#install.packages("forecast")
#install.packages("graphics")
#install.packages("TTR")
#install.packages("TSA")Metode Pemulusan Deret Waktu: Moving Average, Exponential Smoothing, dan Holt-Winters
Pengantar Pemulusan Deret Waktu
Pemulusan deret waktu (time series smoothing) merupakan teknik untuk mengenali pola umum dalam data dengan cara mengurangi variasi acak atau fluktuasi lokal. Tujuannya adalah agar pola utama seperti tren, siklus, atau musiman menjadi lebih mudah diamati.
Prinsip dasar pemulusan adalah melakukan penghalusan nilai data melalui proses rata-rata, sehingga nilai ekstrem (naik-turun tajam) dapat dikurangi tanpa mengubah arah pola secara keseluruhan.
Metode yang akan dibahas: • Single Moving Average • Double Moving Average • Single Exponential Smoothing • Double Exponential Smoothing • MetodeWinter untuk musiman aditif • MetodeWinter untuk musiman multiplikatif
1 Library/Packages
Package R yang akan digunakan pada perkuliahan Analisis Deret Waktu sesi UTS adalah: forecast, graphics, TTR, TSA . Jika package tersebut belum ada, silakan install terlebih dahulu.
Jika sudah ada, silakan panggil library package tersebut.
library("forecast")Warning: package 'forecast' was built under R version 4.5.1Registered S3 method overwritten by 'quantmod':
method from
as.zoo.data.frame zoo library("graphics")
library("TTR")Warning: package 'TTR' was built under R version 4.5.1library("TSA")Warning: package 'TSA' was built under R version 4.5.1Registered S3 methods overwritten by 'TSA':
method from
fitted.Arima forecast
plot.Arima forecast
Attaching package: 'TSA'The following objects are masked from 'package:stats':
acf, arimaThe following object is masked from 'package:utils':
tar2 Impor Data
library(rio) #install packages jika belum adaWarning: package 'rio' was built under R version 4.5.1data1 <- import("https://raw.githubusercontent.com/windipngsti/Praktikum-STA1341/refs/heads/main/Pertemuan%202/Data_1.csv")
data2 <- import("https://raw.githubusercontent.com/windipngsti/Praktikum-STA1341/refs/heads/main/Pertemuan%202/Data_2.csv")
#via csv gunakan read.csv("namafile.csv")
#data1 <- read.csv("Data_1.csv")
# menampilkan 5 data pertama
head(data1)head(data2)3 Eksplorasi Data
View() : menampilkan data dalam bentuk tabel, str() : menampilkan struktur data, dim() : menampilkan dimensi data
View(data1)
str(data1)'data.frame': 50 obs. of 2 variables:
$ Periode: int 1 2 3 4 5 6 7 8 9 10 ...
$ Yt : num 48.7 45.8 46.4 46.2 44 53.8 47.6 47 47.6 51.1 ...dim(data1)[1] 50 2View(data2)
str(data2)'data.frame': 120 obs. of 2 variables:
$ Period: int 1 2 3 4 5 6 7 8 9 10 ...
$ Sales : num 10618 10538 10209 10553 9935 ...dim(data2)[1] 120 2Mengubah data agar terbaca sebagai data deret waktu dengan fungsi ts() .
data1.ts <- ts(data1$Yt)
data2.ts <- ts(data2$Sales)Menampilkan ringkasan data
summary(data1.ts) Min. 1st Qu. Median Mean 3rd Qu. Max.
43.30 46.40 48.65 48.92 51.55 54.80 summary(data2.ts) Min. 1st Qu. Median Mean 3rd Qu. Max.
9815 10210 10392 10379 10535 10827 Membuat plot data deret waktu
ts.plot(data1.ts, xlab="Time Period ", ylab="Reading",
main = "Time Series Plot")
points(data1.ts)
#menyimpan plot
dev.copy(png, "plot_data1.png")png
3 dev.off()png
2 ts.plot(data2.ts, xlab="Time Period ", ylab="Sales",
main = "Time Series Plot of Sales")
points(data2.ts)
#menyimpan plot
dev.copy(png, "plot_data2.png")png
3 dev.off()png
2 4 Single Moving Average & Double Moving Average
4.1 Pembagian Data
Pembagian data latih dan data uji dilakukan dengan perbandingan 80% data latih dan 20% data uji.
# hitung jumlah data
n <- nrow(data2)
# tentukan batas 80%
n_train <- floor(0.8 * n)
# bagi data
training_ma <- data2[1:n_train, ]
testing_ma <- data2[(n_train+1):n, ]
# ubah ke time series
train_ma.ts <- ts(training_ma$Sales)
test_ma.ts <- ts(testing_ma$Sales)Eksplorasi data dilakukan pada keseluruhan data, data latih serta data uji menggunakan plot data deret waktu.
#eksplorasi keseluruhan data
plot(data2.ts, col="red",main="Plot semua data")
points(data2.ts)
#eksplorasi data latih
plot(train_ma.ts, col="blue",main="Plot data latih")
points(train_ma.ts)
#eksplorasi data uji
plot(test_ma.ts, col="blue",main="Plot data uji")
points(test_ma.ts)
Eksplorasi data juga dapat dilakukan menggunakan package ggplot2 dengan terlebih dahulu memanggil library package ggplot2.
library(ggplot2)Warning: package 'ggplot2' was built under R version 4.5.1ggplot() +
geom_line(data = training_ma, aes(x = Period, y = Sales, col = "Data Latih")) +
geom_line(data = testing_ma, aes(x = Period, y = Sales, col = "Data Uji")) +
labs(x = "Periode Waktu", y = "Sales", color = "Legend") +
scale_colour_manual(name="Keterangan:", breaks = c("Data Latih", "Data Uji"),
values = c("blue", "red")) +
theme_bw() + theme(legend.position = "bottom",
plot.caption = element_text(hjust=0.5, size=12))
4.2 Single Moving Average (SMA)
Ide dasar dari Single Moving Average (SMA) adalah data suatu periode dipengaruhi oleh data periode sebelumnya. Metode pemulusan ini cocok digunakan untuk pola data stasioner atau konstan. Prinsip dasar metode pemulusan ini adalah data pemulusan pada periode ke-t merupakan rata rata dari m buah data pada periode ke-t hingga periode ke (t-m+1).
\[ S_t = \frac{1}{m} \sum_{i=t-m+1}^{t} X_i \]
Data pemulusan pada periode ke-t selanjutnya digunakan sebagai nilai peramalan pada periode ke t+1.
\[ F_t = S_{t-1}, F_{n,h} = S_n \]
Pemulusan menggunakan metode SMA dilakukan dengan fungsi SMA(). Dalam hal ini akan dilakukan pemulusan dengan parameter m=4.
