Chapter 7 Exercises

1

Consider the pigs series - the number of pigs slaughtered in Victoria each month.

1.a

Use the ses() function in R to find the optimal values of ?? and l(0), and generate forecasts for the next four months.

##  Time-Series [1:188] from 1980 to 1996: 76378 71947 33873 96428 105084 ...
## 
## Forecast method: Simple exponential smoothing
## 
## Model Information:
## Simple exponential smoothing 
## 
## Call:
##  ses(y = pigs, h = 4) 
## 
##   Smoothing parameters:
##     alpha = 0.2971 
## 
##   Initial states:
##     l = 77260.0561 
## 
##   sigma:  10308.58
## 
##      AIC     AICc      BIC 
## 4462.955 4463.086 4472.665 
## 
## Error measures:
##                    ME    RMSE      MAE       MPE     MAPE      MASE
## Training set 385.8721 10253.6 7961.383 -0.922652 9.274016 0.7966249
##                    ACF1
## Training set 0.01282239
## 
## Forecasts:
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Sep 1995       98816.41 85605.43 112027.4 78611.97 119020.8
## Oct 1995       98816.41 85034.52 112598.3 77738.83 119894.0
## Nov 1995       98816.41 84486.34 113146.5 76900.46 120732.4
## Dec 1995       98816.41 83958.37 113674.4 76092.99 121539.8

From the above, we see that our alpha value is .2971 while our initial states is 77260.0561. ### 1.b Compute a 95% prediction interval for the first forecast using yhat plus or minus 1.96s where s is the standard deviation of the residuals. Compare your interval with the interval produced by R.

##      95% 
## 119020.8
##      95% 
## 78611.97
## [1] 118952.8
## [1] 78679.97

The interval generated by R is [78611.97, 119020.8], while the computed interval is [78679.97. 118952.8]. There is a slight but not significant difference. I’d like to plot the forecast and actual data.

Unlike with the example in the text, there is no apparent change in level over time. Because our alpha was on the smaller side, the changes are smaller over time(smoother fitted values).

5

Data set books contains the daily sales of paperback and hardcover books at the same store. The task is to forecast the next four days sales for paperback and hardcover books.

5.a

Plot the series and discuss the main features of the data.

##  Time-Series [1:30, 1:2] from 1 to 30: 199 172 111 209 161 119 195 195 131 183 ...
##  - attr(*, "dimnames")=List of 2
##   ..$ : NULL
##   ..$ : chr [1:2] "Paperback" "Hardcover"

There seems to be a dip every 3 days(seasonality?) and there seems to be an overall upward trend.

5.b

Use the ses() function to forecast each series and plot the forecasts.

## 
## Forecast method: Simple exponential smoothing
## 
## Model Information:
## Simple exponential smoothing 
## 
## Call:
##  ses(y = books[, "Paperback"], h = 4) 
## 
##   Smoothing parameters:
##     alpha = 0.1685 
## 
##   Initial states:
##     l = 170.8271 
## 
##   sigma:  34.8183
## 
##      AIC     AICc      BIC 
## 318.9747 319.8978 323.1783 
## 
## Error measures:
##                    ME     RMSE     MAE       MPE     MAPE      MASE
## Training set 7.175981 33.63769 27.8431 0.4736071 15.57784 0.7021303
##                    ACF1
## Training set -0.2117522
## 
## Forecasts:
##    Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 31       207.1097 162.4882 251.7311 138.8670 275.3523
## 32       207.1097 161.8589 252.3604 137.9046 276.3147
## 33       207.1097 161.2382 252.9811 136.9554 277.2639
## 34       207.1097 160.6259 253.5935 136.0188 278.2005
## 
## Forecast method: Simple exponential smoothing
## 
## Model Information:
## Simple exponential smoothing 
## 
## Call:
##  ses(y = books[, "Hardcover"], h = 4) 
## 
##   Smoothing parameters:
##     alpha = 0.3283 
## 
##   Initial states:
##     l = 149.2861 
## 
##   sigma:  33.0517
## 
##      AIC     AICc      BIC 
## 315.8506 316.7737 320.0542 
## 
## Error measures:
##                    ME     RMSE      MAE      MPE     MAPE      MASE
## Training set 9.166735 31.93101 26.77319 2.636189 13.39487 0.7987887
##                    ACF1
## Training set -0.1417763
## 
## Forecasts:
##    Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 31       239.5601 197.2026 281.9176 174.7799 304.3403
## 32       239.5601 194.9788 284.1414 171.3788 307.7414
## 33       239.5601 192.8607 286.2595 168.1396 310.9806
## 34       239.5601 190.8347 288.2855 165.0410 314.0792

The alpha value for the paperback books is 0.1685 and the initial states is 170.8271. For the hardcover books, the alpha is .3283 and the initial states is 170.8271. I would expect the forecast for the paperback book to be smoother than for the hardcover books as it has a smaller alpha value.

5.c

Compute the RMSE values for the training data in each case.

The RMSE(root mean squared error) is calculated by taking square root of the mean of the residuals squared.

## [1] "The RMSE value for paperback sales is 33.637686782912."
## [1] "The RMSE value for hardcover sales is 31.9310149844547."

The hardcover sales have a smaller variance than the paperback sales, based on the RMSE values.

6.a

Now apply Host’s linear method to the paperback and hardback series and compute the four-day forecast for each case.

## 
## Forecast method: Holt's method
## 
## Model Information:
## Holt's method 
## 
## Call:
##  holt(y = pb, h = 4) 
## 
##   Smoothing parameters:
##     alpha = 1e-04 
##     beta  = 1e-04 
## 
##   Initial states:
##     l = 170.699 
##     b = 1.2621 
## 
##   sigma:  33.4464
## 
##      AIC     AICc      BIC 
## 318.3396 320.8396 325.3456 
## 
## Error measures:
##                     ME     RMSE      MAE       MPE     MAPE      MASE
## Training set -3.717178 31.13692 26.18083 -5.508526 15.58354 0.6602122
##                    ACF1
## Training set -0.1750792
## 
## Forecasts:
##    Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 31       209.4668 166.6035 252.3301 143.9130 275.0205
## 32       210.7177 167.8544 253.5811 145.1640 276.2715
## 33       211.9687 169.1054 254.8320 146.4149 277.5225
## 34       213.2197 170.3564 256.0830 147.6659 278.7735
## 
## Forecast method: Holt's method
## 
## Model Information:
## Holt's method 
## 
## Call:
##  holt(y = hc, h = 4) 
## 
##   Smoothing parameters:
##     alpha = 1e-04 
##     beta  = 1e-04 
## 
##   Initial states:
##     l = 147.7935 
##     b = 3.303 
## 
##   sigma:  29.2106
## 
##      AIC     AICc      BIC 
## 310.2148 312.7148 317.2208 
## 
## Error measures:
##                      ME     RMSE      MAE       MPE    MAPE      MASE
## Training set -0.1357882 27.19358 23.15557 -2.114792 12.1626 0.6908555
##                     ACF1
## Training set -0.03245186
## 
## Forecasts:
##    Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 31       250.1739 212.7390 287.6087 192.9222 307.4256
## 32       253.4765 216.0416 290.9113 196.2248 310.7282
## 33       256.7791 219.3442 294.2140 199.5274 314.0308
## 34       260.0817 222.6468 297.5166 202.8300 317.3334

6.b

Compare the RMSE measures of Holt’s method for the two series to those of simple exponential smoothing in the previous questions. (Remember that Holt’s method is using one more parameter that SES.) Discuss the merits of the two forecasting methods for these data sets.

