I. Clean the enviroment and install required packages

rm(list = ls())
cat("\f")
packages <- c("fpp2", "seasonal", "urca", "tseries", "Quandl")
for (i in 1:length(packages)){
  if(!packages[i] %in% installed.packages()){
    install.packages(packages[i])
  }
  library(packages[i], character.only = T)
}
## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo
## -- Attaching packages ---------------------------------------------- fpp2 2.4 --
## v ggplot2   3.3.5     v fma       2.4  
## v forecast  8.15      v expsmooth 2.3
## 
## 载入需要的程辑包：xts
## 载入需要的程辑包：zoo
## 
## 载入程辑包：'zoo'
## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric
rm(packages)

setwd("C:/Users/admin/Desktop/study/HW2")
  1. Problem Solving
  1. Chapter 7
    1. Optimal value of alpha is 0.2971, and initial states is 77260.0561.
    2. My 95% confidence interval is narrower than the interval produced by R.
pigs_ses <- ses(pigs)

summary(pigs_ses)
## 
## Forecast method: Simple exponential smoothing
## 
## Model Information:
## Simple exponential smoothing 
## 
## Call:
##  ses(y = pigs) 
## 
##   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 121539.8
## Jan 1996       98816.41 83448.52 114184.3 75313.25 122319.6
## Feb 1996       98816.41 82955.06 114677.8 74558.56 123074.2
## Mar 1996       98816.41 82476.49 115156.3 73826.66 123806.2
## Apr 1996       98816.41 82011.54 115621.3 73115.58 124517.2
## May 1996       98816.41 81559.12 116073.7 72423.66 125209.2
## Jun 1996       98816.41 81118.26 116514.6 71749.42 125883.4
fpigs <- forecast(pigs_ses, h = 4)

pigs %>%
  autoplot() +
  autolayer(fpigs) +
  ylab("number of pigs slaughtered") +
  ggtitle("Pigs Slaughter Forecast")

manual_upper <- fpigs$mean + sd(pigs_ses$residuals)*1.96
manual_lower <- fpigs$mean - sd(pigs_ses$residuals)*1.96

data.frame(manual_upper = manual_upper
           , R_upper = fpigs$upper[,2]
           , manual_lower = manual_lower
           , R_lower = fpigs$lower[,2])
##   manual_upper  R_upper manual_lower  R_lower
## 1     118952.8 119020.8     78679.97 78611.97
## 2     118952.8 119894.0     78679.97 77738.83
## 3     118952.8 120732.4     78679.97 76900.46
## 4     118952.8 121539.8     78679.97 76092.99
pig_alpha <- fpigs$model$par[1]
pig_init <- fpigs$model$par[2]
  1. Yes, the results are virtually the same.
fses <- function(ts, par = c(alpha, initial)) {
  t = length(ts)
  alpha = par[1]
  yhat = par[2]
  for(i in 1 : t) {
    yhat =  alpha * ts[i] + (1 - alpha) * yhat 
  }
  return(yhat)
}
pig_alpha <- fpigs$model$par[1]
pig_init <- fpigs$model$par[2]

fses(pigs, par = c(pig_alpha, pig_init))
##    alpha 
## 98816.41
forecast(fpigs, h = 1)$mean
##           Sep
## 1995 98816.41
  1. No, the results are not exactly the same but very close to the original result.
error_ses <- function(ts, par = c(alpha, initial)) {
  t = length(ts)
  alpha = par[1]
  yhat = par[2]
  err = 0
  sse = 0
  for(i in 1 : t) {
    err = ts[i] - yhat
    sse = sse + err^2
    yhat =  alpha * ts[i] + (1 - alpha) * yhat 
  }
  return(sse)
}

optim(ts = pigs, par = c(0, pigs[1]), fn=error_ses)
## $par
## [1]     0.297086 77272.075070
## 
## $value
## [1] 19765613447
## 
## $counts
## function gradient 
##      179       NA 
## 
## $convergence
## [1] 0
## 
## $message
## NULL
  1. Results are very close.
pig_fun <- function(ts) {
  pig_error = function(ts, par = c(alpha, initial)) {
     t = length(ts)
     alpha = par[1]
     yhat = par[2]
     err = 0
     sse = 0
     for(i in 1 : t) {
       err = ts[i] - yhat
       sse = sse + err^2
       yhat =  alpha * ts[i] + (1 - alpha) * yhat 
  }
  return(sse)
  }
  palpha = optim(ts = ts, par = c(0, ts[1]), fn=pig_error)$par[1]
  y_hat = ts[1]
  for ( j in 1:length(ts)) {
    y_hat = palpha * ts[j] + (1-palpha) * y_hat
  }
  return(y_hat)

}

pig_fun(pigs)
## [1] 98816.47
  1. Both have a strong uptrend, but paperback books have a more clear seasonal pattern than hardcover books.
autoplot(books)

paper_back <- books[,1]
hard_cover <- books[,2]
fp<- ses(paper_back)
fh <- ses(hard_cover)
forecast(fp, h = 4) %>%
  autoplot() +
  ylab("Paperback Books Sale")

forecast(fh, h = 4) %>%
  autoplot() +
  ylab("Hardcover Books Sale")

rmse_fun <- function(fit, actual) {
  error = 0
  for (i in 1:length(actual)) {
    error = error + (actual[i] - fit[i])^2
  }
  return(sqrt(error/length(actual)))
}

rmse_fun(fp$fitted, paper_back)
## [1] 33.63769
rmse_fun(fh$fitted, hard_cover)
## [1] 31.93101
  1. Holt’s linear method outperforms the previous ses method as it has a lower rmse. Since Holt’s linear method consider the trend in the series and the books data set has a clear uptrend, it is expected that Holt’s linear method would have a better performance than the ses.
  2. I think forecast from Holt’s linear method is more accurate because it shows an expected uptrend as before.
  3. Generated prediction interval is wider than the R interval.
hp <- holt(paper_back, h = 4)
hh <- holt(hard_cover, h = 4)

rmse_fun(hp$fitted, paper_back)
## [1] 31.13692
rmse_fun(hh$fitted, hard_cover)
## [1] 27.19358
fp %>% autoplot(series = "SES") + 
  autolayer(hp, series="Holt's Linear Method")

fh %>% autoplot(series = "SES") + 
  autolayer(hh, series="Holt's Linear Method")

pf_holt <- forecast(hp, h = 1)
pf_ses <- forecast(fp, h = 1)
hf_holt <- forecast(hh, h = 1)
hf_ses <- forecast(fh, h = 1)

pf_holt_ci <- c(pf_holt$lower[2], pf_holt$upper[2])
manual_phci <- c(pf_holt$mean[1]-1.96*sd(pf_holt$residuals)
                      , pf_holt$mean[1]+1.96*sd(pf_holt$residuals))
data.frame(pf_holt_ci, manual_phci)
##   pf_holt_ci manual_phci
## 1   143.9130    147.8390
## 2   275.0205    271.0945
pf_ses_ci <- c(pf_ses$lower[2], pf_ses$upper[2])
manual_psci <- c(pf_ses$mean[1]-1.96*sd(pf_ses$residuals)
                      , pf_ses$mean[1]+1.96*sd(pf_ses$residuals))
data.frame(pf_ses_ci, manual_psci)
##   pf_ses_ci manual_psci
## 1  138.8670    141.5964
## 2  275.3523    272.6230
hf_holt_ci <- c(hf_holt$lower[2], hf_holt$upper[2])
manual_hhci <- c(hf_holt$mean[1]-1.96*sd(hf_holt$residuals)
                      , hf_holt$mean[1]+1.96*sd(hf_holt$residuals))
data.frame(hf_holt_ci, manual_hhci)
##   hf_holt_ci manual_hhci
## 1   192.9222    195.9640
## 2   307.4256    304.3838
hf_ses_ci <- c(hf_ses$lower[2], hf_ses$upper[2])
manual_hsci <- c(hf_ses$mean[1]-1.96*sd(hf_ses$residuals)
                      , hf_ses$mean[1]+1.96*sd(hf_ses$residuals))
data.frame(hf_ses_ci, manual_hsci)
##   hf_ses_ci manual_hsci
## 1  174.7799    178.5848
## 2  304.3403    300.5354
  1. From the three plots, we can see that Holt’s linear approach generate a straight line along with previous trend. However, Holt’s linear approach with damped will dampen and replace the previous trend with a rather constant flat forecast just as what SES method does. Furthermore, with the boxcox transformation, forecast’s trend is also dampened but with a slower rate. RMSE for each model shows that Holt’s linear method with boxcox transformation is the best model among them due to its lowest RMSE.
autoplot(eggs)

heggs <- holt(eggs, h = 100)
hdeggs <- holt(eggs, damped = T, h = 100)
hbeggs <- holt(eggs, lambda = BoxCox.lambda(eggs), h = 100)
forecast(heggs) %>% autoplot()

forecast(hdeggs) %>% autoplot()

forecast(hbeggs) %>% autoplot()

(h_rmse <- rmse_fun(heggs$fitted, eggs))
## [1] 26.58219
(hd_rmse <- rmse_fun(hdeggs$fitted, eggs))
## [1] 26.54019
(hb_rmse <- rmse_fun(hbeggs$fitted, eggs))
## [1] 26.39376
  1. Multiplicative seasonality is necessary because this series’s seasonality is changing proportionally along the time. Model without damping is better.
retaildata <- readxl::read_excel("retail.xlsx", skip=1)
myts <- ts(retaildata[,"A3349873A"],
  frequency=12, start=c(1982,4))
autoplot(myts)

seasonplot(myts)

mretail <- holt(myts, seasonal = "multiplicative")
mdretail <- holt(myts, seasonal = "multiplicative", damped = T)

accuracy(mretail)
##                     ME     RMSE      MAE       MPE     MAPE     MASE     ACF1
## Training set 0.2014937 42.81252 26.89987 -2.619164 11.36189 1.420687 0.264695
accuracy(mdretail)
##                    ME     RMSE      MAE        MPE     MAPE     MASE     ACF1
## Training set 3.091964 42.90522 26.01247 -0.8997451 10.72444 1.373819 0.330954
checkresiduals(mretail)

## 
##  Ljung-Box test
## 
## data:  Residuals from Holt's method
## Q* = 734.49, df = 20, p-value < 2.2e-16
## 
## Model df: 4.   Total lags used: 24
ts_train <- window(myts, end=c(2010, 12))
ts_test <- window(myts, start=c(2011,1))
autoplot(myts)+autolayer(ts_train, series="Training")+autolayer(ts_test, series="Test")

