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1 Forecasting Forecasting Terminology Simple Moving Average Weighted Moving Average Exponential Smoothing Simple Linear Regression Model Holts Trend Model Seasonal Model (No Trend) Winters Model for Data with Trend and Seasonal Components Slide 2 2 Evaluating Forecasts Visual Review Errors Errors Measure MPE and MAPE Tracking Signal Slide 3 3 Historical Data 0 50 100 150 200 250 300 350 400 01020304050 Forecasting Terminology Initialization ExPost Forecast Historical Data Slide 4 4 We are now looking at a future from here, and the future we were looking at in February now includes some of our past, and we can incorporate the past into our forecast. 1993, the first half, which is now the past and was the future when we issued our first forecast, is now over Laura DAndrea Tyson, Head of the Presidents Council of Economic Advisors, quoted in November of 1993 in the Chicago Tribune, explaining why the Administration reduced its projections of economic growth to 2 percent from the 3.1percent it predicted in February. Forecasting Terminology Slide 5 5 Forecasting Problem Suppose your fraternity/sorority house consumed the following number of cases of beer for the last 6 weekends: 8, 5, 7, 3, 6, 9 How many cases do you think your fraternity / sorority will consume this weekend? Slide 6 6 0 1 2 3 4 5 6 7 8 9 10 012345678 Week Cases Forecasting: Simple Moving Average Method Using a three period moving average, we would get the following forecast: Slide 7 7 0 1 2 3 4 5 6 7 8 9 10 012345678 Week Cases Forecasting: Simple Moving Average Method What if we used a two period moving average? Slide 8 8 The number of periods used in the moving average forecast affects the responsiveness of the forecasting method: 0 1 2 3 4 5 6 7 8 9 10 012345678 Week Cases Forecasting: Simple Moving Average Method 2 Periods 3 Periods 1 Period Slide 9 9 Forecasting Terminology Applying this terminology to our problem using the Moving Average forecast: Initialization ExPost Forecast Model Evaluation Slide 10 10 Rather than equal weights, it might make sense to use weights which favor more recent consumption values. With the Weighted Moving Average, we have to select weights that are individually greater than zero and less than 1, and as a group sum to 1: Valid Weights: (.5,.3,.2), (.6,.3,.1), ( 1/2, 1/3, 1/6) Invalid Weights: (.5,.2,.1), (.6, -.1,.5), (.5,.4,.3,.2) Forecasting: Weighted Moving Average Method Slide 11 11 Forecasting: Weighted Moving Average Method A Weighted Moving Average forecast with weights of (1/6, 1/3, 1/2), is performed as follows: How do you make the Weighted Moving Average forecast more responsive? Slide 12 12 Exponential Smoothing is designed to give the benefits of the Weighted Moving Average forecast without the cumbersome problem of specifying weights. In Exponential Smoothing, there is only one parameter ( ): = smoothing constant (between 0 and 1) Forecasting: Exponential Smoothing Slide 13 13 Initialization: Forecasting: Exponential Smoothing Slide 14 14 tA(t)F(t) 18 256.5 375.9 436.34 565 695.4 7 6.84 8 9 10 6.84 Forecasting: Exponential Smoothing Using = 0.4, Initialization ExPost Forecast Slide 15 15 Forecasting: Exponential Smoothing Slide 16 16 Forecasting: Exponential Smoothing 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1234567 Period Weight = 0.1 = 0.3 = 0.5 = 0.7 = 0.9 Slide 17 17 Outliers (eloping point) Outlier Slide 18 18 Data with Trends Slide 19 19 0 1 2 3 4 5 6 7 8 9 10 1234567 A(t) Data with Trends = 0.3 = 0.5 = 0.7 = 0.9 Slide 20 20 Forecasting: Simple Linear Regression Model Simple linear regression can be used to forecast data with trends D is the regressed forecast value or dependent variable in the model, a is the intercept value of the regression line, and b is the slope of the regression line. a D 0 1 2 3 4 5 I b Slide 21 21 