Assignment Homework Time Series
Jacob Eye
12/7/2021
library(forecast)## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoolibrary(fpp)## Loading required package: fma## Loading required package: expsmooth## Loading required package: lmtest## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric## Loading required package: tserieslibrary(readr)
library(caret)## Loading required package: ggplot2## Loading required package: latticelibrary(neuralnet)goldprice <- read.csv("Downloads/1979-2021.csv")summary(goldprice)## Date USD Europe.EUR. Japan.JPY.
## Length:511 Min. : 233.7 Min. : 144.8 Min. : 27938
## Class :character 1st Qu.: 354.2 1st Qu.: 306.9 1st Qu.: 43096
## Mode :character Median : 415.1 Median : 364.1 Median : 70456
## Mean : 693.7 Mean : 572.9 Mean : 82948
## 3rd Qu.:1166.7 3rd Qu.: 929.2 3rd Qu.:124847
## Max. :1964.9 Max. :1661.7 Max. :207845
##
## United.Kingdom.GBP. Canada.CAD. Switzerland.CHF. India.INR.
## Min. : 116.2 Min. : 267.1 Min. : 379.3 Min. : 1841
## 1st Qu.: 218.4 1st Qu.: 454.3 1st Qu.: 501.6 1st Qu.: 6274
## Median : 256.4 Median : 532.3 Median : 698.2 Median : 13027
## Mean : 450.1 Mean : 843.5 Mean : 826.1 Mean : 33124
## 3rd Qu.: 742.1 3rd Qu.:1327.0 3rd Qu.:1127.0 3rd Qu.: 61885
## Max. :1497.1 Max. :2632.0 Max. :1787.5 Max. :146999
##
## China.CNY. Turkey.TRY. Saudi.Arabia.SAR. Indonesia.IDR.
## Min. : 821.1 Min. : 0.100 Min. : 745.2 Min. : 139739
## 1st Qu.: 2171.3 1st Qu.: 2.475 1st Qu.:1311.3 1st Qu.: 697974
## Median : 3287.8 Median : 446.050 Median :1502.6 Median : 2536803
## Mean : 4883.3 Mean : 1739.876 Mean :2580.5 Mean : 6207427
## 3rd Qu.: 7991.1 3rd Qu.: 2595.025 3rd Qu.:4375.9 3rd Qu.:12021637
## Max. :13714.0 Max. :16105.400 Max. :7369.2 Max. :28687539
## NA's :72 NA's :47
## United.Arab.Emirates.AED. Thailand.THB. Vietnam.VND. Egypt.EGP.
## Min. : 856.1 Min. : 4511 Min. : 1571728 Min. : 870.1
## 1st Qu.:1297.5 1st Qu.: 9612 1st Qu.: 4090357 1st Qu.: 1176.1
## Median :1528.9 Median :12140 Median : 7502084 Median : 2578.9
## Mean :2549.2 Mean :21660 Mean :15126333 Mean : 6759.6
## 3rd Qu.:4285.0 3rd Qu.:38774 3rd Qu.:27547811 3rd Qu.: 9149.5
## Max. :7217.4 Max. :61270 Max. :45539503 Max. :31379.5
## NA's :131 NA's :119
## South.Korean.KRW. Australia.AUD. South.Africa.ZAR.
## Min. : 108236 Min. : 197.2 Min. : 194.5
## 1st Qu.: 301252 1st Qu.: 474.6 1st Qu.: 972.5
## Median : 374980 Median : 554.1 Median : 1864.7
## Mean : 717692 Mean : 869.9 Mean : 5950.8
## 3rd Qu.:1304939 3rd Qu.:1372.6 3rd Qu.: 9549.4
## Max. :2340982 Max. :2739.9 Max. :33459.8
## gold.ts <- ts(goldprice$USD, start = 1979, end = 2021, frequency = 12)
plot(gold.ts, ylab="Monthly Gold Price", main="Monthly Gold Price, USA, 1979-1921")
gold.train=ts(goldprice$USD, start = 1979, end = 2000, frequency = 12)
plot(gold.train, ylab="Monthly Gold Price", main="Monthly Gold Price, USA, 1979-2000")
gold.test=ts(goldprice$USD, start = 2000, end = 2021, frequency = 12)
plot(gold.test, ylab="Monthly Gold Price", main="Monthly Gold Price, USA, 2000-2021")
plot(decompose(gold.train))
acf(gold.train)
pacf(gold.train)
Model #1
fit1<-auto.arima(gold.train, seasonal = TRUE)
fit1## Series: gold.train
