1. a. Is the Holt-Winters model a good model of the N2O data? Justify
b. Using the Holt-Winters model, provide a 95% prediction interval for the N2O level in May, 2020.
c. Comment on the diagnostic plots for the model (SF.fit1). Why was this model rejected?
d. Comment on the diagnostic plots for the model SF.fit2. Are the underlying assumptions satisfied? Briefly explain.
e. Using model SF.fit2, write an equation to calculate the estimated N2O for November, 2019 . (not the model equation, so use the estimated values for the coecients and substitute appropriate values for variables).- do not need final value.
The following data are the monthly average global atmospheric concentrations of N20 from January 2008 to October 2019 in parts per billion (ppb). Sep > N20.ts = ts (N20. df$120, start=2008, frequency=12) > N20.ts Jan Feb Mar Apr May Jun Jul Aug Oct Nov Dec 2008 321.3 321.4 321.4 321.4 321.4 321.5 321.4 321.4 321.5 321.6 321.8 322.0 2009 322.2 322.3 322.3 322.1 322.0 322.0 322.0 322.1 322.3 322.5 322.7 322.8 2010 322.9 322.9 322.9 323.0 323.0 323.0 323.0 323.1 323.3 323.5 323.7 323.9 2011 324.0 324.1 324.1 324.1 324.1 324.0 324.1 324.1 324.2 324.4 324.6 324.8 2012 324.9 325.0 325.0 324.9 324.9 324.9 324.9 325.0 325.1 325.3 325.4 325.5 2013 325.6 325.6 325.7 325.7 325.8 325.9 326.0 326.0 326.1 326.2 326.4 326.5 2014 326.6 326.7 326.7 326.8 326.8 326.9 327.0 327.2 327.3 327.5 327.7 327.9 2015 328.0 328.1 328.0 328.0 327.9 328.0 328.0 328.1 328.2 328.4 328.6 328.8 2016 328.9 328.9 328.9 328.9 328.9 328.9 328.9 328.8 328.9 329.0 329.2 329.4 2017 329.5 329.5 329.5 329.5 329.5 329.6 329.7 329.8 329.9 330.0 330.2 330.3 2018 330.4 330.6 330.7 330.7 330.7 330.7 330.7 330.9 331.1 331.3 331.5 331.7 2019 331.8 331.7 331.7 331.6 331.6 331.7 331.9 331.9 331.9 332.1 > plot(N20.ts,xlab="month",ylab="ppb", main="Global atmospheric concentration of Nitrous Oxide - Jan 2008 to Oct 2019") Global atmospheric concentration of Nitrous Oxide - Jan 2008 to Oct 2019 ppb 322 324 326 328 330 332 2008 2010 2012 2014 2016 2018 2020 month > HW. fit - HoltWinters (N20.00) > Hw.fit Holt-Winters exponential smoothing with trend and additive seasonal component. > HW.pred - predict (HW.fit, n.ahead=12, prediction.interval=T) > plot (Hw.fit,w.pred, main="Global atmospheric concentration of Nitrous Oxide") Call: HaltWinters (x - N20.ts) Global atmospheric concentration of Nitrous Oxide Smoothing parameters: alpha: 0.8527811 beta : 0.01102914 garana: 1 OCE Coefficients: 1.1) a. 332.17880227 b 0.07663861 0.01925176 B2 0.10884812 0.18119490 54 0.22432467 85 0.23304180 86 0.14142845 87 0.01506802 -0.08650202 89 -0.14807969 510 -0.18645879 811 -0.16506783 B12 -0.07880227 1 2014 2010 2012 2016 2018 2020 Time > HW pred fit upr lwr Nay AD 19 132 2747 337.47653 332.0726 Dec 2019 332.4409 332.7077 332.1741 Jan 2020 332.5899 332.9096 332.2702 Feb 2020 332.7097 333.0756 332.3437 Mar 2020 332.7950 383.2028 332.3873 Apr 2020 332.7801 333.2266 332.3336 May 2020 332.7303 333.2132 332.2475 Jun 2020 332.7054 333.2227 332.1881 Jul 2020 332.7205 333.2707 332.1703 Aug 2020 332.7587 333. 