data.sma<-SMA(train_ma.ts, n=4)
data.smaTime Series:
Start = 1
End = 96
Frequency = 1
[1] NA NA NA 10479.58 10308.78 10307.93 10304.73 10294.43
[9] 10333.10 10292.28 10303.00 10293.15 10477.55 10544.95 10614.55 10535.00
[17] 10320.50 10225.40 10280.20 10349.15 10529.10 10571.60 10387.95 10314.38
[25] 10166.18 10130.75 10202.43 10315.98 10429.90 10409.80 10421.15 10393.70
[33] 10352.63 10351.20 10322.40 10298.05 10406.28 10445.30 10514.13 10542.85
[41] 10445.60 10467.88 10515.33 10435.05 10481.18 10526.15 10526.05 10543.60
[49] 10455.85 10321.53 10120.80 10171.55 10174.38 10195.58 10278.43 10264.88
[57] 10332.28 10447.40 10460.58 10499.60 10497.68 10410.30 10386.83 10347.20
[65] 10253.23 10222.80 10261.73 10254.73 10403.13 10477.90 10416.28 10444.90
[73] 10251.15 10145.05 10186.45 10158.55 10274.90 10398.93 10534.13 10436.08
[81] 10364.73 10336.43 10329.00 10451.00 10487.53 10415.68 10278.30 10284.35
[89] 10273.18 10435.98 10477.90 10386.75 10436.85 10291.75 10328.58 10361.03Data pemulusan pada periode ke-t selanjutnya digunakan sebagai nilai peramalan pada periode ke t+1 sehingga hasil peramalan 1 periode kedepan adalah sebagai berikut.
data.ramal<-c(NA,data.sma)
data.ramal #forecast 1 periode ke depan [1] NA NA NA NA 10479.58 10308.78 10307.93 10304.73
[9] 10294.43 10333.10 10292.28 10303.00 10293.15 10477.55 10544.95 10614.55
[17] 10535.00 10320.50 10225.40 10280.20 10349.15 10529.10 10571.60 10387.95
[25] 10314.38 10166.18 10130.75 10202.43 10315.98 10429.90 10409.80 10421.15
[33] 10393.70 10352.63 10351.20 10322.40 10298.05 10406.28 10445.30 10514.13
[41] 10542.85 10445.60 10467.88 10515.33 10435.05 10481.18 10526.15 10526.05
[49] 10543.60 10455.85 10321.53 10120.80 10171.55 10174.38 10195.58 10278.43
[57] 10264.88 10332.28 10447.40 10460.58 10499.60 10497.68 10410.30 10386.83
[65] 10347.20 10253.23 10222.80 10261.73 10254.73 10403.13 10477.90 10416.28
[73] 10444.90 10251.15 10145.05 10186.45 10158.55 10274.90 10398.93 10534.13
[81] 10436.08 10364.73 10336.43 10329.00 10451.00 10487.53 10415.68 10278.30
[89] 10284.35 10273.18 10435.98 10477.90 10386.75 10436.85 10291.75 10328.58
[97] 10361.03Selanjutnya dilakukan peramalan sebanyak 24 periode sesuai dengan jumlah data uji. Pada metode SMA, seluruh hasil peramalan untuk 24 periode ke depan akan memiliki nilai yang sama dengan hasil ramalan satu periode ke depan.
data.gab<-cbind(
aktual=c(data2.ts),
pemulusan=c(data.sma,rep(NA,24)),
ramalan=c(data.ramal,rep(data.ramal[length(data.ramal)],23)))
data.gab #forecast 24 periode ke depan aktual pemulusan ramalan
[1,] 10618.1 NA NA
[2,] 10537.9 NA NA
[3,] 10209.3 NA NA
[4,] 10553.0 10479.58 NA
[5,] 9934.9 10308.78 10479.58
[6,] 10534.5 10307.93 10308.78
[7,] 10196.5 10304.73 10307.93
[8,] 10511.8 10294.43 10304.73
[9,] 10089.6 10333.10 10294.43
[10,] 10371.2 10292.28 10333.10
[11,] 10239.4 10303.00 10292.28
[12,] 10472.4 10293.15 10303.00
[13,] 10827.2 10477.55 10293.15
[14,] 10640.8 10544.95 10477.55
[15,] 10517.8 10614.55 10544.95
[16,] 10154.2 10535.00 10614.55
[17,] 9969.2 10320.50 10535.00
[18,] 10260.4 10225.40 10320.50
[19,] 10737.0 10280.20 10225.40
[20,] 10430.0 10349.15 10280.20
[21,] 10689.0 10529.10 10349.15
[22,] 10430.4 10571.60 10529.10
[23,] 10002.4 10387.95 10571.60
[24,] 10135.7 10314.38 10387.95
[25,] 10096.2 10166.18 10314.38
[26,] 10288.7 10130.75 10166.18
[27,] 10289.1 10202.43 10130.75
[28,] 10589.9 10315.98 10202.43
[29,] 10551.9 10429.90 10315.98
[30,] 10208.3 10409.80 10429.90
[31,] 10334.5 10421.15 10409.80
[32,] 10480.1 10393.70 10421.15
[33,] 10387.6 10352.63 10393.70
[34,] 10202.6 10351.20 10352.63
[35,] 10219.3 10322.40 10351.20
[36,] 10382.7 10298.05 10322.40
[37,] 10820.5 10406.28 10298.05
[38,] 10358.7 10445.30 10406.28
[39,] 10494.6 10514.13 10445.30
[40,] 10497.6 10542.85 10514.13
[41,] 10431.5 10445.60 10542.85
[42,] 10447.8 10467.88 10445.60
[43,] 10684.4 10515.33 10467.88
[44,] 10176.5 10435.05 10515.33
[45,] 10616.0 10481.18 10435.05
[46,] 10627.7 10526.15 10481.18
[47,] 10684.0 10526.05 10526.15
[48,] 10246.7 10543.60 10526.05
[49,] 10265.0 10455.85 10543.60
[50,] 10090.4 10321.53 10455.85
[51,] 9881.1 10120.80 10321.53
[52,] 10449.7 10171.55 10120.80
[53,] 10276.3 10174.38 10171.55
[54,] 10175.2 10195.58 10174.38
[55,] 10212.5 10278.43 10195.58
[56,] 10395.5 10264.88 10278.43
[57,] 10545.9 10332.28 10264.88
[58,] 10635.7 10447.40 10332.28
[59,] 10265.2 10460.58 10447.40
[60,] 10551.6 10499.60 10460.58
[61,] 10538.2 10497.68 10499.60
[62,] 10286.2 10410.30 10497.68
[63,] 10171.3 10386.83 10410.30
[64,] 10393.1 10347.20 10386.83
[65,] 10162.3 10253.23 10347.20
[66,] 10164.5 10222.80 10253.23
[67,] 10327.0 10261.73 10222.80
[68,] 10365.1 10254.73 10261.73
[69,] 10755.9 10403.13 10254.73
[70,] 10463.6 10477.90 10403.13
[71,] 10080.5 10416.28 10477.90
[72,] 10479.6 10444.90 10416.28
[73,] 9980.9 10251.15 10444.90
[74,] 10039.2 10145.05 10251.15
[75,] 10246.1 10186.45 10145.05
[76,] 10368.0 10158.55 10186.45
[77,] 10446.3 10274.90 10158.55
[78,] 10535.3 10398.93 10274.90
[79,] 10786.9 10534.13 10398.93
[80,] 9975.8 10436.08 10534.13
[81,] 10160.9 10364.73 10436.08
[82,] 10422.1 10336.43 10364.73
[83,] 10757.2 10329.00 10336.43
[84,] 10463.8 10451.00 10329.00
[85,] 10307.0 10487.53 10451.00
[86,] 10134.7 10415.68 10487.53
[87,] 10207.7 10278.30 10415.68
[88,] 10488.0 10284.35 10278.30
[89,] 10262.3 10273.18 10284.35
[90,] 10785.9 10435.98 10273.18
[91,] 10375.4 10477.90 10435.98
[92,] 10123.4 10386.75 10477.90
[93,] 10462.7 10436.85 10386.75
[94,] 10205.5 10291.75 10436.85
[95,] 10522.7 10328.58 10291.75
[96,] 10253.2 10361.03 10328.58
[97,] 10428.7 NA 10361.03
[98,] 10615.8 NA 10361.03
[99,] 10417.3 NA 10361.03
[100,] 10445.4 NA 10361.03
[101,] 10690.6 NA 10361.03
[102,] 10271.8 NA 10361.03
[103,] 10524.8 NA 10361.03
[104,] 9815.0 NA 10361.03
[105,] 10398.5 NA 10361.03
[106,] 10553.1 NA 10361.03
[107,] 10655.8 NA 10361.03
[108,] 10199.1 NA 10361.03
[109,] 10416.6 NA 10361.03
[110,] 10391.3 NA 10361.03
[111,] 10210.1 NA 10361.03
[112,] 10352.5 NA 10361.03
[113,] 10423.8 NA 10361.03
[114,] 10519.3 NA 10361.03
[115,] 10596.7 NA 10361.03
[116,] 10650.0 NA 10361.03
[117,] 10741.6 NA 10361.03
[118,] 10246.0 NA 10361.03
[119,] 10354.4 NA 10361.03
[120,] 10155.4 NA 10361.03Adapun plot data deret waktu dari hasil peramalan yang dilakukan adalah sebagai berikut.
