G1401221060_Muhammad Rizqa Salas_Latihan Pert 4
Muhammad Rizqa Salas
2024-09-16
Proses MA(2)
Bangkitkan model MA(2) dengan 𝜃1 = 0.4 dan 𝜃2 = 0.6 sebanyak
300 data (Manual)
set.seed(1234)
n <- 300
theta1 <- 0.4
theta2 <- 0.6
e <- rnorm(n)
Y <- numeric(n)
for (t in 3:n) {
Y[t] <- e[t] + theta1 * e[t-1] + theta2 * e[t-2]
}
Y <- ts(Y) #ubah ke time series
Y## Time Series:
## Start = 1
## End = 300
## Frequency = 1
## [1] 0.000000000 0.000000000 0.471173424 -1.745463687 0.141510314
## [6] -0.729712854 -0.114842790 -0.472894305 -1.127948717 -1.443797742
## [11] -1.171879031 -1.723286222 -1.461924092 -0.845074607 0.519525249
## [16] 0.312187420 0.020572732 -1.181770516 -1.508255550 1.534249256
## [21] 0.598119283 1.012450498 -0.556369299 -0.011041246 -0.774213194
## [26] -1.449939345 -0.420758391 -1.662676381 -0.079747157 -1.556197355
## [31] 0.718835126 -0.596243221 -0.238298741 -1.070389923 -2.255260716
## [36] -2.120011486 -3.624543435 -2.913580609 -2.138714925 -1.388210999
## [41] 1.086560934 -0.768382742 -0.413123964 -1.263954490 -1.619808057
## [46] -1.534624150 -2.091327966 -2.276021753 -1.689013389 -1.457572736
## [51] -2.319068111 -1.602598402 -2.425338748 -1.807763414 -1.233628102
## [56] -0.110845196 1.775654086 0.223607057 2.285258743 -0.979456751
## [61] 1.157010823 2.116941328 1.378789117 0.845856907 -0.296314421
## [66] 1.372262398 -0.432336811 1.978634754 1.193531021 1.688995021
## [71] 0.939220832 -0.250827924 -0.544575725 0.228395751 2.109671129
## [76] 1.063681873 -0.209897795 -1.371901203 -0.865591517 -0.647903476
## [81] -0.149656546 -0.431345529 -1.546973491 -0.824704371 -0.042663743
## [86] 0.933429312 1.339180190 0.235832192 -0.022688150 -1.512804573
## [91] -0.645926233 -0.482784254 1.776147116 1.837016456 0.928600262
## [96] 0.758224871 -1.289737991 0.637690588 0.643433378 3.037205983
## [101] 1.846720429 0.963761203 0.124820225 -0.760911469 -0.987393604
## [106] -0.464896825 -1.325068067 -0.090126895 -0.115516360 0.190793420
## [111] -0.003795709 -0.758706669 -1.486955903 0.015925459 -0.303788132
## [116] 1.349650149 -0.898414089 0.169995644 -0.005795838 0.344405995
## [121] 0.201997065 -0.874229340 1.630564028 1.104622260 3.359306076
## [126] 1.260043834 0.495139539 -1.717816055 -1.620260658 -0.936111272
## [131] 0.722551062 0.794007194 -0.462634050 0.351585084 -2.085784191
## [136] -0.624664025 -1.452519789 -2.294204728 -0.048911894 -1.423796214
## [141] -0.031150775 0.887810276 0.994311578 0.983326803 0.852394746
## [146] 0.539717900 2.130822837 1.180599067 1.614343853 0.715597066
## [151] 0.065546716 0.155255589 1.451449842 -0.161522938 0.754769796
## [156] 0.885479176 0.383287177 -0.329952846 -1.431909381 -1.370070157
## [161] -1.525271048 -0.833655639 -1.027085507 -0.505202316 0.085183955
## [166] 0.677625089 2.172292654 0.977368521 0.658606072 1.301453560
## [171] 2.100227737 1.607579227 0.707237935 -1.929351923 0.482773840
## [176] -1.366418725 -0.612038329 2.091711308 0.778269986 1.887009445
## [181] -2.604510182 -1.212633564 -0.703216214 0.744407256 1.663488769
## [186] 2.596982415 2.509994691 1.149145711 1.202150481 -0.221986412
## [191] -0.194929191 -3.190036735 -2.254792788 -1.541499299 1.889370282
## [196] 1.660596411 2.121307573 -0.417402360 0.148419546 -2.592717585
## [201] -0.248475371 -0.356083018 0.755357519 1.193000349 0.703282784
## [206] 1.305574883 2.333657188 2.305625709 1.583087270 -0.433965677
## [211] -0.008123389 -0.901681487 1.584233701 -1.249824841 -0.162488303
## [216] -0.708540906 -0.232384260 -0.413640706 -0.638223437 0.143957003
## [221] 0.964153779 1.396003084 3.074671761 2.479018560 2.121198870
## [226] 0.159412341 2.827857930 1.449209496 1.338129934 0.319213246
## [231] -0.660388674 -0.292566289 -1.238021008 -0.421378522 -0.194203628
## [236] -0.879770820 -3.472672839 -2.194660779 -2.012323310 0.453156115
## [241] 0.314555376 0.793617226 -0.896931516 0.520105724 0.344695528
## [246] 0.965911715 1.889268102 1.098368887 1.240516320 