3 months ► Purchasing, job scheduling, workforce levels, job assignments, production levels 2. Medium-range forecast ► 3 months to 3 years ► Sales and production planning, budgeting 3. Long-range forecast ► 3+ years ► New product planning, facility location, research and development Forecasting Time Horizons
- 5. Distinguishing Differences 1. Medium/longrange forecasts deal with more comprehensive issues and support management decisions regarding planning and products, plants and processes 2. Short-term forecasting usually employs different methodologies than longer-term forecasting 3. Short-term forecasts tend to be more accurate than longer-term forecasts
- 6. Types of Forecasts 1.Economic forecasts ► Address business cycle – inflation rate, money supply, housing starts, etc. 2. Technological forecasts ► Predict rate of technological progress ► Impacts development of new products 3. Demand forecasts ► Predict sales of existing products and services
- 7. Strategic Importance of Forecasting ►Supply-Chain Management – Good supplier relations, advantages in product innovation, cost and speed to market ► Human Resources – Hiring, training, laying off workers ► Capacity – Capacity shortages can result in undependable delivery, loss of customers, loss of market share
- 8. Seven Steps inForecasting 1. Determine the use of the forecast 2. Select the items to be forecasted 3. Determine the time horizon of the forecast 4. Select the forecasting model(s) 5. Gather the data needed to make the forecast 6. Make the forecast 7. Validate and implement results
- 9. The Realities! ► Forecastsare seldom perfect, unpredictable outside factors may impact the forecast ► Most techniques assume an underlying stability in the system ► Product family and aggregated forecasts are more accurate than individual product forecasts
- 10. Forecasting Approaches ► Usedwhen situation is vague and little data exist ► New products ► New technology ► Involves intuition, experience ► e.g., forecasting sales on Internet Qualitative Methods
- 11. Forecasting Approaches ► Usedwhen situation is ‘stable’ and historical data exist ► Existing products ► Current technology ► Involves mathematical techniques ► e.g., forecasting sales of color televisions Quantitative Methods
- 12. Overview of QualitativeMethods 1. Jury of executive opinion ► Pool opinions of high-level experts, sometimes augment by statistical models 2. Delphi method ► Panel of experts, queried iteratively
- 13. Overview of QualitativeMethods 3. Sales force composite ► Estimates from individual salespersons are reviewed for reasonableness, then aggregated 4. Market Survey ► Ask the customer
- 14. ► Involves smallgroup of high-level experts and managers ► Group estimates demand by working together ► Combines managerial experience with statistical models ► Relatively quick ► ‘Group-think’ disadvantage Jury of Executive Opinion
- 15. Delphi Method ► Iterativegroup process, continues until consensus is reached ► 3 types of participants ► Decision makers ► Staff ► Respondents Staff (Administering survey) Decision Makers (Evaluate responses and make decisions) Respondents (People who can make valuable judgments)
- 16. Sales Force Composite ►Each salesperson projects his or her sales ► Combined at district and national levels ► Sales reps know customers’ wants ► May be overly optimistic
- 17. Overview of Quantitative Approaches 1.Naive approach 2. Moving averages 3. Exponential smoothing 4. Trend projection 5. Linear regression Time-series models Associative model
- 18. ► Set ofevenly spaced numerical data ► Obtained by observing response variable at regular time periods ► Forecast based only on past values, no other variables important ► Assumes that factors influencing past and present will continue influence in future Time-Series Forecasting
- 19.