  1. Boxcox transformation does not seem to have a better performance.
myts %>% stlm( s.window = 13
               , robust = TRUE
               , method = "ets"
               , lambda = BoxCox.lambda(myts) ) %>%
  forecast( h = 36, lambda = BoxCox.lambda(myts) ) %>%
  autoplot()
## Warning in InvBoxCox(fcast$mean, lambda, biasadj, fcast): biasadj information
## not found, defaulting to FALSE.

myts %>% stlm( s.window = 13
               , robust = TRUE
               ,  method = "ets" ) %>%
  forecast(h = 36) %>%
  autoplot()

  1. Damped ETS (A,A,N) is the best model.
autoplot(ukcars)

ukcars %>% stl(s.window = 4, robust = TRUE) %>% seasadj() %>% autoplot()

ukcars %>% stlf(h = 8, etsmodel = "AAN", damped = TRUE) %>% autoplot()

ukcars %>% stlf(h = 8, etsmodel = "AAN", damped = TRUE) %>% autoplot()

ukcars %>% ets() %>% forecast(h=8) %>% autoplot()

ukcars %>% stlf(h = 8, etsmodel = "AAN", damped = TRUE) %>% accuracy()
##                    ME     RMSE      MAE        MPE     MAPE      MASE
## Training set 1.040088 22.81499 17.73846 -0.1296296 5.821059 0.5780878
##                    ACF1
## Training set 0.01590553
ukcars %>% stlf(h = 8, etsmodel = "AAN", damped = TRUE) %>% accuracy()
##                    ME     RMSE      MAE        MPE     MAPE      MASE
## Training set 1.040088 22.81499 17.73846 -0.1296296 5.821059 0.5780878
##                    ACF1
## Training set 0.01590553
ukcars %>% ets() %>% forecast(h=8) %>% accuracy()
##                    ME     RMSE      MAE        MPE     MAPE      MASE
## Training set 1.313884 25.23244 20.17907 -0.1570979 6.629003 0.6576259
##                    ACF1
## Training set 0.02573334
ukcars %>% stlf(h = 8, etsmodel = "AAN", damped = TRUE) %>% checkresiduals()
## Warning in checkresiduals(.): The fitted degrees of freedom is based on the
## model used for the seasonally adjusted data.

## 
##  Ljung-Box test
## 
## data:  Residuals from STL +  ETS(A,Ad,N)
## Q* = 23.825, df = 3, p-value = 2.717e-05
## 
## Model df: 5.   Total lags used: 8
autoplot(visitors)

ggseasonplot(visitors)

vtrain <- subset(visitors, end = length(visitors) - 24)
vtest <- subset(visitors, start = length(visitors) - 23)
hw(vtrain, h = 24,seasonal = "multiplicative") %>% autoplot()

hw(vtrain, h = 24,seasonal = "multiplicative") %>% accuracy()
##                      ME     RMSE      MAE        MPE     MAPE      MASE
## Training set -0.9749466 14.06539 10.35763 -0.5792169 4.223204 0.3970304
##                   ACF1
## Training set 0.1356528
forecast(ets(vtrain), h = 24) %>% autoplot()

forecast(ets(vtrain), h = 24) %>% accuracy()
##                     ME    RMSE      MAE       MPE     MAPE     MASE        ACF1
## Training set 0.7640074 14.5348 10.57657 0.1048224 3.994788 0.405423 -0.05311217
forecast(ets(vtrain
             , lambda = BoxCox.lambda(vtrain)
             , additive.only = TRUE)
         , h = 24) %>%
  autoplot()

forecast(ets(vtrain
             , lambda = BoxCox.lambda(vtrain)
             , additive.only = TRUE)
         , h = 24) %>%
  accuracy()
##                    ME     RMSE      MAE       MPE     MAPE      MASE
## Training set 1.001363 14.97096 10.82396 0.1609336 3.974215 0.4149057
##                     ACF1
## Training set -0.02535299
snaive(vtrain, h = 24) %>% autoplot()

snaive(vtrain, h = 24) %>% accuracy()
##                    ME     RMSE      MAE      MPE     MAPE MASE      ACF1
## Training set 17.29363 31.15613 26.08775 7.192445 10.28596    1 0.6327669
vtrain %>%
  stlm(lambda = BoxCox.lambda(vtrain),
    s.window = 13,
    robust = TRUE,
    method = "ets") %>%
  forecast(h = 24) %>%
  autoplot()

vtrain %>%
  stlm(lambda = BoxCox.lambda(vtrain),
    s.window = 13,
    robust = TRUE,
    method = "ets") %>%
  forecast(h = 24) %>%
  accuracy()
##                     ME     RMSE      MAE        MPE   MAPE      MASE
## Training set 0.5803348 13.36431 9.551391 0.08767744 3.5195 0.3661256
##                     ACF1
## Training set -0.05924203
fets <- function(y, h) {
 forecast(ets(y),
          h = h)
}

#tsCV(qcement,fets, h = 4)
#tsCV(qcement, snaive, h = 4)
  1. ETS models have better predict on these series.
boxcox_brisk <- BoxCox(bricksq, BoxCox.lambda(bricksq))
brick_train    <- window(boxcox_brisk, end = c(1991,1))
brick_test  <- window(boxcox_brisk, start = c(1991,2))

boxcox_a10 <- BoxCox(a10, BoxCox.lambda(a10))
a10_train <- window(boxcox_a10 , end = c(2005,6))
a10_test <- window(boxcox_a10 , start = c(2005,7))

ausbeer_train <- window(ausbeer, end = c(2007,2))
ausbeer_test  <- window(ausbeer, start = c(2007,3))

dole_train <- window(dole, end=c(1989,7))
dole_test <- window(dole, start=c(1989,8))

h02_train <- window(h02, end = c(2005,7))
h02_test <- window(h02, start = c(2005,8))

usmelec_train <- window(usmelec, end = c(2010,6))
usmelec_test <- window(usmelec, start = c(2010,7))

brick_ets <- ets(brick_train)
a10_ets <- ets(a10_train)
ausbeer_ets <- ets(ausbeer_train)
dole_ets <- ets(dole_train)
h02_ets <- ets(h02_train)
usmelec_ets <- ets(usmelec_train)