0.616470695
## [251] 0.702766701 1.395722479 1.138398305 1.480149118 -0.599779870
## [256] -0.427127769 0.646946451 -0.855414425 0.654688641 -0.226112090
## [261] 1.292645476 0.284446275 0.604556738 -1.909130933 -0.271530716
## [266] -1.589079651 -0.302115008 -1.683162950 -0.233358766 0.107894475
## [271] 0.192979110 2.103113218 1.217795463 2.311862960 2.831191653
## [276] 0.737827135 1.777815630 0.919360158 0.873633013 0.908532408
## [281] -0.887903437 0.256870201 -1.119996454 0.815395259 -0.906539838
## [286] 0.483034891 -1.205622129 -0.392733748 -2.159077875 -0.640167479
## [291] 0.105101596 0.654485299 -1.101664171 -1.217076868 0.348092030
## [296] 2.099902382 0.542511208 0.249165265 -0.498077915 0.240584445Bangkitkan model MA(2) dengan 𝜃1 = 0.4 dan 𝜃2 = 0.6 sebanyak
300 data (arima.sim)
set.seed(1234)
Y_arima <- arima.sim(model = list(ma = c(0.4, 0.6)), n = 300)
Y_arima## Time Series:
## Start = 1
## End = 300
## Frequency = 1
## [1] 0.471173424 -1.745463687 0.141510314 -0.729712854 -0.114842790
## [6] -0.472894305 -1.127948717 -1.443797742 -1.171879031 -1.723286222
## [11] -1.461924092 -0.845074607 0.519525249 0.312187420 0.020572732
## [16] -1.181770516 -1.508255550 1.534249256 0.598119283 1.012450498
## [21] -0.556369299 -0.011041246 -0.774213194 -1.449939345 -0.420758391
## [26] -1.662676381 -0.079747157 -1.556197355 0.718835126 -0.596243221
## [31] -0.238298741 -1.070389923 -2.255260716 -2.120011486 -3.624543435
## [36] -2.913580609 -2.138714925 -1.388210999 1.086560934 -0.768382742
## [41] -0.413123964 -1.263954490 -1.619808057 -1.534624150 -2.091327966
## [46] -2.276021753 -1.689013389 -1.457572736 -2.319068111 -1.602598402
## [51] -2.425338748 -1.807763414 -1.233628102 -0.110845196 1.775654086
## [56] 0.223607057 2.285258743 -0.979456751 1.157010823 2.116941328
## [61] 1.378789117 0.845856907 -0.296314421 1.372262398 -0.432336811
## [66] 1.978634754 1.193531021 1.688995021 0.939220832 -0.250827924
## [71] -0.544575725 0.228395751 2.109671129 1.063681873 -0.209897795
## [76] -1.371901203 -0.865591517 -0.647903476 -0.149656546 -0.431345529
## [81] -1.546973491 -0.824704371 -0.042663743 0.933429312 1.339180190
## [86] 0.235832192 -0.022688150 -1.512804573 -0.645926233 -0.482784254
## [91] 1.776147116 1.837016456 0.928600262 0.758224871 -1.289737991
## [96] 0.637690588 0.643433378 3.037205983 1.846720429 0.963761203
## [101] 0.124820225 -0.760911469 -0.987393604 -0.464896825 -1.325068067
## [106] -0.090126895 -0.115516360 0.190793420 -0.003795709 -0.758706669
## [111] -1.486955903 0.015925459 -0.303788132 1.349650149 -0.898414089
## [116] 0.169995644 -0.005795838 0.344405995 0.201997065 -0.874229340
## [121] 1.630564028 1.104622260 3.359306076 1.260043834 0.495139539
## [126] -1.717816055 -1.620260658 -0.936111272 0.722551062 0.794007194
## [131] -0.462634050 0.351585084 -2.085784191 -0.624664025 -1.452519789
## [136] -2.294204728 -0.048911894 -1.423796214 -0.031150775 0.887810276
## [141] 0.994311578 0.983326803 0.852394746 0.539717900 2.130822837
## [146] 1.180599067 1.614343853 0.715597066 0.065546716 0.155255589
## [151] 1.451449842 -0.161522938 0.754769796 0.885479176 0.383287177
## [156] -0.329952846 -1.431909381 -1.370070157 -1.525271048 -0.833655639
## [161] -1.027085507 -0.505202316 0.085183955 0.677625089 2.172292654
## [166] 0.977368521 0.658606072 1.301453560 2.100227737 1.607579227
## [171] 0.707237935 -1.929351923 0.482773840 -1.366418725 -0.612038329
## [176] 2.091711308 0.778269986 1.887009445 -2.604510182 -1.212633564
## [181] -0.703216214 0.744407256 1.663488769 2.596982415 2.509994691
## [186] 1.149145711 1.202150481 -0.221986412 -0.194929191 -3.190036735
## [191] -2.254792788 -1.541499299 1.889370282 1.660596411 2.121307573
## [196] -0.417402360 0.148419546 -2.592717585 -0.248475371 -0.356083018
## [201] 0.755357519 1.193000349 0.703282784 1.305574883 2.333657188
## [206] 2.305625709 1.583087270 -0.433965677 -0.008123389 -0.901681487
## [211] 1.584233701 -1.249824841 -0.162488303 -0.708540906 -0.232384260
## [216] -0.413640706 -0.638223437 0.143957003 0.964153779 1.396003084
## [221] 3.074671761 2.479018560 2.121198870 0.159412341 2.827857930
## [226] 1.449209496 1.338129934 0.319213246 -0.660388674 -0.292566289
## [231] -1.238021008 -0.421378522 -0.194203628 -0.879770820 -3.472672839
## [236] -2.194660779 -2.012323310 0.453156115 0.314555376 0.793617226
## [241] -0.896931516 0.520105724 0.344695528 0.965911715 1.889268102
## [246] 1.098368887 1.240516320 0.616470695 0.702766701 1.395722479
## [251] 1.138398305 1.480149118 -0.599779870 -0.427127769 0.646946451
## [256] -0.855414425 0.654688641 -0.226112090 1.292645476 0.284446275
## [261] 0.604556738 -1.909130933 -0.271530716 -1.589079651 -0.302115008
## [266] -1.683162950 -0.233358766 0.107894475 0.192979110 2.103113218
## [271] 1.217795463 2.311862960 2.831191653 0.737827135 1.777815630
## [276] 0.919360158 0.873633013 0.908532408 -0.887903437 0.256870201
## [281] -1.119996454 0.815395259 -0.906539838 0.483034891 -1.205622129
## [286] -0.392733748 -2.159077875 -0.640167479 0.105101596 0.654485299
## [291] -1.101664171 -1.217076868 0.348092030 2.099902382 0.542511208
## [296] 0.249165265 -0.498077915 0.240584445 -0.240119231 -0.993767765Buat plot time series, plot ACF, plot PACF dan plot EACF. Idenifikasi
apakah data tersebut stasioner
Plot Time Series
plot(Y, main = "MA(2) Time Series", ylab = "Yt", xlab = "Time")
Plot ACF
acf(Y, main = "ACF of MA(2)")
Plot PACF
pacf(Y, main = "PACF of MA(2)")
Plot EACF
library(TSA)## Warning: package 'TSA' was built under R version 4.3.3##
## Attaching package: 'TSA'## The following objects are masked from 'package:stats':
##
## acf, arima## The following object is masked from 'package:utils':
##
## tareacf(Y)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x o o o o o o o o o o x x
## 1 x x x o o o o o o o o o o o
## 2 x x x x o o o o o o o o o o
## 3 x x x x o o o o o o o o o o
## 4 x x x x x o o o o o o o o o
## 5 x x x x x o x o o o o o o o
## 6 x x x o x o o x o o o o o o
## 7 x o x o x o o x o o o o o oIdentifikasi Stasioner
polyroot(c(1, -0.4, -0.6)) # contoh model MA(2) dengan 𝜃1 = 0.4 dan 𝜃2 = 0.6## [1] 1.000000-0i -1.666667+0iBuat scatterplot antara 𝑌𝑡 dengan 𝑌𝑡−1, 𝑌𝑡 dengan 𝑌𝑡−2 serta 𝑌𝑡
dengan 𝑌𝑡−3
Scatterplot Yt dengan Yt-1
plot(Y[2:300], Y[1:299], main = "Scatterplot Yt vs Yt-1", xlab = "Yt-1", ylab = "Yt")
Scatterplot Yt dengan Yt-2
plot(Y[3:300], Y[1:298], main = "Scatterplot Yt vs Yt-2", xlab = "Yt-2", ylab = "Yt")
Scatterplot Yt dengan Yt-3
plot(Y[4:300], Y[1:297], main = "Scatterplot Yt vs Yt-3", xlab = "Yt-3", ylab = "Yt")
Hitung autokorelasi masing masing baik dari data bangkitan maupun secara teoritis
Autokorelasi Empiris
acf(Y)
Autokorelasi Teoritis
rho1 <- theta1 / (1 + theta1^2 + theta2^2)
rho2 <- theta2 / (1 + theta1^2 + theta2^2)
rho1 # Lag 1 autokorelasi## [1] 0.2631579rho2 # Lag 2 autokorelasi## [1] 0.3947368Proses AR(2)
Bangkitkan model AR(2) dengan 𝜙1 = 0.5 dan 𝜙2 = 0.2 sebanyak
300 data (Manual)
set.seed(2045)
n <- 300
phi1 <- 0.5
phi2 <- 0.2
e <- rnorm(n)
Y <- numeric(n)
for (t in 3:n) {
Y[t] <- phi1 * Y[t-1] + phi2 * Y[t-2] + e[t]
}
Y <- ts(Y) # ubah ke time series
Y## Time Series:
## Start = 1
## End = 300
## Frequency = 1
## [1] 0.000000000 0.000000000 0.349020405 1.270425965 0.017362695
## [6] 1.474514746 2.596405374 2.319602862 2.153949875 0.278376633
## [11] 0.912608295 -0.366786480 -1.573453956 1.437720486 -0.552850198
## [16] -1.345251829 0.825378964 1.088179363 0.589119857 0.350507953
## [21] -0.961247294 -0.384974917 -0.462455657 -1.106286974 -0.398312256
## [26] -2.548595257 -2.023777688 -1.529759184 -1.252899760 -0.098629642
## [31] -1.217998097 -2.068706922 -2.612921715 -1.605997949 -1.206676609
## [36] 0.100573712 0.296889935 2.213252575 1.304099028 1.818767455
## [41] 0.718669521 -0.042567024 1.131770309 -0.063905989 0.110665093
## [46] -0.536175954 -0.464512163 -1.754010488 -0.179975774 -0.148920498