ts.plot(data2.ts, xlab="Time Period ", ylab="Sales", main= "SMA N=4 Data Sales")
points(data2.ts)
lines(data.gab[,2],col="green",lwd=2)
lines(data.gab[,3],col="red",lwd=2)
legend("topleft",c("data aktual","data pemulusan","data peramalan"), lty=8, col=c("black","green","red"), cex=0.5)
Selanjutnya perhitungan akurasi dilakukan dengan ukuran akurasi Sum Squares Error (SSE), Mean Square Error (MSE) dan Mean Absolute Percentage Error (MAPE). Perhitungan akurasi dilakukan baik pada data latih maupun pada data uji.
\[ SSE = \sum_{t=1}^{n} (y_t - \hat{y}_t)^2 \] \[ MSE = \frac{1}{n} \sum_{t=1}^{n} (y_t - \hat{y}_t)^2 \] \[ MAPE = \frac{1}{n} \sum_{t=1}^{n} \left|\frac{y_t - \hat{y}_t}{y_t}\right| * 100 \]
#Menghitung nilai keakuratan data latih
error_train.sma = train_ma.ts-data.ramal[1:length(train_ma.ts)]
SSE_train.sma = sum(error_train.sma[5:length(train_ma.ts)]^2)
MSE_train.sma = mean(error_train.sma[5:length(train_ma.ts)]^2)
MAPE_train.sma = mean(abs((error_train.sma[5:length(train_ma.ts)]/train_ma.ts[5:length(train_ma.ts)])*100))
akurasi_train.sma <- matrix(c(SSE_train.sma, MSE_train.sma, MAPE_train.sma))
row.names(akurasi_train.sma)<- c("SSE", "MSE", "MAPE")
colnames(akurasi_train.sma) <- c("Akurasi m = 4")
akurasi_train.sma Akurasi m = 4
SSE 6.396116e+06
MSE 6.952300e+04
MAPE 2.049321e+00Dalam hal ini nilai MAPE data latih pada metode pemulusan SMA sekitar 2%, nilai ini dapat dikategorikan sebagai nilai akurasi yang sangat baik. Selanjutnya dilakukan perhitungan nilai MAPE data uji pada metode pemulusan SMA.
#Menghitung nilai keakuratan data uji
error_test.sma = test_ma.ts-data.gab[97:120,3]
SSE_test.sma = sum(error_test.sma^2)
MSE_test.sma = mean(error_test.sma^2)
MAPE_test.sma = mean(abs((error_test.sma/test_ma.ts*100)))
akurasi_test.sma <- matrix(c(SSE_test.sma, MSE_test.sma, MAPE_test.sma))
row.names(akurasi_test.sma)<- c("SSE", "MSE", "MAPE")
colnames(akurasi_test.sma) <- c("Akurasi m = 4")
akurasi_test.sma Akurasi m = 4
SSE 1.068022e+06
MSE 4.450094e+04
MAPE 1.591089e+00Perhitungan akurasi menggunakan data latih menghasilkan nilai MAPE yang kurang dari 10% sehingga nilai akurasi ini dapat dikategorikan sebagai sangat baik.
4.3 Double Moving Average (DMA)
Metode pemulusan Double Moving Average (DMA) pada dasarnya mirip dengan SMA. Namun demikian, metode ini lebih cocok digunakan untuk pola data trend. Proses pemulusan dengan rata rata dalam metode ini dilakukan sebanyak 2 kali.