forecast(brick_ets, h=12)
##         Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 1991 Q2       14.39012 14.06821 14.71204 13.89780 14.88245
## 1991 Q3       14.55275 14.09235 15.01316 13.84862 15.25689
## 1991 Q4       14.29232 13.73850 14.84614 13.44532 15.13932
## 1992 Q1       13.82491 13.20629 14.44354 12.87881 14.77102
## 1992 Q2       14.39013 13.67014 15.11011 13.28900 15.49125
## 1992 Q3       14.55275 13.75508 15.35043 13.33281 15.77270
## 1992 Q4       14.29232 13.44609 15.13855 12.99812 15.58652
## 1993 Q1       13.82491 12.94978 14.70005 12.48651 15.16332
## 1993 Q2       14.39013 13.42387 15.35638 12.91237 15.86788
## 1993 Q3       14.55276 13.52265 15.58286 12.97734 16.12817
## 1993 Q4       14.29232 13.23119 15.35345 12.66947 15.91517
## 1994 Q1       13.82491 12.75277 14.89706 12.18521 15.46462
forecast(a10_ets, h=36)
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Jul 2005       3.392676 3.305704 3.479648 3.259664 3.525688
## Aug 2005       3.391047 3.303038 3.479056 3.256449 3.525645
## Sep 2005       3.416806 3.327771 3.505840 3.280639 3.552972
## Oct 2005       3.513518 3.423468 3.603568 3.375798 3.651238
## Nov 2005       3.538857 3.447801 3.629912 3.399599 3.678114
## Dec 2005       3.760049 3.667998 3.852101 3.619268 3.900830
## Jan 2006       3.852071 3.759033 3.945109 3.709782 3.994361
## Feb 2006       3.177780 3.083764 3.271795 3.033995 3.321564
## Mar 2006       3.328305 3.233321 3.423289 3.183040 3.473570
## Apr 2006       3.333654 3.237710 3.429597 3.186920 3.480387
## May 2006       3.435400 3.338504 3.532295 3.287211 3.583588
## Jun 2006       3.429342 3.331503 3.527182 3.279710 3.578974
## Jul 2006       3.544173 3.445397 3.642949 3.393108 3.695238
## Aug 2006       3.542544 3.442840 3.642249 3.390060 3.695029
## Sep 2006       3.568303 3.467678 3.668928 3.414410 3.722196
## Oct 2006       3.665015 3.563476 3.766554 3.509725 3.820305
## Nov 2006       3.690354 3.587909 3.792799 3.533677 3.847031
## Dec 2006       3.911546 3.808201 4.014892 3.753494 4.069599
## Jan 2007       4.003569 3.899330 4.107807 3.844150 4.162987
## Feb 2007       3.329277 3.224152 3.434402 3.168502 3.490052
## Mar 2007       3.479802 3.373797 3.585808 3.317681 3.641924
## Apr 2007       3.485151 3.378271 3.592031 3.321692 3.648609
## May 2007       3.586897 3.479149 3.694645 3.422110 3.751683
## Jun 2007       3.580840 3.472229 3.689450 3.414734 3.746945
## Jul 2007       3.695670 3.586202 3.805139 3.528253 3.863088
## Aug 2007       3.694042 3.583722 3.804361 3.525322 3.862761
## Sep 2007       3.719800 3.608635 3.830966 3.549788 3.889813
## Oct 2007       3.816512 3.704507 3.928518 3.645215 3.987810
## Nov 2007       3.841851 3.729010 3.954692 3.669276 4.014426
## Dec 2007       4.063044 3.949373 4.176715 3.889199 4.236888
## Jan 2008       4.155066 4.040570 4.269562 3.979959 4.330172
## Feb 2008       3.480774 3.365458 3.596090 3.304413 3.657135
## Mar 2008       3.631299 3.515168 3.747431 3.453691 3.808908
## Apr 2008       3.636648 3.519706 3.753591 3.457800 3.815496
## May 2008       3.738394 3.620645 3.856143 3.558313 3.918475
## Jun 2008       3.732337 3.613786 3.850888 3.551029 3.913644
forecast(ausbeer_ets, h=12)
##         Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 2007 Q3       404.3218 385.3540 423.2895 375.3131 433.3305
## 2007 Q4       484.3093 460.9469 507.6717 448.5796 520.0390
## 2008 Q1       424.9750 403.7708 446.1792 392.5459 457.4040
## 2008 Q2       385.2270 365.2353 405.2187 354.6524 415.8017
## 2008 Q3       403.1265 379.8780 426.3750 367.5710 438.6820
## 2008 Q4       482.8764 453.8000 511.9528 438.4079 527.3449
## 2009 Q1       423.7166 396.9820 450.4513 382.8295 464.6038
## 2009 Q2       384.0855 358.6189 409.5521 345.1376 423.0334
## 2009 Q3       401.9449 372.4788 431.4111 356.8803 447.0095
## 2009 Q4       481.4600 444.4034 518.5166 424.7868 538.1332
## 2010 Q1       422.4728 388.2859 456.6596 370.1885 474.7570
## 2010 Q2       382.9571 350.3454 415.5687 333.0819 432.8323
forecast(dole_ets, h=36)
##          Point Forecast      Lo 80    Hi 80       Lo 95     Hi 95
## Aug 1989       357550.0  312945.06 402154.9  289332.642  425767.3
## Sep 1989       342308.5  286561.31 398055.7  257050.527  427566.5
## Oct 1989       331606.7  264605.91 398607.6  229137.813  434075.6
## Nov 1989       326881.1  247817.71 405944.6  205964.050  447798.2
## Dec 1989       340124.0  244214.27 436033.7  193442.733  486805.2
## Jan 1990       374863.2  254107.59 495618.9  190183.411  559543.1
## Feb 1990       384381.2  245185.54 523576.9  171499.791  597262.7
## Mar 1990       379612.5  227069.19 532155.7  146317.666  612907.3
## Apr 1990       382797.3  213919.40 551675.2  124520.833  641073.8
## May 1990       386541.1  200981.73 572100.4  102752.547  670329.6
## Jun 1990       379469.8  182734.40 576205.1   78588.990  680350.5
## Jul 1990       365033.3  161951.09 568115.4   54445.888  675620.6
## Aug 1990       343352.8  134570.12 552135.6   24047.226  662658.5
## Sep 1990       330385.0  117585.66 543184.4    4936.488  655833.5
## Oct 1990       321501.8  102954.77 540048.8  -12737.036  655740.6
## Nov 1990       318193.3   90658.21 545728.3  -29791.564  666178.1
## Dec 1990       332265.8   83052.34 581479.3  -48873.310  713405.0
## Jan 1991       367363.1   79110.65 655615.6  -73480.995  808207.2
## Feb 1991       377750.1   68399.23 687101.1  -95361.262  850861.6
## Mar 1991       373994.3   55017.20 692971.5 -113839.088  861827.7
## Apr 1991       377966.1   42875.37 713056.9 -134510.977  890443.3
## May 1991       382410.7   30607.00 734214.3 -155626.608  920447.9
## Jun 1991       376066.7   17631.09 734502.3 -172113.270  924246.7
## Jul 1991       362315.9    5054.39 719577.4 -184068.427  908700.2
## Aug 1991       341464.2  -10319.53 693248.0 -196542.599  879471.1
## Sep 1991       328960.6  -20518.82 678440.0 -205522.031  863443.2
## Oct 1991       320454.4  -30267.22 671175.9 -215928.007  856836.7
## Nov 1991       317453.5  -40145.69 675052.7 -229447.261  864354.2
## Dec 1991       331767.8  -52558.50 716094.2 -256008.572  919544.2
## Jan 1992       367081.1  -69869.52 804031.8 -301177.236 1035339.5
## Feb 1992       377704.7  -83937.83 839347.1 -328316.595 1083725.9
## Mar 1992       374163.4  -95080.06 843406.9 -343482.547 1091809.4
## Apr 1992       378328.5 -108201.75 864858.7 -365755.280 1122412.2
## May 1992       382948.5 -121740.51 887637.6 -388906.744 1154803.8
## Jun 1992       376744.6 -131799.79 885289.0 -401006.938 1154496.2
## Jul 1992       363095.9 -138637.24 864829.1 -404238.730 1130430.6
forecast(h02_ets, h=36)
##          Point Forecast     Lo 80     Hi 80     Lo 95     Hi 95
## Aug 2005      0.9261277 0.8532823 0.9989732 0.8147203 1.0375352
## Sep 2005      0.9788350 0.8988398 1.0588302 0.8564929 1.1011771
## Oct 2005      1.0390566 0.9510645 1.1270488 0.9044843 1.1736290
## Nov 2005      1.0478122 0.9560802 1.1395442 0.9075202 1.1881041
## Dec 2005      1.2201827 1.1099771 1.3303883 1.0516377 1.3887276
## Jan 2006      1.1511696 1.0440984 1.2582409 0.9874183 1.3149209
## Feb 2006      0.6254695 0.5656567 0.6852824 0.5339937 0.7169454
## Mar 2006      0.7010367 0.6322100 0.7698635 0.5957753 0.8062981
## Apr 2006      0.6925375 0.6228230 0.7622521 0.5859183 0.7991567
## May 2006      0.7414197 0.6649838 0.8178555 0.6245211 0.8583182
## Jun 2006      0.7888606 0.7056609 0.8720603 0.6616176 0.9161036
## Jul 2006      0.9004823 0.8034177 0.9975469 0.7520348 1.0489298
## Aug 2006      0.9296094 0.8272828 1.0319359 0.7731144 1.0861043
## Sep 2006      0.9824397 0.8721057 1.0927738 0.8136983 1.1511811
## Oct 2006      1.0428051 0.9234085 1.1622017 0.8602037 1.2254064
## Nov 2006      1.0515151 0.9288607 1.1741696 0.8639313 1.2390990
## Dec 2006      1.2244069 1.0789989 1.3698148 1.0020246 1.4467892
## Jan 2007      1.1550737 1.0155006 1.2946468 0.9416151 1.3685324
## Feb 2007      0.6275476 0.5504357 0.7046594 0.5096152 0.7454800
## Mar 2007      0.7033184 0.6154814 0.7911554 0.5689833 0.8376535
## Apr 2007      0.6947457 0.6066032 0.7828882 0.5599434 0.8295480
## May 2007      0.7437356 0.6479262 0.8395450 0.5972077 0.8902634
## Jun 2007      0.7912745 0.6878187 0.8947304 0.6330525 0.9494966
## Jul 2007      0.9031818 0.7833808 1.0229827 0.7199621 1.0864015
## Aug 2007      0.9323395 0.8069204 1.0577586 0.7405275 1.1241515
## Sep 2007      0.9852664 0.8509072 1.1196256 0.7797817 1.1907510
## Oct 2007      1.0457445 0.9012304 1.1902585 0.8247293 1.2667596
## Nov 2007      1.0544189 0.9068064 1.2020313 0.8286651 1.2801726
## Dec 2007      1.2277194 1.0536606 1.4017783 0.9615194 1.4939194
## Jan 2008      1.1581352 0.9919042 1.3243662 0.9039069 1.4123636
## Feb 2008      0.6291771 0.5377750 0.7205792 0.4893897 0.7689645
## Mar 2008      0.7051076 0.6014623 0.8087529 0.5465958 0.8636195
## Apr 2008      0.6964772 0.5929158 0.8000387 0.5380936 0.8548608
## May 2008      0.7455516 0.6334383 0.8576649 0.5740892 0.9170141
## Jun 2008      0.7931675 0.6725728 0.9137622 0.6087339 0.9776011
## Jul 2008      0.9052987 0.7661628 1.0444345 0.6925088 1.1180886
forecast(usmelec_ets, h = 36)
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Jul 2010       398.3597 383.9893 412.7301 376.3821 420.3373
## Aug 2010       402.4829 387.5680 417.3977 379.6725 425.2932
## Sep 2010       344.9220 331.8103 358.0336 324.8694 364.9745
## Oct 2010       323.8277 311.2164 336.4390 304.5403 343.1151
## Nov 2010       312.2226 299.7797 324.6654 293.1929 331.2523
## Dec 2010       350.9904 336.6915 365.2893 329.1221 372.8587
## Jan 2011       360.3726 345.3794 375.3658 337.4425 383.3027
## Feb 2011       318.6280 305.1018 332.1543 297.9414 339.3147
## Mar 2011       322.7679 308.7983 336.7375 301.4033 344.1325
## Apr 2011       302.5312 289.1918 315.8706 282.1304 322.9321
## May 2011       333.2581 318.2985 348.2176 310.3794 356.1367
## Jun 2011       373.3013 356.2527 390.3498 347.2278 399.3748
## Jul 2011       404.5701 384.9434 424.1968 374.5537 434.5865
## Aug 2011       408.7495 388.6252 428.8737 377.9721 439.5269
## Sep 2011       350.2854 332.7910 367.7799 323.5299 377.0409
## Oct 2011       328.8567 312.2027 345.5107 303.3866 354.3268
## Nov 2011       317.0651 300.7901 333.3401 292.1746 341.9556
## Dec 2011       356.4273 337.8901 374.9644 328.0772 384.7773
## Jan 2012       365.9476 346.6708 385.2244 336.4663 395.4289
## Feb 2012       323.5509 306.2943 340.8075 297.1592 349.9426
## Mar 2012       327.7484 310.0550 345.4418 300.6887 354.8081
## Apr 2012       307.1935 290.4128 323.9741 281.5297 332.8573
## May 2012       338.3873 319.6886 357.0859 309.7901 366.9844
## Jun 2012       379.0395 357.8578 400.2212 346.6449 411.4341
## Jul 2012       410.7809 386.8705 434.6912 374.2132 447.3486
## Aug 2012       415.0164 390.6141 439.4187 377.6963 452.3365
## Sep 2012       355.6492 334.5297 376.7687 323.3497 387.9487
## Oct 2012       333.8859 313.8658 353.9061 303.2677 364.5041
## Nov 2012       321.9079 302.4219 341.3938 292.1067 351.7091
## Dec 2012       361.8644 339.7551 383.9736 328.0512 395.6775
## Jan 2013       371.5229 348.6157 394.4301 336.4894 406.5564
## Feb 2013       328.4741 308.0393 348.9088 297.2219 359.7263
## Mar 2013       332.7291 311.8474 353.6108 300.7933 364.6649
## Apr 2013       311.8560 292.1151 331.5968 281.6650 342.0470
## May 2013       343.5168 321.5873 365.4462 309.9786 377.0550
## Jun 2013       384.7780 360.0100 409.5460 346.8986 422.6574
snaive(brick_train, h= 12)