## [51] -0.119640196 0.349312384 1.222631169 -0.474740673 -1.350357255
## [56] -0.436715506 -1.637800653 -0.024276175 -0.397876145 0.430292208
## [61] -1.176418631 -0.278187905 0.528451376 0.396331205 0.756909308
## [66] 1.294856282 1.155846487 -0.822637995 -1.278652933 -1.693699790
## [71] -1.648627171 -0.775305761 -1.773598338 -1.557341727 -0.767990769
## [76] -1.485018115 -1.652698099 -2.548366279 -0.627212756 -2.371863661
## [81] -1.311007654 -1.287429345 1.560994051 2.209376465 1.540482426
## [86] -0.017561349 -0.566208334 -0.202630946 1.051230863 0.276965668
## [91] 1.325417529 0.845468334 0.220523476 1.631390439 0.537581507
## [96] 0.726615580 -0.264136582 -0.712478821 -1.499169555 -0.338776967
## [101] -2.259311778 -1.539737525 -3.047990681 -1.264124635 -0.955427324
## [106] -0.181308526 -1.187961743 0.508909767 0.799150781 0.466130367
## [111] 0.990659051 0.912713747 0.713904156 0.624013578 1.307770701
## [116] 0.879917311 1.773883285 0.692647268 1.050279788 1.793408686
## [121] 1.900245965 -0.189361354 -0.082761708 -0.241123849 -0.262185962
## [126] -0.144812882 0.429232814 -0.770622830 -1.212353609 -0.970526724
## [131] -0.329333629 0.216480677 0.077560660 0.960968454 -0.798930504
## [136] 0.228057490 -0.758582354 -1.431085323 -1.627825976 -1.220806258
## [141] -1.100277332 -0.004178275 1.154885048 2.485164950 1.795042409
## [146] 1.138274537 1.289111387 1.023972788 -0.504250538 0.428296980
## [151] -0.281177386 1.152998624 1.149967889 -0.098607504 0.673731822
## [156] 2.045183555 2.224055194 1.485682007 0.424982863 -0.434563725
## [161] -0.725783878 -0.726177063 -1.132231088 -0.009038700 0.674843742
## [166] -0.321853748 0.939558171 -0.856933542 -2.484562816 -1.179468992
## [171] -2.869532925 -2.383501302 0.013953327 0.504269014 0.420271938
## [176] -0.807755968 -1.592064335 1.308270116 -0.106099037 -1.205737648
## [181] 0.705752391 -0.297404289 1.374343783 -0.573189714 -0.085318873
## [186] 0.244207194 -1.599592854 -0.349845558 -0.056991413 0.315291394
## [191] 1.048847433 0.662831772 0.959636404 0.294246989 0.034575587
## [196] 0.137871293 -0.883018780 -0.867146848 0.248802076 -0.232012155
## [201] 1.002976050 0.484213173 1.626911963 0.860741733 0.351823963
## [206] 1.459381221 0.844250125 2.912962318 3.477182751 3.271074924
## [211] 0.486826876 0.854291297 0.371641491 1.270969422 0.715391739
## [216] 2.073519283 0.660422091 1.320845178 -0.650887420 -0.224463500
## [221] -0.599988230 0.790442044 1.472566343 0.675874551 0.428479628
## [226] 1.827851685 1.261117783 1.045867094 0.882852385 -2.334116955
## [231] -2.103603557 -0.658874256 -1.942833888 0.615523016 -0.006842329
## [236] 0.995076257 1.694676891 0.433146991 2.125971999 1.478629905
## [241] 3.083769633 1.915815714 2.425950795 3.182572762 2.738938326
## [246] 2.872239794 1.131842641 1.258204028 1.504419774 0.420148676
## [251] 0.012389324 -0.330958229 -1.558435697 -0.486396443 -0.704596653
## [256] -1.772139555 0.409627494 0.449945403 0.830116243 0.666561563
## [261] 2.349910446 1.825761859 1.701351434 2.603741729 0.937504850
## [266] -0.099971818 0.178723265 0.122184822 1.686884383 -0.385524912
## [271] 0.229673936 0.856964087 -0.105920342 -1.174953821 0.051775014
## [276] 0.591220110 -0.558995867 -0.197546961 1.176973506 0.057601942
## [281] 0.956296804 -1.343840432 -0.485971570 0.494059269 -0.155562760
## [286] 0.102526731 -1.133036582 0.729583434 0.370684823 0.888165023
## [291] 1.007059306 1.256546000 0.924997834 -0.550435936 0.534597153
## [296] 2.580043778 2.337655235 0.998758660 0.652649717 0.941998246Bangkitkan model AR(2) dengan 𝜙1 = 0.5 dan 𝜙2 = 0.2 sebanyak
300 data (arima.sim)
set.seed(2045)
Y_arima <- arima.sim(model = list(ar = c(0.5, 0.2)), n = 300)
Y_arima## Time Series:
