Penerapan Metode Arima pada Data Harga Emas UK
Kelompok F
Muhamad Naufal Zaky (22305141012)
Raihan Hijri Firjatullah (22305141034)
Fadhila Asma’ul Karimah (22305144031)
Library
library(readxl)
library(tseries)
library(forecast)
library(TTR)
library(TSA)
library(graphics)
library(dplyr)
library(ggplot2)Import Data
Data yang digunakan adalah data harga emas bulanan di United Kingdom (dalam USD) dari Bulan Januari tahun 1979 sampai Desember tahun 2020 yang merupakan data deret waktu.
data_emas <- read_excel("C:/Users/ASUS/Downloads/harga emas negara.xlsx")
View(data_emas)data.emas <- data_emas %>%
select(Time, UK)
data.table::data.table(data.emas)## Time UK
## <POSc> <num>
## 1: 1979-01-01 117.4
## 2: 1979-02-01 124.2
## 3: 1979-03-01 116.2
## 4: 1979-04-01 118.8
## 5: 1979-05-01 132.7
## ---
## 500: 2020-08-01 1461.9
## 501: 2020-09-01 1459.6
## 502: 2020-10-01 1455.4
## 503: 2020-11-01 1320.2
## 504: 2020-12-01 1380.9emas.ts <- ts(data.emas$UK)
emas.ts.month <- ts(data.emas$UK, start = c(1979, 1), frequency = 12)Plot Data
# Plot Data
data.emas$Time <- as.Date(data.emas$Time)
ggplot(data.emas, aes(x = Time, y = UK)) + geom_line()
Berdasarkan plot deret waktu diatas, terlihat bahwa data memiliki pola tidak stasioner.
Partisi Data
Data harga emas UK di partisi menjadi dua bagian yaitu data training (Januari 1979 - Juli 2012) dan data testing (Agustus 2012 - Desember 2020).
# Partisi Data
data.emas.training <- window(emas.ts, start = 1, end = 403)
data.emas.training ## Time Series:
## Start = 1
## End = 403
## Frequency = 1
## [1] 117.4 124.2 116.2 118.8 132.7 127.3 131.9 139.9 180.4 184.0
## [11] 189.0 230.6 288.1 280.3 228.5 229.4 228.4 277.3 262.2 263.4
## [21] 279.3 258.2 262.8 246.7 214.0 221.8 228.9 225.5 231.5 220.7
## [31] 220.5 229.9 237.5 229.6 212.0 208.2 205.7 199.1 179.6 201.4
## [41] 181.7 182.1 197.3 239.7 234.3 252.4 268.6 282.5 328.6 269.6
## [51] 279.6 275.1 272.7 271.9 277.5 277.3 270.5 255.4 276.3 263.5
## [61] 266.7 264.5 269.3 268.7 277.3 275.0 261.8 266.1 278.3 273.1
## [71] 274.6 266.0 271.1 266.4 266.1 258.6 244.1 242.6 232.5 239.2
## [81] 231.2 225.6 218.5 226.1 248.1 233.7 231.8 222.9 233.1 226.3
## [91] 239.5 258.5 292.5 285.3 271.7 262.2 264.6 262.5 262.3 273.0
## [101] 276.7 277.2 290.5 277.6 282.7 272.2 269.7 257.7 258.8 240.3
## [111] 242.0 238.9 247.8 255.6 255.4 254.0 234.6 233.2 228.4 226.8
## [121] 225.1 221.8 227.0 223.5 230.1 240.8 221.0 228.7 226.9 237.8
## [131] 260.1 247.2 247.1 241.2 223.7 224.3 216.5 201.9 200.2 204.9
## [141] 218.0 195.2 198.4 200.1 186.3 189.9 204.5 207.5 212.1 227.5
## [151] 215.3 206.7 202.5 205.1 207.5 188.8 197.8 201.0 196.9 189.6
## [161] 184.5 180.4 186.4 171.5 195.9 217.1 220.8 219.9 222.2 230.2
## [171] 224.4 225.8 241.8 253.4 270.5 249.8 237.7 248.6 249.9 264.8
## [181] 252.1 256.8 262.2 248.3 256.4 251.6 249.9 251.1 250.4 235.4
## [191] 244.8 245.0 236.2 238.2 240.7 242.2 241.9 243.3 239.6 246.8
## [201] 242.6 242.5 253.6 249.3 268.4 261.7 259.7 260.8 252.1 245.9
## [211] 247.5 247.3 242.4 233.1 221.0 215.8 215.6 219.8 212.0 209.6
## [221] 211.3 201.0 199.3 200.6 205.6 185.7 176.2 176.4 186.5 180.6
## [231] 179.8 185.8 180.1 177.6 176.6 163.3 172.9 174.5 178.5 173.0
## [241] 173.7 179.2 173.1 178.0 167.5 165.6 157.8 158.5 181.6 182.3
## [251] 182.9 180.1 174.8 186.0 173.5 175.8 182.1 190.3 184.8 190.4
## [261] 185.1 182.2 189.8 183.7 181.0 184.9 181.3 183.9 188.3 192.4
## [271] 186.6 188.2 199.4 191.7 193.2 190.0 199.8 209.9 211.7 211.5
## [281] 223.2 209.0 195.0 202.2 205.8 202.6 205.0 215.7 223.6 220.6
## [291] 211.8 210.7 220.6 209.7 220.7 237.5 233.5 227.6 231.6 232.5
## [301] 219.6 213.3 230.5 219.1 214.5 218.3 215.2 226.4 229.7 232.3
## [311] 237.2 226.9 223.8 226.1 226.2 228.1 227.4 243.9 243.7 240.9
## [321] 267.5 265.9 286.4 298.8 320.0 317.5 335.5 354.3 349.0 331.7
## [331] 338.8 327.8 320.8 316.6 328.8 322.9 332.3 338.9 337.4 338.5
## [341] 333.2 324.2 327.5 333.2 364.7 380.1 381.1 418.8 464.4 488.4
## [351] 469.7 439.8 448.2 467.4 463.4 456.7 496.2 452.3 530.8 604.9
## [361] 637.8 667.9 639.4 596.1 604.9 567.5 566.4 586.3 622.6 630.9
## [371] 716.4 673.4 673.1 728.0 735.4 770.4 831.1 831.5 746.4 810.7
## [381] 829.4 842.4 888.4 897.7 828.5 867.5 897.7 920.6 933.5 937.7
## [391] 992.1 1113.8 1039.9 1066.9 1110.1 985.1 1105.2 1108.0 1040.5 1016.9
## [401] 1012.3 1019.2 1035.2data.emas.test <- subset(emas.ts, start = 404, end = 504)
data.emas.test## Time Series:
