The data below is for 25 periods of sales in $millions. t Sales(Y) 1 70 2 75 3 80 4 84 5 77 6 85 7 100 8 92 9 95 10 98 11 93 12 110 13 125 14 120 15 112 16 128 17 135 18 142 19 145 20 140 21 150 22 155 23 163 24 170 25 165 1. Plot the time series data. Forecast for the 26th period using simple exponential smoothing with alpha 0.4 and 0.7 and plot. the actual 25 period values with forecasted values. Also calculate the errors (as well as MPEs) and plot the errors for the two forecasts. Compare the two forecasts using MSE and RMSE. Comment on the error plots (about stationarity). Do you see systematic underestimation? How do you measure it and visualize it? 25 points 2. With the data above, perform the forecast using Holt's two parameter model with alpha = 0.5 and beta = 0.6. Plot the Holt's forecast with actual values. Also calculate the MSE for Holt's forecast. Is it better than the simple exponential smoothing forecasts? Comment on the underestimation or overestimation. forecasts? Comment on the underestimation or overestimation using suitable measure. Plot the errors and MPE for Holt's model. Does it look nearly stationary? 35 points 3. For the data below (Production = Y in million tons) plot the data and comment whether you see a seasonal pattern. Perform Holt-Winters' seasonal forecast using alpha = 0.4, beta = 0.6 and gamma = 0.8. Make forecasts for 2018-q1 and 2018-q2 using Holt-Winters' model. Calculate the errors and MSE and RMSE. Plot the errors and comment on the plot. 40 points Year-q period=t| Y 2012-1 1 65 2012-2 2 80 2012-3 3 75 2012-4 4 73 2013-1 5 76 2013-2 6 99 2013-3 7 96 2013-4 8 86 2014-1 9 90 2014-2 10 105 2014-3 11 100 2014-4 12 95 2015-1 13 102 2015-2 14 132 2015-3 15 118 2015-4 16 115 2016-1 17 131 2016-2 18 150 2016-3 19 146 2016-4 20 140 2017-1 21 145 2017-2 22 170 |2017-3 23 165 2017-4 24 160 2018-1 2018-2 2