## ARIMA(0,1,1)(2,0,1)[12]
##
## Coefficients:
## ma1 sar1 sar2 sma1
## 0.0446 -0.6353 -0.1641 0.4719
## s.e. 0.0719 0.4704 0.1011 0.4676
##
## sigma^2 estimated as 599.2: log likelihood=-1161.69
## AIC=2333.39 AICc=2333.63 BIC=2351.04f.arima=forecast(fit1, h=58)
plot(f.arima, xlab="Time" ,ylab="Monthly Gold Price")
accuracy(f.arima, gold.test)## ME RMSE MAE MPE MAPE MASE
## Training set 0.1997619 24.23525 14.92405 -0.1036188 3.74794 0.2611221
## Test set 153.7410766 187.88654 158.50275 30.8790983 32.78669 2.7732803
## ACF1 Theil's U
## Training set -0.002558343 NA
## Test set 0.886504288 3.769971hist(residuals(f.arima))
plot(residuals(f.arima))
Model #2
fit2=nnetar(gold.train)
summary(fit2)## Length Class Mode
## x 253 ts numeric
## m 1 -none- numeric
## p 1 -none- numeric
## P 1 -none- numeric
## scalex 2 -none- list
## size 1 -none- numeric
## subset 253 -none- numeric
## model 20 nnetarmodels list
## nnetargs 0 -none- list
## fitted 253 ts numeric
## residuals 253 ts numeric
## lags 13 -none- numeric
## series 1 -none- character
## method 1 -none- character
## call 2 -none- callf.nnetar=forecast(fit2, h=58)
plot(f.nnetar,xlab="Time" ,ylab="Monthly Gold Price")
accuracy(f.nnetar,gold.test)## ME RMSE MAE MPE MAPE MASE
## Training set -0.0340266 8.386147 6.127924 -0.08960853 1.692649 0.1072187
## Test set 160.6827786 192.352752 164.663669 32.66580506 34.275485 2.8810762
## ACF1 Theil's U
## Training set -0.01188528 NA
## Test set 0.87761528 3.904474hist(residuals(f.nnetar))
plot(residuals(f.nnetar))
Model #3
lambda <- BoxCox.lambda(gold.train)
lambda## [1] 1.114233fit3<-stlf(gold.train, method="ets", lambda = lambda, h=58)
summary(fit3)##
## Forecast method: STL + ETS(M,Ad,N)
##
## Model Information:
## ETS(M,Ad,N)
##
## Call:
## ets(y = na.interp(x), model = etsmodel, allow.multiplicative.trend = allow.multiplicative.trend)
##
## Smoothing parameters:
## alpha = 0.9999
## beta = 0.0083
## phi = 0.9583
##
## Initial states:
## l = 309.5948
## b = 50.7586
##
## sigma: 0.0563
##
## AIC AICc BIC
## 3237.109 3237.451 3258.309
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -1.650934 22.30299 14.02948 -0.5156365 3.51985 0.24547 0.02375412
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Feb 2000 284.8397 266.3693 303.1742 256.532766 312.8283
## Mar 2000 285.8344 259.5659 311.8295 245.538115 325.4896
## Apr 2000 286.8515 254.5117 318.7790 237.203622 335.5312
## May 2000 285.1887 247.6185 322.2001 227.468826 341.5941
## Jun 2000 283.0770 240.8156 324.6267 218.104406 346.3729
## Jul 2000 282.5521 236.0140 328.2262 210.958655 352.1069
## Aug 2000 277.5108 226.8832 327.0983 199.566457 352.9956
## Sep 2000 289.2072 235.1140 342.1613 205.910277 369.8088
## Oct 2000 285.9549 228.2327 342.3659 197.010024 371.7913
## Nov 2000 283.3053 222.1133 343.0108 188.951016 374.1281
## Dec 2000 283.0429 218.5740 345.8607 183.578956 378.5772
## Jan 2001 281.6730 214.0042 347.5144 177.206941 381.7807
## Feb 2001 283.2810 212.6060 351.9744 174.121972 387.7060
## Mar 2001 284.3413 210.7331 355.8084 170.595285 392.9631
## Apr 2001 285.4213 208.9612 359.5800 167.210542 398.1148
## May 2001 283.8173 204.4764 360.6664 161.072686 400.5736
## Jun 2001 281.7617 199.5790 361.2515 154.531508 402.5030
## Jul 2001 281.2914 196.3832 