## Start = 404
## End = 504
## Frequency = 1
## [1] 1037.9 1099.8 1067.0 1077.0 1019.7 1050.0 1046.5 1052.6 943.9 919.8
## [11] 785.9 867.1 901.6 819.1 824.1 764.9 727.3 761.2 791.5 774.8
## [21] 763.1 745.5 769.1 761.3 774.2 750.4 727.7 755.2 773.5 839.1
## [31] 785.6 799.6 768.0 780.8 744.6 703.9 738.0 735.4 739.7 705.4
## [41] 719.2 783.8 886.1 860.6 877.6 832.8 988.0 1010.8 999.7 1018.1
## [51] 1041.9 942.9 927.4 964.0 1009.0 995.5 978.9 980.8 956.4 961.5
## [61] 1018.0 956.4 956.5 945.7 954.4 945.9 956.5 943.7 953.4 981.0
## [71] 947.1 930.8 925.1 910.4 950.9 954.3 1004.2 1005.9 991.8 994.1
## [81] 983.6 1027.9 1107.1 1165.9 1255.0 1205.3 1167.7 1128.8 1143.4 1201.8
## [91] 1260.4 1297.6 1349.9 1398.3 1431.0 1497.1 1461.9 1459.6 1455.4 1320.2
## [101] 1380.9Plot Data Partisi
# Data Training
plot.ts(data.emas.training, main = "Data Training Harga Emas", xlab = "Bulan", ylab = "UK")
# Data Testing
plot.ts(data.emas.test, main = "Data Testing Harga Emas", xlab = "Bulan", ylab = "UK")
Stasioneritas Data
Plot ACF
# Stasioneritas Data
acf(data.emas.training, main = "Plot ACF Data Harga Emas UK")
Uji ADF
adf.test(data.emas.training)## Warning in adf.test(data.emas.training): p-value greater than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: data.emas.training
## Dickey-Fuller = 1.5662, Lag order = 7, p-value = 0.99
## alternative hypothesis: stationaryBerdasarkan hasil uji ADF diperoleh p-value sebesar 0.99 > 0.05 yang berarti bahwa H0 diterima atau data tidak stasioner. Karena data tidak stasioner maka selanjutnya dilakukan differencing untuk membuat data stasioner sebelum dilakukan identifikasi model.
Differencing
# Differencing Ordo 1
data.emas.diff <- diff(data.emas.training, difference = 1)
plot.ts(data.emas.diff, lty = 1, xlab = "Waktu", ylab = "Data Harga Emas UK Differencing")
points(data.emas.diff)
acf(data.emas.diff, main = "Plot ACF Data Harga Emas UK Differencing", lag.max = 10)
adf.test(data.emas.diff)## Warning in adf.test(data.emas.diff): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: data.emas.diff
## Dickey-Fuller = -7.6743, Lag order = 7, p-value = 0.01
## alternative hypothesis: stationaryBerdasarkan hasil uji ADF, diperoleh nilai p-value sebesar 0.01 < 0.05 yang berarti H0 ditolak, sehingga data harga emas telah stasioner setelah dilakukan differencing ordo 1. Karena data telah stasioner selanjutnya akan dilakukan identifikasi model.
Identifikasi Model
Plot ACF
acf(data.emas.diff, main="Plot ACF Data Harga Emas UK", lag.max = 10) 
Plot PACF
pacf(data.emas.diff, main="Plot PACF Data Harga Emas UK", lag.max = 10) 
Plot EACF
eacf(data.emas.diff, ar.max = 10, ma.max = 10)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10
## 0 x o x x o x x o o o x
## 1 x o o x o x x o o o x
## 2 x x o o o o o o o o x
## 3 x x o o o o o o o o x
## 4 x x x x o o o o o o x
## 5 x x x o o o o o o o o
## 6 x x o o o x o o o o o
## 7 x x x o o x o o o o o
## 8 x x o o x x o o o o o
## 9 x x x x x o o o o o o
## 10 x x x o x o x o o o oKandidat Model
Berdasarkan plot ACF (cut off setelah lag ke-1), PACF (cut off setelah lag ke-4), dan EACF, diperoleh kandidat model:
ARIMA (4,1,1)
ARIMA (4,1,0)
ARIMA (1,1,1)
ARIMA (1,1,0)
ARIMA (2,1,1)
Pendugaan Parameter dan Penentuan Model Terbaik
# Uji Signifikansi Parameter
arima411 <- arima(data.emas.diff, order = c(4,1,1),include.mean = FALSE, method = "ML")
arima411##
## Call:
## arima(x = data.emas.diff, order = c(4, 1, 1), include.mean = FALSE, method = "ML")
##
## Coefficients:
## ar1 ar2 ar3 ar4 ma1
## -0.1341 -0.1478 0.1109 -0.1694 -0.9693
## s.e. 0.0501 0.0508 0.0507 0.0500 0.0123
##
## sigma^2 estimated as 405.1: log likelihood = -1774.61, aic = 3559.22arima410 <- arima(data.emas.diff, order = c(4,1,0),include.mean = FALSE, method = "ML")
arima410##
## Call:
## arima(x = data.emas.diff, order = c(4, 1, 0), include.mean = FALSE, method = "ML")
##
## Coefficients:
## ar1 ar2 ar3 ar4
## -0.8739 -0.7659 -0.3336 -0.2471
## s.e. 0.0485 0.0636 0.0637 0.0489
##
## sigma^2 estimated as 500.2: log likelihood = -1815.7, aic = 3639.4arima111 <- arima(data.emas.diff, order = c(1,1,1),include.mean = FALSE, method = "ML")
arima111##