The RMSE for paperback books using SES is 33.63769, using Holt’s, it’s 31.13692. The RMSE value for hardcover books using SES is 31.93101 and using Holt’s, it’s 27.19358. We see that the value decrease for both series using Holt’s, which indicates a better fit.

6.c

Compare the forecasts for the two series using both methods. Which do you think is best?

The forecast for the hardcover books using Holt’s seems to be the best as it has the lowest RMSE value.

6.d

Calculate the 95% prediction interval for the first forecast for each series using the RMSE values and assuming normal errors. Compare your intervals with those produced using ses and holt.

## [1] "The interval from SES is [275.352305883445, 138.867024106993]"
## [1] "The interval from Holt's is [275.020545221845, 143.912985820256]"
## [1] "The interval that is calculated is [270.495134632871, 148.438396409231]"

Both the SES and Holt method give approximately the same upper value. It’s a little more different with the calucated value. The lower value is the different for all methods.

7

For this exercise use data set eggs, the price of a dozen eggs in the United States from 1900 - 1993. Experiment with the various options in the holt() function to see how much the forecasts change with damped trend, or with a Box-Cox transformation. Try to develop an intuition of what each argument is doing to the forecasts.

[Hint: use h = 100 when calling holt() so you can clearly see the difference between the various options when plotting the forecasts] Which model gives the best RMSE?

I plan to start without setting any parameters, then perhaps build and see what peeks my interest to explore. I’ll also see what he difference is when damp is set to TRUE vs FALSE. Then I’ll perform a Box-Cox on the data and see how it changes our forecast. I took a look at the R documentation before beginning. From there I saw the arguments that Holt takes, giving me a better idea of what to play with.

## 
## Forecast method: Holt's method
## 
## Model Information:
## Holt's method 
## 
## Call:
##  holt(y = eggs) 
## 
##   Smoothing parameters:
##     alpha = 0.8124 
##     beta  = 1e-04 
## 
##   Initial states:
##     l = 314.7232 
##     b = -2.7222 
## 
##   sigma:  27.1665
## 
##      AIC     AICc      BIC 
## 1053.755 1054.437 1066.472 
## 
## Error measures:
##                      ME     RMSE      MAE       MPE     MAPE      MASE
## Training set 0.04499087 26.58219 19.18491 -1.142201 9.653791 0.9463626
##                    ACF1
## Training set 0.01348202
## 
## Forecasts:
##      Point Forecast      Lo 80     Hi 80       Lo 95    Hi 95
## 1994       59.78553  24.970286  94.60078    6.540207 113.0309
## 1995       57.06373  12.206005 101.92145  -11.540238 125.6677
## 1996       54.34192   1.308668 107.37517  -26.765440 135.4493
## 1997       51.62011  -8.488401 111.72863  -40.307926 143.5482
## 1998       48.89831 -17.537664 115.33428  -52.706742 150.5034
## 1999       46.17650 -26.035964 118.38896  -64.262933 156.6159
## 2000       43.45469 -34.106500 121.01589  -75.164916 162.0743
## 2001       40.73289 -41.832449 123.29822  -85.539898 167.0057
## 2002       38.01108 -49.273098 125.29526  -95.478551 171.5007
## 2003       35.28927 -56.472472 127.05102 -105.048206 175.6268

Our data is not seasonal (from plots) but there is a clear downward trend. This tells us that we can use holt for smoothing. The result without setting any parameters are: + Smoothing parameters: + alpha = 0.8124 + beta = 1e-04

  • Initial states:
    • l = 314.7232
    • b = -2.7222

Now I will use h = 100 as suggested in the question. The h argument tells holt how far to forecast. Since this data is in years, setting h to 100 should forecast 100 years in the future.