- Tahap I:
\[ S_{1,t} = \frac{1}{m} \sum_{i=t-m+1}^{t} X_i \]
- Tahap II:
\[ S_{2,t} = \frac{1}{m} \sum_{i=t-m+1}^{t} S_{1,i} \]
Forecast \(h\) langkah ke depan dihitung dengan:
\[ F_{2,t,t+h} = A_t + B_t \,(h) \]
dengan komponen level (\(A_t\)) dan tren (\(B_t\)):
\[ A_t = 2S_{1,t} - S_{2,t} \qquad\text{dan}\qquad B_t = \frac{2}{m-1}\,\big(S_{1,t} - S_{2,t}\big) \]
dma <- SMA(data.sma, n = 4)
At <- 2*data.sma - dma
Bt <- 2/(4-1)*(data.sma - dma)
data.dma<- At+Bt
data.ramal2<- c(NA, data.dma)
t = 1:24
f = c()
for (i in t) {
f[i] = At[length(At)] + Bt[length(Bt)]*(i)
}
data.gab2 <- cbind(aktual = c(data2.ts),
pemulusan1 = c(data.sma,rep(NA,24)),
pemulusan2 = c(dma, rep(NA,24)),
At = c(At, rep(NA,24)),
Bt = c(Bt,rep(NA,24)),
ramalan = c(data.ramal2, f[-1]))
data.gab2 aktual pemulusan1 pemulusan2 At Bt ramalan
[1,] 10618.1 NA NA NA NA NA
[2,] 10537.9 NA NA NA NA NA
[3,] 10209.3 NA NA NA NA NA
[4,] 10553.0 10479.58 NA NA NA NA
[5,] 9934.9 10308.78 NA NA NA NA
[6,] 10534.5 10307.93 NA NA NA NA
[7,] 10196.5 10304.73 10350.25 10259.200 -30.3500000 NA
[8,] 10511.8 10294.43 10303.96 10284.888 -6.3583333 10228.850
[9,] 10089.6 10333.10 10310.04 10356.156 15.3708333 10278.529
[10,] 10371.2 10292.28 10306.13 10278.419 -9.2375000 10371.527
[11,] 10239.4 10303.00 10305.70 10300.300 -1.8000000 10269.181
[12,] 10472.4 10293.15 10305.38 10280.919 -8.1541667 10298.500
[13,] 10827.2 10477.55 10341.49 10613.606 90.7041667 10272.765
[14,] 10640.8 10544.95 10404.66 10685.238 93.5250000 10704.310
[15,] 10517.8 10614.55 10482.55 10746.550 88.0000000 10778.763
[16,] 10154.2 10535.00 10543.01 10526.988 -5.3416667 10834.550
[17,] 9969.2 10320.50 10503.75 10137.250 -122.1666667 10521.646
[18,] 10260.4 10225.40 10423.86 10026.938 -132.3083333 10015.083
[19,] 10737.0 10280.20 10340.28 10220.125 -40.0500000 9894.629
[20,] 10430.0 10349.15 10293.81 10404.488 36.8916667 10180.075
[21,] 10689.0 10529.10 10345.96 10712.238 122.0916667 10441.379
[22,] 10430.4 10571.60 10432.51 10710.688 92.7250000 10834.329
[23,] 10002.4 10387.95 10459.45 10316.450 -47.6666667 10803.413
[24,] 10135.7 10314.38 10450.76 10177.994 -90.9208333 10268.783
[25,] 10096.2 10166.18 10360.03 9972.325 -129.2333333 10087.073
[26,] 10288.7 10130.75 10249.81 10011.688 -79.3750000 9843.092
[27,] 10289.1 10202.43 10203.43 10201.419 -0.6708333 9932.313
[28,] 10589.9 10315.98 10203.83 10428.119 74.7625000 10200.748
[29,] 10551.9 10429.90 10269.76 10590.038 106.7583333 10502.881
[30,] 10208.3 10409.80 10339.53 10480.075 46.8500000 10696.796
[31,] 10334.5 10421.15 10394.21 10448.094 17.9625000 10526.925
[32,] 10480.1 10393.70 10413.64 10373.763 -13.2916667 10466.056
[33,] 10387.6 10352.63 10394.32 10310.931 -27.7958333 10360.471
[34,] 10202.6 10351.20 10379.67 10322.731 -18.9791667 10283.135
[35,] 10219.3 10322.40 10354.98 10289.819 -21.7208333 10303.752
[36,] 10382.7 10298.05 10331.07 10265.031 -22.0125000 10268.098
[37,] 10820.5 10406.28 10344.48 10468.069 41.1958333 10243.019
[38,] 10358.7 10445.30 10368.01 10522.594 51.5291667 10509.265
[39,] 10494.6 10514.13 10415.94 10612.313 65.4583333 10574.123
[40,] 10497.6 10542.85 10477.14 10608.563 43.8083333 10677.771
[41,] 10431.5 10445.60 10486.97 10404.231 -27.5791667 10652.371
[42,] 10447.8 10467.88 10492.61 10443.138 -16.4916667 10376.652
[43,] 10684.4 10515.33 10492.91 10537.738 14.9416667 10426.646
[44,] 10176.5 10435.05 10465.96 10404.138 -20.6083333 10552.679
[45,] 10616.0 10481.18 10474.86 10487.494 4.2125000 10383.529
[46,] 10627.7 10526.15 10489.42 10562.875 24.4833333 10491.706
[47,] 10684.0 10526.05 10492.11 10559.994 22.6291667 10587.358
[48,] 10246.7 10543.60 10519.24 10567.956 16.2375000 10582.623
[49,] 10265.0 10455.85 10512.91 10398.788 -38.0416667 10584.194
[50,] 10090.4 10321.53 10461.76 10181.294 -93.4875000 10360.746
[51,] 9881.1 10120.80 10360.44 9881.156 -159.7625000 10087.806
[52,] 10449.7 10171.55 10267.43 10075.669 -63.9208333 9721.394
[53,] 10276.3 10174.38 10197.06 10151.688 -15.1250000 10011.748
[54,] 10175.2 10195.58 10165.57 10225.575 20.0000000 10136.563
[55,] 10212.5 10278.43 10204.98 10351.869 48.9625000 10245.575
[56,] 10395.5 10264.88 10228.31 10301.438 24.3750000 10400.831
[57,] 10545.9 10332.28 10267.79 10396.763 42.9916667 10325.813
[58,] 10635.7 10447.40 10330.74 10564.056 77.7708333 10439.754
[59,] 10265.2 10460.58 10376.28 10544.869 56.1958333 10641.827
[60,] 10551.6 10499.60 10434.96 10564.238 43.0916667 10601.065
[61,] 10538.2 10497.68 10476.31 10519.038 14.2416667 10607.329
[62,] 10286.2 10410.30 10467.04 10353.563 -37.8250000 10533.279
[63,] 10171.3 10386.83 10448.60 10325.050 -41.1833333 10315.738
[64,] 10393.1 10347.20 10410.50 10283.900 -42.2000000 10283.867
[65,] 10162.3 10253.23 10349.39 10157.063 -64.1083333 10241.700
[66,] 10164.5 10222.80 10302.51 10143.088 -53.1416667 10092.954
[67,] 10327.0 10261.73 10271.24 10252.213 -6.3416667 10089.946
[68,] 10365.1 10254.73 10248.12 10261.331 4.4041667 10245.871
[69,] 10755.9 10403.13 10285.59 10520.656 78.3541667 10265.735
[70,] 10463.6 10477.90 10349.37 10606.431 85.6875000 10599.010
[71,] 10080.5 10416.28 10388.01 10444.544 18.8458333 10692.119
[72,] 10479.6 10444.90 10435.55 10454.250 6.2333333 10463.390
[73,] 9980.9 10251.15 10397.56 10104.744 -97.6041667 10460.483
[74,] 10039.2 10145.05 10314.34 9975.756 -112.8625000 10007.140
[75,] 10246.1 10186.45 10256.89 10116.013 -46.9583333 9862.894
[76,] 10368.0 10158.55 10185.30 10131.800 -17.8333333 10069.054