## Call:
## arima(x = data.emas.diff, order = c(1, 1, 1), include.mean = FALSE, method = "ML")
##
## Coefficients:
## ar1 ma1
## -0.1500 -0.9769
## s.e. 0.0499 0.0100
##
## sigma^2 estimated as 434.9: log likelihood = -1788.76, aic = 3581.53arima110 <- arima(data.emas.diff, order = c(1,1,0),include.mean = FALSE, method = "ML")
arima110##
## Call:
## arima(x = data.emas.diff, order = c(1, 1, 0), include.mean = FALSE, method = "ML")
##
## Coefficients:
## ar1
## -0.5126
## s.e. 0.0428
##
## sigma^2 estimated as 738: log likelihood = -1893.25, aic = 3788.5arima211 <- arima(data.emas.diff, order = c(2,1,1),include.mean = FALSE, method = "ML")
arima211##
## Call:
## arima(x = data.emas.diff, order = c(2, 1, 1), include.mean = FALSE, method = "ML")
##
## Coefficients:
## ar1 ar2 ma1
## -0.1778 -0.1500 -0.9707
## s.e. 0.0502 0.0501 0.0116
##
## sigma^2 estimated as 425.4: log likelihood = -1784.35, aic = 3574.71AIC
# AIC
aic.arima <- data.frame(
"Model" = c("ARIMA(4,1,1)", "ARIMA(4,1,0)", "ARIMA(1,1,1)", "ARIMA(1,1,0)", "ARIMA(2,1,1)"),
"AIC" = c(arima411$aic, arima410$aic, arima111$aic, arima110$aic, arima211$aic)
)
aic.arima## Model AIC
## 1 ARIMA(4,1,1) 3559.218
## 2 ARIMA(4,1,0) 3639.398
## 3 ARIMA(1,1,1) 3581.526
## 4 ARIMA(1,1,0) 3788.498
## 5 ARIMA(2,1,1) 3574.706Berdasarkan hasil di atas dapat disimpulkan bahwa ARIMA (4,1,1) merupakan model terbaik karena model tersebut memiliki nilai AIC terkecil yaitu 3559.218.
Diagnostik Model
# Residuals
arima.411<-arima(data.emas.training, order=c(4,1,1), include.mean = FALSE, method="ML")
residual.arima <- arima.411$residualsUji Non-Autokorelasi
# Uji Non-Autokorelasi
Box.test(residual.arima, type = "Ljung-Box")##
## Box-Ljung test
##
## data: residual.arima
## X-squared = 0.070551, df = 1, p-value = 0.7905Berdasarkan hasil uji Ljung-Box di atas dapat disimpulkan bahwa H0 tidak ditolak karena p-value sebesar 0.7905 > 0.05 yang berarti tidak terdapat autokorelasi pada residuals dari model ARIMA
Uji Normalitas
# Uji Normalitas
shapiro.test(residual.arima)##
## Shapiro-Wilk normality test
##
## data: residual.arima
## W = 0.84748, p-value < 2.2e-16hist(residual.arima, main = "Histogram Residual", xlab = "Residual", breaks = 20)
qqnorm(residual.arima)
qqline(residual.arima, col = "red")
Berdasarkan hasil uji Shapiro-Wilk di atas dapat disimpulkan bahwa H0 ditolak karena p-value = 2.2e-16 < 0.05 sehingga sisaan dari model ARIMA tidak berdistribusi Normal.
Ukuran Keakuratan Ramalan Model Arima (4,1,1)
# Ukuran Keakuratan
dugaan <- fitted(arima411)
cbind(data.emas.training, dugaan)## Time Series:
## Start = 1
## End = 403
## Frequency = 1
## data.emas.training dugaan
## 1 117.4 NA
## 2 124.2 6.79320001
## 3 116.2 1.42794490
## 4 118.8 0.37263602
## 5 132.7 3.57387873
## 6 127.3 -0.92663116
## 7 131.9 3.14208300
## 8 139.9 4.57322774
## 9 180.4 1.53569416
## 10 184.0 4.55333060
## 11 189.0 3.37951058
## 12 230.6 12.94832238
## 13 288.1 2.12992616
## 14 280.3 4.14396088
## 15 228.5 11.92110883
## 16 229.4 18.99886777
## 17 228.4 7.34380120
## 18 277.3 6.40073427
## 19 262.2 14.85475394
## 20 263.4 5.42847481
## 21 279.3 18.32743065
## 22 258.2 -1.97153792
## 23 262.8 12.53960931
## 24 246.7 12.76777912
## 25 214.0 3.57778088
## 26 221.8 16.42734491
## 27 228.9 6.46119477
## 28 225.5 2.21487855
## 29 231.5 10.77907427
## 30 220.7 3.84757417
## 31 220.5 3.09671504
## 32 229.9 6.87334634
## 33 237.5 0.69375354
## 34 229.6 3.72718420
## 35 212.0 4.81840376
## 36 208.2 5.74974374
## 37 205.7 3.55922677
## 38 199.1 2.64961519
## 39 179.6 5.77162681
## 40 201.4 5.10667723
## 41 181.7 1.29982087
## 42 182.1 -0.67955181
## 43 197.3 9.57648757
## 44 239.7 -6.66590022
## 45 234.3 -1.62268251
## 46 252.4 -1.07806783
## 47 268.6 3.99964221
## 48 282.5 -8.68566820
## 49 328.6 3.42307237
## 50 269.6 -3.45875208
## 51 279.6 4.15720502
## 52 275.1 14.55771587
## 53 272.7 -11.40784440
## 54 271.9 16.18008124
## 55 277.5 1.81825397
## 56 277.3 3.52938457
## 57 270.5 3.05363293
## 58 255.4 4.90161948
## 59 276.3 4.64493451
## 60 263.5 1.79494243
