Lecture 4

Lecture 4 - 540:453 Production Control Lecture 4:...

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Unformatted text preview: 540:453 Production Control Lecture 4: Forecasting (Ch. 2) Prof. T. Boucher 1 Winter’s Method: Exponential Smoothing w/ Trend and Seasonality • Basic Data Pattern 2 Winter’s Method: Exponential Smoothing w/ Trend and Seasonality • Data Generating Process X t ,t (at bt )C t t • The Model , let N be the number of seasons ˆ ˆˆˆ (a b )C F X t ,t t ,t t t N • Steps ˆ – Initialize the model parameters: at bt ˆ ˆ Ct – Forecast – Revise ˆˆˆ a b C based on new data t t t 3 Winter’s Method: Procedure • Obtain initial estimates of the slope, and the seasonal factors (a minimum of two seasons of data) • Assume we have two seasons of data, each with N data points 1. Calculate the sample means (V1 and V2) for the two separate seasons 2. Define the initial trend b0 = (V2 – V1)/N 3. Estimate the initial interception a0 = V2 + b0[(N-1)/2] 4. Calculate the initial seasonal factors 4 Example • quarterly sales data • Initialization S1 30 48 60 35 4 43.25 54.5 43.25 4 ˆ b ˆ a8 S2 S2 ˆ n 1) b( 2 42 58 74 44 4 2 .8 3 54.5 2.8( ) 2 58.7 54.5 Example • quarterly sales data • Initialization Ct Xt ˆ a0 ˆ bt Forecasting from period 8 ˆ X 8, 9 [58.7 2.8(1)]0.8 49.4 ˆ X 8,10 [58.7 2.8(2)]1.12 72.0 ˆ X 8,11 [58.7 2.8(3)]1.33 89.2 ˆ X 8,12 [58.7 2.8(4)]0.75 52.4 7 Updating Model Parameters • Assume the following forecast and actual for period t = 9 ˆ X 8, 9 [58.7 2.8(1)]0.8 X9 49.4 48.0 • Update at by smoothing two adjustments to the new level ˆ at X ˆ [ t ] (1 a)[at Ct N 1 ˆ bt 1 ] ˆ a9 0.2[ 48 (0.8)[58.7 2.8] 0 .8 61.2 • Update Ct by smoothing two adjustments to the new level Xt [ ] (1 ˆ at ˆ Ct ˆ )Ct ˆ C1 N 0.2[ 48 (0.8)(0.8) 61.2 0.8 • Update bt by smoothing two adjustments to the new level ˆ bt ˆ [at ˆ at 1 ] (1 ˆ )bt 1 ˆ b9 0.2[61.2 58.7] (0.8)2.8 2.74 8 Updating the forecast for period 10 • Prior Forecast ˆ X 8,10 [58.7 2.8(2)]1.12 72.0 • Updated Forecast ˆ X 9,10 [61.2 2.74(1)]1.12 71.6 Confidence Interval of Forecast 10 Time Series Models and Tracking Signals • • Because time series models are not causal, but depend on the same variable moving through time, it is possible that the model parameters will change. A tracking signal is used to monitor the performance of the model and will signal when the model should be re-evaluated. Tt t t i0 Linear Regression • Used in causal models in which there is a structural relationship between variables – amount of advertising and subsequent sales – amount of fertilizer used per acre and crop yield • Fundamentally different for time series in which there is no cause and effect relationship necessary Linear Regression • Data Generating Process: Yi a bX i e ~ NID( ei 0, e ) • Model: need to estimate the parameters a and b Yi a bX i Linear Regression • If we have data with paired observations of X and Y, we can empirically derive the parameters of the model by finding the linear function that minimizes the sum of squared errors. ei L n ei 2 i1 2 n i1 (Yi (Yi ˆ Yi ) 2 ˆ Yi ) 2 n (Yi ˆˆ a bX i ) 2 i1 • Transform the data set and the model by moving the reference point to and Y X L n i1 {(Yi ˆ Y ) b( X i X )}2 Linear Regression • The regression line goes through the mean of X and Y. Yi ˆ a L n ei n 2 i1 L i1 n i1 {(Yi (Yi ˆ Yi ) n 2 ˆ b( X i Y Y (Yi ˆˆ a bX i ) 2 i1 ˆ Y ) b( X i ˆ bX X )}2 X) Linear Regression n L ˆ Y ) b( X i {(Yi X )}2 i1 • Taking partial derivatives: L Y n 2 [Yi Y ˆ b( X i X )] 0 i1 L ˆ b 2 n [Yi Y ˆ b( X i X )]( X i X) 0 i1 • Which Yields: n ˆ b • Also: Yi ( X i X) i1 n (X i i1 X) 2 ˆ a Y ˆ bX Linear Regression • Example: X 400 Y 60 • Model: • Computations: n ˆ b i1 n Yi ( X i (X i X) X )2 19000 280000 ˆ Yi 0.068 32.8 0.068 X i • Estimation: i1 ˆ a Y ˆ bX 60 0.068(400) 32.8 ˆ Yi 32.8 0.068(500) 66.8 Linear Regression Prediction Interval: it can be shown that the 95% confidence interval around a forecast is given by s s2 2 1 n2 1 (35.72) 5 [Yi ˆ Yi ] 7.144 2 CI CI ˆ Yk ˆ Yk 1 t0.025 s n 1 t 0.025 s n ( X k X )2 ˆ 1; Y700 ( X i X )2 X k2 X i2 1 2.571(2.67) 1 7 90000 1 8.31 280000 Linear Regression • Coefficient of Multiple Determination, R2: • If you had no function, Y would be the “best guess” at any given value of X. ˆ • b( X i X ) is the amount of variation “explained” by the function. ˆ [b( X i R 2 1 X )] 2 i (Yi i Y) 2 0 R2 1 Linear Regression • Example: ˆ [b( X i R2 1 X )] 2 i (Yi i Y) 2 1 1294.72 1325 0.98 Quantitative Forecasting Applications Small and Large Firms Low Sales Technique Moving average Straight line projection Naive Exponential smoothing Regression Simulation Classical decomposition Box-Jenkins Number of Firms High Sales < $100M > $500M 29.6% 14.8% 18.5% 14.8% 22.2% 3.7% 3.7% 3.7% 27 29.2% 14.6% 14.6% 20.8% 27.1% 10.4% 8.3% 6.3% 48 Source: Nada Sanders and Karl Mandrodt (1994) “Practitioners Continue to Rely on Judgmental Forecasting Methods Instead of Quantitative Methods,” Interfaces, vol. 24, no. 2, pp. 92-100. 21 Time Series Forecasting using Excel • Excel can be used to develop forecasts: – Moving average – Exponential smoothing – Linear trend line 12-22 ...
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