- 20. Components of DemandDemandforproductorservice || | | 1 2 3 4 Time (years) Average demand over 4 years Trend component Actual demand line Random variation Figure 4.1 Seasonal peaks
- 21. ► Persistent, overallupward or downward pattern ► Changes due to population, technology, age, culture, etc. ► Typically several years duration Trend Component
- 22. ► Regular patternof up and down fluctuations ► Due to weather, customs, etc. ► Occurs within a single year Seasonal Component PERIOD LENGTH “SEASON” LENGTH NUMBER OF “SEASONS” IN PATTERN Week Day 7 Month Week 4 – 4.5 Month Day 28 – 31 Year Quarter 4 Year Month 12 Year Week 52
- 23. ► Repeating upand down movements ► Affected by business cycle, political, and economic factors ► Multiple years duration ► Often causal or associative relationships Cyclical Component 0 5 10 15 20
- 24. ► Erratic, unsystematic,‘residual’ fluctuations ► Due to random variation or unforeseen events ► Short duration and nonrepeating Random Component M T W T F
- 25. Naive Approach ► Assumesdemand in next period is the same as demand in most recent period ► e.g., If January sales were 68, then February sales will be 68 ► Sometimes cost effective and efficient ► Can be good starting point
- 26. ► MA isa series of arithmetic means ► Used if little or no trend ► Used often for smoothing ► Provides overall impression of data over time Moving Average Method Moving average = demand in previous n periodså n
- 27. Moving Average Example MONTHACTUAL SHED SALES 3-MONTH MOVING AVERAGE January 10 February 12 March 13 April 16 May 19 June 23 July 26 August 30 September 28 October 18 November 16 December 14 (10 + 12 + 13)/3 = 11 2/3 (12 + 13 + 16)/3 = 13 2/3 (13 + 16 + 19)/3 = 16 (16 + 19 + 23)/3 = 19 1/3 (19 + 23 + 26)/3 = 22 2/3 (23 + 26 + 30)/3 = 26 1/3 (29 + 30 + 28)/3 = 28 (30 + 28 + 18)/3 = 25 1/3 (28 + 18 + 16)/3 = 20 2/3 10 12 13
- 28. ► Used whensome trend might be present ► Older data usually less important ► Weights based on experience and intuition Weighted Moving Average = Weight for period n( ) Demand in period n( )( )å Weightså Weighted moving average
- 29. Weighted Moving Average MONTHACTUAL SHED SALES 3-MONTH WEIGHTED MOVING AVERAGE January 10 February 12 March 13 April 16 May 19 June 23 July 26 August 30 September 28 October 18 November 16 December 14 WEIGHTS APPLIED PERIOD 3 Last month 2 Two months ago 1 Three months ago 6 Sum of the weights Forecast for this month = 3 x Sales last mo. + 2 x Sales 2 mos. ago + 1 x Sales 3 mos. ago Sum of the weights [(3 x 13) + (2 x 12) + (10)]/6 = 12 1/6 10 12 13
- 30. Weighted Moving Average MONTHACTUAL SHED SALES 3-MONTH WEIGHTED MOVING AVERAGE January 10 February 12 March 13 April 16 May 19 June 23 July 26 August 30 September 28 October 18 November 16 December 14 [(3 x 13) + (2 x 12) + (10)]/6 = 12 1/6 10 12 13 [(3 x 16) + (2 x 13) + (12)]/6 = 14 1/3 [(3 x 19) + (2 x 16) + (13)]/6 = 17 [(3 x 23) + (2 x 19) + (16)]/6 = 20 1/2 [(3 x 26) + (2 x 23) + (19)]/6 = 23 5/6 [(3 x 30) + (2 x 26) + (23)]/6 = 27 1/2 [(3 x 28) + (2 x 30) + (26)]/6 = 28 1/3 [(3 x 18) + (2 x 28) + (30)]/6 = 23 1/3 [(3 x 16) + (2 x 18) + (28)]/6 = 18 2/3