## 
## Forecast method: Holt's method
## 
## Model Information:
## Holt's method 
## 
## Call:
##  holt(y = eggs, h = 100) 
## 
##   Smoothing parameters:
##     alpha = 0.8124 
##     beta  = 1e-04 
## 
##   Initial states:
##     l = 314.7232 
##     b = -2.7222 
## 
##   sigma:  27.1665
## 
##      AIC     AICc      BIC 
## 1053.755 1054.437 1066.472 
## 
## Error measures:
##                      ME     RMSE      MAE       MPE     MAPE      MASE
## Training set 0.04499087 26.58219 19.18491 -1.142201 9.653791 0.9463626
##                    ACF1
## Training set 0.01348202
## 
## Forecasts:
##      Point Forecast       Lo 80     Hi 80       Lo 95    Hi 95
## 1994    59.78553313   24.970286  94.60078    6.540207 113.0309
## 1995    57.06372643   12.206005 101.92145  -11.540238 125.6677
## 1996    54.34191973    1.308668 107.37517  -26.765440 135.4493
## 1997    51.62011302   -8.488401 111.72863  -40.307926 143.5482
## 1998    48.89830632  -17.537664 115.33428  -52.706742 150.5034
## 1999    46.17649962  -26.035964 118.38896  -64.262933 156.6159
## 2000    43.45469292  -34.106500 121.01589  -75.164916 162.0743
## 2001    40.73288622  -41.832449 123.29822  -85.539898 167.0057
## 2002    38.01107951  -49.273098 125.29526  -95.478551 171.5007
## 2003    35.28927281  -56.472472 127.05102 -105.048206 175.6268
## 2004    32.56746611  -63.464327 128.59926 -114.300487 179.4354
## 2005    29.84565941  -70.275216 129.96654 -123.276007 182.9673
## 2006    27.12385271  -76.926479 131.17418 -132.007398 186.2551
## 2007    24.40204600  -83.435566 132.23966 -140.521350 189.3254
## 2008    21.68023930  -89.816966 133.17744 -148.840022 192.2005
## 2009    18.95843260  -96.082866 133.99973 -156.982051 194.8989
## 2010    16.23662590 -102.243631 134.71688 -164.963291 197.4365
## 2011    13.51481920 -108.308165 135.33780 -172.797358 199.8270
## 2012    10.79301249 -114.284184 135.87021 -180.496053 202.0821
## 2013     8.07120579 -120.178426 136.32084 -188.069681 204.2121
## 2014     5.34939909 -125.996817 136.69562 -195.527304 206.2261
## 2015     2.62759239 -131.744601 136.99979 -202.876944 208.1321
## 2016    -0.09421431 -137.426446 137.23802 -210.125737 209.9373
## 2017    -2.81602102 -143.046526 137.41448 -217.280071 211.6480
## 2018    -5.53782772 -148.608596 137.53294 -224.345686 213.2700
## 2019    -8.25963442 -154.116046 137.59678 -231.327766 214.8085
## 2020   -10.98144112 -159.571946 137.60906 -238.231008 216.2681
## 2021   -13.70324782 -164.979092 137.57260 -245.059687 217.6532
## 2022   -16.42505453 -170.340036 137.48993 -251.817706 218.9676
## 2023   -19.14686123 -175.657116 137.36339 -258.508640 220.2149
## 2024   -21.86866793 -180.932478 137.19514 -265.135773 221.3984
## 2025   -24.59047463 -186.168101 136.98715 -271.702130 222.5212
## 2026   -27.31228133 -191.365811 136.74125 -278.210504 223.5859
## 2027   -30.03408804 -196.527300 136.45912 -284.663482 224.5953
## 2028   -32.75589474 -201.654137 136.14235 -291.063466 225.5517
## 2029   -35.47770144 -206.747783 135.79238 -297.412688 226.4573
## 2030   -38.19950814 -211.809598 135.41058 -303.713228 227.3142
## 2031   -40.92131484 -216.840851 134.99822 -309.967029 228.1244
## 2032   -43.64312155 -221.842733 134.55649 -316.175908 228.8897
## 2033   -46.36492825 -226.816355 134.08650 -322.341569 229.6117
## 2034   -49.08673495 -231.762762 133.58929 -328.465610 230.2921
## 2035   -51.80854165 -236.682939 133.06586 -334.549533 230.9324
## 2036   -54.53034835 -241.577809 132.51711 -340.594753 231.5341
## 2037   -57.25215506 -246.448244 131.94393 -346.602603 232.0983
## 2038   -59.97396176 -251.295068 131.34714 -352.574344 232.6264
## 2039   -62.69576846 -256.119059 130.72752 -358.511164 233.1196
## 2040   -65.41757516 -260.920954 130.08580 -364.414190 233.5790
## 2041   -68.13938186 -265.701450 129.42269 -370.284491 234.0057
## 2042   -70.86118857 -270.461210 128.73883 -376.123079 234.4007
## 2043   -73.58299527 -275.200863 128.03487 -381.930915 234.7649
## 2044   -76.30480197 -279.921006 127.31140 -387.708914 235.0993
## 2045   -79.02660867 -284.622209 126.56899 -393.457946 235.4047
## 2046   -81.74841537 -289.305013 125.80818 -399.178839 235.6820
## 2047   -84.47022208 -293.969936 125.02949 -404.872385 235.9319
## 2048   -87.19202878 -298.617469 124.23341 -410.539337 236.1553
## 2049   -89.91383548 -303.248085 123.42041 -416.180415 236.3527
## 2050   -92.63564218 -307.862233 122.59095 -421.796308 236.5250
## 2051   -95.35744888 -312.460345 121.74545 -427.387675 236.6728
## 2052   -98.07925559 -317.042831 120.88432 -432.955146 236.7966
## 2053  -100.80106229 -321.610088 120.00796 -438.499326 236.8972
## 2054  -103.52286899 -326.162494 119.11676 -444.020793 236.9751
## 2055  -106.24467569 -330.700413 118.21106 -449.520103 237.0308
## 2056  -108.96648239 -335.224193 117.29123 -454.997790 237.0648
## 2057  -111.68828910 -339.734170 116.35759 -460.454367 237.0778
## 2058  -114.41009580 -344.230666 115.41047 -465.890327 237.0701
## 2059  -117.13190250 -348.713991 114.45019 -471.306143 237.0423
## 2060  -119.85370920 -353.184443 113.47702 -476.702272 236.9949
## 2061  -122.57551590 -357.642310 112.49128 -482.079153 236.9281
## 2062  -125.29732261 -362.087868 111.49322 -487.437211 236.8426
## 2063  -128.01912931 -366.521384 110.48313 -492.776852 236.7386
## 2064  -130.74093601 -370.943117 109.46124 -498.098471 236.6166
## 2065  -133.46274271 -375.353314 108.42783 -503.402447 236.4770
## 2066  -136.18454941 -379.752215 107.38312 -508.689149 236.3200
## 2067  -138.90635612 -384.140052 106.32734 -513.958929 236.1462
## 2068  -141.62816282 -388.517050 105.26072 -519.212132 235.9558
## 2069  -144.34996952 -392.883424 104.18349 -524.449088 235.7491
## 2070  -147.07177622 -397.239385 103.09583 -529.670117 235.5266
## 2071  -149.79358292 -401.585134 101.99797 -534.875530 235.2884
## 2072  -152.51538963 -405.920870 100.89009 -540.065628 235.0348
## 2073  -155.23719633 -410.246781  99.77239 -545.240700 234.7663
## 2074  -157.95900303 -414.563051  98.64505 -550.401029 234.4830
## 2075  -160.68080973 -418.869861  97.50824 -555.546889 234.1853
## 2076  -163.40261643 -423.167382  96.36215 -560.678543 233.8733
## 2077  -166.12442314 -427.455784  95.20694 -565.796249 233.5474
## 2078  -168.84622984 -431.735228  94.04277 -570.900257 233.2078
## 2079  -171.56803654 -436.005874  92.86980 -575.990809 232.8547
## 2080  -174.28984324 -440.267874  91.68819 -581.068139 232.4885
## 2081  -177.01164994 -444.521379  90.49808 -586.132476 232.1092
## 2082  -179.73345665 -448.766534  89.29962 -591.184043 231.7171
## 2083  -182.45526335 -453.003480  88.09295 -596.223054 231.3125
## 2084  -185.17707005 -457.232353  86.87821 -601.249720 230.8956
## 2085  -187.89887675 -461.453288  85.65553 -606.264245 230.4665
## 2086  -190.62068345 -465.666414  84.42505 -611.266828 230.0255
## 2087  -193.34249016 -469.871857  83.18688 -616.257661 229.5727
## 2088  -196.06429686 -474.069741  81.94115 -621.236934 229.1083
## 2089  -198.78610356 -478.260186  80.68798 -626.204828 228.6326
## 2090  -201.50791026 -482.443308  79.42749 -631.161524 228.1457
## 2091  -204.22971696 -486.619220  78.15979 -636.107193 227.6478
## 2092  -206.95152367 -490.788035  76.88499 -641.042007 227.1390
## 2093  -209.67333037 -494.949859  75.60320 -645.966131 226.6195

Somehow, I don’t think this is a good way to go because we have values that are below 0. I’m not sure what that implies but I’m going to go back and plot my previous holt work. The previous plot has a confidence interval that is negative, but no actual values fell below 0. I’m going to try the damped option next.