Forecasting: Simple Linear Regression Model In linear regression, the squared errors are minimized Error Slide 22 22 Forecasting: Simple Linear Regression Model Slide 23 23 0 50 100 150 200 250 0246810121416 Limitations in Linear Regression Model As with the simple moving average model, all data points count equally with simple linear regression. Slide 24 24 Forecasting: Holts Trend Model To forecast data with trends, we can use an exponential smoothing model with trend, frequently known as Holts model: L(t) = A(t) + (1- ) F(t) T(t) = L(t) - L(t-1) ] + (1- ) T(t-1) F(t+1) = L(t) + T(t) We could use linear regression to initialize the model Slide 25 25 Holts Trend Model: Initialization First, well initialize the model: L(4) = 20.5+4(9.9)=60.1 T(4) = 9.9 Slide 26 26 Holts Trend Model: Updating 52 L(t) = A(t) + (1- ) F(t) L(5) = 0.3 (52) + 0.7 (70)=64.6 T(t) = [L(t) - L(t-1) ] + (1- ) T(t-1) T(5) = 0.4 [64.6 60.1] + 0.6 (9.9) = 7.74 F(t+1) = L(t) + T(t) F(6) = 64.6 + 7.74 = 72.34 64.6 7.74 72.346 Slide 27 27 Holts Trend Model: Updating 63 L(6) = 0.3 (63) + 0.7 (72.34)=69.54 T(6) = 0.4 [69.54 64.60] + 0.6 (7.74) = 6.62 F(7) = 69.54 + 6.62 = 76.16 69.54 6.62 76.167 72 Slide 28 28 Holts Model Results Initialization ExPost Forecast Slide 29 29 Regression 0 50 100 150 200 250 300 350 05101520 Initialization ExPost Forecast Forecast Holts Model Results Slide 30 30 Forecasting: Seasonal Model (No Trend) Slide 31 31 L(t) = A(t) / S(t-p) + (1- ) L(t-1) S(t) = A(t) / L(t)] + (1- ) S(t-p) Seasonal Model Formulas p is the number of periods in a season Quarterly data: p = 4 Monthly data: p = 12 F(t+1) = L(t) * S(t+1-p) Slide 32 32 Seasonal Model Initialization S(5) = 0.60 S(6) = 1.00 S(7) = 1.55 S(8) = 0.85 L(8) = 26.5 Quarter Average 16.0 26.5 41.0 22.5 Seasonal Factor S(t) 0.60 1.00 1.55 0.85 Average Sales per Quarter =26.5 A(t) 2003Spring16 Summer27 Fall39 Winter22 2004Spring16 Summer26 Fall43 Winter23 Slide 33 33 Seasonal Model Forecasting 26.711.0325.18 26.621.5541.32 25.180.5916.002005Spring14 Summer29 Fall41 Winter22 26.340.8422.60 2006Spring Summer Fall Winter 15.53 27.02 40.69 22.25 A(t)L(t) Seasonal Factor S(t)F(t) 2004Spring160.60 Summer261.00 Fall431.55 Winter2326.500.85 Slide 34 34 0 5 10 15 20 25 30 35 40 45 50 0246810121416 Seasonal Model Forecasting Slide 35 35 Forecasting: Winters Model for Data with Trend and Seasonal Components L(t) = A(t) / S(t-p) + (1- )[L(t-1)+T(t-1)] T(t) = L(t) - L(t-1)] + (1- ) T(t-1) S(t) = A(t) / L(t)] + (1- ) S(t-p) F(t+1) = L(t) + T(t)] S(t+1-p) Slide 36 36 Seasonal-Trend Model Decomposition To initialize Winters Model, we will use Decomposition Forecasting, which itself can be used to make forecasts. Slide 37 37 Decomposition Forecasting There are two ways to decompose forecast data with trend and seasonal components: Use regression to get the trend, use the trend line to get seasonal factors Use averaging to get seasonal factors, de- seasonalize the data, then use regression to get the trend. Slide 38 38 Decomposition Forecasting The following data contains trend and seasonal components: Slide 39 39 Decomposition Forecasting The seasonal factors are obtained by the same method used for the Seasonal Model forecast: PeriodQuarterSales 1Spring90 2Summer157 3Fall123 4Winter93 5Spring128 6Summer211 7Fall163 8Winter122 Average =135.9 Average to 1 Qtr. Ave. 109 184 143 107.5 Seas. Factor 0.80 1.35 1.05 0.79 1.00 Slide 40 40 Decomposition Forecasting With the seasonal factors, the data can be de- seasonalized by dividing the data by the seasonal factors: Regression on the De-seasonalized data will give the trend Slide 41 41 Decomposition Forecasting Regression Results Slide 42 42 Decomposition Forecast Regression on the de-seasonalized data produces the following results: Slope (m) = 7.71 Intercept (b) = 101.2 Forecasts can be performed using the following equation [mx + b](seasonal