##         Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 1991 Q2       15.43122 14.73297 16.12946 14.36335 16.49909
## 1991 Q3       15.17199 14.47375 15.87023 14.10412 16.23986
## 1991 Q4       14.79913 14.10089 15.49737 13.73126 15.86700
## 1992 Q1       13.82491 13.12667 14.52315 12.75704 14.89278
## 1992 Q2       15.43122 14.44375 16.41868 13.92102 16.94141
## 1992 Q3       15.17199 14.18452 16.15945 13.66179 16.68218
## 1992 Q4       14.79913 13.81167 15.78659 13.28894 16.30932
## 1993 Q1       13.82491 12.83745 14.81237 12.31472 15.33511
## 1993 Q2       15.43122 14.22183 16.64061 13.58161 17.28082
## 1993 Q3       15.17199 13.96260 16.38138 13.32238 17.02159
## 1993 Q4       14.79913 13.58974 16.00852 12.94953 16.64873
## 1994 Q1       13.82491 12.61552 15.03430 11.97531 15.67451
snaive(a10_train, h = 36)
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Jul 2005       3.284117 3.062268 3.505965 2.944828 3.623405
## Aug 2005       3.287069 3.065220 3.508918 2.947780 3.626357
## Sep 2005       3.357387 3.135538 3.579236 3.018098 3.696676
## Oct 2005       3.405230 3.183381 3.627079 3.065941 3.744519
## Nov 2005       3.485739 3.263890 3.707588 3.146451 3.825028
## Dec 2005       3.584657 3.362808 3.806506 3.245368 3.923946
## Jan 2006       3.727330 3.505481 3.949179 3.388041 4.066619
## Feb 2006       2.956087 2.734238 3.177936 2.616798 3.295375
## Mar 2006       3.092632 2.870783 3.314481 2.753343 3.431921
## Apr 2006       3.199894 2.978045 3.421743 2.860605 3.539183
## May 2006       3.232556 3.010707 3.454405 2.893267 3.571845
## Jun 2006       3.317933 3.096085 3.539782 2.978645 3.657222
## Jul 2006       3.284117 2.970375 3.597858 2.804290 3.763943
## Aug 2006       3.287069 2.973327 3.600811 2.807242 3.766895
## Sep 2006       3.357387 3.043645 3.671129 2.877560 3.837214
## Oct 2006       3.405230 3.091488 3.718972 2.925403 3.885057
## Nov 2006       3.485739 3.171998 3.799481 3.005913 3.965566
## Dec 2006       3.584657 3.270915 3.898399 3.104830 4.064484
## Jan 2007       3.727330 3.413588 4.041072 3.247503 4.207157
## Feb 2007       2.956087 2.642345 3.269828 2.476260 3.435913
## Mar 2007       3.092632 2.778890 3.406374 2.612805 3.572459
## Apr 2007       3.199894 2.886152 3.513636 2.720067 3.679721
## May 2007       3.232556 2.918814 3.546298 2.752729 3.712383
## Jun 2007       3.317933 3.004192 3.631675 2.838107 3.797760
## Jul 2007       3.284117 2.899863 3.668370 2.696451 3.871782
## Aug 2007       3.287069 2.902815 3.671322 2.699403 3.874734
## Sep 2007       3.357387 2.973133 3.741641 2.769722 3.945052
## Oct 2007       3.405230 3.020976 3.789484 2.817565 3.992895
## Nov 2007       3.485739 3.101486 3.869993 2.898074 4.073405
## Dec 2007       3.584657 3.200403 3.968911 2.996992 4.172323
## Jan 2008       3.727330 3.343076 4.111584 3.139665 4.314995
## Feb 2008       2.956087 2.571833 3.340340 2.368421 3.543752
## Mar 2008       3.092632 2.708378 3.476886 2.504967 3.680297
## Apr 2008       3.199894 2.815640 3.584148 2.612229 3.787559
## May 2008       3.232556 2.848302 3.616810 2.644891 3.820221
## Jun 2008       3.317933 2.933680 3.702187 2.730268 3.905599
snaive(ausbeer_train, h = 12)
##         Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 2007 Q3            405 379.7918 430.2082 366.4474 443.5526
## 2007 Q4            491 465.7918 516.2082 452.4474 529.5526
## 2008 Q1            427 401.7918 452.2082 388.4474 465.5526
## 2008 Q2            383 357.7918 408.2082 344.4474 421.5526
## 2008 Q3            405 369.3502 440.6498 350.4784 459.5216
## 2008 Q4            491 455.3502 526.6498 436.4784 545.5216
## 2009 Q1            427 391.3502 462.6498 372.4784 481.5216
## 2009 Q2            383 347.3502 418.6498 328.4784 437.5216
## 2009 Q3            405 361.3381 448.6619 338.2250 471.7750
## 2009 Q4            491 447.3381 534.6619 424.2250 557.7750
## 2010 Q1            427 383.3381 470.6619 360.2250 493.7750
## 2010 Q2            383 339.3381 426.6619 316.2250 449.7750
snaive(dole_train, h= 36)
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Aug 1989         458767 386507.9 531026.1 348256.3 569277.7
## Sep 1989         441201 368941.9 513460.1 330690.3 551711.7
## Oct 1989         428578 356318.9 500837.1 318067.3 539088.7
## Nov 1989         419052 346792.9 491311.1 308541.3 529562.7
## Dec 1989         420900 348640.9 493159.1 310389.3 531410.7
## Jan 1990         448572 376312.9 520831.1 338061.3 559082.7
## Feb 1990         441100 368840.9 513359.1 330589.3 551610.7
## Mar 1990         409708 337448.9 481967.1 299197.3 520218.7
## Apr 1990         393323 321063.9 465582.1 282812.3 503833.7
## May 1990         391918 319658.9 464177.1 281407.3 502428.7
## Jun 1990         390001 317741.9 462260.1 279490.3 500511.7
## Jul 1990         383839 311579.9 456098.1 273328.3 494349.7
## Aug 1990         458767 356577.2 560956.8 302481.2 615052.8
## Sep 1990         441201 339011.2 543390.8 284915.2 597486.8
## Oct 1990         428578 326388.2 530767.8 272292.2 584863.8
## Nov 1990         419052 316862.2 521241.8 262766.2 575337.8
## Dec 1990         420900 318710.2 523089.8 264614.2 577185.8
## Jan 1991         448572 346382.2 550761.8 292286.2 604857.8
## Feb 1991         441100 338910.2 543289.8 284814.2 597385.8
## Mar 1991         409708 307518.2 511897.8 253422.2 565993.8
## Apr 1991         393323 291133.2 495512.8 237037.2 549608.8
## May 1991         391918 289728.2 494107.8 235632.2 548203.8
## Jun 1991         390001 287811.2 492190.8 233715.2 546286.8
## Jul 1991         383839 281649.2 486028.8 227553.2 540124.8
## Aug 1991         458767 333610.6 583923.4 267356.8 650177.2
## Sep 1991         441201 316044.6 566357.4 249790.8 632611.2
## Oct 1991         428578 303421.6 553734.4 237167.8 619988.2
## Nov 1991         419052 293895.6 544208.4 227641.8 610462.2
## Dec 1991         420900 295743.6 546056.4 229489.8 612310.2
## Jan 1992         448572 323415.6 573728.4 257161.8 639982.2
## Feb 1992         441100 315943.6 566256.4 249689.8 632510.2
## Mar 1992         409708 284551.6 534864.4 218297.8 601118.2
## Apr 1992         393323 268166.6 518479.4 201912.8 584733.2
## May 1992         391918 266761.6 517074.4 200507.8 583328.2
## Jun 1992         390001 264844.6 515157.4 198590.8 581411.2
## Jul 1992         383839 258682.6 508995.4 192428.8 575249.2
snaive(h02_train, h = 36)
##          Point Forecast     Lo 80     Hi 80     Lo 95     Hi 95
## Aug 2005      0.9948643 0.9029552 1.0867735 0.8543014 1.1354272
## Sep 2005      1.1344320 1.0425228 1.2263412 0.9938691 1.2749949
## Oct 2005      1.1810110 1.0891018 1.2729202 1.0404481 1.3215739
## Nov 2005      1.2160370 1.1241278 1.3079462 1.0754741 1.3565999
## Dec 2005      1.2572380 1.1653288 1.3491472 1.1166751 1.3978009
## Jan 2006      1.1706900 1.0787808 1.2625992 1.0301271 1.3112529
## Feb 2006      0.5976390 0.5057298 0.6895482 0.4570761 0.7382019
## Mar 2006      0.6525900 0.5606808 0.7444992 0.5120271 0.7931529
## Apr 2006      0.6705050 0.5785958 0.7624142 0.5299421 0.8110679
## May 2006      0.6952480 0.6033388 0.7871572 0.5546851 0.8358109
## Jun 2006      0.8422630 0.7503538 0.9341722 0.7017001 0.9828259
## Jul 2006      0.8743360 0.7824268 0.9662452 0.7337731 1.0148989
## Aug 2006      0.9948643 0.8648852 1.1248435 0.7960783 1.1936503
## Sep 2006      1.1344320 1.0044528 1.2644112 0.9356460 1.3332180
## Oct 2006      1.1810110 1.0510318 1.3109902 0.9822250 1.3797970
## Nov 2006      1.2160370 1.0860578 1.3460162 1.0172510 1.4148230
## Dec 2006      1.2572380 1.1272588 1.3872172 1.0584520 1.4560240
## Jan 2007      1.1706900 1.0407108 1.3006692 0.9719040 1.3694760
## Feb 2007      0.5976390 0.4676598 0.7276182 0.3988530 0.7964250
## Mar 2007      0.6525900 0.5226108 0.7825692 0.4538040 0.8513760
## Apr 2007      0.6705050 0.5405258 0.8004842 0.4717190 0.8692910
## May 2007      0.6952480 0.5652688 0.8252272 0.4964620 0.8940340
## Jun 2007      0.8422630 0.7122838 0.9722422 0.6434770 1.0410490
## Jul 2007      0.8743360 0.7443568 1.0043152 0.6755500 1.0731220
## Aug 2007      0.9948643 0.8356730 1.1540556 0.7514022 1.2383264
## Sep 2007      1.1344320 0.9752407 1.2936233 0.8909699 1.3778941
## Oct 2007      1.1810110 1.0218197 1.3402023 0.9375489 1.4244731
## Nov 2007      1.2160370 1.0568457 1.3752283 0.9725749 1.4594991
## Dec 2007      1.2572380 1.0980467 1.4164293 1.0137759 1.5007001
## Jan 2008      1.1706900 1.0114987 1.3298813 0.9272279 1.4141521
## Feb 2008      0.5976390 0.4384477 0.7568303 0.3541769 0.8411011
## Mar 2008      0.6525900 0.4933987 0.8117813 0.4091279 0.8960521
## Apr 2008      0.6705050 0.5113137 0.8296963 0.4270429 0.9139671
## May 2008      0.6952480 0.5360567 0.8544393 0.4517859 0.9387101
## Jun 2008      0.8422630 0.6830717 1.0014543 0.5988009 1.0857251
## Jul 2008      0.8743360 0.7151447 1.0335273 0.6308739 1.1177981
snaive(usmelec_train, h= 36)
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Jul 2010        372.542 357.6945 387.3895 349.8348 395.2492
## Aug 2010        381.221 366.3735 396.0685 358.5138 403.9282
## Sep 2010        327.401 312.5535 342.2485 304.6938 350.1082
## Oct 2010        307.040 292.1925 321.8875 284.3328 329.7472
## Nov 2010        296.635 281.7875 311.4825 273.9278 319.3422
## Dec 2010        350.507 335.6595 365.3545 327.7998 373.2142
## Jan 2011        360.957 346.1095 375.8045 338.2498 383.6642
## Feb 2011        319.735 304.8875 334.5825 297.0278 342.4422
## Mar 2011        312.168 297.3205 327.0155 289.4608 334.8752
## Apr 2011        287.800 272.9525 302.6475 265.0928 310.5072
## May 2011        327.936 313.0885 342.7835 305.2288 350.6432
## Jun 2011        375.759 360.9115 390.6065 353.0518 398.4662
## Jul 2011        372.542 351.5445 393.5395 340.4291 404.6549
## Aug 2011        381.221 360.2235 402.2185 349.1081 413.3339
## Sep 2011        327.401 306.4035 348.3985 295.2881 359.5139
## Oct 2011        307.040 286.0425 328.0375 274.9271 339.1529
## Nov 2011        296.635 275.6375 317.6325 264.5221 328.7479
## Dec 2011        350.507 329.5095 371.5045 318.3941 382.6199
## Jan 2012        360.957 339.9595 381.9545 328.8441 393.0699
## Feb 2012        319.735 298.7375 340.7325 287.6221 351.8479
## Mar 2012        312.168 291.1705 333.1655 280.0551 344.2809
## Apr 2012        287.800 266.8025 308.7975 255.6871 319.9129
## May 2012        327.936 306.9385 348.9335 295.8231 360.0489
## Jun 2012        375.759 354.7615 396.7565 343.6461 407.8719
## Jul 2012        372.542 346.8255 398.2585 333.2119 411.8721
## Aug 2012        381.221 355.5045 406.9375 341.8909 420.5511
## Sep 2012        327.401 301.6845 353.1175 288.0709 366.7311
## Oct 2012        307.040 281.3235 332.7565 267.7099 346.3701
## Nov 2012        296.635 270.9185 322.3515 257.3049 335.9651
## Dec 2012        350.507 324.7905 376.2235 311.1769 389.8371
## Jan 2013        360.957 335.2405 386.6735 321.6269 400.2871
## Feb 2013        319.735 294.0185 345.4515 280.4049 359.0651
## Mar 2013        312.168 286.4515 337.8845 272.8379 351.4981
## Apr 2013        287.800 262.0835 313.5165 248.4699 327.1301
## May 2013        327.936 302.2195 353.6525 288.6059 367.2661
## Jun 2013        375.759 350.0425 401.4755 336.4289 415.0891
stlf(brick_train, h= 12)