## 
## Forecast method: Damped Holt's method
## 
## Model Information:
## Damped Holt's method 
## 
## Call:
##  holt(y = eggs, h = 100, damped = TRUE) 
## 
##   Smoothing parameters:
##     alpha = 0.8462 
##     beta  = 0.004 
##     phi   = 0.8 
## 
##   Initial states:
##     l = 276.9842 
##     b = 4.9966 
## 
##   sigma:  27.2755
## 
##      AIC     AICc      BIC 
## 1055.458 1056.423 1070.718 
## 
## Error measures:
##                     ME     RMSE     MAE       MPE     MAPE      MASE
## Training set -2.891496 26.54019 19.2795 -2.907633 10.01894 0.9510287
##                      ACF1
## Training set -0.003195358
## 
## Forecasts:
##      Point Forecast        Lo 80     Hi 80       Lo 95    Hi 95
## 1994       62.84884   27.8938665  97.80381    9.389822 116.3079
## 1995       62.79968   16.9363788 108.66299   -7.342188 132.9416
## 1996       62.76036    8.0760016 117.44472  -20.872149 146.3929
## 1997       62.72890    0.4263979 125.03140  -32.554554 158.0124
## 1998       62.70373   -6.4067675 131.81423  -42.991656 168.3991
## 1999       62.68360  -12.6402008 138.00740  -52.514212 177.8814
## 2000       62.66749  -18.4083657 143.74335  -61.327332 186.6623
## 2001       62.65461  -23.8016345 149.11085  -69.568803 194.8780
## 2002       62.64430  -28.8842950 154.17289  -77.336605 202.6252
## 2003       62.63605  -33.7040638 158.97616  -84.703440 209.9755
## 2004       62.62945  -38.2975438 163.55645  -91.725068 216.9840
## 2005       62.62417  -42.6935603 167.94191  -98.445402 223.6938
## 2006       62.61995  -46.9153057 172.15521 -104.899769 230.1397
## 2007       62.61657  -50.9817746 176.21492 -111.117108 236.3503
## 2008       62.61387  -54.9087594 180.13650 -117.121483 242.3492
## 2009       62.61171  -58.7095595 183.93298 -122.933160 248.1566
## 2010       62.60998  -62.3954996 187.61546 -128.569404 253.7894
## 2011       62.60860  -65.9763173 191.19351 -134.045059 259.2623
## 2012       62.60749  -69.4604571 194.67544 -139.373005 264.5880
## 2013       62.60660  -72.8552982 198.06851 -144.564498 269.7777
## 2014       62.60590  -76.1673328 201.37913 -149.629443 274.8412
## 2015       62.60533  -79.4023080 204.61297 -154.576610 279.7873
## 2016       62.60488  -82.5653397 207.77509 -159.413810 284.6236
## 2017       62.60451  -85.6610050 210.87003 -164.148030 289.3571
## 2018       62.60422  -88.6934183 213.90187 -168.785552 293.9940
## 2019       62.60399  -91.6662933 216.87428 -173.332049 298.5400
## 2020       62.60381  -94.5829961 219.79061 -177.792663 303.0003
## 2021       62.60366  -97.4465882 222.65390 -182.172070 307.3794
## 2022       62.60354 -100.2598639 225.46694 -186.474541 311.6816
## 2023       62.60344 -103.0253816 228.23227 -190.703985 315.9109
## 2024       62.60337 -105.7454907 230.95223 -194.863993 320.0707
## 2025       62.60331 -108.4223544 233.62897 -198.957870 324.1645
## 2026       62.60326 -111.0579701 236.26449 -202.988671 328.1952
## 2027       62.60322 -113.6541863 238.86062 -206.959220 332.1657
## 2028       62.60319 -116.2127174 241.41909 -210.872140 336.0785
## 2029       62.60316 -118.7351575 243.94148 -214.729866 339.9362
## 2030       62.60314 -121.2229914 246.42928 -218.534669 343.7410
## 2031       62.60313 -123.6776050 248.88386 -222.288668 347.4949
## 2032       62.60311 -126.1002941 251.30652 -225.993844 351.2001
## 2033       62.60310 -128.4922724 253.69848 -229.652054 354.8583
## 2034       62.60310 -130.8546789 256.06087 -233.265039 358.4712
## 2035       62.60309 -133.1885838 258.39476 -236.834435 362.0406
## 2036       62.60308 -135.4949941 260.70116 -240.361782 365.5679
## 2037       62.60308 -137.7748593 262.98102 -243.848533 369.0547
## 2038       62.60308 -140.0290751 265.23523 -247.296057 372.5022
## 2039       62.60307 -142.2584882 267.46464 -250.705648 375.9118
## 2040       62.60307 -144.4638997 269.67004 -254.078533 379.2847
## 2041       62.60307 -146.6460685 271.85221 -257.415871 382.6220
## 2042       62.60307 -148.8057141 274.01185 -260.718763 385.9249
## 2043       62.60307 -150.9435200 276.14965 -263.988255 389.1944
## 2044       62.60307 -153.0601355 278.26627 -267.225338 392.4315
## 2045       62.60307 -155.1561786 280.36231 -270.430959 395.6371
## 2046       62.60307 -157.2322378 282.43837 -273.606018 398.8121
## 2047       62.60306 -159.2888738 284.49500 -276.751371 401.9575
## 2048       62.60306 -161.3266220 286.53275 -279.867837 405.0740
## 2049       62.60306 -163.3459932 288.55212 -282.956199 408.1623
## 2050       62.60306 -165.3474759 290.55360 -286.017203 411.2233
## 2051       62.60306 -167.3315372 292.53766 -289.051562 414.2577
## 2052       62.60306 -169.2986243 294.50475 -292.059962 417.2661
## 2053       62.60306 -171.2491656 296.45529 -295.043058 420.2492
## 2054       62.60306 -173.1835715 298.38970 -298.001476 423.2076
## 2055       62.60306 -175.1022361 300.30836 -300.935821 426.1419
## 2056       62.60306 -177.0055375 302.21166 -303.846669 429.0528
## 2057       62.60306 -178.8938390 304.09997 -306.734577 431.9407
## 2058       62.60306 -180.7674896 305.97362 -309.600078 434.8062
## 2059       62.60306 -182.6268253 307.83295 -312.443686 437.6498
## 2060       62.60306 -184.4721691 309.67830 -315.265896 440.4720
## 2061       62.60306 -186.3038323 311.50996 -318.067183 443.2733
## 2062       62.60306 -188.1221147 313.32824 -320.848006 446.0541
## 2063       62.60306 -189.9273054 315.13343 -323.608807 448.8149
## 2064       62.60306 -191.7196831 316.92581 -326.350012 451.5561
## 2065       62.60306 -193.4995169 318.70564 -329.072033 454.2782
## 2066       62.60306 -195.2670664 320.47319 -331.775267 456.9814
## 2067       62.60306 -197.0225826 322.22871 -334.460097 459.6662
## 2068       62.60306 -198.7663080 323.97243 -337.126895 462.3330
## 2069       62.60306 -200.4984769 325.70460 -339.776019 464.9821
## 2070       62.60306 -202.2193162 327.42544 -342.407816 467.6139
## 2071       62.60306 -203.9290453 329.13517 -345.022621 470.2287
## 2072       62.60306 -205.6278766 330.83400 -347.620759 472.8269
## 2073       62.60306 -207.3160160 332.52214 -350.202545 475.4087
## 2074       62.60306 -208.9936627 334.19979 -352.768284 477.9744
## 2075       62.60306 -210.6610101 335.86714 -355.318272 480.5244
## 2076       62.60306 -212.3182455 337.52437 -357.852795 483.0589
## 2077       62.60306 -213.9655507 339.17168 -360.372131 485.5783
## 2078       62.60306 -215.6031021 340.80923 -362.876550 488.0827
## 2079       62.60306 -217.2310709 342.43720 -365.366313 490.5724
## 2080       62.60306 -218.8496235 344.05575 -367.841676 493.0478
## 2081       62.60306 -220.4589213 345.66505 -370.302884 495.5090
## 2082       62.60306 -222.0591213 347.26525 -372.750179 497.9563
## 2083       62.60306 -223.6503761 348.85650 -375.183793 500.3899
## 2084       62.60306 -225.2328340 350.43896 -377.603954 502.8101
## 2085       62.60306 -226.8066394 352.01277 -380.010881 505.2170
## 2086       62.60306 -228.3719326 353.57806 -382.404791 507.6109
## 2087       62.60306 -229.9288503 355.13498 -384.785891 509.9920
## 2088       62.60306 -231.4775255 356.68365 -387.154385 512.3605
## 2089       62.60306 -233.0180877 358.22421 -389.510472 514.7166
## 2090       62.60306 -234.5506631 359.75679 -391.854344 517.0605
## 2091       62.60306 -236.0753747 361.28150 -394.186189 519.3923
## 2092       62.60306 -237.5923423 362.79847 -396.506191 521.7123
## 2093       62.60306 -239.1016828 364.30781 -398.814528 524.0207