factor) Slide 43 43 0 50 100 150 200 250 300 123456789101112 Decomposition Forecasting Slide 44 44 Winters Model Initialization We can use the decomposition forecast to define the following Winters Model parameters: L(n) = b + m (n) T(n) = m S(j) = S(j-p) L(8) = 101.2 + 8 (7.71) = 162.88 T(8) = 7.71 S(5) = 0.80 S(6) = 1.35 S(7) = 1.05 S(8) = 0.79 So from our previous model, we have Slide 45 45 Winters Model Example 176.4110.040.81136.47 197.8514.601.39251.71 215.0015.621.06223.07 9Spring152 10Summer303 11Fall232 12Winter171 226.3713.920.78182.19 13Spring 14Summer 15Fall 16Winter 195.19 352.41 283.09 220.87 = 0.3 = 0.4 = 0.2 PeriodQuarterSalesL(t)T(t)S(t)F(t) 1Spring90 2Summer157 3Fall123 4Winter93 5Spring1280.8 6Summer2111.35 7Fall1621.05 8Winter122162.887.710.79 Slide 46 46 0 50 100 150 200 250 300 350 400 12345678910111213141516 Winters Model Example Slide 47 47 Evaluating Forecasts Trust, but Verify Ronald W. Reagan Computer software gives us the ability to mess up more data on a greater scale more efficiently While software like SAP can automatically select models and model parameters for a set of data, and usually does so correctly, when the data is important, a human should review the model results One of the best tools is the human eye Slide 48 48 0 10 20 30 40 50 60 123456789101112131415 Visual Review How would you evaluate this forecast? Slide 49 49 0 50 100 150 200 250 300 350 400 01020304050 Forecast Evaluation Initialization ExPost Forecast Where Forecast is Evaluated Do not include initialization data in evaluation Slide 50 50 0 100 150 200 250 300 350 400 2025303540 Errors All error measures compare the forecast model to the actual data for the ExPost Forecast region Slide 51 51 Errors Measure All error measures are based on the comparison of forecast values to actual values in the ExPost Forecast regiondo not include data from initialization. Slide 52 52 Bias and MAD Slide 53 53 Bias tells us whether we have a tendency to over- or under-forecast. If our forecasts are in the middle of the data, then the errors should be equally positive and negative, and should sum to 0. MAD (Mean Absolute Deviation) is the average error, ignoring whether the error is positive or negative. Errors are bad, and the closer to zero an error is, the better the forecast is likely to be. Error measures tell how well the method worked in the ExPost forecast region. How well the forecast will work in the future is uncertain. Bias and MAD Slide 54 54 Absolute vs. Relative Measures Forecasts were made for two sets of data. Which forecast was better? Data Set 1 Bias = 18.72 MAD = 43.99 Data Set 2 Bias = 182 MAD = 912.5 Data Set 1 Data Set 2 Slide 55 55 MPE and MAPE When the numbers in a data set are larger in magnitude, then the error measures are likely to be large as well, even though the fit might not be as good. Mean Percentage Error (MPE) and Mean Absolute Percentage Error (MAPE) are relative forms of the Bias and MAD, respectively. MPE and MAPE can be used to compare forecasts for different sets of data. Slide 56 56 MPE and MAPE Mean Percentage Error (MPE) Mean Absolute Percentage Error (MAPE) Slide 57 57 MPE and MAPE Data Set 1 Slide 58 58 MPE and MAPE Data Set 2 Slide 59 59 MPE and MAPE Data Set 2 Data Set 1 Slide 60 60 0 10 20 30 40 50 60 123456789101112131415 Tracking Signal Whats happened in this situation? How could we detect this in an automatic forecasting environment? Slide 61 61 Tracking Signal The tracking signal can be calculated after each actual sales value is recorded. The tracking signal is calculated as: The tracking signal is a relative measure, like MPE and MAPE, so it can be compared to a set value (typically 4 or 5) to identify when forecasting parameters and/or models need to be changed. Slide 62 62 Tracking Signal Slide 63 63 0 10 20 30 40 50 60 123456789101112131415 Tracking Signal TS = -5.78