Homework5
Justin Lynch
7/29/2020
This series contains the daily pageviews for one year for the Hyndsight blog, beginning 30 April 2014. It is from the fpp2 package.
Setup
library(fGarch)## Loading required package: timeDate## Loading required package: timeSeries## Loading required package: fBasicslibrary(forecast)## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoolibrary(fpp2)## Loading required package: ggplot2## Loading required package: fma## Loading required package: expsmoothlibrary(tseries)Data
autoplot(hyndsight, xlab="Weeks", main = "Pageviews")
decomp = stl(hyndsight,s.window='periodic')
autoplot(decomp)
trainData = head(hyndsight, n = 294)
testData = tail(hyndsight, n = 71)Holt Winters Model
HWmod = HoltWinters(trainData)
HWfc = forecast(HWmod, h = 71)
autoplot(HWfc)
checkresiduals(HWfc)## Warning in modeldf.default(object): Could not find appropriate degrees of
## freedom for this model.
Arima
adf.test(trainData)##
## Augmented Dickey-Fuller Test
##
## data: trainData
## Dickey-Fuller = -3.1753, Lag order = 6, p-value = 0.0924
## alternative hypothesis: stationaryarimaMod = auto.arima(trainData)
arimaFC = forecast(arimaMod, h=71)
autoplot(arimaFC)
checkresiduals(arimaFC)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(1,0,0)(2,1,1)[7]
## Q* = 19.671, df = 10, p-value = 0.03252
##
## Model df: 4. Total lags used: 14GARCH model
pacf(trainData)
acf(trainData)
garchMod = garchFit(trainData, formula = ~ garch(1,1), cond.dist = 'QMLE')##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(0, 0)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 0 0
## Max ARMA Order: 0
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 1
## Conditional Dist: QMLE
## h.start: 2
## llh.start: 1
## Length of Series: 294
## Recursion Init: mci
## Series Scale: 401.5943
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -29.95077825 29.95078 2.995078 TRUE
## omega 0.00000100 100.00000 0.100000 TRUE
## alpha1 0.00000001 1.00000 0.100000 TRUE
## gamma1 -0.99999999 1.00000 0.100000 FALSE
## beta1 0.00000001 1.00000 0.800000 TRUE
## delta 0.00000000 2.00000 2.000000 FALSE
## skew 0.10000000 10.00000 1.000000 FALSE
## shape 1.00000000 10.00000 4.000000 FALSE
## Index List of Parameters to be Optimized:
## mu omega alpha1 beta1
## 1 2 3 5
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 412.25139: 2.99508 0.100000 0.100000 0.800000
## 1: 411.98846: 2.96187 0.104621 0.102008 0.804406
## 2: 411.84934: 2.95540 0.101297 0.0989035 0.801188
## 3: 411.83051: 2.94110 0.103588 0.100145 0.803666
## 4: 411.80214: 2.93757 0.101094 0.0979813 0.801493
## 5: 411.79270: 2.92769 0.100886 0.0987226 0.803850
## 6: 411.78526: 2.93376 0.0981615 0.0990060 0.806176
## 7: 411.78513: 2.93370 0.0980369 0.0988736 0.806173
## 8: 411.78502: 2.93364 0.0980545 0.0988533 0.806353
## 9: 411.78480: 2.93365 0.0978493 0.0986067 0.806529
## 10: 411.78283: 2.93427 0.0966428 0.0970869 0.809468
## 11: 411.78184: 2.93280 0.0938219 0.0975861 0.811467
## 12: 411.78161: 2.93300 0.0939818 0.0976826 0.811656
## 13: 411.78145: 2.93328 0.0938752 0.0974735 0.811762
## 14: 411.78129: 2.93341 0.0936296 0.0974088 0.812245
## 15: 411.78112: 2.93317 0.0928343 0.0973111 0.812987
## 16: 411.78089: 2.93424 0.0933771 0.0964447 0.813150
## 17: 411.78088: 2.93421 0.0934234 0.0964939 0.813236
## 18: 411.78085: 2.93420 0.0933375 0.0964438 0.813283
## 19: 411.78082: 2.93409 0.0932437 0.0964438 0.813479
## 20: 411.78054: 2.93434 0.0917358 0.0957139 0.815615
## 21: 411.78054: 2.93433 0.0915393 0.0958682 0.815680
## 22: 411.78054: 2.93415 0.0915536 0.0958368 0.815680
## 23: 411.78054: 2.93425 0.0915792 0.0958288 0.815667
## 24: 411.78054: 2.93425 0.0915735 0.0958326 0.815670
##
## Final Estimate of the Negative LLH:
## LLH: 2174.441 norm LLH: 7.396056
## mu omega alpha1 beta1
## 1.178377e+03 1.476878e+04 9.583263e-02 8.156697e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu omega alpha1 beta1
## mu -1.398228e-03 9.706655e-07 -1.562787e-01 1.002592e-01
## omega 9.706655e-07 -1.931015e-07 -2.401451e-02 -2.891337e-02
## alpha1 -1.562787e-01 -2.401451e-02 -3.741445e+03 -3.818930e+03
## beta1 1.002592e-01 -2.891337e-02 -3.818930e+03 -4.506027e+03
## attr(,"time")
## Time difference of 0.01150608 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.04597807 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.checkresiduals(garchMod@residuals)## Warning in modeldf.default(object): Could not find appropriate degrees of
## freedom for this model.
garchFC = predict(garchMod, n.ahead=71, plot = TRUE)
summary(garchFC)## meanForecast meanError standardDeviation lowerInterval
## Min. :1178 Min. :404.4 Min. :404.4 Min. :481.0
## 1st Qu.:1178 1st Qu.:407.7 1st Qu.:407.7 1st Qu.:481.0
## Median :1178 Median :408.4 Median :408.4 Median :481.2
## Mean :1178 Mean :407.9 Mean :407.9 Mean :482.1
## 3rd Qu.:1178 3rd Qu.:408.5 3rd Qu.:408.5 3rd Qu.:482.4
## Max. :1178 Max. :408.5 Max. :408.5 Max. :488.0
## upperInterval
## Min. :1972
## 1st Qu.:1979
## Median :1980
## Mean :1979
## 3rd Qu.:1980
## Max. :1980Comparison
accuracy(garchFC$meanForecast, testData)## ME RMSE MAE MPE MAPE ACF1 Theil's U
## Test set 452.1019 611.9964 514.5392 22.18546 28.77365 0.5247578 1.210584accuracy(HWfc, testData)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -3.039342 244.0232 175.3343 -3.422528 15.45622 0.794019 0.2465167
## Test set 253.524421 334.2907 262.6859 14.157454 15.03539 1.189599 0.4400235
## Theil's U
## Training set NA
## Test set 0.6745651accuracy(arimaFC, testData)## ME RMSE MAE MPE MAPE MASE
## Training set 20.09961 221.2017 150.8014 -1.247853 12.74527 0.682919
## Test set 251.77245 338.0943 263.0560 13.756835 14.79158 1.191275
## ACF1 Theil's U
## Training set -0.03521556 NA
## Test set 0.46563461 0.675819We can see that the arima and holt winters model outperforn the garch model in this instance. I'm not sure why the garch was returning flat forecasts, but that is likely the reason why the forecasts performed so poorly against the test set. I will note that the holt winters and arima models also were quite biased and inaccurate. We also see in the garch fit that the model still contains lots of autocorrelation.