3406 332.1768 Sep 2020 332.8568 333.4693 332.2442 Oct 2020 333.0197 333.6620 332.3773 > acf (residuals(SF fit1)) Note: Define Month as a factor for month of the year coded as 1 for January, 2 for February, etc. Define Tine es el nuencric variable with valucs 1 for January 2008, 2 for February 1980, 142 for October 2019 Series residuals(SF.fit1) > Tame = 1:142 > Month - factor (e(rep(1:12,11),(1:10))) 9 > SF.fiti = ln (20.ts Time+Month) > plot.ts (residuals (SF. fiti), nain="Residual Series) ACF Residual Series 0 5 10 15 20 Lag -0,3 -0.1 > SF fit2 = ln (N20.ta (-1) "Time (-1) +Month (-1)+N20.ta(-142]) > plot.ta(residuals (SF.fit2),main="Residual Series") 0 20 40 60 100 120 140 Residual Series Time 0 20 40 60 80 100 120 140 Time > acf (residuals(SF fit2)) > summary (SF.fit2) Series residuals(SF.fit2) Call: 1m(formula - 120.ts [-1] - Time[-1] + Month (-1] + N20.ts (-142]) Residuals: Min 10 Median 30 Max -0.163659 -0.042180 0.005527 0.041432 0.157619 ACH -0.2 0 5 10 15 20 Coefficients: Estimate Std. Error t value Pr>t) (Intercept) 30.135131 11.598702 2.598 0.010480. Time (-1) 0.007490 0.002867 2.613 0.010067 * Month(-1)2 -0.045989 0.025966 -1.771 0.078939 Month(-1)3 -0.098020 0.025925 -3.781 0.000239 *** Month[-1]4 -0.129731 0.025956 -4.998 1.88e-06 *** Month(-1) 5 -0.130448 0.026398 -4.941 2.40e-06 *** Month[-1]6 -0.088718 0.027186 -3.263 0.001414 +. Month(-1)7 -0.092309 0.027628 -3.341 0.001096 ** Month(-1) 8 -0.070900 0.028129 -2.521 0.012958 * Month(-1) -0.022152 -0.782 0.435404 Month(-1) 10 0.031276 0.027794 1.125 0.262593 Month[-1]11 0.063619 0.027393 2.322 0.021800 - Month(-1) 12 0.016722 0.026635 1.754 0.081815 N20.ts (-142] 0.906416 0.036144 25.078 normcheck (SF.tit2, main - "") Signif. codes: O'***' 0.001 '**' 0.01 's' 0.05'.' 0.1'' 1 Sample Quantiles Residual standard error: 0.06205 on 127 degrees of freedom Multiple R-squared: 0.9997,Adjusted R-squared: 0.9996 F-statistic: 2.938e+04 on 13 and 127 DF, p-value: N20.ts(142) (1) 332.1 Theoretical Quantiles Andror i Ti-1]+ MWOK-12 The following data are the monthly average global atmospheric concentrations of N20 from January 2008 to October 2019 in parts per billion (ppb). Sep > N20.ts = ts (N20. df$120, start=2008, frequency=12) > N20.ts Jan Feb Mar Apr May Jun Jul Aug Oct Nov Dec 2008 321.3 321.4 321.4 321.4 321.4 321.5 321.4 321.4 321.5 321.6 321.8 322.0 2009 322.2 322.3 322.3 322.1 322.0 322.0 322.0 322.1 322.3 322.5 322.7 322.8 2010 322.9 322.9 322.9 323.0 323.0 323.0 323.0 323.1 323.3 323.5 323.7 323.9 2011 324.0 324.1 324.1 324.1 324.1 324.0 324.1 324.1 324.2 324.4 324.6 324.8 2012 324.9 325.0 325.0 324.9 324.9 324.9 324.9 325.0 325.1 325.3 325.4 325.5 2013 325.6 325.6 325.7 325.7 325.8 325.9 326.0 326.0 326.1 326.2 326.4 326.5 2014 326.6 326.7 326.7 326.8 326.8 326.9 327.0 327.2 327.3 327.5 327.7 327.9 2015 328.0 328.1 328.0 328.0 327.9 328.0 328.0 328.1 328.2 328.4 328.6 328.8 2016 328.9 328.9 328.9 328.9 328.9 328.9 328.9 328.8 328.9 329.0 329.2 329.4 2017 329.5 329.5 329.5 329.5 329.5 329.6 329.7 329.8 329.9 330.0 330.2 330.3 2018 330.4 330.6 330.7 330.7 330.7 330.7 330.7 330.9 331.1 331.3 331.5 331.7 2019 331.8 331.7 331.7 331.6 331.6 331.7 331.9 331.9 331.9 332.1 > plot(N20.ts,xlab="month",ylab="ppb", main="Global atmospheric concentration of Nitrous Oxide - Jan 2008 to Oct 2019") Global