##         Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 1991 Q2       14.53531 14.24229 14.82832 14.08718 14.98344
## 1991 Q3       14.60765 14.19326 15.02205 13.97390 15.24141
## 1991 Q4       14.29050 13.78295 14.79804 13.51427 15.06672
## 1992 Q1       13.82496 13.23886 14.41106 12.92860 14.72132
## 1992 Q2       14.53531 13.87999 15.19062 13.53309 15.53753
## 1992 Q3       14.60765 13.88975 15.32556 13.50971 15.70560
## 1992 Q4       14.29050 13.51502 15.06597 13.10451 15.47648
## 1993 Q1       13.82496 12.99589 14.65403 12.55701 15.09292
## 1993 Q2       14.53531 13.65589 15.41472 13.19036 15.88026
## 1993 Q3       14.60765 13.68061 15.53470 13.18986 16.02545
## 1993 Q4       14.29050 13.31814 15.26285 12.80341 15.77758
## 1994 Q1       13.82496 12.80931 14.84062 12.27165 15.37827
stlf(a10_train, h = 36)
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Jul 2005       3.415153 3.340177 3.490128 3.300488 3.529818
## Aug 2005       3.395244 3.318998 3.471489 3.278637 3.511851
## Sep 2005       3.418283 3.340787 3.495778 3.299763 3.536802
## Oct 2005       3.533661 3.454933 3.612388 3.413257 3.654064
## Nov 2005       3.547334 3.467392 3.627276 3.425073 3.669594
## Dec 2005       3.704706 3.623567 3.785845 3.580614 3.828797
## Jan 2006       3.855858 3.773538 3.938179 3.729960 3.981757
## Feb 2006       3.139824 3.056338 3.223310 3.012143 3.267505
## Mar 2006       3.315041 3.230404 3.399678 3.185600 3.444482
## Apr 2006       3.324589 3.238815 3.410363 3.193409 3.455768
## May 2006       3.452448 3.365552 3.539345 3.319551 3.585345
## Jun 2006       3.435097 3.347091 3.523104 3.300503 3.569692
## Jul 2006       3.564846 3.475742 3.653950 3.428574 3.701118
## Aug 2006       3.544937 3.454748 3.635126 3.407005 3.682869
## Sep 2006       3.567976 3.476714 3.659238 3.428402 3.707549
## Oct 2006       3.683354 3.591030 3.775678 3.542157 3.824551
## Nov 2006       3.697027 3.603652 3.790402 3.554222 3.839832
## Dec 2006       3.854399 3.759984 3.948814 3.710003 3.998795
## Jan 2007       4.005552 3.910106 4.100997 3.859580 4.151523
## Feb 2007       3.289517 3.193051 3.385983 3.141985 3.437049
## Mar 2007       3.464734 3.367258 3.562211 3.315657 3.613811
## Apr 2007       3.474282 3.375804 3.572760 3.323673 3.624891
## May 2007       3.602141 3.502671 3.701611 3.450015 3.754268
## Jun 2007       3.584791 3.484337 3.685244 3.431160 3.738421
## Jul 2007       3.714539 3.613111 3.815968 3.559418 3.869661
## Aug 2007       3.694630 3.592235 3.797025 3.538030 3.851230
## Sep 2007       3.717669 3.614315 3.821023 3.559602 3.875735
## Oct 2007       3.833047 3.728742 3.937352 3.673526 3.992568
## Nov 2007       3.846720 3.741472 3.951968 3.685756 4.007684
## Dec 2007       4.004092 3.897908 4.110276 3.841697 4.166487
## Jan 2008       4.155245 4.048131 4.262358 3.991429 4.319060
## Feb 2008       3.439210 3.331175 3.547245 3.273985 3.604435
## Mar 2008       3.614427 3.505478 3.723377 3.447803 3.781052
## Apr 2008       3.623975 3.514117 3.733833 3.455961 3.791989
## May 2008       3.751834 3.641075 3.862594 3.582442 3.921227
## Jun 2008       3.734484 3.622828 3.846139 3.563721 3.905246
stlf(ausbeer_train, h = 12)
##         Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 2007 Q3       406.1670 388.7321 423.6019 379.5027 432.8313
## 2007 Q4       483.2273 464.9265 501.5280 455.2386 511.2159
## 2008 Q1       426.0710 406.7446 445.3974 396.5138 455.6281
## 2008 Q2       385.4522 364.9441 405.9603 354.0877 416.8166
## 2008 Q3       405.2659 383.4262 427.1056 371.8649 438.6668
## 2008 Q4       482.3262 459.0127 505.6396 446.6714 517.9809
## 2009 Q1       425.1699 400.2492 450.0906 387.0569 463.2829
## 2009 Q2       384.5511 357.8979 411.2042 343.7886 425.3135
## 2009 Q3       404.3648 375.8623 432.8673 360.7740 447.9555
## 2009 Q4       481.4251 450.9640 511.8861 434.8389 528.0112
## 2010 Q1       424.2688 391.7468 456.7908 374.5307 474.0069
## 2010 Q2       383.6500 348.9709 418.3290 330.6129 436.6870
stlf(dole_train, h= 36)
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Aug 1989       374079.4 366021.1 382137.7 361755.4 386403.5
## Sep 1989       365745.7 352329.3 379162.1 345227.1 386264.3
## Oct 1989       354372.2 335660.9 373083.5 325755.8 382988.6
## Nov 1989       348635.7 324621.6 372649.8 311909.2 385362.2
## Dec 1989       372192.6 342887.8 401497.3 327374.7 417010.4
## Jan 1990       416820.5 382266.7 451374.3 363975.1 469665.9
## Feb 1990       416350.0 376614.2 456085.8 355579.3 477120.7
## Mar 1990       390535.4 345703.6 435367.1 321971.1 459099.6
## Apr 1990       378456.7 328628.6 428284.8 302251.2 454662.1
## May 1990       376417.3 321701.6 431132.9 292736.8 460097.7
## Jun 1990       373115.1 313626.0 432604.2 282134.4 464095.8
## Jul 1990       370885.8 306740.3 435031.3 272783.7 468987.9
## Aug 1990       362840.5 294156.3 431524.8 257797.1 467884.0
## Sep 1990       355994.5 282888.4 429100.5 244188.5 467800.5
## Oct 1990       345911.7 268498.9 423324.5 227519.0 464304.3
## Nov 1990       341295.1 259687.8 422902.4 216487.5 466102.6
## Dec 1990       365823.6 280130.9 451516.3 234767.9 496879.3
## Jan 1991       411294.6 321621.8 500967.3 274151.9 548437.2
## Feb 1991       411555.6 318004.2 505106.9 268481.2 554629.9
## Mar 1991       386375.5 289043.2 483707.9 237518.6 535232.5
## Apr 1991       374847.5 273827.6 475867.3 220351.0 529344.0
## May 1991       373285.8 268667.9 477903.6 213286.6 533285.0
## Jun 1991       370398.1 262267.8 478528.5 205027.1 535769.2
## Jul 1991       368528.5 256967.4 480089.5 197910.6 539146.4
## Aug 1991       360795.2 245881.4 475709.0 185049.8 536540.7
## Sep 1991       354219.9 236027.8 472412.0 173460.7 534979.1
## Oct 1991       344372.0 222972.7 465771.4 158707.7 530036.3
## Nov 1991       339959.2 215420.3 464498.1 149493.4 530425.0
## Dec 1991       364664.5 237050.8 492278.3 169496.1 559833.0
## Jan 1992       410288.9 279662.0 540915.9 210512.2 610065.7
## Feb 1992       410683.0 277101.7 544264.4 206388.0 614978.0
## Mar 1992       385618.5 249139.1 522097.9 176891.2 594345.8
## Apr 1992       374190.6 234866.8 513514.5 161113.2 587268.1
## May 1992       372715.9 230598.9 514832.9 155366.8 590065.0
## Jun 1992       369903.7 225042.7 514764.7 148357.9 591449.5
## Jul 1992       368099.5 220541.4 515657.5 142428.9 593770.1
stlf(h02_train, h = 36)
##          Point Forecast     Lo 80     Hi 80     Lo 95     Hi 95
## Aug 2005      0.9280346 0.8754631 0.9806060 0.8476334 1.0084357
## Sep 2005      0.9985831 0.9427200 1.0544463 0.9131478 1.0840184
## Oct 2005      1.0623864 1.0034134 1.1213595 0.9721949 1.1525779
## Nov 2005      1.0668152 1.0048866 1.1287437 0.9721036 1.1615268
## Dec 2005      1.1502578 1.0855069 1.2150087 1.0512299 1.2492858
## Jan 2006      1.1170493 1.0495927 1.1845060 1.0138833 1.2202154
## Feb 2006      0.6294279 0.5593684 0.6994873 0.5222812 0.7365746
## Mar 2006      0.7108066 0.6382362 0.7833769 0.5998198 0.8217933
## Apr 2006      0.7146153 0.6396168 0.7896138 0.5999150 0.8293156
## May 2006      0.7741045 0.6967528 0.8514563 0.6558052 0.8924039
## Jun 2006      0.8280947 0.7484580 0.9077315 0.7063008 0.9498887
## Jul 2006      0.9443360 0.8624768 1.0261953 0.8191431 1.0695290
## Aug 2006      0.9640223 0.8799982 1.0480465 0.8355185 1.0925262
## Sep 2006      1.0345709 0.9484351 1.1207067 0.9028376 1.1663043
## Oct 2006      1.0983742 1.0101762 1.1865723 0.9634870 1.2332615
## Nov 2006      1.1028029 1.0125887 1.1930172 0.9648321 1.2407737
## Dec 2006      1.1862456 1.0940581 1.2784331 1.0452570 1.3272342
## Jan 2007      1.1530371 1.0589167 1.2471575 1.0090924 1.2969818
## Feb 2007      0.6654157 0.5694003 0.7614311 0.5185728 0.8122586
## Mar 2007      0.7467943 0.6489195 0.8446692 0.5971078 0.8964809
## Apr 2007      0.7506031 0.6509026 0.8503036 0.5981243 0.9030819
## May 2007      0.8100923 0.7085979 0.9115867 0.6548701 0.9653146
## Jun 2007      0.8640825 0.7608245 0.9673406 0.7061630 1.0220021
## Jul 2007      0.9803238 0.8753308 1.0853169 0.8197508 1.1408968
## Aug 2007      1.0000101 0.8933094 1.1067109 0.8368254 1.1631948
## Sep 2007      1.0705587 0.9621762 1.1789412 0.9048020 1.2363154
## Oct 2007      1.1343620 1.0243226 1.2444015 0.9660712 1.3026528
## Nov 2007      1.1387907 1.0271180 1.2504634 0.9680021 1.3095794
## Dec 2007      1.2222334 1.1089501 1.3355167 1.0489815 1.3954853
## Jan 2008      1.1890249 1.0741527 1.3038971 1.0133430 1.3647068
## Feb 2008      0.7014035 0.5849632 0.8178437 0.5233234 0.8794835
## Mar 2008      0.7827821 0.6647938 0.9007705 0.6023345 0.9632298
## Apr 2008      0.7865909 0.6670737 0.9061081 0.6038050 0.9693767
## May 2008      0.8460801 0.7250525 0.9671077 0.6609843 1.0311759
## Jun 2008      0.9000703 0.7775501 1.0225905 0.7126918 1.0874488
## Jul 2008      1.0163116 0.8923160 1.1403073 0.8266766 1.2059466
stlf(usmelec_train, h= 36)
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Jul 2010       400.5899 390.0856 411.0941 384.5250 416.6548
## Aug 2010       403.9813 392.5917 415.3709 386.5624 421.4002
## Sep 2010       347.7292 335.5157 359.9426 329.0503 366.4080
## Oct 2010       327.0296 314.0421 340.0171 307.1670 346.8922
## Nov 2010       315.6826 301.9625 329.4028 294.6995 336.6658
## Dec 2010       353.0671 338.6493 367.4849 331.0170 375.1172
## Jan 2011       361.6913 346.6061 376.7766 338.6204 384.7622
## Feb 2011       323.6885 307.9622 339.4149 299.6371 347.7399
## Mar 2011       327.5662 311.2220 343.9105 302.5699 352.5626
## Apr 2011       306.9165 289.9750 323.8579 281.0068 332.8262
## May 2011       335.0546 317.5346 352.5747 308.2600 361.8493
## Jun 2011       373.3414 355.2594 391.4233 345.6874 400.9953
## Jul 2011       406.0252 387.3967 424.6537 377.5353 434.5151
## Aug 2011       409.4166 390.2555 428.5778 380.1122 438.7211
## Sep 2011       353.1645 333.4835 372.8455 323.0651 383.2639
## Oct 2011       332.4649 312.2760 352.6539 301.5886 363.3413
## Nov 2011       321.1180 300.4320 341.8040 289.4815 352.7545
## Dec 2011       358.5024 337.3296 379.6753 326.1213 390.8835
## Jan 2012       367.1266 345.4764 388.7769 334.0155 400.2378
## Feb 2012       329.1239 307.0051 351.2426 295.2962 362.9516
## Mar 2012       333.0016 310.4227 355.5805 298.4701 367.5330
## Apr 2012       312.3518 289.3205 335.3831 277.1285 347.5751
## May 2012       340.4900 317.0137 363.9663 304.5861 376.3939
## Jun 2012       378.7767 354.8624 402.6910 342.2029 415.3505
## Jul 2012       411.4605 387.1147 435.8063 374.2268 448.6942
## Aug 2012       414.8520 390.0809 439.6230 376.9679 452.7360
## Sep 2012       358.5998 333.4094 383.7902 320.0744 397.1252
## Oct 2012       337.9003 312.2961 363.5044 298.7421 377.0584
## Nov 2012       326.5533 300.5408 352.5658 286.7706 366.3360
## Dec 2012       363.9378 337.5220 390.3535 323.5383 404.3372
## Jan 2013       372.5620 345.7478 399.3762 331.5532 413.5708
## Feb 2013       334.5592 307.3512 361.7672 292.9482 376.1702
## Mar 2013       338.4369 310.8396 366.0342 296.2305 380.6433
## Apr 2013       317.7871 289.8048 345.7695 274.9919 360.5824
## May 2013       345.9253 317.5620 374.2886 302.5474 389.3032
## Jun 2013       384.2120 355.4717 412.9524 340.2575 428.1666
ets(bicoal) %>% forecast() %>% autoplot