[77,] 10446.3 10274.90 10191.24 10358.563 55.7750000 10113.967
[78,] 10535.3 10398.93 10254.71 10543.144 96.1458333 10414.338
[79,] 10786.9 10534.13 10341.63 10726.625 128.3333333 10639.290
[80,] 9975.8 10436.08 10411.01 10461.144 16.7125000 10854.958
[81,] 10160.9 10364.73 10433.46 10295.988 -45.8250000 10477.856
[82,] 10422.1 10336.43 10417.84 10255.013 -54.2750000 10250.163
[83,] 10757.2 10329.00 10366.56 10291.444 -25.0375000 10200.738
[84,] 10463.8 10451.00 10370.29 10531.713 53.8083333 10266.406
[85,] 10307.0 10487.53 10400.99 10574.063 57.6916667 10585.521
[86,] 10134.7 10415.68 10420.80 10410.550 -3.4166667 10631.754
[87,] 10207.7 10278.30 10408.13 10148.475 -86.5500000 10407.133
[88,] 10488.0 10284.35 10366.46 10202.238 -54.7416667 10061.925
[89,] 10262.3 10273.18 10312.88 10233.475 -26.4666667 10147.496
[90,] 10785.9 10435.98 10317.95 10554.000 78.6833333 10207.008
[91,] 10375.4 10477.90 10367.85 10587.950 73.3666667 10632.683
[92,] 10123.4 10386.75 10393.45 10380.050 -4.4666667 10661.317
[93,] 10462.7 10436.85 10434.37 10439.331 1.6541667 10375.583
[94,] 10205.5 10291.75 10398.31 10185.188 -71.0416667 10440.985
[95,] 10522.7 10328.58 10360.98 10296.169 -21.6041667 10114.146
[96,] 10253.2 10361.03 10354.55 10367.500 4.3166667 10274.565
[97,] 10428.7 NA NA NA NA 10371.817
[98,] 10615.8 NA NA NA NA 10376.133
[99,] 10417.3 NA NA NA NA 10380.450
[100,] 10445.4 NA NA NA NA 10384.767
[101,] 10690.6 NA NA NA NA 10389.083
[102,] 10271.8 NA NA NA NA 10393.400
[103,] 10524.8 NA NA NA NA 10397.717
[104,] 9815.0 NA NA NA NA 10402.033
[105,] 10398.5 NA NA NA NA 10406.350
[106,] 10553.1 NA NA NA NA 10410.667
[107,] 10655.8 NA NA NA NA 10414.983
[108,] 10199.1 NA NA NA NA 10419.300
[109,] 10416.6 NA NA NA NA 10423.617
[110,] 10391.3 NA NA NA NA 10427.933
[111,] 10210.1 NA NA NA NA 10432.250
[112,] 10352.5 NA NA NA NA 10436.567
[113,] 10423.8 NA NA NA NA 10440.883
[114,] 10519.3 NA NA NA NA 10445.200
[115,] 10596.7 NA NA NA NA 10449.517
[116,] 10650.0 NA NA NA NA 10453.833
[117,] 10741.6 NA NA NA NA 10458.150
[118,] 10246.0 NA NA NA NA 10462.467
[119,] 10354.4 NA NA NA NA 10466.783
[120,] 10155.4 NA NA NA NA 10471.100Hasil pemulusan menggunakan metode DMA divisualisasikan sebagai berikut
ts.plot(data2.ts, xlab="Time Period ", ylab="Sales", main= "DMA N=4 Data Sales")
points(data2.ts)
lines(data.gab2[,3],col="green",lwd=2)
lines(data.gab2[,6],col="red",lwd=2)
legend("topleft",c("data aktual","data pemulusan","data peramalan"), lty=8, col=c("black","green","red"), cex=0.8)
Selanjutnya perhitungan akurasi dilakukan baik pada data latih maupun data uji. Perhitungan akurasi dilakukan dengan ukuran akurasi SSE, MSE dan MAPE.
#Menghitung nilai keakuratan data latih
error_train.dma = train_ma.ts-data.ramal2[1:length(train_ma.ts)]
SSE_train.dma = sum(error_train.dma[8:length(train_ma.ts)]^2)
MSE_train.dma = mean(error_train.dma[8:length(train_ma.ts)]^2)
MAPE_train.dma = mean(abs((error_train.dma[8:length(train_ma.ts)]/train_ma.ts[8:length(train_ma.ts)])*100))
akurasi_train.dma <- matrix(c(SSE_train.dma, MSE_train.dma, MAPE_train.dma))
row.names(akurasi_train.dma)<- c("SSE", "MSE", "MAPE")
colnames(akurasi_train.dma) <- c("Akurasi m = 4")
akurasi_train.dma Akurasi m = 4
SSE 9.856243e+06
MSE 1.107443e+05
MAPE 2.524911e+00Perhitungan akurasi pada data latih menggunakan nilai MAPE menghasilkan nilai MAPE yang kurang dari 10% sehingga dikategorikan sangat baik. Selanjutnya, perhitungan nilai akurasi dilakukan pada data uji.
#Menghitung nilai keakuratan data uji
error_test.dma = test_ma.ts-data.gab2[97:120,6]
SSE_test.dma = sum(error_test.dma^2)
MSE_test.dma = mean(error_test.dma^2)
MAPE_test.dma = mean(abs((error_test.dma/test_ma.ts*100)))
akurasi_test.dma <- matrix(c(SSE_test.dma, MSE_test.dma, MAPE_test.dma))
row.names(akurasi_test.dma)<- c("SSE", "MSE", "MAPE")
colnames(akurasi_test.dma) <- c("Akurasi m = 4")
akurasi_test.dma Akurasi m = 4
SSE 1.022231e+06
MSE 4.259296e+04
MAPE 1.551109e+00Perhitungan akurasi menggunakan data latih menghasilkan nilai MAPE yang kurang dari 10% sehingga nilai akurasi ini dapat dikategorikan sebagai sangat baik.
Pada data latih, metode SMA lebih baik dibandingkan dengan metode DMA, sedangkan pada data uji, metode DMA lebih baik dibandingkan SMA.
5 Single Exponential Smoothing & Double Exponential Smoothing
Metode Exponential Smoothing merupakan metode pemulusan deret waktu dengan memberikan bobot yang menurun secara eksponensial pada data historis, di mana nilai terbaru mendapat bobot lebih besar dibanding nilai yang lebih lama. Metode ini menggunakan satu atau lebih parameter pemulusan yang secara langsung menentukan besar kecilnya bobot setiap pengamatan. Pemilihan parameter yang tepat akan sangat berpengaruh terhadap hasil ramalan. Secara umum, Exponential Smoothing dibedakan menjadi dua jenis, yaitu model tunggal (single) yang digunakan untuk data tanpa tren maupun musiman, serta model ganda (double) yang mampu menangkap adanya tren pada data.
5.1 Pembagian Data
Pembagian data latih dan data uji dilakukan dengan perbandingan 80% data latih dan 20% data uji.
# jumlah baris data
n <- nrow(data1)
# hitung batas 80%
n_train <- floor(0.8 * n)
# bagi data
training <- data1[1:n_train, ]
testing <- data1[(n_train+1):n, ]
# ubah ke time series
train.ts <- ts(training$Yt)
test.ts <- ts(testing$Yt)5.2 Eksplorasi
Eksplorasi dilakukan dengan membuat plot data deret waktu untuk keseluruhan data, data latih, dan data uji.
