Data 624 -Assignment 5
Anthony Munoz
3/8/2020
suppressMessages(
library(fpp2) )7.1
a-)
summary(ses(pigs,h=4))##
## 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 ACF1
## Training set 385.8721 10253.6 7961.383 -0.922652 9.274016 0.7966249 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 121539.8b-)
se <- ses(pigs, h=4)
paste('lower',98816.41 - (sd(se$residuals * 1.96)))## [1] "lower 78679.9711418255"paste('upper',98816.41 + (sd(se$residuals * 1.96)))## [1] "upper 118952.848858174"se$lower[, '95%'][1]## [1] 78611.97se$upper[, '95%'][1]## [1] 119020.8I see that the R produce interval are slighty bigger range than the interval calculated.
7.5
a-)
autoplot(books)
The 2 types Books sales seem increasing but you cant see a seasonal sales or cycle.
b-)
paperback
ses(books[,1]) %>% autoplot() + autolayer(fitted(ses(books[,1])), series='paperback-fitted') 
hardcover
ses(books[,2]) %>% autoplot() + autolayer(fitted(ses(books[,2])), series='hardcover-fitted') 
c-)
paperback
accuracy(ses(books[,1]))## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 7.175981 33.63769 27.8431 0.4736071 15.57784 0.7021303 -0.2117522hardcover
accuracy(ses(books[,2]))## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 9.166735 31.93101 26.77319 2.636189 13.39487 0.7987887 -0.14177637.6
a-)
paperback
holt(books[,1])## 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
## 35 214.4707 171.6073 257.3340 148.9169 280.0245
## 36 215.7216 172.8583 258.5850 150.1678 281.2754
## 37 216.9726 174.1093 259.8360 151.4188 282.5264
## 38 218.2236 175.3602 261.0869 152.6697 283.7774
## 39 219.4746 176.6112 262.3379 153.9207 285.0284
## 40 220.7255 177.8621 263.5889 155.1716 286.2794hardcover
holt(books[,2])## 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
## 35 263.3843 225.9494 300.8192 206.1326 320.6360
## 36 266.6869 229.2520 304.1218 209.4351 323.9386
## 37 269.9895 232.5546 307.4244 212.7377 327.2412
## 38 273.2921 235.8572 310.7270 216.0403 330.5438
## 39 276.5947 239.1597 314.0296 219.3429 333.8465
## 40 279.8973 242.4623 317.3322 222.6455 337.1491b-)
paperback
accuracy(holt(books[,1]))## ME RMSE MAE MPE MAPE MASE
## Training set -3.717178 31.13692 26.18083 -5.508526 15.58354 0.6602122
## ACF1
## Training set -0.1750792hardcover
accuracy(holt(books[,2]))## ME RMSE MAE MPE MAPE MASE
## Training set -0.1357882 27.19358 23.15557 -2.114792 12.1626 0.6908555
## ACF1
## Training set -0.03245186c-)
autoplot(books[,1]) +
autolayer(holt(books[,1]),series = 'hw',PI=F) +
autolayer(ses(books[,1]),series = 'ses',PI=F)
autoplot(books[,2]) +
autolayer(holt(books[,2]),series = 'hw',PI=F) +
autolayer(ses(books[,2]),series = 'ses',PI=F)
7.7
default.holt <- holt(eggs , h = 100)
damped.holt <- holt(eggs , h = 100, damped = T)
exp.holt <- holt(eggs , h = 100, exponential = T)
lambda.holt <- holt(eggs , h = 100, lambda = 'auto', biasadj = T)
autoplot(eggs) +
autolayer(default.holt, series='default', PI=F) +
autolayer(damped.holt, series='damped', PI=F) +
autolayer(exp.holt, series='exp', PI=F) +
autolayer(lambda.holt, series='boxcox', PI=F) 
The damped and damped exponential seem very close trend results. the box cox and exponential seem very close trend results. The default result seems by itself trend straight down line.