atmospheric concentration of Nitrous Oxide - Jan 2008 to Oct 2019 ppb 322 324 326 328 330 332 2008 2010 2012 2014 2016 2018 2020 month > HW. fit - HoltWinters (N20.00) > Hw.fit Holt-Winters exponential smoothing with trend and additive seasonal component. > HW.pred - predict (HW.fit, n.ahead=12, prediction.interval=T) > plot (Hw.fit,w.pred, main="Global atmospheric concentration of Nitrous Oxide") Call: HaltWinters (x - N20.ts) Global atmospheric concentration of Nitrous Oxide Smoothing parameters: alpha: 0.8527811 beta : 0.01102914 garana: 1 OCE Coefficients: 1.1) a. 332.17880227 b 0.07663861 0.01925176 B2 0.10884812 0.18119490 54 0.22432467 85 0.23304180 86 0.14142845 87 0.01506802 -0.08650202 89 -0.14807969 510 -0.18645879 811 -0.16506783 B12 -0.07880227 1 2014 2010 2012 2016 2018 2020 Time > HW pred fit upr lwr Nay AD 19 132 2747 337.47653 332.0726 Dec 2019 332.4409 332.7077 332.1741 Jan 2020 332.5899 332.9096 332.2702 Feb 2020 332.7097 333.0756 332.3437 Mar 2020 332.7950 383.2028 332.3873 Apr 2020 332.7801 333.2266 332.3336 May 2020 332.7303 333.2132 332.2475 Jun 2020 332.7054 333.2227 332.1881 Jul 2020 332.7205 333.2707 332.1703 Aug 2020 332.7587 333. 3406 332.1768 Sep 2020 332.8568 333.4693 332.2442 Oct 2020 333.0197 333.6620 332.3773 > acf (residuals(SF fit1)) Note: Define Month as a factor for month of the year coded as 1 for January, 2 for February, etc. Define Tine es el nuencric variable with valucs 1 for January 2008, 2 for February 1980, 142 for October 2019 Series residuals(SF.fit1) > Tame = 1:142 > Month - factor (e(rep(1:12,11),(1:10))) 9 > SF.fiti = ln (20.ts Time+Month) > plot.ts (residuals (SF. fiti), nain="Residual Series) ACF Residual Series 0 5 10 15 20 Lag -0,3 -0.1 > SF fit2 = ln (N20.ta (-1) "Time (-1) +Month (-1)+N20.ta(-142]) > plot.ta(residuals (SF.fit2),main="Residual Series") 0 20 40 60 100 120 140 Residual Series Time 0 20 40 60 80 100 120 140 Time > acf (residuals(SF fit2)) > summary (SF.fit2) Series residuals(SF.fit2) Call: 1m(formula - 120.ts [-1] - Time[-1] + Month (-1] + N20.ts (-142]) Residuals: Min 10 Median 30 Max -0.163659 -0.042180 0.005527 0.041432 0.157619 ACH -0.2 0 5 10 15 20 Coefficients: Estimate Std. Error t value Pr>t) (Intercept) 30.135131 11.598702 2.598 0.010480. Time (-1) 0.007490 0.002867 2.613 0.010067 * Month(-1)2 -0.045989 0.025966 -1.771 0.078939 Month(-1)3 -0.098020 0.025925 -3.781 0.000239 *** Month[-1]4 -0.129731 0.025956 -4.998 1.88e-06 *** Month(-1) 5 -0.130448 0.026398 -4.941 2.40e-06 *** Month[-1]6 -0.088718 0.027186 -3.263 0.001414 +. Month(-1)7 -0.092309 0.027628 -3.341 0.001096 ** Month(-1) 8 -0.070900 0.028129 -2.521 0.012958 * Month(-1) -0.022152 -0.782 0.435404 Month(-1) 10 0.031276 0.027794 1.125 0.262593 Month[-1]11 0.063619 0.027393 2.322 0.021800 - Month(-1) 12 0.016722 0.026635 1.754 0.081815 N20.ts (-142] 0.906416 0.036144 25.078 normcheck (SF.tit2, main - "") Signif. codes: O'***' 0.001 '**' 0.01 's' 0.05'.' 0.1'' 1 Sample Quantiles Residual standard error: 0.06205 on 127 degrees of freedom Multiple R-squared: 0.9997,Adjusted R-squared: 0.9996 F-statistic: 2.938e+04 on 13 and 127 DF, p-value: N20.ts(142) (1) 332.1 Theoretical Quantiles Andror i Ti-1]+ MWOK-12