#eksplorasi data
plot(data1.ts, col="black",main="Plot semua data")
points(data1.ts)
plot(train.ts, col="red",main="Plot data latih")
points(train.ts)
plot(test.ts, col="blue",main="Plot data uji")
points(test.ts)
#Eksplorasi dengan GGPLOT
library(ggplot2)
ggplot() +
geom_line(data = training, aes(x = Periode, y = Yt, col = "Data Latih")) +
geom_line(data = testing, aes(x = Periode, y = Yt, col = "Data Uji")) +
labs(x = "Periode Waktu", y = "Membaca", color = "Legend") +
scale_colour_manual(name="Keterangan:", breaks = c("Data Latih", "Data Uji"),
values = c("blue", "red")) +
theme_bw() + theme(legend.position = "bottom",
plot.caption = element_text(hjust=0.5, size=12))
6 Single Exponential Smoothing
Single Exponential Smoothing (SES) adalah metode peramalan deret waktu yang dikembangkan untuk mengatasi kelemahan Moving Average. Jika pada Moving Average setiap data periode sebelumnya dianggap memiliki bobot yang sama, maka SES memberikan bobot yang semakin mengecil secara eksponensial terhadap data lama, sehingga data terbaru diberi bobot lebih besar.
Single Exponential Smoothing merupakan metode pemulusan yang tepat digunakan untuk data dengan pola stasioner atau konstan.
Nilai pemulusan pada periode ke-t didapat dari persamaan:
\[ \tilde{y}_T=\lambda y_t+(1-\lambda)\tilde{y}_{T-1} \]
Nilai parameter \(\lambda\) adalah nilai antara 0 dan 1.
Nilai pemulusan periode ke-t bertindak sebagai nilai ramalan pada periode ke-\((T+\tau)\).
\[ \tilde{y}_{T+\tau}(T)=\tilde{y}_T \]
Pemulusan dengan metode SES dapat dilakukan dengan dua fungsi dari packages berbeda, yaitu (1) fungsi ses() dari packages forecast dan (2) fungsi HoltWinters dari packages stats .
#Cara 1 (fungsi ses)
ses.1 <- ses(train.ts, h = 10, alpha = 0.2)
plot(ses.1)
ses.1 Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
41 51.2589 47.18575 55.33206 45.02955 57.48826
42 51.2589 47.10508 55.41272 44.90618 57.61162
43 51.2589 47.02596 55.49185 44.78517 57.73264
44 51.2589 46.94828 55.56953 44.66637 57.85143
45 51.2589 46.87198 55.64583 44.54968 57.96812
46 51.2589 46.79698 55.72082 44.43499 58.08282
47 51.2589 46.72323 55.79458 44.32219 58.19562
48 51.2589 46.65065 55.86715 44.21119 58.30661
49 51.2589 46.57920 55.93860 44.10192 58.41589
50 51.2589 46.50883 56.00898 43.99429 58.52352ses.2<- ses(train.ts, h = 10, alpha = 0.7)
plot(ses.2)
ses.2 Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
41 51.64682 46.89683 56.39681 44.38234 58.91130
42 51.64682 45.84872 57.44492 42.77939 60.51425
43 51.64682 44.96299 58.33065 41.42479 61.86885
44 51.64682 44.18162 59.11201 40.22979 63.06385
45 51.64682 43.47463 59.81901 39.14853 64.14511
46 51.64682 42.82411 60.46953 38.15364 65.14000
47 51.64682 42.21836 61.07528 37.22723 66.06640
48 51.64682 41.64925 61.64439 36.35685 66.93679
49 51.64682 41.11083 62.18281 35.53342 67.76022
50 51.64682 40.59863 62.69501 34.75006 68.54357Pada fungsi ses() , terdapat beberapa argumen yang umum digunakan, yaitu
y: nilai data deret waktualpha: parameter pemulusan utama (0–1), mengatur bobot data terbaru vs data lama.beta: parameter pemulusan tren.gamma: parameter pemulusan musiman.h: jumlah periode ke depan yang ingin diramalkan (forecast horizon).Kasus di atas merupakan contoh inisialisasi nilai parameter \(\lambda\) dengan nilai
alpha0,2 dan 0,7 dan banyak periode data yang akan diramalkan adalah sebanyak 10 periode.
Untuk mendapatkan gambar hasil pemulusan pada data latih dengan fungsi ses() , perlu digunakan fungsi autoplot() dan autolayer() dari library packages ggplot2 .
autoplot(ses.1) +
autolayer(fitted(ses.1), series="Fitted") +
ylab("Membaca") + xlab("Periode")
Selanjutnya akan digunakan fungsi HoltWinters() dengan nilai inisialisasi parameter dan panjang periode peramalan yang sama dengan fungsi ses() .
#Cara 2 (fungsi Holtwinter)
ses1<- HoltWinters(train.ts, gamma = FALSE, beta = FALSE, alpha = 0.2)
plot(ses1)
#ramalan
ramalan1<- forecast(ses1, h=10)
ramalan1 Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
41 51.25907 47.18474 55.33340 45.02791 57.49023
42 51.25907 47.10405 55.41409 44.90451 57.61363
43 51.25907 47.02490 55.49324 44.78346 57.73468
44 51.25907 46.94720 55.57094 44.66463 57.85351
45 51.25907 46.87088 55.64726 44.54791 57.97023
46 51.25907 46.79586 55.72228 44.43318 58.08496
47 51.25907 46.72208 55.79606 44.32035 58.19779
48 51.25907 46.64949 55.86865 44.20932 58.30882
49 51.25907 46.57802 55.94012 44.10002 58.41812
50 51.25907 46.50762 56.01052 43.99236 58.52579ses2<- HoltWinters(train.ts, gamma = FALSE, beta = FALSE, alpha = 0.7)
plot(ses2)
#ramalan
ramalan2<- forecast(ses2, h=10)
ramalan2 Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
41 51.64682 46.89554 56.39810 44.38036 58.91328
42 51.64682 45.84714 57.44650 42.77697 60.51667
43 51.64682 44.96117 58.33247 41.42200 61.87164
44 51.64682 44.17959 59.11405 40.22668 63.06696
45 51.64682 43.47240 59.82124 39.14513 64.14851
46 51.64682 42.82170 60.47194 38.14997 65.14367
47 51.64682 42.21579 61.07785 37.22331 66.07033
48 51.64682 41.64652 61.64711 36.35269 66.94095
49 51.64682 41.10796 62.18568 35.52903 67.76461
50 51.64682 40.59562 62.69802 34.74546 68.54818Fungsi HoltWinters memiliki argumen yang sama dengan fungsi ses() . Argumen-argumen kedua fungsi dapat dilihat lebih lanjut dengan ?ses() atau ?HoltWinters .
Nilai parameter \(\alpha\) dari kedua fungsi dapat dioptimalkan menyesuaikan dari error-nya paling minimumnya. Caranya adalah dengan membuat parameter \(\alpha =\) NULL .