ets(chicken) %>% forecast() %>% autoplot

ets(dole) %>% forecast() %>% autoplot

ets(usdeaths) %>% forecast() %>% autoplot

ets(lynx) %>% forecast() %>% autoplot

ets(ibmclose) %>% forecast() %>% autoplot

ets(eggs) %>% forecast() %>% autoplot

15)

ets(usdeaths,model = "MAM") %>% forecast(h=1)
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Jan 1979       8233.107 7883.699 8582.515 7698.734 8767.481
hw(usdeaths, seasonal = 'multiplicative', h=1)
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Jan 1979        8217.64 7845.352 8589.928 7648.274 8787.005

When H is 2, a^2(h - 1) is equal to the forecast variance, sigma^2.

ets(pigs, model = "ANN") %>% forecast(h = 1)
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Sep 1995       98815.93 85604.96 112026.9 78611.49 119020.4
pigs_ets <- ets(pigs, model = "ANN") %>% forecast(h = 1) 

c(pigs_ets$lower[2], pigs_ets$upper[2])
## [1]  78611.49 119020.38
  1. Chapter 8

Yes, three series are white noise as the ACF are inside the critical value. The scale of the critical value varies due to each residuals’ variance.

  1. ACF plots clearly show that ibm close price is autocorrelated. Past data is highly correlated with current data, so differencing is necessary.
autoplot(ibmclose)

acf(ibmclose)

pacf(ibmclose)

  1. ACF plots clearly show that ibm close price is autocorrelated. Past data is highly correlated with current data, so differencing is necessary.
Box.test(usnetelec, lag = 10, type = "Ljung-Box")
## 
##  Box-Ljung test
## 
## data:  usnetelec
## X-squared = 329.22, df = 10, p-value < 2.2e-16
usnetelec %>% ur.kpss() -> ur_lec

usnetelec %>% diff() %>% ur.kpss() -> ur_diff_lec


Box.test(usgdp, lag =10, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  usgdp
## X-squared = 2078.3, df = 10, p-value < 2.2e-16
usgdp %>% ur.kpss()  -> ur_gdp

usgdp %>% diff() %>% ur.kpss() -> ur_diff_gdp


Box.test(mcopper, lag =10, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  mcopper
## X-squared = 3819, df = 10, p-value < 2.2e-16
mcopper %>% ur.kpss()  -> ur_cop

mcopper %>% diff() %>% ur.kpss() -> ur_diff_cop


Box.test(enplanements, lag =10, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  enplanements
## X-squared = 2122.7, df = 10, p-value < 2.2e-16
enplanements %>% ur.kpss()  -> ur_pl

enplanements %>% diff() %>% ur.kpss() -> ur_diff_pl


Box.test(visitors, lag =10, type = 'Ljung-Box')
## 
##  Box-Ljung test
## 
## data:  visitors
## X-squared = 1522.6, df = 10, p-value < 2.2e-16
visitors %>% ur.kpss()  -> ur_vis

visitors %>% diff() %>% ur.kpss() -> ur_diff_vis

summary(ur_lec)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 1.464 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur_diff_lec)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.1585 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur_gdp)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 4 lags. 
## 
## Value of test-statistic is: 4.6556 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur_diff_gdp)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 4 lags. 
## 
## Value of test-statistic is: 1.7909 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur_cop)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 6 lags. 
## 
## Value of test-statistic is: 5.01 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur_diff_cop)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 6 lags. 
## 
## Value of test-statistic is: 0.1843 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur_pl)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 5 lags. 
## 
## Value of test-statistic is: 4.4423 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur_diff_pl)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 5 lags. 
## 
## Value of test-statistic is: 0.0086 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur_vis)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 4 lags. 
## 
## Value of test-statistic is: 4.6025 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur_diff_vis)
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 4 lags. 
## 
## Value of test-statistic is: 0.0191 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739