accuracy(default.holt)## ME RMSE MAE MPE MAPE MASE
## Training set 0.04499087 26.58219 19.18491 -1.142201 9.653791 0.9463626
## ACF1
## Training set 0.01348202accuracy(damped.holt)## ME RMSE MAE MPE MAPE MASE
## Training set -2.891496 26.54019 19.2795 -2.907633 10.01894 0.9510287
## ACF1
## Training set -0.003195358accuracy(exp.holt)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 0.4918791 26.49795 19.29399 -1.263235 9.766049 0.9517436 0.0103908accuracy(lambda.holt)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -0.2015298 26.38689 18.99362 -1.63043 9.713172 0.9369265 0.0383996The result of RMSE all of them are pretty close with just decimal difference. if we have to select one will be the Lambda one which has the best RMSE
7.8
a-)
retaildata <- readxl::read_excel("retail.xlsx", skip=1)
myts <- ts(retaildata[,"A3349709X"],
frequency=12, start=c(1982,4))
autoplot(myts)
because the seasonality is increasing.
b-)
summary(hw(myts, seasonal = "multi",damped = F))##
## Forecast method: Holt-Winters' multiplicative method
##
## Model Information:
## Holt-Winters' multiplicative method
##
## Call:
## hw(y = myts, seasonal = "multi", damped = F)
##
## Smoothing parameters:
## alpha = 0.8439
## beta = 0.001
## gamma = 0.0569
##
## Initial states:
## l = 60.0587
## b = 0.8487
## s = 0.9671 0.9175 1.0584 1.1371 1.0343 1.0717
## 1.0187 1.0045 0.9912 0.9301 0.9602 0.9093
##
## sigma: 0.0668
##
## AIC AICc BIC
## 4326.889 4328.574 4393.916
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.6269268 16.20066 11.21058 -0.09885034 4.83248 0.3420854
## ACF1
## Training set -0.01791788
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 2014 629.5695 575.6504 683.4887 547.1073 712.0318
## Feb 2014 584.5749 519.0182 650.1315 484.3146 684.8351
## Mar 2014 616.2703 533.9845 698.5561 490.4250 742.1155
## Apr 2014 596.6718 506.0038 687.3398 458.0071 735.3365
## May 2014 599.2660 498.3401 700.1918 444.9132 753.6188
## Jun 2014 570.8063 466.1013 675.5113 410.6738 730.9388
## Jul 2014 602.9768 483.9860 721.9676 420.9961 784.9576
## Aug 2014 617.9231 487.9412 747.9051 419.1328 816.7134
## Sep 2014 621.6684 483.2650 760.0718 409.9986 833.3381
## Oct 2014 656.2616 502.5046 810.0187 421.1105 891.4128
## Nov 2014 663.7379 500.8422 826.6336 414.6104 912.8654
## Dec 2014 742.4375 552.3069 932.5682 451.6578 1033.2172
## Jan 2015 643.3577 471.0944 815.6210 379.9037 906.8117
## Feb 2015 597.3548 431.5215 763.1880 343.7346 850.9749
## Mar 2015 629.7191 448.8949 810.5433 353.1723 906.2659
## Apr 2015 609.6697 428.9629 790.3766 333.3025 886.0370
## May 2015 612.2972 425.3074 799.2870 326.3210 898.2734
## Jun 2015 583.1967 399.9905 766.4029 303.0070 863.3864
## Jul 2015 616.0424 417.2621 814.8227 312.0341 920.0506
## Aug 2015 631.2889 422.3305 840.2473 311.7145 950.8633
## Sep 2015 635.0914 419.7029 850.4799 305.6832 964.4997
## Oct 2015 670.4067 437.6987 903.1146 314.5106 1026.3028
## Nov 2015 678.0190 437.3753 918.6626 309.9863 1046.0516
## Dec 2015 758.3839 483.4110 1033.3567 337.8492 1178.9185summary(hw(myts, seasonal = "multi", damped = T))##
## Forecast method: Damped Holt-Winters' multiplicative method
##
## Model Information:
## Damped Holt-Winters' multiplicative method
##
## Call:
## hw(y = myts, seasonal = "multi", damped = T)
##
## Smoothing parameters:
## alpha = 0.7177
## beta = 0.0073
## gamma = 1e-04
## phi = 0.98
##
## Initial states:
## l = 58.5098
## b = 0.2133
## s = 0.997 0.9329 0.9988 1.1725 1.0477 1.0367
## 0.9938 0.9952 0.9644 0.9179 0.9725 0.9706
##
## sigma: 0.0633
##
## AIC AICc BIC
## 4283.222 4285.111 4354.192
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 1.482515 15.65027 10.92983 0.3859787 4.551222 0.3335188 0.09915935