#SES
ses.opt <- ses(train.ts, h = 10, alpha = NULL)
plot(ses.opt)
ses.opt Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
41 50.98117 46.91751 55.04483 44.76633 57.19601
42 50.98117 46.86658 55.09576 44.68845 57.27389
43 50.98117 46.81628 55.14606 44.61153 57.35081
44 50.98117 46.76658 55.19575 44.53552 57.42682
45 50.98117 46.71746 55.24487 44.46039 57.50194
46 50.98117 46.66890 55.29344 44.38613 57.57621
47 50.98117 46.62088 55.34146 44.31269 57.64965
48 50.98117 46.57339 55.38895 44.24005 57.72229
49 50.98117 46.52640 55.43594 44.16818 57.79416
50 50.98117 46.47990 55.48244 44.09707 57.86527#Lamda Optimum Holt Winter
HWopt<- HoltWinters(train.ts, gamma = FALSE, beta = FALSE,alpha = NULL)
HWoptHolt-Winters exponential smoothing without trend and without seasonal component.
Call:
HoltWinters(x = train.ts, alpha = NULL, beta = FALSE, gamma = FALSE)
Smoothing parameters:
alpha: 0.161055
beta : FALSE
gamma: FALSE
Coefficients:
[,1]
a 51.001plot(HWopt)
#ramalan
ramalanopt<- forecast(HWopt, h=10)
ramalanopt Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
41 51.001 46.94040 55.06161 44.79084 57.21117
42 51.001 46.88807 55.11394 44.71082 57.29119
43 51.001 46.83640 55.16561 44.63179 57.37021
44 51.001 46.78537 55.21664 44.55374 57.44827
45 51.001 46.73494 55.26707 44.47662 57.52539
46 51.001 46.68511 55.31690 44.40041 57.60160
47 51.001 46.63584 55.36617 44.32506 57.67695
48 51.001 46.58712 55.41489 44.25055 57.75146
49 51.001 46.53894 55.46307 44.17686 57.82515
50 51.001 46.49126 55.51074 44.10395 57.89806Setelah dilakukan peramalan, akan dilakukan perhitungan keakuratan hasil peramalan. Perhitungan akurasi ini dilakukan baik pada data latih dan data uji.
6.0.1 Akurasi Data Latih
Perhitungan akurasi data dapat dilakukan dengan cara langsung maupun manual. Secara langsung, nilai akurasi dapat diambil dari objek yang tersimpan pada hasil SES, yaitu sum of squared errors (SSE). Nilai akurasi lain dapat dihitung pula dari nilai SSE tersebut.
#Keakuratan Metode
#Pada data training
# SES dengan alpha = 0.2
SSE1<-ses1$SSE
MSE1<-ses1$SSE/length(train.ts)
RMSE1<-sqrt(MSE1)
akurasi1 <- matrix(c(SSE1,MSE1,RMSE1))
row.names(akurasi1)<- c("SSE", "MSE", "RMSE")
colnames(akurasi1) <- c("Akurasi lamda=0.2")
akurasi1 Akurasi lamda=0.2
SSE 388.280675
MSE 9.707017
RMSE 3.115609# SES dengan alpha = 0.7
SSE2<-ses2$SSE
MSE2<-ses2$SSE/length(train.ts)
RMSE2<-sqrt(MSE2)
akurasi2 <- matrix(c(SSE2,MSE2,RMSE2))
row.names(akurasi2)<- c("SSE", "MSE", "RMSE")
colnames(akurasi2) <- c("Akurasi lamda=0.7")
akurasi2 Akurasi lamda=0.7
SSE 522.770369
MSE 13.069259
RMSE 3.615143#Cara Manual
# Alpha = 0.2
fitted1<-ramalan1$fitted
sisaan1<-ramalan1$residuals
head(sisaan1)Time Series:
Start = 1
End = 6
Frequency = 1
[1] NA -2.90000 -1.72000 -1.57600 -3.46080 7.03136resid1<-training$Yt-ramalan1$fitted
head(resid1)Time Series:
Start = 1
End = 6
Frequency = 1
[1] NA -2.90000 -1.72000 -1.57600 -3.46080 7.03136SSE.1=sum(sisaan1[2:length(train.ts)]^2)
SSE.1[1] 388.2807MSE.1 = SSE.1/length(train.ts)
MSE.1[1] 9.707017MAPE.1 = sum(abs(sisaan1[2:length(train.ts)]/train.ts[2:length(train.ts)])*
100)/length(train.ts)
MAPE.1[1] 5.249856akurasi.1 <- matrix(c(SSE.1,MSE.1,MAPE.1))
row.names(akurasi.1)<- c("SSE", "MSE", "MAPE")
colnames(akurasi.1) <- c("Akurasi lamda=0.2")
akurasi.1 Akurasi lamda=0.2
SSE 388.280675
MSE 9.707017
MAPE 5.249856# Alpha = 0.7
fitted2<-ramalan2$fitted
sisaan2<-ramalan2$residuals
head(sisaan2)Time Series:
Start = 1
End = 6
Frequency = 1
[1] NA -2.90000 -0.27000 -0.28100 -2.28430 9.11471resid2<-training$Yt-ramalan2$fitted
head(resid2)Time Series:
Start = 1
End = 6
Frequency = 1
[1] NA -2.90000 -0.27000 -0.28100 -2.28430 9.11471SSE.2=sum(sisaan2[2:length(train.ts)]^2)
SSE.2[1] 522.7704MSE.2 = SSE.2/length(train.ts)
MSE.2[1] 13.06926MAPE.2 = sum(abs(sisaan2[2:length(train.ts)]/train.ts[2:length(train.ts)])*
100)/length(train.ts)
MAPE.2[1] 5.767118akurasi.2 <- matrix(c(SSE.2,MSE.2,MAPE.2))
row.names(akurasi.2)<- c("SSE", "MSE", "MAPE")
colnames(akurasi.2) <- c("Akurasi lamda=0.7")
akurasi.2 Akurasi lamda=0.7
SSE 522.770369
MSE 13.069259
MAPE 5.767118Berdasarkan nilai SSE, MSE, RMSE, dan MAPE di antara kedua parameter, nilai parameter \(\lambda=0,2\) menghasilkan akurasi yang lebih baik dibanding \(\lambda=0,7\) . Hal ini dilihat dari nilai masing-masing ukuran akurasi yang lebih kecil. Berdasarkan nilai MAPE-nya, hasil ini dapat dikategorikan sebagai peramalan sangat baik.
6.0.2 Akurasi Data Uji
Akurasi data uji dapat dihitung dengan cara yang hampir sama dengan perhitungan akurasi data latih.