With damped set to TRUE, and keep h set ot 100, we do not get any values below zero, although our interval still is. Now let’s see what a Box-Cox transformation will do to our forecasts. We’ll do the box-cox first then add the damped argument.

## 
## Forecast method: Holt's method
## 
## Model Information:
## Holt's method 
## 
## Call:
##  holt(y = eggs, h = 100, lambda = BoxCox.lambda(eggs)) 
## 
##   Box-Cox transformation: lambda= 0.3956 
## 
##   Smoothing parameters:
##     alpha = 0.809 
##     beta  = 1e-04 
## 
##   Initial states:
##     l = 21.0322 
##     b = -0.1144 
## 
##   sigma:  1.0549
## 
##      AIC     AICc      BIC 
## 443.0310 443.7128 455.7475 
## 
## Error measures:
##                     ME     RMSE      MAE       MPE     MAPE      MASE
## Training set 0.7736844 26.39376 18.96387 -1.072416 9.620095 0.9354593
##                    ACF1
## Training set 0.03887152
## 
## Forecasts:
##      Point Forecast         Lo 80     Hi 80         Lo 95     Hi 95
## 1994     61.4107603  4.640814e+01  79.02212  3.947196e+01  89.44945
## 1995     60.0421240  4.147646e+01  82.90348  3.327908e+01  96.84743
## 1996     58.6920872  3.751269e+01  85.83783  2.853596e+01 102.79296
## 1997     57.3605616  3.414325e+01  88.19844  2.467300e+01 107.85505
## 1998     56.0474589  3.119333e+01  90.15981  2.142386e+01 112.29946
## 1999     54.7526902  2.856371e+01  91.82084  1.863743e+01 116.27667
## 2000     53.4761662  2.619126e+01  93.24432  1.621764e+01 119.88213
## 2001     52.2177974  2.403270e+01  94.47323  1.409858e+01 123.18111
## 2002     50.9774937  2.205666e+01  95.53860  1.223246e+01 126.22071
## 2003     49.7551646  2.023944e+01  96.46378  1.058319e+01 129.03638
## 2004     48.5507193  1.856257e+01  97.26691  9.122550e+00 131.65559
## 2005     47.3640666  1.701125e+01  97.96245  7.827917e+00 134.10027
## 2006     46.1951148  1.557337e+01  98.56214  6.680721e+00 136.38821
## 2007     45.0437717  1.423883e+01  99.07572  5.665431e+00 138.53416
## 2008     43.9099449  1.299909e+01  99.51139  4.768844e+00 140.55053
## 2009     42.7935413  1.184680e+01  99.87611  3.979562e+00 142.44785
## 2010     41.6944676  1.077559e+01 100.17588  3.287622e+00 144.23520
## 2011     40.6126299  9.779860e+00 100.41591  2.684205e+00 145.92049
## 2012     39.5479337  8.854646e+00 100.60079  2.161419e+00 147.51063
## 2013     38.5002844  7.995507e+00 100.73454  1.712125e+00 149.01174
## 2014     37.4695866  7.198443e+00 100.82076  1.329798e+00 150.42926
## 2015     36.4557446  6.459820e+00 100.86268  1.008416e+00 151.76806
## 2016     35.4586622  5.776316e+00 100.86319  7.423615e-01 153.03254
## 2017     34.4782425  5.144876e+00 100.82490  5.263381e-01 154.22665
## 2018     33.5143883  4.562675e+00 100.75020  3.552964e-01 155.35402
## 2019     32.5670019  4.027081e+00 100.64127  2.243584e-01 156.41793
## 2020     31.6359850  3.535636e+00 100.50010  1.287391e-01 157.42142
## 2021     30.7212388  3.086027e+00 100.32851  6.365231e-02 158.36726
## 2022     29.8226638  2.676069e+00 100.12820  2.417959e-02 159.25801
## 2023     28.9401603  2.303690e+00  99.90074  5.043607e-03 160.09606
## 2024     28.0736276  1.966911e+00  99.64756  4.087608e-05 160.88361
## 2025     27.2229649  1.663838e+00  99.37003 -2.009381e-03 161.62272
## 2026     26.3880704  1.392648e+00  99.06941 -1.480853e-02 162.31531
## 2027     25.5688420  1.151578e+00  98.74686 -4.423448e-02 162.96318
## 2028     24.7651768  9.389142e-01  98.40350 -9.420466e-02 163.56801
## 2029     23.9769715  7.529844e-01  98.04037 -1.677803e-01 164.13138
## 2030     23.2041220  5.921462e-01  97.65843 -2.674858e-01 164.65479
## 2031     22.4465236  4.547786e-01  97.25860 -3.954676e-01 165.13965
## 2032     21.7040710  3.392712e-01  96.84176 -5.535872e-01 165.58728
## 2033     20.9766582  2.440135e-01  96.40872 -7.434830e-01 165.99894
## 2034     20.2641786  1.673812e-01  95.96026 -9.666134e-01 166.37583
## 2035     19.5665248  1.077203e-01  95.49712 -1.224288e+00 166.71908
## 2036     18.8835888  6.332581e-02  95.01999 -1.517692e+00 167.02977
## 