## Start = 1
## End = 300
## Frequency = 1
## [1] -2.548034118 -2.023349904 -1.529433064 -1.252651144 -0.098440109
## [6] -1.217853608 -2.068596771 -2.612837741 -1.605933932 -1.206627806
## [11] 0.100610917 0.296918298 2.213274198 1.304115512 1.818780021
## [16] 0.718679101 -0.042559721 1.131775877 -0.063901745 0.110668328
## [21] -0.536173487 -0.464510283 -1.754009054 -0.179974681 -0.148919665
## [26] -0.119639560 0.349312868 1.222631538 -0.474740392 -1.350357040
## [31] -0.436715343 -1.637800528 -0.024276080 -0.397876073 0.430292263
## [36] -1.176418589 -0.278187873 0.528451401 0.396331223 0.756909322
## [41] 1.294856293 1.155846495 -0.822637989 -1.278652928 -1.693699786
## [46] -1.648627168 -0.775305758 -1.773598336 -1.557341726 -0.767990768
## [51] -1.485018114 -1.652698099 -2.548366279 -0.627212755 -2.371863661
## [56] -1.311007654 -1.287429345 1.560994051 2.209376465 1.540482426
## [61] -0.017561349 -0.566208334 -0.202630946 1.051230864 0.276965668
## [66] 1.325417529 0.845468334 0.220523476 1.631390439 0.537581507
## [71] 0.726615580 -0.264136582 -0.712478821 -1.499169555 -0.338776967
## [76] -2.259311778 -1.539737525 -3.047990681 -1.264124635 -0.955427324
## [81] -0.181308526 -1.187961743 0.508909767 0.799150781 0.466130367
## [86] 0.990659051 0.912713747 0.713904156 0.624013578 1.307770701
## [91] 0.879917311 1.773883285 0.692647268 1.050279788 1.793408686
## [96] 1.900245965 -0.189361354 -0.082761708 -0.241123849 -0.262185962
## [101] -0.144812882 0.429232814 -0.770622830 -1.212353609 -0.970526724
## [106] -0.329333629 0.216480677 0.077560660 0.960968454 -0.798930504
## [111] 0.228057490 -0.758582354 -1.431085323 -1.627825976 -1.220806258
## [116] -1.100277332 -0.004178275 1.154885048 2.485164950 1.795042409
## [121] 1.138274537 1.289111387 1.023972788 -0.504250538 0.428296980
## [126] -0.281177386 1.152998624 1.149967889 -0.098607504 0.673731822
## [131] 2.045183555 2.224055194 1.485682007 0.424982863 -0.434563725
## [136] -0.725783878 -0.726177063 -1.132231088 -0.009038700 0.674843742
## [141] -0.321853748 0.939558171 -0.856933542 -2.484562816 -1.179468992
## [146] -2.869532925 -2.383501302 0.013953327 0.504269014 0.420271938
## [151] -0.807755968 -1.592064335 1.308270116 -0.106099037 -1.205737648
## [156] 0.705752391 -0.297404289 1.374343783 -0.573189714 -0.085318873
## [161] 0.244207194 -1.599592854 -0.349845558 -0.056991413 0.315291394
## [166] 1.048847433 0.662831772 0.959636404 0.294246989 0.034575587
## [171] 0.137871293 -0.883018780 -0.867146848 0.248802076 -0.232012155
## [176] 1.002976050 0.484213173 1.626911963 0.860741733 0.351823963
## [181] 1.459381221 0.844250125 2.912962318 3.477182751 3.271074924
## [186] 0.486826876 0.854291297 0.371641491 1.270969422 0.715391739
## [191] 2.073519283 0.660422091 1.320845178 -0.650887420 -0.224463500
## [196] -0.599988230 0.790442044 1.472566343 0.675874551 0.428479628
## [201] 1.827851685 1.261117783 1.045867094 0.882852385 -2.334116955
## [206] -2.103603557 -0.658874256 -1.942833888 0.615523016 -0.006842329
## [211] 0.995076257 1.694676891 0.433146991 2.125971999 1.478629905
## [216] 3.083769633 1.915815714 2.425950795 3.182572762 2.738938326
## [221] 2.872239794 1.131842641 1.258204028 1.504419774 0.420148676
## [226] 0.012389324 -0.330958229 -1.558435697 -0.486396443 -0.704596653
## [231] -1.772139555 0.409627494 0.449945403 0.830116243 0.666561563
## [236] 2.349910446 1.825761859 1.701351434 2.603741729 0.937504850
## [241] -0.099971818 0.178723265 0.122184822 1.686884383 -0.385524912
## [246] 0.229673936 0.856964087 -0.105920342 -1.174953821 0.051775014
## [251] 0.591220110 -0.558995867 -0.197546961 1.176973506 0.057601942
## [256] 0.956296804 -1.343840432 -0.485971570 0.494059269 -0.155562760
## [261] 0.102526731 -1.133036582 0.729583434 0.370684823 0.888165023
## [266] 1.007059306 1.256546000 0.924997834 -0.550435936 0.534597153
## [271] 2.580043778 2.337655235 0.998758660 0.652649717 0.941998246
## [276] 0.527504110 1.111115683 -0.627950139 -0.941101697 -1.723258979
## [281] -0.684048818 -0.572227212 0.650246462 1.384110378 0.439484383
## [286] -0.961953461 -0.572682964 -0.689930160 -0.834167422 -1.504773911
## [291] 1.062257278 -0.590214172 -0.427143672 -1.854268270 -0.859444018
## [296] -0.245741525 -0.220231163 3.066928007 0.848696962 1.205644524Buat plot time series, plot ACF, plot PACF dan plot EACF. Idenifikasi apakah data tersebut stasioner
Plot Time Series