## 61 266.7 0.73319485
## 62 264.5 9.03613126
## 63 269.3 -2.78931379
## 64 268.7 4.78898248
## 65 277.3 1.00481904
## 66 275.0 2.49524019
## 67 261.8 0.65442423
## 68 266.1 5.22941107
## 69 278.3 1.70214147
## 70 273.1 -0.98076830
## 71 274.6 3.84053043
## 72 266.0 3.34697886
## 73 271.1 0.07358276
## 74 266.4 3.57331344
## 75 266.1 0.35045692
## 76 258.6 4.41552382
## 77 244.1 0.94872617
## 78 242.6 4.61999955
## 79 232.5 2.17444058
## 80 239.2 1.47170143
## 81 231.2 3.27661003
## 82 225.6 -0.74093876
## 83 218.5 4.27782830
## 84 226.1 -0.70217849
## 85 248.1 0.56517921
## 86 233.7 -3.45231578
## 87 231.8 0.84745538
## 88 222.9 3.57073510
## 89 233.1 -4.19723040
## 90 226.3 2.27196748
## 91 239.5 -1.44292043
## 92 258.5 2.14479203
## 93 292.5 -6.18731249
## 94 285.3 -2.72131053
## 95 271.7 -2.29740975
## 96 262.2 4.98193959
## 97 264.6 -2.17813215
## 98 262.5 2.03130013
## 99 262.3 2.28796042
## 100 273.0 3.24751317
## 101 276.7 -0.78105920
## 102 277.2 -0.34234732
## 103 290.5 2.03493940
## 104 277.6 -1.48646113
## 105 282.7 0.61606337
## 106 272.2 4.17325242
## 107 269.7 -1.92035306
## 108 257.7 5.72942360
## 109 258.8 0.49666028
## 110 240.3 3.69207437
## 111 242.0 1.29400346
## 112 238.9 4.55665559
## 113 247.8 -2.41321870
## 114 255.6 2.59570201
## 115 255.4 -2.82544387
## 116 254.0 0.63472726
## 117 234.6 -0.24013201
## 118 233.2 1.06215202
## 119 228.4 2.45522346
## 120 226.8 -1.76484755
## 121 225.1 3.35973883
## 122 221.8 -0.68197665
## 123 227.0 0.39813321
## 124 223.5 -0.91128555
## 125 230.1 -1.24064023
## 126 240.8 0.14525233
## 127 221.0 -3.97886780
## 128 228.7 1.61441842
## 129 226.9 1.36436334
## 130 237.8 -5.60040948
## 131 260.1 2.82468089
## 132 247.2 -5.69691026
## 133 247.1 0.13599397
## 134 241.2 2.72709499
## 135 223.7 -4.48728628
## 136 224.3 4.91014906
## 137 216.5 1.25297592
## 138 201.9 -0.87723584
## 139 200.2 4.82822335
## 140 204.9 -0.09531504
## 141 218.0 -2.04394346
## 142 195.2 -1.06904480
## 143 198.4 0.36157961
## 144 200.1 2.11501091
## 145 186.3 -6.94428361
## 146 189.9 4.11220726
## 147 204.5 -0.51768337
## 148 207.5 -5.56583380
## 149 212.1 -0.81714144
## 150 227.5 -0.87921971
## 151 215.3 -5.21521390
## 152 206.7 -1.18143956
## 153 202.5 3.11415851
## 154 205.1 -3.12333881
## 155 207.5 0.56562064
## 156 188.8 -0.47827760
## 157 197.8 1.83085493
## 158 201.0 0.28012835
## 159 196.9 -5.25281625
## 160 189.6 3.26604509
## 161 184.5 -0.88653070
## 162 180.4 -0.66478634
## 163 186.4 -0.34742162
## 164 171.5 -0.86885124
## 165 195.9 -0.25083455
## 166 217.1 -0.72554246
## 167 220.8 -9.46176819
## 168 219.9 1.66164280
## 169 222.2 -2.22729873
## 170 230.2 -3.23657726
## 171 224.4 -1.67449941
## 172 225.8 0.34153188
## 173 241.8 1.53802352
## 174 253.4 -3.53694620
## 175 270.5 -1.50376767
## 176 249.8 -0.62120263
## 177 237.7 0.05763659
## 178 248.6 5.47368846
## 179 249.9 -3.83978360
## 180 264.8 1.56441268
## 181 252.1 2.66221379
## 182 256.8 -1.07928828
## 183 262.2 3.97859903
## 184 248.3 -4.00882935
## 185 256.4 4.77890642
## 186 251.6 1.91236645
## 187 249.9 -2.07421192
## 188 251.1 5.13825357
## 189 250.4 -0.98788592
## 190 235.4 1.37665741
## 191 244.8 2.86942433
## 192 245.0 1.20888334
## 193 236.2 -2.45860559
## 194 238.2 5.04279750
## 195 240.7 -0.32326251
## 196 242.2 -1.33935958
## 197 241.9 1.53068267
## 198 243.3 0.08930973
## 199 239.6 -0.02808331
## 200 246.8 0.26179825
## 201 242.6 0.26008318
## 202 242.5 -0.81253434
## 203 253.6 2.41740913
## 204 249.3 -2.53553106
## 205 268.4 0.20656941
## 206 261.7 0.47142353
## 207 259.7 -3.35268728
## 208 260.8 5.07623433
## 209 252.1 -2.98236507
## 210 245.9 2.59105572
## 211 247.5 2.98212472
## 212 247.3 -0.08794771
## 213 242.4 0.93494974
## 214 233.1 2.09390812
## 215 221.0 1.50801436
## 216 215.8 1.90023725
## 217 215.6 1.47935556
## 218 219.8 0.17252245
## 219 212.0 0.20671409
## 220 209.6 0.30565542
## 221 211.3 0.91291323
## 222 201.0 -2.48734207
## 223 199.3 0.90854055
## 224 200.6 0.98859779
## 225 205.6 -2.70083567
## 226 185.7 -0.41721698
## 227 176.2 0.65337846
## 228 176.4 2.52951992
## 229 186.5 -3.76843252
## 230 180.6 -0.73202700
## 231 179.8 -0.89421169
## 232 185.8 0.24399039
## 233 180.1 -4.69719520
## 234 