  1. (1-B)^5*yhat

Data is stationary.

ggtsdisplay(myts)

myts %>% BoxCox(BoxCox.lambda(myts)) %>% diff(1) %>% diff(12) %>%
  ggtsdisplay()

myts %>% BoxCox(BoxCox.lambda(myts))%>%
  diff(1) %>% diff(12) %>%
  ur.kpss() %>% summary()
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 5 lags. 
## 
## Value of test-statistic is: 0.0138 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
arima_func <- function(phi){
  y = ts(numeric(100))
  e <- rnorm(100)
  for(i in 2:100)
  y[i] <- phi*y[i-1] + e[i]
  return(y)
}


dl <- list()
i <- 0
phi_range <- c(0.0000001, 0.0001, 0.1) * 6
for (phi in phi_range){
  i <- i + 1
  dl[[i]] <- arima_func(phi)
}
gd <- do.call(cbind, dl)
colnames(gd) <- phi_range
autoplot(gd, series = "") + ylab("")
## Warning: Ignoring unknown parameters: series

par(mfrow=c(1,3))
acf(gd[,1])
acf(gd[,2])
acf(gd[,3])

ma_func <- function(theta){
  y <- ts(numeric(100))
  e <- rnorm(100, sd=1)
  e[1] <- 0
  for(i in 2:100)
  y[i] <- theta*e[i-1] + e[i]
  return(y)}

dl2 <- list()
i <- 0
theta_range <- c(0.0000001, 0.0001, 0.1) * 6
for (theta in theta_range){
  i <- i + 1
  dl2[[i]] <- ma_func(theta)}
gd2 <- do.call(cbind, dl2)
colnames(gd2) <- theta_range
autoplot(gd2)

df1 <- ts(numeric(100))
e <- rnorm(100, sd=1)
for(i in 2:100)
df1[i] <- 0.6*df1[i-1] + 0.6*e[i-1] + e[i]
autoplot(df1)

df2 <- ts(numeric(100))
e <- rnorm(100, sd=1)
for(i in 3:100)
df2[i] <- -0.8*df2[i-1] + 0.3*df2[i-2] + e[i]
autoplot(df2)

  1. No, constant from the the ARIMA model without trend is identical. (1−ϕ1B)(1−B)2yhat=c+(1+θ1B)εt
ggtsdisplay(wmurders)

wmurders %>% diff(1) %>% diff(1) %>% ggtsdisplay()

wmurders %>% diff(1) %>% diff(1) %>% ur.kpss() %>% summary()
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.0458 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
Arima(wmurders, order=c(1,2,1))
## Series: wmurders 
## ARIMA(1,2,1) 
## 
## Coefficients:
##           ar1      ma1
##       -0.2434  -0.8261
## s.e.   0.1553   0.1143
## 
## sigma^2 estimated as 0.04632:  log likelihood=6.44
## AIC=-6.88   AICc=-6.39   BIC=-0.97
Arima(wmurders, order=c(1,2,1)) %>% checkresiduals()

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(1,2,1)
## Q* = 12.419, df = 8, p-value = 0.1335
## 
## Model df: 2.   Total lags used: 10
Arima(wmurders, order=c(1,2,1)) %>% forecast(h = 3) %>% autoplot()

auto.arima(wmurders, seasonal=F, stepwise=F, approximation=F)
## Series: wmurders 
## ARIMA(0,2,3) 
## 
## Coefficients:
##           ma1     ma2      ma3
##       -1.0154  0.4324  -0.3217
## s.e.   0.1282  0.2278   0.1737
## 
## sigma^2 estimated as 0.04475:  log likelihood=7.77
## AIC=-7.54   AICc=-6.7   BIC=0.35
auto.arima(wmurders, seasonal=F, stepwise=F, approximation=F) %>%
  forecast(h = 3) %>% autoplot()

autoplot(austa)

forecast(auto.arima(austa), h = 10) %>% autoplot()

auto.arima(austa) %>% checkresiduals()

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(0,1,1) with drift
## Q* = 2.297, df = 5, p-value = 0.8067
## 
## Model df: 2.   Total lags used: 7
forecast(Arima(austa, order = c(0, 1, 1)), h = 10) %>% autoplot()

forecast(Arima(austa, order = c(0, 1, 0)), h = 10) %>% autoplot()

farima_213 <- forecast(Arima(austa, order = c(2, 1, 3)
                             , include.drift = TRUE),h = 10)

autoplot(farima_213)

dausta <- farima_213$model$coef[6]
ndausta <- farima_213$mean - dausta*seq_len(10)
autoplot(farima_213) +
  autolayer(ndausta)

forecast(Arima(austa, order = c(0, 0, 1), include.constant = TRUE),h = 10) %>%
  autoplot()

forecast(Arima(austa, order = c(0, 0, 0), include.constant = TRUE),h = 10) %>%
  autoplot()

forecast(Arima(austa, order = c(0, 2, 1)),h = 10) %>%
  autoplot()

  1. Auto.arima model has the best performance.
autoplot(usgdp)

autoplot(BoxCox(usgdp, BoxCox.lambda(usgdp)))

auto.arima(usgdp,lambda = BoxCox.lambda(usgdp)) %>% autoplot()

ndiffs(BoxCox(usgdp, BoxCox.lambda(usgdp)))
## [1] 1
ggtsdisplay(diff(BoxCox(usgdp, BoxCox.lambda(usgdp))))

Arima(usgdp, lambda = BoxCox.lambda(usgdp), order = c(1, 1, 0))
## Series: usgdp 
## ARIMA(1,1,0) 
## Box Cox transformation: lambda= 0.366352 
## 
## Coefficients:
##          ar1
##       0.6326
## s.e.  0.0504
## 
## sigma^2 estimated as 0.04384:  log likelihood=34.39
## AIC=-64.78   AICc=-64.73   BIC=-57.85
Arima(usgdp, lambda = BoxCox.lambda(usgdp), order = c(1, 1, 0))$fitted %>%
  autoplot(series = "fit") +
  autolayer(usgdp, series = "Original")

Arima(usgdp, lambda = BoxCox.lambda(usgdp), order = c(1, 1, 0),include.drift = TRUE)
## Series: usgdp 
## ARIMA(1,1,0) with drift 
## Box Cox transformation: lambda= 0.366352 
## 
## Coefficients:
##          ar1   drift
##       0.3180  0.1831
## s.e.  0.0619  0.0179
## 
## sigma^2 estimated as 0.03555:  log likelihood=59.83
## AIC=-113.66   AICc=-113.56   BIC=-103.27
Arima(usgdp, lambda = BoxCox.lambda(usgdp), order = c(1, 1, 0),include.drift = TRUE)$fitted %>%  autoplot(series = "fit") +
  autolayer(usgdp, series = "Original")

auto.arima(usgdp,lambda = BoxCox.lambda(usgdp)) %>% accuracy()
##                    ME    RMSE      MAE         MPE      MAPE      MASE
## Training set 1.195275 39.2224 29.29521 -0.01363259 0.6863491 0.1655687
##                     ACF1
## Training set -0.03824844
Arima(usgdp, lambda = BoxCox.lambda(usgdp), order = c(1, 1, 0)) %>%
  accuracy()
##                    ME     RMSE      MAE       MPE      MAPE     MASE       ACF1
## Training set 15.45449 45.49569 35.08393 0.3101283 0.7815664 0.198285 -0.3381619
Arima(usgdp, lambda = BoxCox.lambda(usgdp), order = c(1, 1, 0),include.drift = TRUE) %>% accuracy()
##                    ME     RMSE     MAE         MPE      MAPE      MASE
## Training set 1.315796 39.90012 29.5802 -0.01678591 0.6834509 0.1671794
##                     ACF1
## Training set -0.08544569
auto.arima(usgdp,lambda = BoxCox.lambda(usgdp)) %>% checkresiduals()

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(2,1,0) with drift
## Q* = 6.5772, df = 5, p-value = 0.254
## 
## Model df: 3.   Total lags used: 8
auto.arima(usgdp,lambda = BoxCox.lambda(usgdp)) %>% forecast() %>%
  autoplot()

forecast(ets(usgdp)) %>% autoplot()

autoplot(austourists)

ggAcf(austourists)

ggPacf(austourists)

ggtsdisplay(diff(austourists, lag = 4))

ggtsdisplay(diff(diff(austourists, lag = 4)))

forecast(auto.arima(austourists)) %>% autoplot()

forecast(auto.arima(austourists)) %>% accuracy()
##                      ME     RMSE      MAE        MPE     MAPE      MASE
## Training set 0.02200144 2.149384 1.620917 -0.7072593 4.388288 0.5378929
##                     ACF1
## Training set -0.06393238
forecast(auto.arima(austourists)) %>% checkresiduals()

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(1,0,0)(1,1,0)[4] with drift
## Q* = 4.0937, df = 5, p-value = 0.536
## 
## Model df: 3.   Total lags used: 8
forecast(auto.arima(austourists))$model
## Series: austourists 
## ARIMA(1,0,0)(1,1,0)[4] with drift 
## 
## Coefficients:
##          ar1     sar1   drift
##       0.4705  -0.5305  0.5489
## s.e.  0.1154   0.1122  0.0864
## 
## sigma^2 estimated as 5.15:  log likelihood=-142.48
## AIC=292.97   AICc=293.65   BIC=301.6
ma(usmelec, order = 12, centre = TRUE) %>%
autoplot(series = "2X12")+
autolayer(usmelec, series = "Original")

BoxCox.lambda(usmelec)
## [1] -0.5738331
ndiffs(usmelec)
## [1] 1
nsdiffs(usmelec)
## [1] 1
ggtsdisplay(diff(BoxCox(usmelec, BoxCox.lambda(usmelec)),lag = 12))

ggtsdisplay(diff(diff(BoxCox(usmelec, BoxCox.lambda(usmelec)),lag = 12)))

Arima(usmelec,lambda = BoxCox.lambda(usmelec)
      ,order = c(0, 1, 2)
      ,seasonal = c(0, 1, 1))$aic
## [1] -5081.506
Arima(usmelec,lambda =BoxCox.lambda(usmelec)
      ,order = c(0, 1, 3)
      ,seasonal = c(0, 1, 1))$aic
## [1] -5080.441
Arima(usmelec,lambda = BoxCox.lambda(usmelec)
      ,order = c(0, 1, 2)
      ,seasonal = c(0, 1, 1)) %>% checkresiduals()

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(0,1,2)(0,1,1)[12]
## Q* = 32.525, df = 21, p-value = 0.05176
## 
## Model df: 3.   Total lags used: 24
auto.arima(usmelec,lambda = BoxCox.lambda(usmelec)) %>% checkresiduals()