2037     18.2152618  3.241169e-02  94.52954 -1.847904e+00 167.30893
## 2038     17.5614344  1.306240e-02  94.02640 -2.215914e+00 167.55753
## 2039     16.9219962  3.144667e-03  93.51118 -2.622629e+00 167.77650
## 2040     16.2968364  9.146118e-05  92.98445 -3.068890e+00 167.96675
## 2041     15.6858432 -5.546778e-04  92.44676 -3.555477e+00 168.12912
## 2042     15.0889039 -5.531563e-03  91.89863 -4.083116e+00 168.26443
## 2043     14.5059053 -1.800422e-02  91.34058 -4.652485e+00 168.37347
## 2044     13.9367330 -4.001572e-02  90.77307 -5.264219e+00 168.45699
## 2045     13.3812721 -7.320011e-02  90.19658 -5.918915e+00 168.51571
## 2046     12.8394066 -1.189430e-01  89.61153 -6.617136e+00 168.55033
## 2047     12.3110197 -1.784566e-01  89.01837 -7.359412e+00 168.56151
## 2048     11.7959937 -2.528229e-01  88.41750 -8.146246e+00 168.54989
## 2049     11.2942098 -3.430209e-01  87.80930 -8.978115e+00 168.51610
## 2050     10.8055486 -4.499463e-01  87.19417 -9.855472e+00 168.46074
## 2051     10.3298892 -5.744249e-01  86.57246 -1.077875e+01 168.38438
## 2052      9.8671102 -7.172233e-01  85.94452 -1.174836e+01 168.28758
## 2053      9.4170888 -8.790568e-01  85.31069 -1.276469e+01 168.17087
## 2054      8.9797013 -1.060596e+00  84.67130 -1.382812e+01 168.03479
## 2055      8.5548230 -1.262474e+00  84.02666 -1.493901e+01 167.87982
## 2056      8.1423278 -1.485286e+00  83.37707 -1.609772e+01 167.70646
## 2057      7.7420887 -1.729598e+00  82.72284 -1.730456e+01 167.51519
## 2058      7.3539773 -1.995947e+00  82.06424 -1.855987e+01 167.30645
## 2059      6.9778642 -2.284847e+00  81.40155 -1.986395e+01 167.08070
## 2060      6.6136186 -2.596785e+00  80.73503 -2.121709e+01 166.83836
## 2061      6.2611083 -2.932230e+00  80.06494 -2.261960e+01 166.57985
## 2062      5.9202000 -3.291630e+00  79.39152 -2.407174e+01 166.30558
## 2063      5.5907588 -3.675416e+00  78.71503 -2.557380e+01 166.01594
## 2064      5.2726484 -4.084002e+00  78.03570 -2.712602e+01 165.71132
## 2065      4.9657312 -4.517788e+00  77.35374 -2.872866e+01 165.39208
## 2066      4.6698679 -4.977159e+00  76.66938 -3.038198e+01 165.05860
## 2067      4.3849175 -5.462487e+00  75.98284 -3.208621e+01 164.71121
## 2068      4.1107377 -5.974131e+00  75.29431 -3.384158e+01 164.35028
## 2069      3.8471841 -6.512440e+00  74.60401 -3.564833e+01 163.97613
## 2070      3.5941109 -7.077752e+00  73.91212 -3.750667e+01 163.58908
## 2071      3.3513701 -7.670394e+00  73.21884 -3.941682e+01 163.18947
## 2072      3.1188120 -8.290685e+00  72.52434 -4.137900e+01 162.77759
## 2073      2.8962849 -8.938933e+00  71.82881 -4.339340e+01 162.35374
## 2074      2.6836349 -9.615441e+00  71.13243 -4.546024e+01 161.91823
## 2075      2.4807060 -1.032050e+01  70.43535 -4.757970e+01 161.47135
## 2076      2.2873400 -1.105440e+01  69.73775 -4.975198e+01 161.01336
## 2077      2.1033762 -1.181741e+01  69.03978 -5.197728e+01 160.54455
## 2078      1.9286514 -1.260982e+01  68.34160 -5.425577e+01 160.06519
## 2079      1.7629999 -1.343187e+01  67.64336 -5.658764e+01 159.57553
## 2080      1.6062531 -1.428385e+01  66.94521 -5.897306e+01 159.07584
## 2081      1.4582396 -1.516598e+01  66.24729 -6.141222e+01 158.56635
## 2082      1.3187848 -1.607854e+01  65.54975 -6.390528e+01 158.04733
## 2083      1.1877111 -1.702175e+01  64.85271 -6.645242e+01 157.51901
## 2084      1.0648371 -1.799587e+01  64.15631 -6.905379e+01 156.98161
## 2085      0.9499781 -1.900111e+01  63.46067 -7.170957e+01 156.43538
## 2086      0.8429454 -2.003772e+01  62.76593 -7.441992e+01 155.88054
## 2087      0.7435459 -2.110591e+01  62.07221 -7.718498e+01 155.31730
## 2088      0.6515824 -2.220590e+01  61.37962 -8.000493e+01 154.74588
## 2089      0.5668525 -2.333792e+01  60.68829 -8.287991e+01 154.16649
## 2090      0.4891490 -2.450218e+01  59.99832 -8.581007e+01 153.57934
## 2091      0.4182588 -2.569888e+01  59.30982 -8.879557e+01 152.98463
## 2092      0.3539627 -2.692823e+01  58.62291 -9.183655e+01 152.38256
## 2093      0.2960349 -2.819043e+01  57.93769 -9.493316e+01 151.77332