177.6 -0.88732917
## 235 176.6 0.25338198
## 236 163.3 -2.90898828
## 237 172.9 0.53749012
## 238 174.5 -0.81357137
## 239 178.5 -4.66960087
## 240 173.0 1.08068808
## 241 173.7 -2.96914186
## 242 179.2 -0.66238928
## 243 173.1 -3.49378587
## 244 178.0 -0.43021595
## 245 167.5 -0.54562023
## 246 165.6 -2.51125240
## 247 157.8 1.81569859
## 248 158.5 -2.53078931
## 249 181.6 0.86350798
## 250 182.3 -4.82637570
## 251 182.9 -3.02114813
## 252 180.1 1.45815853
## 253 174.8 -4.48146362
## 254 186.0 0.11540566
## 255 173.5 -1.74811893
## 256 175.8 -1.03933067
## 257 182.1 2.83469540
## 258 190.3 -5.20712981
## 259 184.8 0.01508233
## 260 190.4 -0.66155928
## 261 185.1 -0.40035252
## 262 182.2 -2.57081048
## 263 189.8 2.26057927
## 264 183.7 -2.42817551
## 265 181.0 -0.14235035
## 266 184.9 2.10600616
## 267 181.3 -2.52514613
## 268 183.9 0.17058424
## 269 188.3 0.67804308
## 270 192.4 -2.31565963
## 271 186.6 -0.38632542
## 272 188.2 -0.03119273
## 273 199.4 0.15135242
## 274 191.7 -2.93817041
## 275 193.2 0.52972528
## 276 190.0 1.92998605
## 277 199.8 -2.67956034
## 278 209.9 0.87706161
## 279 211.7 -2.88177774
## 280 211.5 0.56871680
## 281 223.2 -0.12932232
## 282 209.0 -2.03828958
## 283 195.0 0.48782661
## 284 202.2 5.50338326
## 285 205.8 -2.20609147
## 286 202.6 -0.26800606
## 287 205.0 3.40355845
## 288 215.7 -0.36447512
## 289 223.6 -2.11018597
## 290 220.6 -0.88118791
## 291 211.8 0.90115950
## 292 210.7 1.27559771
## 293 220.6 0.29280569
## 294 209.7 -0.82225365
## 295 220.7 1.86913549
## 296 237.5 2.20175307
## 297 233.5 -5.53590592
## 298 227.6 2.39657889
## 299 231.6 2.40380483
## 300 232.5 -1.88380908
## 301 219.6 0.46758820
## 302 213.3 3.78648750
## 303 230.5 2.61046232
## 304 219.1 -2.07444930
## 305 214.5 1.07205665
## 306 218.3 5.70135748
## 307 215.2 -3.64224050
## 308 226.4 1.65824707
## 309 229.7 0.83243811
## 310 232.3 -2.33447711
## 311 237.2 1.83342442
## 312 226.9 -1.57665507
## 313 223.8 1.11520722
## 314 226.1 2.64058053
## 315 226.2 -1.23372873
## 316 228.1 1.67808085
## 317 227.4 1.14766400
## 318 243.9 0.01470253
## 319 243.7 -0.82975815
## 320 240.9 -1.70632023
## 321 267.5 3.42521603
## 322 265.9 -4.18912521
## 323 286.4 -2.13198263
## 324 298.8 3.46743158
## 325 320.0 -6.54780518
## 326 317.5 1.54921386
## 327 335.5 -1.34015147
## 328 354.3 2.35486868
## 329 349.0 -4.39763624
## 330 331.7 4.97766996
## 331 338.8 6.08101513
## 332 327.8 1.80506148
## 333 320.8 2.98593242
## 334 316.6 9.55791027
## 335 328.8 2.02769242
## 336 322.9 3.23687075
## 337 332.3 2.59301998
## 338 338.9 4.76958809
## 339 337.4 -1.84629289
## 340 338.5 4.42811769
## 341 333.2 2.27163923
## 342 324.2 2.08948885
## 343 327.5 4.85231861
## 344 333.2 2.55170583
## 345 364.7 1.18252294
## 346 380.1 0.28867150
## 347 381.1 -2.71982793
## 348 418.8 4.15973865
## 349 464.4 -3.76236173
## 350 488.4 -7.60157808
## 351 469.7 1.60683149
## 352 439.8 4.56076973
## 353 448.2 7.58260354
## 354 467.4 3.05048893
## 355 463.4 2.42941735
## 356 456.7 9.89300388
## 357 496.2 7.88370830
## 358 452.3 -1.34653748
## 359 530.8 5.33629581
## 360 604.9 9.07297939
## 361 637.8 -23.51112915
## 362 667.9 12.10052414
## 363 639.4 -2.10968679
## 364 596.1 1.52925370
## 365 604.9 17.47022592
## 366 567.5 6.37975272
## 367 566.4 11.82023362
## 368 586.3 21.67043811
## 369 622.6 -0.51662642
## 370 630.9 7.16327944
## 371 716.4 4.70800470
## 372 673.4 -0.76850486
## 373 673.1 -2.12501865
## 374 728.0 24.50286173
## 375 735.4 -15.61117634
## 376 770.4 9.81433086
## 377 831.1 12.79315419
## 378 831.5 -7.88467723
## 379 746.4 7.76681273
## 380 810.7 23.47132367
## 381 829.4 6.28184059
## 382 842.4 -8.56894314
## 383 888.4 30.65332391
## 384 897.7 -2.83015296
## 385 828.5 4.68013391
## 386 867.5 22.99196188
## 387 897.7 10.91268474
## 388 920.6 -5.79563809
## 389 933.5 22.66520774
## 390 937.7 5.47699640
## 391 992.1 8.76380482
## 392 1113.8 4.84458202
## 393 1039.9 -7.28750164
## 394 1066.9 13.99622474
## 395 1110.1 28.73121671
## 396 985.1 -21.00770746
## 397 1105.2 40.29814806
## 398 1108.0 19.43600705
## 399 1040.5 -22.96846712
## 400 1016.9 58.11358488
## 401 1012.3 5.57250176
## 402 1019.2 8.30174214
## 403 1035.2 20.68882071Plot Dugaan
# Plot Dugaan
plot.ts(data.emas.training, xlab="Month", ylab="Data")