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 2014 612.1426 562.4737 661.8116 536.1805 688.1047
## Feb 2014 573.1030 515.6697 630.5363 485.2663 660.9397
## Mar 2014 613.8077 542.3224 685.2929 504.4804 723.1349
## Apr 2014 598.8676 520.4536 677.2816 478.9438 718.7914
## May 2014 601.3891 514.6846 688.0936 468.7860 733.9922
## Jun 2014 568.6949 479.7024 657.6875 432.5926 704.7973
## Jul 2014 598.7143 498.0854 699.3431 444.8158 752.6128
## Aug 2014 619.0468 508.1871 729.9065 449.5015 788.5920
## Sep 2014 619.3910 501.9511 736.8309 439.7822 798.9998
## Oct 2014 647.3120 518.0291 776.5950 449.5908 845.0333
## Nov 2014 655.4658 518.1512 792.7804 445.4612 865.4704
## Dec 2014 734.9023 573.9895 895.8151 488.8074 980.9972
## Jan 2015 627.0841 484.0075 770.1607 408.2674 845.9008
## Feb 2015 586.7808 447.6391 725.9225 373.9819 799.5797
## Mar 2015 628.1321 473.6852 782.5791 391.9259 864.3384
## Apr 2015 612.5342 456.6751 768.3934 374.1683 850.9002
## May 2015 614.8103 453.2102 776.4104 367.6643 861.9564
## Jun 2015 581.1070 423.5771 738.6368 340.1859 822.0280
## Jul 2015 611.4942 440.7749 782.2136 350.4016 872.5869
## Aug 2015 631.9708 450.4995 813.4421 354.4345 909.5072
## Sep 2015 632.0391 445.5886 818.4896 346.8876 917.1905
## Oct 2015 660.2413 460.3639 860.1188 354.5552 965.9275
## Nov 2015 668.2724 460.8634 875.6813 351.0677 985.4770
## Dec 2015 748.9482 510.8565 987.0399 384.8183 1113.0781autoplot(myts) +
autolayer(hw(myts, seasonal = "multi",damped = F),series = 'damped-false',PI=F) +
autolayer(hw(myts, seasonal = "multi",damped = T),series = 'damped-true',PI=F)
c-)
accuracy(hw(myts, seasonal = "multi",damped = F))## ME RMSE MAE MPE MAPE MASE
## Training set 0.6269268 16.20066 11.21058 -0.09885034 4.83248 0.3420854
## ACF1
## Training set -0.01791788accuracy(hw(myts, seasonal = "multi",damped = T))## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 1.482515 15.65027 10.92983 0.3859787 4.551222 0.3335188 0.09915935The damped trend have better RMSE.
d-)
autoplot(residuals(hw(myts, seasonal = "multi",damped = T)))
they seem white noise.
7.9
myts.train <- window(myts, end = 2008)
myts.test <- window(myts, start = 2009)
autoplot(myts.train) +
autolayer(hw(myts.train, seasonal = "multi",damped = F),series = 'damped-false',PI=F) +
autolayer(hw(myts.train, seasonal = "multi",damped = T),series = 'damped-true',PI=F)
accuracy(hw(myts.train, seasonal = "multi",damped = F))## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 0.6060111 13.11679 9.293446 0.108113 4.679442 0.3335598 0.03697905accuracy(hw(myts.test, seasonal = "multi",damped = T))## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 1.212733 19.00978 14.21132 0.125952 3.030403 0.3044195 0.00796861After applied the HW we seem a better improved on the RMSE results.now we got better RMSE on the dumped being False.