## 
## Forecast method: Damped Holt's method
## 
## Model Information:
## Damped Holt's method 
## 
## Call:
##  holt(y = eggs, h = 100, damped = TRUE, lambda = BoxCox.lambda(eggs)) 
## 
##   Box-Cox transformation: lambda= 0.3956 
## 
##   Smoothing parameters:
##     alpha = 0.8356 
##     beta  = 1e-04 
##     phi   = 0.98 
## 
##   Initial states:
##     l = 21.6922 
##     b = -0.1429 
## 
##   sigma:  1.068
## 
##      AIC     AICc      BIC 
## 446.2962 447.2617 461.5560 
## 
## Error measures:
##                      ME     RMSE      MAE       MPE     MAPE      MASE
## Training set -0.8200445 26.53321 19.45654 -2.019718 9.976131 0.9597618
##                     ACF1
## Training set 0.005852382
## 
## Forecasts:
##      Point Forecast       Lo 80     Hi 80         Lo 95     Hi 95
## 1994       62.64655 47.27973262  80.69832  4.017989e+01  91.39126
## 1995       62.38988 42.92322722  86.41174  3.434621e+01 101.08240
## 1996       62.13895 39.55414540  91.14596  3.000254e+01 109.28536
## 1997       61.89363 36.76017601  95.31306  2.652069e+01 116.62820
## 1998       61.65379 34.35554959  99.09862  2.361805e+01 123.39337
## 1999       61.41930 32.23747696 102.60524  2.113849e+01 129.73689
## 2000       61.19001 30.34192101 105.89676  1.898490e+01 135.75574
## 2001       60.96582 28.62573829 109.01614  1.709192e+01 141.51512
## 2002       60.74659 27.05819232 111.99394  1.541311e+01 147.06137
## 2003       60.53221 25.61644715 114.85283  1.391398e+01 152.42884
## 2004       60.32256 24.28297202 117.61016  1.256811e+01 157.64382
## 2005       60.11752 23.04395071 120.27958  1.135471e+01 162.72698
## 2006       59.91700 21.88825720 122.87206  1.025706e+01 167.69492
## 2007       59.72089 20.80676914 125.39657  9.261488e+00 172.56120
## 2008       59.52907 19.79189192 127.86054  8.356605e+00 177.33707
## 2009       59.34145 18.83721946 130.27024  7.532834e+00 182.03200
## 2010       59.15793 17.93728641 132.63098  6.782011e+00 186.65403
## 2011       58.97841 17.08738337 134.94734  6.097110e+00 191.21007
## 2012       58.80280 16.28341653 137.22327  5.472033e+00 195.70610
## 2013       58.63101 15.52179931 139.46222  4.901442e+00 200.14734
## 2014       58.46296 14.79936752 141.66722  4.380631e+00 204.53837
## 2015       58.29854 14.11331203 143.84097  3.905428e+00 208.88326
## 2016       58.13769 13.46112475 145.98583  3.472108e+00 213.18561
## 2017       57.98031 12.84055481 148.10395  3.077331e+00 217.44863
## 2018       57.82633 12.24957272 150.19722  2.718085e+00 221.67523
## 2019       57.67568 11.68634066 152.26738  2.391643e+00 225.86800
## 2020       57.52826 11.14918787 154.31598  2.095523e+00 230.02931
## 2021       57.38401 10.63658995 156.34444  1.827459e+00 234.16130
## 2022       57.24287 10.14715132 158.35403  1.585375e+00 238.26592
## 2023       57.10474  9.67959041 160.34595  1.367360e+00 242.34495
## 2024       56.96958  9.23272695 162.32127  1.171649e+00 246.40003
## 2025       56.83731  8.80547104 164.28097  9.966082e-01 250.43266
## 2026       56.70786  8.39681370 166.22596  8.407190e-01 254.44422
## 2027       56.58117  8.00581871 168.15709  7.025648e-01 258.43599
## 2028       56.45719  7.63161542 170.07514  5.808207e-01 262.40914
## 2029       56.33584  7.27339254 171.98082  4.742430e-01 266.36478
## 2030       56.21708  6.93039260 173.87480  3.816600e-01 270.30390
## 2031       56.10083  6.60190708 175.75769  3.019643e-01 274.22747
## 2032       55.98705  6.28727217 177.63009  2.341044e-01 278.13635
## 2033       55.87569  5.98586488 179.49252  1.770781e-01 282.03138
## 2034       55.76668  5.69709965 181.34550  1.299253e-01 285.91330
## 2035       55.65997  5.42042531 183.18948  9.172068e-02 289.78285
## 2036       55.55552  5.15532234 185.02491  6.156653e-02 293.64068
## 2037       55.45327  4.90130040 186.85221  3.858416e-02 297.48744
## 2038       55.35318  4.65789612 188.67175  2.190372e-02 301.32371
## 2039       55.25520  4.42467111 190.48392  1.065036e-02 305.15005
## 2040       55.15928  4.20121008 192.28903  3.922915e-03 308.96698
## 2041       55.06537  3.98711929 194.08743  7.540037e-04 312.77499
## 2042       54.97344  3.78202495 195.87941  1.700210e-06 316.57455
## 2043       54.88343  3.58557191 197.66526 -4.439016e-04 320.36609
## 2044       54.79531  3.39742238 199.44524 -2.892414e-03 324.15002
## 2045       54.70904  3.21725481 201.21963 -8.310487e-03 327.92675
## 2046       54.62457  3.04476280 202.98864 -1.734558e-02 331.69663
## 2047       54.54187  2.87965417 204.75252 -3.049319e-02 335.46001
## 2048       54.46089  2.72165005 206.51148 -4.815086e-02 339.21723
## 2049       54.38161  2.57048405 208.26573 -7.064573e-02 342.96860
## 2050       54.30398  2.42590154 210.01545 -9.825115e-02 346.71442
## 2051       54.22796  2.28765891 211.76084 -1.311978e-01 350.45496
## 2052       54.15353  2.15552294 213.50207 -1.696814e-01 354.19050
## 2053       54.08065  2.02927018 215.23930 -2.138689e-01 357.92129
## 2054       54.00928  1.90868643 216.97269 -2.639028e-01 361.64757
## 2055       53.93940  1.79356618 218.70239 -3.199046e-01 365.36957
## 2056       53.87097  1.68371211 220.42855 -3.819780e-01 369.08751
## 2057       53.80395  1.57893472 222.15130 -4.502112e-01 372.80159
## 2058       53.73833  1.47905181 223.87077 -5.246786e-01 376.51202
## 2059       53.67406  1.38388814 225.58708 -6.054428e-01 380.21898
## 2060       53.61112  1.29327505 227.30035 -6.925559e-01 383.92266
## 2061       53.54949  1.20705011 229.01069 -7.860607e-01 387.62321
## 2062       53.48913  1.12505679 230.71821 -8.859916e-01 391.32081
## 2063       53.43002  1.04714415 232.42301 -9.923760e-01 395.01561
## 2064       53.37213  0.97316658 234.12519 -1.105234e+00 398.70776
## 2065       53.31543  0.90298349 235.82484 -1.224582e+00 402.39740
## 2066       53.25990  0.83645909 237.52205 -1.350427e+00 406.08466
## 2067       53.20552  0.77346211 239.21690 -1.482777e+00 409.76968
## 2068       53.15226  0.71386560 240.90948 -1.621630e+00 413.45258
## 2069       53.10009  0.65754669 242.59986 -1.766984e+00 417.13347
## 2070       53.04900  0.60438643 244.28811 -1.918833e+00 420.81247
## 2071       52.99896  0.55426953 245.97431 -2.077168e+00 424.48968
## 2072       52.94994  0.50708420 247.65853 -2.241975e+00 428.16522
## 2073       52.90194  0.46272201 249.34083 -2.413241e+00 431.83916
## 2074       52.85492  0.42107764 251.02127 -2.590949e+00 435.51162
## 2075       52.80886  0.38204880 252.69991 -2.775080e+00 439.18268
## 2076       52.76375  0.34553601 254.37680 -2.965613e+00 442.85242
## 2077       52.71956  0.31144248 256.05201 -3.162526e+00 446.52093
## 2078       52.67628  0.27967396 257.72559 -3.365797e+00 450.18828
## 2079       52.63389  0.25013860 259.39758 -3.575400e+00 453.85456
## 2080       52.59236  0.22274678 261.06804 -3.791308e+00 457.51984
## 2081       52.55168  0.19741102 262.73700 -4.013497e+00 461.18417
## 2082       52.51184  0.17404582 264.40452 -4.241937e+00 464.84764
## 2083       52.47281  0.15256753 266.07064 -4.476600e+00 468.51030
## 2084       52.43457  0.13289421 267.73539 -4.717457e+00 472.17221
## 2085       52.39712  0.11494553 269.39882 -4.964479e+00 475.83344
## 2086       52.36043  0.09864257 271.06097 -5.217634e+00 479.49403
## 2087       52.32450  0.08390776 272.72187 -5.476893e+00 483.15405
## 2088       52.28929  0.07066465 274.38155 -5.742224e+00 486.81354
## 2089       52.25480  0.05883783 276.04005 -6.013596e+00 490.47255
## 2090       52.22102  0.04835272 277.69741 -6.290976e+00 494.13113
## 2091       52.18792  0.03913540 279.35364 -6.574335e+00 497.78932
## 2092       52.15550  0.03111240 281.00879 -6.863638e+00 501.44717
## 2093       52.12374  0.02421049 282.66288 -7.158854e+00 505.10472