points(data.emas.training)
par(col="black")
lines(dugaan)
Forecast
# Forecast
ramalan.arima411<- forecast::forecast(data.emas.training, model = arima.411, h = 101)
ramalan.arima411## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 404 1034.423 1008.1852 1060.661 994.2957 1074.550
## 405 1034.805 999.1098 1070.500 980.2140 1089.396
## 406 1036.957 994.8760 1079.038 972.5998 1101.314
## 407 1034.890 984.6673 1085.113 958.0809 1111.699
## 408 1035.228 979.6452 1090.811 950.2213 1120.235
## 409 1035.687 975.1851 1096.188 943.1575 1128.216
## 410 1035.009 969.4412 1100.576 934.7319 1135.285
## 411 1035.348 965.5610 1105.134 928.6181 1142.077
## 412 1035.392 961.4679 1109.316 922.3348 1148.450
## 413 1035.193 957.2679 1113.118 916.0169 1154.368
## 414 1035.349 953.7568 1116.941 910.5646 1160.133
## 415 1035.313 950.1327 1120.494 905.0409 1165.585
## 416 1035.267 946.6465 1123.888 899.7335 1170.801
## 417 1035.325 943.4264 1127.223 894.7783 1175.871
## 418 1035.297 940.2042 1130.390 889.8650 1180.729
## 419 1035.293 937.1184 1133.467 885.1481 1185.437
## 420 1035.310 934.1545 1136.466 880.6059 1190.015
## 421 1035.297 931.2358 1139.358 876.1491 1194.445
## 422 1035.300 928.4172 1142.183 871.8369 1198.763
## 423 1035.304 925.6725 1144.935 867.6371 1202.971
## 424 1035.299 922.9836 1147.615 863.5273 1207.071
## 425 1035.301 920.3666 1150.236 859.5238 1211.079
## 426 1035.302 917.8056 1152.798 855.6069 1214.997
## 427 1035.300 915.2968 1155.304 851.7707 1218.830
## 428 1035.301 912.8426 1157.760 848.0168 1222.586
## 429 1035.301 910.4350 1160.167 844.3349 1226.267
## 430 1035.301 908.0728 1162.529 840.7224 1229.879
## 431 1035.301 905.7546 1164.848 837.1768 1233.426
## 432 1035.301 903.4765 1167.126 833.6928 1236.909
## 433 1035.301 901.2372 1169.365 830.2681 1240.334
## 434 1035.301 899.0349 1171.567 826.8999 1243.702
## 435 1035.301 896.8674 1173.735 823.5850 1247.017
## 436 1035.301 894.7334 1175.869 820.3214 1250.281
## 437 1035.301 892.6314 1177.971 817.1066 1253.496
## 438 1035.301 890.5598 1180.042 813.9384 1256.664
## 439 1035.301 888.5175 1182.085 810.8150 1259.787
## 440 1035.301 886.5032 1184.099 807.7344 1262.868
## 441 1035.301 884.5158 1186.086 804.6949 1265.907
## 442 1035.301 882.5543 1188.048 801.6951 1268.907
## 443 1035.301 880.6176 1189.984 798.7332 1271.869
## 444 1035.301 878.7049 1191.897 795.8080 1274.794
## 445 1035.301 876.8153 1193.787 792.9181 1277.684
## 446 1035.301 874.9480 1195.654 790.0622 1280.540
## 447 1035.301 873.1021 1197.500 787.2392 1283.363
## 448 1035.301 871.2770 1199.325 784.4479 1286.154
## 449 1035.301 869.4720 1201.130 781.6874 1288.915
## 450 1035.301 867.6865 1202.916 778.9567 1291.645
## 451 1035.301 865.9197 1204.682 776.2546 1294.347
## 452 1035.301 864.1712 1206.431 773.5805 1297.022
## 453 1035.301 862.4404 1208.162 770.9335 1299.669
## 454 1035.301 860.7267 1209.875 768.3127 1302.289
## 455 1035.301 859.0297 1211.572 765.7173 1304.885
## 456 1035.301 857.3489 1213.253 763.1467 1307.455
## 457 1035.301 855.6838 1214.918 760.6002 1310.002
## 458 1035.301 854.0340 1216.568 758.0771 1312.525
## 459 1035.301 852.3991 1218.203 755.5767 1315.025
## 460 1035.301 850.7787 1219.823 753.0984 1317.504
## 461 1035.301 849.1724 1221.430 750.6418 1319.960
## 462 1035.301 847.5798 1223.022 748.2061 1322.396
## 463 1035.301 846.0006 1224.601 745.7910 1324.811
## 464 1035.301 844.4345 1226.168 743.3958 1327.206
## 465 1035.301 842.8811 1227.721 741.0201 1329.582
## 466 1035.301 841.3402 1229.262 738.6635 1331.939
## 467 1035.301 839.8114 1230.791 736.3254 1334.277
## 468 1035.301 838.2945 1232.308 734.0055 1336.597
## 469 1035.301 836.7891 1233.813 731.7033 1338.899
## 470 1035.301 835.2951 1235.307 729.4184 1341.184
## 471 1035.301 833.8122 1236.790 727.1505 1343.452
## 472 1035.301 832.3401 1238.262 724.8991 1345.703
## 473 1035.301 830.8787 1239.723 722.6640 1347.938
## 474 1035.301 829.4275 1241.175 720.4447 1350.157
## 475 1035.301 827.9866 1242.615 718.2409 1352.361
## 476 1035.301 826.5556 1244.047 716.0524 1354.550
## 477 1035.301 825.1343 1245.468 713.8788 1356.723
## 478 1035.301 823.7226 1246.879 711.7197 1358.882
## 479 