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(1,1,3)(2,1,1)[12]
## Q* = 25.995, df = 17, p-value = 0.07454
## 
## Model df: 7.   Total lags used: 24
Arima(usmelec, order=c(2,1,3), seasonal=c(2,1,3)
      , lambda=BoxCox.lambda(usmelec)) %>%
  forecast(h = 12*15) %>% autoplot()

tsdisplay(BoxCox(mcopper, lambda = BoxCox.lambda(mcopper)))

auto.arima(mcopper, trace = TRUE, ic ="aic", lambda = BoxCox.lambda(mcopper))
## 
##  Fitting models using approximations to speed things up...
## 
##  ARIMA(2,1,2)(1,0,1)[12] with drift         : -66.32021
##  ARIMA(0,1,0)            with drift         : -13.82248
##  ARIMA(1,1,0)(1,0,0)[12] with drift         : -62.06849
##  ARIMA(0,1,1)(0,0,1)[12] with drift         : -82.25982
##  ARIMA(0,1,0)                               : -12.82906
##  ARIMA(0,1,1)            with drift         : -83.13952
##  ARIMA(0,1,1)(1,0,0)[12] with drift         : -73.91207
##  ARIMA(0,1,1)(1,0,1)[12] with drift         : -71.92506
##  ARIMA(1,1,1)            with drift         : -80.30584
##  ARIMA(0,1,2)            with drift         : -81.16609
##  ARIMA(1,1,0)            with drift         : -71.41152
##  ARIMA(1,1,2)            with drift         : -78.27629
##  ARIMA(0,1,1)                               : -83.33279
##  ARIMA(0,1,1)(1,0,0)[12]                    : -74.05872
##  ARIMA(0,1,1)(0,0,1)[12]                    : -82.603
##  ARIMA(0,1,1)(1,0,1)[12]                    : -72.07316
##  ARIMA(1,1,1)                               : -80.55931
##  ARIMA(0,1,2)                               : -81.33929
##  ARIMA(1,1,0)                               : -71.94456
##  ARIMA(1,1,2)                               : -80.62448
## 
##  Now re-fitting the best model(s) without approximations...
## 
##  ARIMA(0,1,1)                               : -86.0969
## 
##  Best model: ARIMA(0,1,1)
## Series: mcopper 
## ARIMA(0,1,1) 
## Box Cox transformation: lambda= 0.1919047 
## 
## Coefficients:
##          ma1
##       0.3720
## s.e.  0.0388
## 
## sigma^2 estimated as 0.04997:  log likelihood=45.05
## AIC=-86.1   AICc=-86.08   BIC=-77.43
auto.arima(mcopper, trace = TRUE, ic ="aic", lambda = BoxCox.lambda(mcopper)) %>%
  checkresiduals()
## 
##  Fitting models using approximations to speed things up...
## 
##  ARIMA(2,1,2)(1,0,1)[12] with drift         : -66.32021
##  ARIMA(0,1,0)            with drift         : -13.82248
##  ARIMA(1,1,0)(1,0,0)[12] with drift         : -62.06849
##  ARIMA(0,1,1)(0,0,1)[12] with drift         : -82.25982
##  ARIMA(0,1,0)                               : -12.82906
##  ARIMA(0,1,1)            with drift         : -83.13952
##  ARIMA(0,1,1)(1,0,0)[12] with drift         : -73.91207
##  ARIMA(0,1,1)(1,0,1)[12] with drift         : -71.92506
##  ARIMA(1,1,1)            with drift         : -80.30584
##  ARIMA(0,1,2)            with drift         : -81.16609
##  ARIMA(1,1,0)            with drift         : -71.41152
##  ARIMA(1,1,2)            with drift         : -78.27629
##  ARIMA(0,1,1)                               : -83.33279
##  ARIMA(0,1,1)(1,0,0)[12]                    : -74.05872
##  ARIMA(0,1,1)(0,0,1)[12]                    : -82.603
##  ARIMA(0,1,1)(1,0,1)[12]                    : -72.07316
##  ARIMA(1,1,1)                               : -80.55931
##  ARIMA(0,1,2)                               : -81.33929
##  ARIMA(1,1,0)                               : -71.94456
##  ARIMA(1,1,2)                               : -80.62448
## 
##  Now re-fitting the best model(s) without approximations...
## 
##  ARIMA(0,1,1)                               : -86.0969
## 
##  Best model: ARIMA(0,1,1)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(0,1,1)
## Q* = 22.913, df = 23, p-value = 0.4659
## 
## Model df: 1.   Total lags used: 24
auto.arima(mcopper, trace = TRUE, ic ="aic", lambda = BoxCox.lambda(mcopper)) %>%
  accuracy()
## 
##  Fitting models using approximations to speed things up...
## 
##  ARIMA(2,1,2)(1,0,1)[12] with drift         : -66.32021
##  ARIMA(0,1,0)            with drift         : -13.82248
##  ARIMA(1,1,0)(1,0,0)[12] with drift         : -62.06849
##  ARIMA(0,1,1)(0,0,1)[12] with drift         : -82.25982
##  ARIMA(0,1,0)                               : -12.82906
##  ARIMA(0,1,1)            with drift         : -83.13952
##  ARIMA(0,1,1)(1,0,0)[12] with drift         : -73.91207
##  ARIMA(0,1,1)(1,0,1)[12] with drift         : -71.92506
##  ARIMA(1,1,1)            with drift         : -80.30584
##  ARIMA(0,1,2)            with drift         : -81.16609
##  ARIMA(1,1,0)            with drift         : -71.41152
##  ARIMA(1,1,2)            with drift         : -78.27629
##  ARIMA(0,1,1)                               : -83.33279
##  ARIMA(0,1,1)(1,0,0)[12]                    : -74.05872
##  ARIMA(0,1,1)(0,0,1)[12]                    : -82.603
##  ARIMA(0,1,1)(1,0,1)[12]                    : -72.07316
##  ARIMA(1,1,1)                               : -80.55931
##  ARIMA(0,1,2)                               : -81.33929
##  ARIMA(1,1,0)                               : -71.94456
##  ARIMA(1,1,2)                               : -80.62448
## 
##  Now re-fitting the best model(s) without approximations...
## 
##  ARIMA(0,1,1)                               : -86.0969
## 
##  Best model: ARIMA(0,1,1)
##                    ME     RMSE      MAE      MPE     MAPE      MASE        ACF1
## Training set 3.480533 77.27254 44.92858 0.166202 4.303677 0.2021433 -0.08442198
ets(mcopper) %>% checkresiduals()

## 
##  Ljung-Box test
## 
## data:  Residuals from ETS(M,Ad,N)
## Q* = 77.585, df = 19, p-value = 4.832e-09
## 
## Model df: 5.   Total lags used: 24
ets(mcopper) %>% accuracy()
##                    ME     RMSE      MAE       MPE     MAPE      MASE      ACF1
## Training set 2.691483 78.87699 46.76047 0.1883633 4.503026 0.2103854 0.1052005
autoplot(auscafe)

BoxCox.lambda(auscafe)
## [1] 0.109056
nsdiffs(auscafe)
## [1] 1
ndiffs(auscafe)
## [1] 1
kpss.test(diff(auscafe, lag = 4))
## Warning in kpss.test(diff(auscafe, lag = 4)): p-value greater than printed p-
## value
## 
##  KPSS Test for Level Stationarity
## 
## data:  diff(auscafe, lag = 4)
## KPSS Level = 0.29812, Truncation lag parameter = 5, p-value = 0.1
ggtsdisplay(diff(BoxCox(auscafe, BoxCox.lambda(auscafe)), lag = 4))

ggtsdisplay(diff(diff(BoxCox(auscafe, BoxCox.lambda(auscafe)), lag = 4)))

auto.arima(auscafe, lambda = BoxCox.lambda(auscafe))
## Series: auscafe 
## ARIMA(1,0,1)(2,1,1)[12] with drift 
## Box Cox transformation: lambda= 0.109056 
## 
## Coefficients:
##          ar1      ma1    sar1     sar2     sma1   drift
##       0.9718  -0.3190  0.1270  -0.0527  -0.8423  0.0056
## s.e.  0.0131   0.0478  0.0649   0.0585   0.0431  0.0004
## 
## sigma^2 estimated as 0.0005754:  log likelihood=952.96
## AIC=-1891.92   AICc=-1891.65   BIC=-1863.74
auto.arima(auscafe, lambda = BoxCox.lambda(auscafe)) %>% checkresiduals()

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(1,0,1)(2,1,1)[12] with drift
## Q* = 56.069, df = 18, p-value = 8.692e-06
## 
## Model df: 6.   Total lags used: 24
auto.arima(auscafe, lambda = BoxCox.lambda(auscafe)) %>% forecast(h = 8) %>%
  autoplot()

forecast(ets(auscafe), h = 8) %>% autoplot()

stlf(auscafe, lambda = BoxCox.lambda(auscafe),
  s.window = 5, robust = TRUE, method = "arima",
  h = 8) %>% autoplot()

forecast(ets(auscafe), h = 8) %>% autoplot()

forecast(auto.arima(auscafe), h = 8) %>% autoplot()

forecast(auto.arima(myts), h = 36) %>% autoplot()

snaive(myts, h = 36) %>% autoplot()

forecast(ets(myts, lambda=BoxCox.lambda(myts)), h = 36) %>%
  autoplot()

autoplot(sheep)

ggtsdisplay(diff(sheep))

forecast(Arima(sheep, order = c(3, 1, 0)),h = 3) $ mean
## Time Series:
## Start = 1940 
## End = 1942 
## Frequency = 1 
## [1] 1777.996 1718.869 1695.985
autoplot(bicoal)

ggAcf(bicoal, lag.max = 36)

ggPacf(bicoal, lag.max = 36)

forecast(ar(bicoal, 4), h = 3) $ mean
## Time Series:
## Start = 1969 
## End = 1971 
## Frequency = 1 
## [1] 526.2057 514.0658 500.0111
  1. I prefer ETS model.
y <- ibmclose



autoplot(y)

checkresiduals(y)
## Warning in modeldf.default(object): Could not find appropriate degrees of
## freedom for this model.

auto.arima(y) %>% forecast(h = 4 * 12) %>% autoplot()

ets(y) %>% forecast(h = 4*12) %>% autoplot()

ets(y) %>% checkresiduals()

## 
##  Ljung-Box test
## 
## data:  Residuals from ETS(A,N,N)
## Q* = 14.132, df = 8, p-value = 0.07839
## 
## Model df: 2.   Total lags used: 10