- 31. ► Increasing nsmooths the forecast but makes it less sensitive to changes ► Does not forecast trends well ► Requires extensive historical data Potential Problems With Moving Average
- 32. Graph of MovingAverages | | | | | | | | | | | | J F M A M J J A S O N D Salesdemand 30 – 25 – 20 – 15 – 10 – 5 – Month Actual sales Moving average Weighted moving average Figure 4.2
- 33. ► Form ofweighted moving average ► Weights decline exponentially ► Most recent data weighted most ► Requires smoothing constant () ► Ranges from 0 to 1 ► Subjectively chosen ► Involves little record keeping of past data Exponential Smoothing
- 34. Exponential Smoothing New forecast= Last period’s forecast + (Last period’s actual demand – Last period’s forecast) Ft = Ft – 1 + (At – 1 - Ft – 1) where Ft = new forecast Ft – 1 = previous period’s forecast = smoothing (or weighting) constant (0 ≤ ≤ 1) At – 1 = previous period’s actual demand
- 35. Exponential Smoothing Example Predicted demand= 142 Ford Mustangs Actual demand = 153 Smoothing constant = .20
- 36. Exponential Smoothing Example Predicted demand= 142 Ford Mustangs Actual demand = 153 Smoothing constant = .20 New forecast = 142 + .2(153 – 142)
- 37. Exponential Smoothing Example Predicted demand= 142 Ford Mustangs Actual demand = 153 Smoothing constant = .20 New forecast = 142 + .2(153 – 142) = 142 + 2.2 = 144.2 ≈ 144 cars
- 38. Effect of Smoothing Constants ▶Smoothing constant generally .05 ≤ ≤ .50 ▶ As increases, older values become less significant WEIGHT ASSIGNED TO SMOOTHING CONSTANT MOST RECENT PERIOD () 2ND MOST RECENT PERIOD (1 – ) 3RD MOST RECENT PERIOD (1 – )2 4th MOST RECENT PERIOD (1 – )3 5th MOST RECENT PERIOD (1 – )4 = .1 .1 .09 .081 .073 .066 = .5 .5 .25 .125 .063 .031
- 39. Impact of Different 225 – 200 – 175 – 150 – | | | | | | | | | 1 2 3 4 5 6 7 8 9 Quarter Demand = .1 Actual demand = .5
- 40. Impact of Different 225 – 200 – 175 – 150 – | | | | | | | | | 1 2 3 4 5 6 7 8 9 Quarter Demand = .1 Actual demand = .5 ► Chose high values of when underlying average is likely to change ► Choose low values of when underlying average is stable
- 41. Choosing The objectiveis to obtain the most accurate forecast no matter the technique We generally do this by selecting the model that gives us the lowest forecast error Forecast error = Actual demand – Forecast value = At – Ft
- 42. Common Measures ofError Mean Absolute Deviation (MAD) MAD = Actual - Forecastå n
- 43. Determining the MAD QUARTER ACTUAL TONNAGE UNLOADEDFORECAST WITH = .10 FORECAST WITH = .50 1 180 175 175 2 168 175.50 = 175.00 + .10(180 – 175) 177.50 3 159 174.75 = 175.50 + .10(168 – 175.50) 172.75 4 175 173.18 = 174.75 + .10(159 – 174.75) 165.88 5 190 173.36 = 173.18 + .10(175 – 173.18) 170.44 6 205 175.02 = 173.36 + .10(190 – 173.36) 180.22 7 180 178.02 = 175.02 + .10(205 – 175.02) 192.61 8 182 178.22 = 178.02 + .10(180 – 178.02) 186.30 9 ? 178.59 = 178.22 + .10(182 – 178.22) 184.15