A few things I noticed, the damped didn’t have any values below zero but it’s didn’t show the downward trend. The confidence interval is still below zero. When I did the box-cox, my forecast changed in that it now showed the downward trend. I believe the best forecast, though is the box-cox with the damped argument set to TRUE. We got not values below zero, our interal is only ever so slightly zero for the 95% interval. We can also see the downward trend although it seems less subtle that previously. The box-cox forecast without damped had the best RMSE at 26.39.

8

Recall your retail time series data.

8.a

Why is multiplicative seasonlity necessary for this series?

Before answering this, I need to take a look at the data and plot it again.

I had to do some reading to figure out how to tell the difference between additive and multiplicative. When a time series is multiplicative, the peaks increase over time. From the plot, we see that this is the case for our retail data.

8.b

Apply Holt-Winters’ multiplicative method to the data. Experiment with making the trend damped.

## 
## Forecast method: Holt's method
## 
## Model Information:
## Holt's method 
## 
## Call:
##  holt(y = myts_2, seasonal = "Multiplicative") 
## 
##   Smoothing parameters:
##     alpha = 0.0897 
##     beta  = 0.0043 
## 
##   Initial states:
##     l = 100.7823 
##     b = 0.3451 
## 
##   sigma:  53.7986
## 
##      AIC     AICc      BIC 
## 5306.944 5307.104 5326.658 
## 
## Error measures:
##                     ME     RMSE     MAE       MPE     MAPE     MASE
## Training set -0.403039 53.51548 30.7876 -1.918889 10.30805 1.558513
##                   ACF1
## Training set 0.1776805
## 
## Forecasts:
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Jan 2014       441.7094 372.7636 510.6551 336.2660 547.1527
## Feb 2014       441.3878 372.1377 510.6380 335.4789 547.2967
## Mar 2014       441.0663 371.4846 510.6480 334.6502 547.4824
## Apr 2014       440.7448 370.8033 510.6862 333.7786 547.7110
## May 2014       440.4233 370.0931 510.7534 332.8626 547.9839
## Jun 2014       440.1017 369.3532 510.8502 331.9012 548.3022
## Jul 2014       439.7802 368.5828 510.9776 330.8932 548.6672
## Aug 2014       439.4587 367.7812 511.1361 329.8375 549.0798
## Sep 2014       439.1371 366.9479 511.3264 328.7332 549.5410
## Oct 2014       438.8156 366.0823 511.5490 327.5795 550.0517

Now let’s try it with damped to see what differences we spot.

## 
## Forecast method: Damped Holt's method
## 
## Model Information:
## Damped Holt's method 
## 
## Call:
##  holt(y = myts_2, damped = TRUE, seasonal = "Multiplicative") 
## 
##   Smoothing parameters:
##     alpha = 0.0178 
##     beta  = 0.0178 
##     phi   = 0.9406 
## 
##   Initial states:
##     l = 97.7553 
##     b = 1.5277 
## 
##   sigma:  52.8742
## 
##      AIC     AICc      BIC 
## 5294.725 5294.950 5318.382 
## 
## Error measures:
##                    ME     RMSE      MAE        MPE     MAPE     MASE
## Training set 3.351375 52.52613 28.70861 -0.6761809 9.513276 1.453272
##                   ACF1
## Training set 0.1898293
## 
## Forecasts:
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Jan 2014       439.5875 371.8265 507.3486 335.9559 543.2191
## Feb 2014       443.4637 375.6622 511.2652 339.7703 547.1571
## Mar 2014       447.1098 379.2227 514.9969 343.2854 550.9341
## Apr 2014       450.5394 382.5090 518.5697 346.4959 554.5828
## May 2014       453.7654 385.5244 522.0063 349.3998 558.1309
## Jun 2014       456.7998 388.2736 525.3260 351.9981 561.6016
## Jul 2014       459.6541 390.7631 528.5452 354.2943 565.0139
## Aug 2014       462.3390 393.0001 531.6778 356.2944 568.3835
## Sep 2014       464.8644 394.9935 534.7353 358.0061 571.7228
## Oct 2014       467.2399 396.7525 537.7273 359.4388 575.0411

The RMSE for the damped, multiplicative had a better RMSE at 52.52, compared to 53.51.

8.c

Compare the RMSE of the one-step forecasts from the two methods. Which do you prefer?

I compared the RMSE above. I prefer the RMSE ofr the damped forecast.

8.d

Check the residuals from the best method look like white noise? 

## 
##  Ljung-Box test
## 
## data:  Residuals from Damped Holt's method
## Q* = 833.83, df = 19, p-value < 2.2e-16
## 
## Model df: 5.   Total lags used: 24

I believe the question is asking to check and see if the residuals represent white noise. White noise occurs when all the residuals are random and unpredictable. My retail data shows that the residual is not white noise.

8.e

Now find the test set RMSE, while training the model to the end of 2010. Can you beat the seaonsonal naive approach from exercise 8?

9

For the same retail data, try an STL decomposition applied to the Box-Cox transformed series, followed by ETS on the seasonally adjusted data. How does that compare with your best previous forecasts on the test set?