363.3202 149.761770 405.8665
## Aug 2001 276.3003 188.5178 360.9491 140.183957 404.8182
## Sep 2001 288.0527 198.1839 374.7712 148.751773 419.7262
## Oct 2001 284.8471 192.2810 374.0329 141.246155 420.2361
## Nov 2001 282.2426 187.0368 373.8390 134.425104 421.2615
## Dec 2001 282.0244 184.3265 375.9178 130.242431 424.5075
## Jan 2002 280.6965 180.4841 376.8860 124.890357 426.6382
## Feb 2002 282.3459 179.8062 380.6975 122.849396 431.5528
## Mar 2002 283.4456 178.5847 383.9428 120.255385 435.8907
## Apr 2002 284.5633 177.4147 387.1727 117.728201 440.1958
## May 2002 282.9946 173.4356 387.7797 112.262570 441.9000
## Jun 2002 280.9727 168.9965 387.9237 106.308960 443.1336
## Jul 2002 280.5352 166.2564 389.5696 102.143961 445.8323
## Aug 2002 275.5741 158.7241 386.8375 92.892338 444.2075
## Sep 2002 287.3601 169.0319 400.1979 102.575432 458.4117
## Oct 2002 284.1825 163.4501 399.1313 95.416585 458.3995
## Nov 2002 281.6052 158.5152 398.6240 88.923582 458.9270
## Dec 2002 281.4135 156.1409 400.3902 85.154411 461.6807
## Jan 2003 280.1107 152.5934 401.0697 80.120529 463.3544
## Feb 2003 281.7850 152.2538 404.5871 78.536750 467.8090
## Mar 2003 282.9083 151.3434 407.5560 76.343877 471.7135
## Apr 2003 284.0487 150.4704 410.5222 74.195858 475.6053
## May 2003 282.5011 146.7206 410.8943 68.920326 476.9369
## Jun 2003 280.4994 142.4880 410.8168 63.090821 477.8180
## Jul 2003 280.0816 139.9750 412.2410 59.124716 480.1666
## Aug 2003 275.1385 132.5603 409.3241 49.682048 478.2420
## Sep 2003 286.9447 143.3324 422.4066 60.451108 492.0290
## Oct 2003 283.7840 137.8912 421.1606 53.233846 491.7288
## Nov 2003 281.2228 133.0979 420.4775 46.670106 491.9752
## Dec 2003 281.0471 130.9069 422.0609 42.993900 494.4408
## Jan 2004 279.7594 127.5097 422.5706 37.911472 495.8450
## Feb 2004 281.4485 127.3784 425.9075 36.554051 500.0183
## Mar 2004 282.5861 126.6570 428.7051 34.515020 503.6553
## Apr 2004 283.7400 125.9665 431.5057 32.506595 507.2885
## May 2004 282.2052 122.3327 431.7322 27.014462 508.3884
## Jun 2004 280.2156 118.1979 431.5163 20.803256 509.0486
## Jul 2004 279.8095 115.8124 432.7982 16.596701 511.1731
## Aug 2004 274.8773 108.4074 429.7675 5.567460 509.0615
## Sep 2004 286.6956 119.5876 442.6530 18.762279 522.5581
## Oct 2004 283.5449 114.1951 441.2920 10.640766 522.0722
## Nov 2004 280.9935 109.4558 440.4945 2.652755 522.1349plot(fit3,xlab="Time" ,ylab="Monthly Gold Price")
accuracy(fit3,gold.test)## ME RMSE MAE MPE MAPE MASE
## Training set -1.650934 22.30299 14.02948 -0.5156365 3.51985 0.245470
## Test set 155.814565 189.59717 160.53840 31.3776255 33.28105 2.808898
## ACF1 Theil's U
## Training set 0.02375412 NA
## Test set 0.88538664 3.811567hist(residuals(fit3))
plot(residuals(fit3))
- Based on all three models, the NNAR(13,1,7) is the best performer compared to the other two models. For instance, the RMSE, MAE, MPE, MAPE, MASE, and ME statistics are the lowest when analyzing the training set. The ARIMA(0,1,1)(2,0,1) and STL + ETS (M,Ad,N) performed relatively similar since the RMSE, MAE, MPE, MAPE, MASE, and ME were very close values. When comparing the three models, we can see that the NNAR(13,1,7) model shows an increasing values in gold prices.