- 44. Determining the MAD QUARTER ACTUAL TONNAGE UNLOADED FORECAST WITH = .10 ABSOLUTE DEVIATION FOR a = .10 FORECAST WITH = .50 ABSOLUTE DEVIATION FOR a = .50 1 180 175 5.00 175 5.00 2 168 175.50 7.50 177.50 9.50 3 159 174.75 15.75 172.75 13.75 4 175 173.18 1.82 165.88 9.12 5 190 173.36 16.64 170.44 19.56 6 205 175.02 29.98 180.22 24.78 7 180 178.02 1.98 192.61 12.61 8 182 178.22 3.78 186.30 4.30 Sum of absolute deviations: 82.45 98.62 MAD = Σ|Deviations| 10.31 12.33 n
- 45. Common Measures ofError Mean Squared Error (MSE) MSE = Forecast errors( ) 2 å n
- 46. Determining the MSE QUARTER ACTUAL TONNAGE UNLOADED FORECASTFOR = .10 (ERROR)2 1 180 175 52 = 25 2 168 175.50 (–7.5)2 = 56.25 3 159 174.75 (–15.75)2 = 248.06 4 175 173.18 (1.82)2 = 3.31 5 190 173.36 (16.64)2 = 276.89 6 205 175.02 (29.98)2 = 898.80 7 180 178.02 (1.98)2 = 3.92 8 182 178.22 (3.78)2 = 14.29 Sum of errors squared = 1,526.52 MSE = Forecast errors( ) 2 å n =1,526.52 / 8 =190.8
- 47. Common Measures ofError Mean Absolute Percent Error (MAPE) MAPE = 100 Actuali -Forecasti i=1 n å / Actuali n
- 48. Determining the MAPE QUARTER ACTUAL TONNAGE UNLOADED FORECASTFOR = .10 ABSOLUTE PERCENT ERROR 100(ERROR/ACTUAL) 1 180 175.00 100(5/180) = 2.78% 2 168 175.50 100(7.5/168) = 4.46% 3 159 174.75 100(15.75/159) = 9.90% 4 175 173.18 100(1.82/175) = 1.05% 5 190 173.36 100(16.64/190) = 8.76% 6 205 175.02 100(29.98/205) = 14.62% 7 180 178.02 100(1.98/180) = 1.10% 8 182 178.22 100(3.78/182) = 2.08% Sum of % errors = 44.75% MAPE = absolute percent errorå n = 44.75% 8 = 5.59%
- 49. Comparison of ForecastError Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded = .10 = .10 = .50 = .50 1 180 175 5.00 175 5.00 2 168 175.5 7.50 177.50 9.50 3 159 174.75 15.75 172.75 13.75 4 175 173.18 1.82 165.88 9.12 5 190 173.36 16.64 170.44 19.56 6 205 175.02 29.98 180.22 24.78 7 180 178.02 1.98 192.61 12.61 8 182 178.22 3.78 186.30 4.30 82.45 98.62
- 50. Comparison of ForecastError Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded a = .10 a = .10 = .50 = .50 1 180 175 5.00 175 5.00 2 168 175.5 7.50 177.50 9.50 3 159 174.75 15.75 172.75 13.75 4 175 173.18 1.82 165.88 9.12 5 190 173.36 16.64 170.44 19.56 6 205 175.02 29.98 180.22 24.78 7 180 178.02 1.98 192.61 12.61 8 182 178.22 3.78 186.30 4.30 82.45 98.62 MAD = ∑ |deviations| n = 82.45/8 = 10.31 For = .10 = 98.62/8 = 12.33 For = .50
- 51. Comparison of ForecastError Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded a = .10 a = .10 = .50 = .50 1 180 175 5.00 175 5.00 2 168 175.5 7.50 177.50 9.50 3 159 174.75 15.75 172.75 13.75 4 175 173.18 1.82 165.88 9.12 5 190 173.36 16.64 170.44 19.56 6 205 175.02 29.98 180.22 24.78 7 180 178.02 1.98 192.61 12.61 8 182 178.22 3.78 186.30 4.30 82.45 98.62 MAD 10.31 12.33 = 1,526.54/8 = 190.82 For = .10 = 1,561.91/8 = 195.24 For = .50 MSE = ∑ (forecast errors)2 n