plot(Y, main = "AR(2) Time Series", ylab = "Yt", xlab = "Time")
Plot ACF
acf(Y, main = "ACF of AR(2)")
Plot PACF
pacf(Y, main = "PACF of AR(2)")
Plot EACF
library(TSA)
eacf(Y)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x o o o o o o o o x
## 1 x o o o o o o o o o o o o o
## 2 x x o o o o o o o o o o o o
## 3 x x o o o o o o o o o o o o
## 4 x x x o o o o o o o o o o o
## 5 x x o o o o o o o o o o o o
## 6 x x o o x o o o o o o o o o
## 7 x x o o x x o o o o o o o oIdentifikasi Stasioneritas
polyroot(c(1, -0.5, -0.2)) # Untuk 𝜙1 = 0.5 dan 𝜙2 = 0.2## [1] 1.311738+0i -3.811738+0iBuat scatterplot antara 𝑌𝑡 dengan 𝑌𝑡−1, 𝑌𝑡 dengan 𝑌𝑡−2 serta 𝑌𝑡
dengan 𝑌𝑡−3
Scatterplot Yt dengan Yt-1
plot(Y[2:300], Y[1:299], main = "Scatterplot Yt vs Yt-1", xlab = "Yt-1", ylab = "Yt")
Scatterplot Yt dengan Yt-2
plot(Y[3:300], Y[1:298], main = "Scatterplot Yt vs Yt-2", xlab = "Yt-2", ylab = "Yt")
Scatterplot Yt dengan Yt-3
plot(Y[4:300], Y[1:297], main = "Scatterplot Yt vs Yt-3", xlab = "Yt-3", ylab = "Yt")
Hitung autokorelasi masing masing baik dari data bangkitan maupun secara teoritis
Autokorelasi Teoritis menggunakan persamaan Yule-Walker
phi1 <- 0.5
phi2 <- 0.2Lag 1 dan Lag 2
rho1 <- phi1 / (1 - phi2)
rho2 <- (phi1 * rho1 + phi2) / (1 - phi2)
rho1 # Lag 1 autokorelasi teoritis## [1] 0.625rho2 # Lag 2 autokorelasi teoritis## [1] 0.640625Proses ARMA (2,2)
Bangkitkan model ARMA(2,2) dengan 𝜃1 = 0.4, 𝜃2 = 0.6 ,𝜙1 = 0.5
dan 𝜙2 = 0.2 sebanyak 300 data (Manual)
set.seed(2045)
n <- 300
phi1 <- 0.5
phi2 <- 0.2
theta1 <- 0.4
theta2 <- 0.6
e <- rnorm(n)
Y <- numeric(n)
for (t in 3:n) {
Y[t] <- phi1 * Y[t-1] + phi2 * Y[t-2] + e[t] - theta1 * e[t-1] - theta2 * e[t-2]
}
Y <- ts(Y) # convert to time series
Y## Time Series:
## Start = 1
## End = 300
## Frequency = 1
## [1] 0.00000000 0.00000000 -0.07756096 0.95904978 -0.87142022 0.58536034
## [7] 1.90196493 0.32523265 -0.38612749 -2.01638137 -0.52269905 -0.92293243
## [13] -1.99266002 2.27318079 -0.19453374 -1.99487654 1.68899002 1.56045248
## [19] -0.34498242 -0.54079446 -1.45701645 -0.21237717 0.26706567 -0.69124755
## [25] 0.32096863 -1.72603738 -0.76576329 0.80859567 0.57303163 1.32020365
## [31] -0.42694523 -1.52243574 -1.05472078 0.68033338 1.00342870 1.54680737
## [37] 0.98063916 2.03413160 0.24064820 -0.03083578 -0.79130608 -1.42130232
## [43] 0.71759006 -0.49107798 -0.54283781 -0.54210077 -0.31644264 -1.24650143
## [49] 0.80033467 0.97547530 0.04791286 0.48652030 1.15468998 -1.17338084
## [55] -1.89403989 0.38827164 -0.65290022 0.89287330 0.59451465 0.60400832
## [61] -1.10980987 -0.06579581 1.34557769 0.35186338 0.28130599 0.75429383
## [67] 0.18375838 -2.06189036 -1.64310563 -0.68865582 -0.20395550 0.90036498
## [73] -0.47429973 -0.38271894 0.91910492 -0.24341677 -0.59789639 -0.99627617
## [79] 1.38375262 -0.59195879 0.01406546 0.66009191 2.86257038 2.35743645
## [85] -0.27986459 -1.95938020 -1.48347325 0.03438920 1.47200824 -0.02194811
## [91] 0.58389274 0.14912192 -0.91291438 1.03590005 -0.24728875 -0.46725129
## [97] -0.87733172 -1.04279354 -1.05569608 0.68837815 -1.22429926 -0.43274663
## [103] -1.07650860 0.87891415 1.37901694 0.95933718 -0.54218194 1.09287958
## [109] 1.30836392 -0.15887581 0.32471644 0.23677191 -0.24557677 -0.20917633
## [115] 0.62982278 -0.01759912 0.63725394 -0.54485643 -0.29110909 0.95770841
## [121] 0.55271462 -2.02550495 -1.14716475 -0.09440235 -0.11607940 0.10473581
## [127] 0.64446954 -0.85542823 -1.16164417 -0.02321158 0.78628923 0.93053016
## [133] 0.18856857 0.80005578 -1.22985428 -0.02895138 -0.37044705 -1.26448688
## [139] -0.60024243 0.28897533 0.36474076 1.16841641 1.81672276 2.02571790
## [145] 0.10804540 -1.07084140 -0.24322387 -0.17463649 -1.68730649 0.01561352
## [151] -0.14994585 1.00849139 