1035.301 822.3202 1248.282 709.5750 1361.027
## 480 1035.301 820.9270 1249.675 707.4443 1363.158
## 481 1035.301 819.5429 1251.059 705.3274 1365.275
## 482 1035.301 818.1675 1252.435 703.2239 1367.378
## 483 1035.301 816.8008 1253.801 701.1337 1369.468
## 484 1035.301 815.4426 1255.160 699.0565 1371.546
## 485 1035.301 814.0927 1256.509 696.9921 1373.610
## 486 1035.301 812.7510 1257.851 694.9401 1375.662
## 487 1035.301 811.4173 1259.185 692.9005 1377.702
## 488 1035.301 810.0916 1260.510 690.8729 1379.729
## 489 1035.301 808.7736 1261.828 688.8572 1381.745
## 490 1035.301 807.4632 1263.139 686.8532 1383.749
## 491 1035.301 806.1604 1264.442 684.8606 1385.741
## 492 1035.301 804.8649 1265.737 682.8793 1387.723
## 493 1035.301 803.5766 1267.025 680.9091 1389.693
## 494 1035.301 802.2954 1268.307 678.9497 1391.652
## 495 1035.301 801.0213 1269.581 677.0011 1393.601
## 496 1035.301 799.7541 1270.848 675.0630 1395.539
## 497 1035.301 798.4936 1272.108 673.1353 1397.467
## 498 1035.301 797.2398 1273.362 671.2178 1399.384
## 499 1035.301 795.9926 1274.609 669.3104 1401.292
## 500 1035.301 794.7518 1275.850 667.4128 1403.189
## 501 1035.301 793.5175 1277.085 665.5250 1405.077
## 502 1035.301 792.2893 1278.313 663.6467 1406.955
## 503 1035.301 791.0674 1279.535 661.7779 1408.824
## 504 1035.301 789.8515 1280.751 659.9184 1410.684plot(ramalan.arima411)
gabungan <- cbind(data.emas.test, ramalan.arima411)
df.gabungan <- as.data.frame(gabungan, digits=3)
df.gabungan %>%
rename(
"Data Aktual Harga Emas UK" = data.emas.test,
Ramalan ="ramalan.arima411.Point Forecast",
"Lo 80"="ramalan.arima411.Lo 80",
"Hi 80"="ramalan.arima411.Hi 80",
"Lo 95"="ramalan.arima411.Lo 95",
"Hi 95"="ramalan.arima411.Hi 95"
)## Data Aktual Harga Emas UK Ramalan Lo 80 Hi 80 Lo 95 Hi 95
## 1 1037.9 1034.423 1008.1852 1060.661 994.2957 1074.550
## 2 1099.8 1034.805 999.1098 1070.500 980.2140 1089.396
## 3 1067.0 1036.957 994.8760 1079.038 972.5998 1101.314
## 4 1077.0 1034.890 984.6673 1085.113 958.0809 1111.699
## 5 1019.7 1035.228 979.6452 1090.811 950.2213 1120.235
## 6 1050.0 1035.687 975.1851 1096.188 943.1575 1128.216
## 7 1046.5 1035.009 969.4412 1100.576 934.7319 1135.285
## 8 1052.6 1035.348 965.5610 1105.134 928.6181 1142.077
## 9 943.9 1035.392 961.4679 1109.316 922.3348 1148.450
## 10 919.8 1035.193 957.2679 1113.118 916.0169 1154.368
## 11 785.9 1035.349 953.7568 1116.941 910.5646 1160.133
## 12 867.1 1035.313 950.1327 1120.494 905.0409 1165.585
## 13 901.6 1035.267 946.6465 1123.888 899.7335 1170.801
## 14 819.1 1035.325 943.4264 1127.223 894.7783 1175.871
## 15 824.1 1035.297 940.2042 1130.390 889.8650 1180.729
## 16 764.9 1035.293 937.1184 1133.467 885.1481 1185.437
## 17 727.3 1035.310 934.1545 1136.466 880.6059 1190.015
## 18 761.2 1035.297 931.2358 1139.358 876.1491 1194.445
## 19 791.5 1035.300 928.4172 1142.183 871.8369 1198.763
## 20 774.8 1035.304 925.6725 1144.935 867.6371 1202.971
## 21 763.1 1035.299 922.9836 1147.615 863.5273 1207.071
## 22 745.5 1035.301 920.3666 1150.236 859.5238 1211.079
## 23 769.1 1035.302 917.8056 1152.798 855.6069 1214.997
## 24 761.3 1035.300 915.2968 1155.304 851.7707 1218.830
## 25 774.2 1035.301 912.8426 1157.760 848.0168 1222.586
## 26 750.4 1035.301 910.4350 1160.167 844.3349 1226.267
## 27 727.7 1035.301 908.0728 1162.529 840.7224 1229.879
## 28 755.2 1035.301 905.7546 1164.848 837.1768 1233.426
## 29 773.5 1035.301 903.4765 1167.126 833.6928 1236.909
## 30 839.1 1035.301 901.2372 1169.365 830.2681 1240.334
## 31 785.6 1035.301 899.0349 1171.567 826.8999 1243.702
## 32 799.6 1035.301 896.8674 1173.735 823.5850 1247.017
## 33 768.0 1035.301 894.7334 1175.869 820.3214 1250.281
## 34 780.8 1035.301 892.6314 1177.971 817.1066 1253.496
## 35 744.6 1035.301 890.5598 1180.042 813.9384 1256.664
## 36 703.9 1035.301 888.5175 1182.085 810.8150 1259.787
## 37 738.0 1035.301 886.5032 1184.099 807.7344 1262.868
## 38 735.4 1035.301 884.5158 1186.086 804.6949 1265.907
## 39 739.7 1035.301 882.5543 1188.048 801.6951 1268.907
## 40 705.4 1035.301 880.6176 1189.984 798.7332 1271.869
## 41 719.2 1035.301 878.7049 1191.897 795.8080 1274.794
## 42 783.8 1035.301 876.8153 1193.787 792.9181 1277.684
## 43 886.1 1035.301 874.9480 