- 52. Comparison of ForecastError Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded a = .10 a = .10 a = .50 = .50 1 180 175 5.00 175 5.00 2 168 175.5 7.50 177.50 9.50 3 159 174.75 15.75 172.75 13.75 4 175 173.18 1.82 165.88 9.12 5 190 173.36 16.64 170.44 19.56 6 205 175.02 29.98 180.22 24.78 7 180 178.02 1.98 192.61 12.61 8 182 178.22 3.78 186.30 4.30 82.45 98.62 MAD 10.31 12.33 MSE 190.82 195.24 = 44.75/8 = 5.59% For = .10 = 54.05/8 = 6.76% For = .50 MAPE = ∑100|deviationi|/actuali n n i = 1
- 53. Comparison of ForecastError Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded = .10 = .10 = .50 = .50 1 180 175 5.00 175 5.00 2 168 175.5 7.50 177.50 9.50 3 159 174.75 15.75 172.75 13.75 4 175 173.18 1.82 165.88 9.12 5 190 173.36 16.64 170.44 19.56 6 205 175.02 29.98 180.22 24.78 7 180 178.02 1.98 192.61 12.61 8 182 178.22 3.78 186.30 4.30 82.45 98.62 MAD 10.31 12.33 MSE 190.82 195.24 MAPE 5.59% 6.76%
- 54. Exponential Smoothing with TrendAdjustment When a trend is present, exponential smoothing must be modified MONTH ACTUAL DEMAND FORECAST (Ft) FOR MONTHS 1 – 5 1 100 Ft = 100 (given) 2 200 Ft = F1 + (A1 – F1) = 100 + .4(100 – 100) = 100 3 300 Ft = F2 + (A2 – F2) = 100 + .4(200 – 100) = 140 4 400 Ft = F3 + (A3 – F3) = 140 + .4(300 – 140) = 204 5 500 Ft = F4 + (A4 – F4) = 204 + .4(400 – 204) = 282
- 55. Exponential Smoothing with TrendAdjustment Forecast including (FITt) = trend Exponentially Exponentially smoothed (Ft) + smoothed (Tt) forecast trend Ft = (At - 1) + (1 - )(Ft - 1 + Tt - 1) Tt = b(Ft - Ft - 1) + (1 - b)Tt - 1 where Ft = exponentially smoothed forecast average Tt = exponentially smoothed trend At = actual demand = smoothing constant for average (0 ≤ ≤ 1) b = smoothing constant for trend (0 ≤ b ≤ 1)
- 56. Exponential Smoothing with TrendAdjustment Step 1: Compute Ft Step 2: Compute Tt Step 3: Calculate the forecast FITt = Ft + Tt
- 57. Exponential Smoothing with TrendAdjustment Example MONTH (t) ACTUAL DEMAND (At) MONTH (t) ACTUAL DEMAND (At) 1 12 6 21 2 17 7 31 3 20 8 28 4 19 9 36 5 24 10 ? = .2 b = .4
- 58. Exponential Smoothing with TrendAdjustment Example TABLE 4.1 Forecast with - .2 and b = .4 MONTH ACTUAL DEMAND SMOOTHED FORECAST AVERAGE, Ft SMOOTHED TREND, Tt FORECAST INCLUDING TREND, FITt 1 12 11 2 13.00 2 17 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10 — F2 = A1 + (1 – )(F1 + T1) F2 = (.2)(12) + (1 – .2)(11 + 2) = 2.4 + (.8)(13) = 2.4 + 10.4 = 12.8 units Step 1: Average for Month 2 12.80
- 59. Exponential Smoothing with TrendAdjustment Example TABLE 4.1 Forecast with - .2 and b = .4 MONTH ACTUAL DEMAND SMOOTHED FORECAST AVERAGE, Ft SMOOTHED TREND, Tt FORECAST INCLUDING TREND, FITt 1 12 11 2 13.00 2 17 12.80 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10 — T2 = b(F2 - F1) + (1 - b)T1 T2 = (.4)(12.8 - 11) + (1 - .4)(2) = .72 + 1.2 = 1.92 units Step 2: Trend for Month 2 1.92