0.85747487 -1.25039383 0.02319409 1.83485533
## [157] 1.00174268 -0.63105020 -1.50372306 -1.49596608 -0.80694811 -0.17512528
## [163] -0.40628994 0.87955997 1.35779787 -0.58636803 0.66339343 -1.03964456
## [169] -2.70552430 0.32851626 -0.90700764 -0.52800674 2.68907360 1.92878846
## [175] 0.21019234 -1.27842615 -1.52112511 2.42974943 0.32583152 -1.94826010
## [181] 1.25170687 0.14373734 1.06985406 -0.94448465 -0.68064926 0.62224857
## [187] -1.64608441 0.14346727 1.04270252 0.54799529 0.95692572 0.05411796
## [193] 0.06519524 -0.48730664 -0.65890505 -0.05250713 -0.95891265 -0.59666211
## [199] 1.12547208 0.18875512 0.94649967 0.22223005 0.83144106 -0.08055096
## [205] -0.96861991 0.80220660 0.04940326 1.69963354 1.80544775 0.13242443
## [211] -2.90791274 -1.30308441 -0.26217115 0.60973805 -0.01598092 1.02478093
## [217] -0.59822067 -0.18743523 -1.57547875 -0.75661564 -0.11967038 1.16511544
## [223] 1.51638246 -0.38741721 -0.72541000 1.25093510 0.27288933 -0.55529103
## [229] -0.29216512 -3.31477817 -1.69966821 1.58303734 -0.41712205 1.78798112
## [235] 0.91264880 0.62849938 1.30075179 -0.84176952 0.93590707 0.36835291
## [241] 1.21673447 -0.20487008 -0.19063727 1.06270302 0.01033874 -0.13287919
## [247] -1.66041627 -0.91787690 0.32203258 -0.93654165 -1.05832201 -0.58800316
## [253] -1.43348600 0.33555277 0.42502334 -1.19846303 1.54124131 1.34937814
## [259] 0.40436159 0.06454782 1.58521607 0.48586074 -0.43889958 0.82774404
## [265] -1.12480270 -2.03721880 -0.34379092 0.11067861 1.53077650 -1.13358956
## [271] -0.62824673 0.99640946 -0.58651034 -1.64676414 0.58530875 1.27548240
## [277] -0.82654892 -0.32868068 1.59138981 -0.29465928 0.22707192 -1.76092032
## [283] -0.52221348 1.49475216 -0.06160353 -0.13168373 -1.08070962 1.12128203
## [289] 0.75867340 0.30214103 0.42938240 0.32082326 -0.18185615 -1.67436267
## [295] 0.19977283 2.69646648 0.98487943 -1.48432970 -1.14944689 0.08168316Bangkitkan model ARMA(2,2) dengan 𝜃1 = 0.4, 𝜃2 = 0.6 ,𝜙1 = 0.5
dan 𝜙2 = 0.2 sebanyak 300 data (arima.sim)
set.seed(2045)
Y_arima <- arima.sim(model = list(ar = c(0.5, 0.2), ma = c(0.4, 0.6)), n = 300)
Y_arima## Time Series:
## Start = 1
## End = 300
## Frequency = 1
## [1] -3.868114041 -3.078831148 -1.517462932 -2.009050969 -2.614978100
## [6] -4.171122651 -3.892329273 -3.416781922 -1.345659950 -0.386859291
## [11] 2.392373554 2.367549859 3.668370686 2.228645125 1.336168275
## [16] 1.545950562 0.363265998 0.764167992 -0.530251140 -0.612581682
## [21] -2.261519548 -1.160286217 -1.273316299 -0.287193249 0.212104473
## [26] 1.290572360 0.223899495 -0.806674617 -1.261702655 -2.622701089
## [31] -0.941425649 -1.390266937 0.256576098 -1.243027394 -0.490580002
## [36] -0.288674941 0.440799030 1.232512629 1.835418738 2.127934592
## [41] 0.416614375 -0.914200234 -2.698743756 -3.093298844 -2.450976501
## [46] -3.072896943 -2.731964518 -2.455086462 -2.726619458 -2.707499806
## [51] -4.100456388 -2.638178127 -4.151768531 -2.636080772 -3.234950603
## [56] 0.259417721 2.061316479 3.360829442 1.924257500 0.351056582
## [61] -0.439651089 0.630453485 0.575879446 2.066942315 1.541814747
## [66] 1.353961327 2.226880830 1.322451768 1.920482446 0.349058555
## [71] -0.382164105 -1.942643033 -1.365932081 -3.294324297 -2.646728416
## [76] -5.019472758 -3.407163423 -3.289871586 -1.321954236 -1.833741547
## [81] -0.075060045 0.289937643 1.091136540 1.656601666 1.588655587
## [86] 1.673385086 1.457203489 1.985718626 1.777433739 2.910512630
## [91] 1.930150969 2.391668666 2.629108962 3.247777312 1.646782243
## [96] 0.981641329 -0.387845345 -0.408292527 -0.394361577 0.213996083
## [101] -0.685817434 -1.263063053 -1.917841866 -1.444956484 -0.497568809
## [106] -0.033447247 1.121881124 -0.368006726 0.485066361 -1.146717660
## [111] -1.597683770 -2.655409517 -2.730587842 -2.565295420 -1.176772962
## [116] 0.493047339 2.944612004 3.482039418 3.347390470 2.821446647
## [121] 2.222582064 0.678805409 0.840980437 -0.412408916 1.297505858
## [126] 