1000 1500 2000 Quarterly sales of videos in the Leeds "Disney" store are shown in figure 1. Below is the code and output for an analysis of these data in R, with the sales data stored in the time series object X. Explain what is being done at points (i)-(iv) in the R code. Explain what is the difference between (v) and (vi) in the R code. Explain, giving reasons, which of (v) and (vi) is preferable. Write out the model with estimated parameters in full. (The relevant points in the R code are denoted #2#2#3#23 (i) #### etc.) Given that the sales for the four quarters of 2018 were 721, 935, 649, and 1071, use model-based forecasting to predict sales for the first quarter of 2019. (A point forecast is sufficient; you do not need to calculate a prediction interval.) Suggest one change to the fitted model which would improve the analysis. (You can assume that the choice of stochastic process at (v) in the R code is the correct one for these data.) 2010 2012 2014 Time 2016 Figure 1: Quarterly video sales in Leeds "Disney" store. 2018 >tt 1:32 > trend.1m = 1m (sales tt) #### (i) #### > summary(trend. 1m) Coefficients: Estimate Std. Error t value Pr(>|t|) 57.997 36.33 < 2e-16 *** 3.067 -14.18 7.72e-15 *** (Intercept) 2107.220 tt -43.500 > trend = ts (fitted (trend.lm), start-start (sales), freq-frequency (sales)) trend # # # #23 (ii) as.numeric((1:32 %% 4) > X = sales > q1 = > q2 > 93 - as.numeric((1:32 %% 4) as.numeric((1:32 %% 4) #### == 1) == 2) == 3) == > q4 as.numeric((1:32 %% 4) 0) = > season.1m = 1m (resid (trend. 1m) > summary(season.1m) Coefficients: 0q1q2q3 + q4) #2#3##23 (iii) #### Estimate Std. Error t value Pr(>|t|) q1 -38.41 92 18.80 43.27 -0.888 0.38232 43.27 0.435 0.66719 q3 -134.78 43.27 -3.115 0.00422 ** 94 154.38 43.27 3.568 0.00132 ** > season = ts (fitted (season.1m), start-start (sales), freq-frequency (sales)) > Y X season %23%23%23%23 (iv) #### >ar (Y, aic=FALSE, order.max=1) %#23%23%23%23 (v) #### Coefficients: 1 0.5704 Order selected 1 sigma^2 estimated as 9431 > ar(Y, aic=FALSE, order.max=2) #23 #23 #23 #23 (vi) #2### Coefficients: 1 0.5574 2 0.0105 Order selected 2 sigma^2 estimated as 9437