- 60. Exponential Smoothing with TrendAdjustment Example TABLE 4.1 Forecast with - .2 and b = .4 MONTH ACTUAL DEMAND SMOOTHED FORECAST AVERAGE, Ft SMOOTHED TREND, Tt FORECAST INCLUDING TREND, FITt 1 12 11 2 13.00 2 17 12.80 1.92 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10 — FIT2 = F2 + T2 FIT2 = 12.8 + 1.92 = 14.72 units Step 3: Calculate FIT for Month 2 14.72
- 61. Exponential Smoothing with TrendAdjustment Example TABLE 4.1 Forecast with - .2 and b = .4 MONTH ACTUAL DEMAND SMOOTHED FORECAST AVERAGE, Ft SMOOTHED TREND, Tt FORECAST INCLUDING TREND, FITt 1 12 11 2 13.00 2 17 12.80 1.92 14.72 3 20 15.18 2.10 17.28 4 19 17.82 2.32 20.14 5 24 19.91 2.23 22.14 6 21 22.51 2.38 24.89 7 31 24.11 2.07 26.18 8 28 27.14 2.45 29.59 9 36 29.28 2.32 31.60 10 — 32.48 2.68 35.16
- 62. Exponential Smoothing with TrendAdjustment Example Figure 4.3 | | | | | | | | | 1 2 3 4 5 6 7 8 9 Time (months) Productdemand 40 – 35 – 30 – 25 – 20 – 15 – 10 – 5 – 0 – Actual demand (At) Forecast including trend (FITt) with = .2 and b = .4
- 63. Trend Projections Fitting atrend line to historical data points to project into the medium to long-range Linear trends can be found using the least squares technique y = a + bx^ where y = computed value of the variable to be predicted (dependent variable) a = y-axis intercept b = slope of the regression line x = the independent variable ^
- 64. Least Squares Method Figure4.4 Deviation1 (error) Deviation5 Deviation7 Deviation2 Deviation6 Deviation4 Deviation3 Actual observation (y-value) Trend line, y = a + bx^ Time period ValuesofDependentVariable(y-values) | | | | | | | 1 2 3 4 5 6 7 Least squares method minimizes the sum of the squared errors (deviations)
- 65. Least Squares Method Equationsto calculate the regression variables ˆy = a+bx b = xy - nxyå x2 - nx2 å a = y -bx
- 66. Least Squares Example YEAR ELECTRICAL POWERDEMAND YEAR ELECTRICAL POWER DEMAND 1 74 5 105 2 79 6 142 3 80 7 122 4 90
- 67. Least Squares Example YEAR(x) ELECTRICAL POWER DEMAND (y) x2 xy 1 74 1 74 2 79 4 158 3 80 9 240 4 90 16 360 5 105 25 525 6 142 36 852 7 122 49 854 Σx = 28 Σy = 692 Σx2 = 140 Σxy = 3,063 x = xå n = 28 7 = 4 y = yå n = 692 7 = 98.86
- 68. Least Squares Example YEAR(x) ELECTRICAL POWER DEMAND (y) x2 xy 1 74 1 74 2 79 4 158 3 80 9 240 4 90 16 360 5 105 25 525 6 142 36 852 7 122 49 854 Σx = 28 Σy = 692 Σx2 = 140 Σxy = 3,063 x = xå n = 28 7 = 4 y = yå n = 692 7 = 98.86 Demand in year 8 = 56.70 + 10.54(8) = 141.02, or 141 megawatts b = xy -nxyå x2 - nx2 å = 3,063- 7( ) 4( ) 98.86( ) 140- 7( ) 42 ( ) = 295 28 =10.54 a = y -bx = 98.86-10.54 4( )= 56.70 Thus, ˆy = 56.70+10.54x
- 69. Least Squares Example || | | | | | | | 1 2 3 4 5 6 7 8 9 160 – 150 – 140 – 130 – 120 – 110 – 100 – 90 – 80 – 70 – 60 – 50 – Year Powerdemand(megawatts) Trend line, y = 56.70 + 10.54x^ Figure 4.5