1195.654 790.0622 1280.540
## 44 860.6 1035.301 873.1021 1197.500 787.2392 1283.363
## 45 877.6 1035.301 871.2770 1199.325 784.4479 1286.154
## 46 832.8 1035.301 869.4720 1201.130 781.6874 1288.915
## 47 988.0 1035.301 867.6865 1202.916 778.9567 1291.645
## 48 1010.8 1035.301 865.9197 1204.682 776.2546 1294.347
## 49 999.7 1035.301 864.1712 1206.431 773.5805 1297.022
## 50 1018.1 1035.301 862.4404 1208.162 770.9335 1299.669
## 51 1041.9 1035.301 860.7267 1209.875 768.3127 1302.289
## 52 942.9 1035.301 859.0297 1211.572 765.7173 1304.885
## 53 927.4 1035.301 857.3489 1213.253 763.1467 1307.455
## 54 964.0 1035.301 855.6838 1214.918 760.6002 1310.002
## 55 1009.0 1035.301 854.0340 1216.568 758.0771 1312.525
## 56 995.5 1035.301 852.3991 1218.203 755.5767 1315.025
## 57 978.9 1035.301 850.7787 1219.823 753.0984 1317.504
## 58 980.8 1035.301 849.1724 1221.430 750.6418 1319.960
## 59 956.4 1035.301 847.5798 1223.022 748.2061 1322.396
## 60 961.5 1035.301 846.0006 1224.601 745.7910 1324.811
## 61 1018.0 1035.301 844.4345 1226.168 743.3958 1327.206
## 62 956.4 1035.301 842.8811 1227.721 741.0201 1329.582
## 63 956.5 1035.301 841.3402 1229.262 738.6635 1331.939
## 64 945.7 1035.301 839.8114 1230.791 736.3254 1334.277
## 65 954.4 1035.301 838.2945 1232.308 734.0055 1336.597
## 66 945.9 1035.301 836.7891 1233.813 731.7033 1338.899
## 67 956.5 1035.301 835.2951 1235.307 729.4184 1341.184
## 68 943.7 1035.301 833.8122 1236.790 727.1505 1343.452
## 69 953.4 1035.301 832.3401 1238.262 724.8991 1345.703
## 70 981.0 1035.301 830.8787 1239.723 722.6640 1347.938
## 71 947.1 1035.301 829.4275 1241.175 720.4447 1350.157
## 72 930.8 1035.301 827.9866 1242.615 718.2409 1352.361
## 73 925.1 1035.301 826.5556 1244.047 716.0524 1354.550
## 74 910.4 1035.301 825.1343 1245.468 713.8788 1356.723
## 75 950.9 1035.301 823.7226 1246.879 711.7197 1358.882
## 76 954.3 1035.301 822.3202 1248.282 709.5750 1361.027
## 77 1004.2 1035.301 820.9270 1249.675 707.4443 1363.158
## 78 1005.9 1035.301 819.5429 1251.059 705.3274 1365.275
## 79 991.8 1035.301 818.1675 1252.435 703.2239 1367.378
## 80 994.1 1035.301 816.8008 1253.801 701.1337 1369.468
## 81 983.6 1035.301 815.4426 1255.160 699.0565 1371.546
## 82 1027.9 1035.301 814.0927 1256.509 696.9921 1373.610
## 83 1107.1 1035.301 812.7510 1257.851 694.9401 1375.662
## 84 1165.9 1035.301 811.4173 1259.185 692.9005 1377.702
## 85 1255.0 1035.301 810.0916 1260.510 690.8729 1379.729
## 86 1205.3 1035.301 808.7736 1261.828 688.8572 1381.745
## 87 1167.7 1035.301 807.4632 1263.139 686.8532 1383.749
## 88 1128.8 1035.301 806.1604 1264.442 684.8606 1385.741
## 89 1143.4 1035.301 804.8649 1265.737 682.8793 1387.723
## 90 1201.8 1035.301 803.5766 1267.025 680.9091 1389.693
## 91 1260.4 1035.301 802.2954 1268.307 678.9497 1391.652
## 92 1297.6 1035.301 801.0213 1269.581 677.0011 1393.601
## 93 1349.9 1035.301 799.7541 1270.848 675.0630 1395.539
## 94 1398.3 1035.301 798.4936 1272.108 673.1353 1397.467
## 95 1431.0 1035.301 797.2398 1273.362 671.2178 1399.384
## 96 1497.1 1035.301 795.9926 1274.609 669.3104 1401.292
## 97 1461.9 1035.301 794.7518 1275.850 667.4128 1403.189
## 98 1459.6 1035.301 793.5175 1277.085 665.5250 1405.077
## 99 1455.4 1035.301 792.2893 1278.313 663.6467 1406.955
## 100 1320.2 1035.301 791.0674 1279.535 661.7779 1408.824
## 101 1380.9 1035.301 789.8515 1280.751 659.9184 1410.684Akurasi
# Akurasi
accuracy(ramalan.arima411, data.emas.test)## ME RMSE MAE MPE MAPE MASE
## Training set 2.449186 20.4481 12.44926 0.4258411 3.915648 0.998604
## Test set -63.501670 204.8856 166.60784 -10.4824622 18.195143 13.364265
## ACF1 Theil's U
## Training set -0.01318205 NA
## Test set 0.94906751 4.641143Intepretasi Nilai MAPE
## Nilai MAPE Interpretasi
## 1 <10% Kemampuan peramalan sangat baik
## 2 10%-20% Kemampuan peramalan baik
## 3 20%-50% Kemampuan peramalan layak
## 4 >50% Kemampuan peramalan burukBerdasarkan hasil di atas diperoleh nilai MAPE pada data testing sebesar 18.2% yang berarti model ARIMA (4,1,1) memiliki kemampuan peramalan baik.
Peramalan Harga Emas United Kingdom
# Ramalan Harga Emas UK 12 Bulan ke Depan
ramalan_arima411 <- forecast::forecast(emas.ts,model = arima.411, h = 12)
plot(ramalan_arima411)