1.442460908 1.053178826 1.324269554 2.255511782 3.446367710
## [131] 3.602414218 2.353688783 0.626838624 -0.644619650 -1.277228849
## [136] -1.858172240 -0.897637373 -0.008110391 -0.057339471 1.215722917
## [141] -0.674222523 -2.263601330 -2.687454243 -4.832058211 -4.238995867
## [146] -2.661166949 -0.920250436 0.630351540 -0.337085784 -1.663003559
## [151] 0.186790802 -0.538029591 -0.463215192 0.159797910 -0.738545921
## [156] 1.678833502 -0.201894774 0.510011511 -0.133834183 -1.553101300
## [161] -0.843158383 -1.156685349 0.082587494 1.140769142 1.271545581
## [166] 1.854077573 1.075800614 0.728056225 0.328249722 -0.807124910
## [171] -1.137631584 -0.627867931 -0.652779433 1.059452434 0.746196301
## [176] 2.422382863 1.802034422 1.672267834 2.116555846 1.639096991
## [181] 4.126291100 5.148917753 6.409725416 3.881566496 3.011667002
## [186] 1.005454135 1.932200796 1.446764402 3.122257632 1.919064847
## [191] 2.829125585 0.273703906 0.307688639 -1.080306082 0.415768652
## [196] 1.428750222 1.739166315 1.582369254 2.404768267 2.249346234
## [201] 2.647025218 2.057869892 -1.353455745 -2.507538908 -2.900785852
## [206] -3.468545724 -0.556935093 -0.926333456 1.361653134 2.088601996
## [211] 1.708063501 3.316036930 2.588906900 4.950804794 4.036501510
## [216] 5.042538860 5.302442508 5.467537908 5.877358781 3.924101554
## [221] 3.434284960 2.686806969 1.776839002 1.083100658 -0.073913294
## [226] -1.683385394 -1.308345659 -1.834216649 -2.345816082 -0.721986320
## [231] -0.449487332 1.255870901 1.268575302 3.114604817 3.165662975
## [236] 3.841602445 4.379739419 2.999812402 1.837275159 0.701237447
## [241] 0.133691037 1.842992270 0.362539734 1.087594601 0.717518714
## [246] 0.374669655 -0.703143505 -0.481758719 -0.093042176 -0.291442814
## [251] -0.066413242 0.762557202 0.409863167 1.685521684 -0.926760546
## [256] -0.449729660 -0.506633618 -0.249521994 0.336737189 -1.185363545
## [261] 0.337884840 -0.017303753 1.474189013 1.584736209 2.192268736
## [266] 2.031851818 0.573490797 0.869421479 2.463621078 3.690431038
## [271] 3.481847021 2.454746322 1.802313329 1.295893239 1.887316275
## [276] 0.132998601 -0.525612343 -2.476469741 -1.938013428 -1.879802127
## [281] 0.010926286 1.300872636 1.383276411 0.044306519 -0.693773719
## [286] -1.496175422 -1.453749265 -2.252398976 -0.040152740 -1.068175607
## [291] -0.025874974 -2.379254242 -1.857437530 -1.702080094 -0.834194184
## [296] 2.831390626 1.943329467 3.385280112 1.976098970 0.569419977Buat plot time series, plot ACF, plot PACF dan plot EACF. Idenifikasi
apakah data tersebut stasioner
Plot Time Series
plot(Y, main = "ARMA(2,2) Time Series", ylab = "Yt", xlab = "Time")
Plot ACF
acf(Y, main = "ACF of ARMA(2,2)")
Plot PACF
pacf(Y, main = "PACF of ARMA(2,2)")
Plot EACF (gunakan paket TSA)
library(TSA)
eacf(Y)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x o x o o o o o o o o o o
## 1 x x o o o o o o o o o o o o
## 2 o x o o o o o o o o o o o o
## 3 o x o o o o o o o o o o o o
## 4 o x x o o o o o o o o o o o
## 5 o x o o x x o o o o o o o o
## 6 o o x x x x o o o o o o o o
## 7 x o x x x x o o o o o o o oIdentifikasi Stasioneritas
library(tseries)## Warning: package 'tseries' was built under R version 4.3.2## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zooadf.test(Y)## Warning in adf.test(Y): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: Y
## Dickey-Fuller = -8.5259, Lag order = 6, p-value = 0.01
## alternative hypothesis: stationaryBuat scatterplot antara 𝑌𝑡 dengan 𝑌𝑡−1, 𝑌𝑡 dengan 𝑌𝑡−2 serta 𝑌𝑡dengan 𝑌𝑡−3
Scatterplot Yt dengan Yt-1
plot(Y[2:300], Y[1:299], main = "Scatterplot Yt vs Yt-1", xlab = "Yt-1", ylab = "Yt")
Scatterplot Yt dengan Yt-2
plot(Y[3:300], Y[1:298], main = "Scatterplot Yt vs Yt-2", xlab = "Yt-2", ylab = "Yt")
Scatterplot Yt dengan Yt-3
plot(Y[4:300], Y[1:297], main = "Scatterplot Yt vs Yt-3", xlab = "Yt-3", ylab = "Yt")
Hitung autokorelasi masing masing baik dari data bangkitan maupun secara teoritis
acf(Y)