# jumlah observasi uji
n_test <- nrow(testing)
# error (ramalan - aktual), samakan panjang dan tipe numeric
e1 <- as.numeric(ramalan1$mean)[1:n_test] - as.numeric(testing$Yt)
e2 <- as.numeric(ramalan2$mean)[1:n_test] - as.numeric(testing$Yt)
eopt <- as.numeric(ramalanopt$mean)[1:n_test] - as.numeric(testing$Yt)
# SSE / MSE / RMSE untuk masing-masing model (abaikan NA)
SSEtesting1 <- sum(e1^2, na.rm = TRUE)
MSEtesting1 <- mean(e1^2, na.rm = TRUE)
RMSEtesting1 <- sqrt(MSEtesting1)
SSEtesting2 <- sum(e2^2, na.rm = TRUE)
MSEtesting2 <- mean(e2^2, na.rm = TRUE)
RMSEtesting2 <- sqrt(MSEtesting2)
SSEtestingopt <- sum(eopt^2, na.rm = TRUE)
MSEtestingopt <- mean(eopt^2, na.rm = TRUE)
RMSEtestingopt <- sqrt(MSEtestingopt)
# Tabel ringkas
akurasitesting_SSE <- matrix(c(SSEtesting1, SSEtesting2, SSEtestingopt),
nrow = 3,
dimnames = list(c("SSE1","SSE2","SSEopt"), "Nilai"))
akurasitesting_MSE <- matrix(c(MSEtesting1, MSEtesting2, MSEtestingopt),
nrow = 3,
dimnames = list(c("MSE1","MSE2","MSEopt"), "Nilai"))
akurasitesting_RMSE <- matrix(c(RMSEtesting1, RMSEtesting2, RMSEtestingopt),
nrow = 3,
dimnames = list(c("RMSE1","RMSE2","RMSEopt"), "Nilai"))
akurasitesting_SSE Nilai
SSE1 95.28706
SSE2 108.02806
SSEopt 88.47392akurasitesting_MSE Nilai
MSE1 9.528706
MSE2 10.802806
MSEopt 8.847392akurasitesting_RMSE Nilai
RMSE1 3.086860
RMSE2 3.286762
RMSEopt 2.974457Selain dengan cara di atas, perhitungan nilai akurasi dapat menggunakan fungsi accuracy() dari package forecast . Penggunaannya yaitu dengan menuliskan accuracy(hasil ramalan, kondisi aktual) . Contohnya adalah sebagai berikut.
accuracy(ramalan1, testing$Yt) ME RMSE MAE MPE MAPE MASE
Training set 0.328086 3.155299 2.647471 0.285487 5.384468 0.7924126
Test set -1.449071 3.086860 2.381814 -3.229724 4.986852 0.7128991
ACF1
Training set -0.1059246
Test set NAaccuracy(ramalan2, testing$Yt) ME RMSE MAE MPE MAPE MASE
Training set 0.1079421 3.661198 2.898138 -0.1459534 5.914992 0.8674395
Test set -1.8368194 3.286762 2.508728 -4.0106046 5.271903 0.7508855
ACF1
Training set -0.3955538
Test set NAaccuracy(ramalanopt, testing$Yt) ME RMSE MAE MPE MAPE MASE
Training set 0.3663352 3.149004 2.666641 0.357541 5.419489 0.7981504
Test set -1.1910047 2.974457 2.330201 -2.710010 4.861054 0.6974508
ACF1
Training set -0.07140176
Test set NA7 Double Exponential Smoothing (DES)
Metode pemulusan Double Exponential Smoothing (DES) digunakan untuk data yang memiliki pola tren. Metode DES adalah metode semacam SES, hanya saja dilakukan dua kali, yaitu pertama untuk tahapan ‘level’ dan kedua untuk tahapan ‘tren’. Pemulusan menggunakan metode ini akan menghasilkan peramalan tidak konstan untuk periode berikutnya.
Pemulusan dengan metode DES kali ini akan menggunakan fungsi HoltWinters() . Jika sebelumnya nilai argumen beta dibuat FALSE , kali ini argumen tersebut akan diinisialisasi bersamaan dengan nilai alpha .
#Lamda=0.2 dan gamma=0.2
des.1<- HoltWinters(train.ts, gamma = FALSE, beta = 0.2, alpha = 0.2)
plot(des.1)
#ramalan
ramalandes1<- forecast(des.1, h=10)
ramalandes1 Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
41 52.93449 46.89794 58.97104 43.70238 62.16660
42 53.23737 47.02939 59.44534 43.74309 62.73164
43 53.54024 47.10629 59.97420 43.70036 63.38013
44 53.84312 47.12544 60.56081 43.56931 64.11693
45 54.14600 47.08556 61.20645 43.34798 64.94402
46 54.44888 46.98696 61.91080 43.03685 65.86090
47 54.75176 46.83121 62.67230 42.63833 66.86519
48 55.05464 46.62073 63.48854 42.15608 67.95319
49 55.35751 46.35839 64.35663 41.59454 69.12048
50 55.66039 46.04729 65.27350 40.95842 70.36236#Lamda=0.6 dan gamma=0.3
des.2<- HoltWinters(train.ts, gamma = FALSE, beta = 0.3, alpha = 0.6)
plot(des.2)
#ramalan
ramalandes2<- forecast(des.2, h=10)
ramalandes2 Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
41 51.35923 46.08547 56.63298 43.29371 59.42474
42 51.24032 44.55201 57.92864 41.01142 61.46922
43 51.12142 42.73300 59.50984 38.29244 63.95040
44 51.00252 40.68212 61.32292 35.21883 66.78621
45 50.88362 38.43488 63.33236 31.84491 69.92232
46 50.76472 36.01516 65.51427 28.20722 73.32221
47 50.64581 33.43979 67.85184 24.33147 76.96015
48 50.52691 30.72117 70.33265 20.23665 80.81717
49 50.40801 27.86889 72.94712 15.93741 84.87861
50 50.28911 24.89063 75.68759 11.44548 89.13273Selanjutnya jika ingin membandingkan plot data latih dan data uji adalah sebagai berikut.
plot(data1.ts)
lines(des.1$fitted[,1], lty=2, col="blue")
lines(ramalandes1$mean, col="red")
Untuk mendapatkan nilai parameter optimum dari DES, argumen alpha dan beta dapat dibuat NULL seperti berikut.
#Lamda dan gamma optimum
des.opt<- HoltWinters(train.ts, gamma = FALSE)
des.optHolt-Winters exponential smoothing with trend and without seasonal component.
Call:
HoltWinters(x = train.ts, gamma = FALSE)
Smoothing parameters:
alpha: 0.4635085
beta : 0.2628024
gamma: FALSE
Coefficients:
[,1]
a 51.81211440
b -0.03605837plot(des.opt)
#ramalan
ramalandesopt<- forecast(des.opt, h=10)
ramalandesopt Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
41 51.77606 46.66859 56.88352 43.96486 59.58725
42 51.74000 45.82194 57.65805 42.68912 60.79088
43 51.70394 44.77088 58.63700 41.10074 62.30714
44 51.66788 43.54431 59.79145 39.24395 64.09181
45 51.63182 42.16755 61.09610 37.15746 66.10618
46 51.59576 40.66041 62.53112 34.87158 68.31995
47 51.55971 39.03793 64.08148 32.40931 70.71011
48 51.52365 37.31150 65.73580 29.78804 73.25926
49 51.48759 35.48986 67.48532 27.02118 75.95400
50 51.45153 33.57990 69.32316 24.11923 78.78383Selanjutnya akan dilakukan perhitungan akurasi pada data latih maupun data uji dengan ukuran akurasi SSE, MSE dan MAPE.
7.0.1 Akurasi Data Latih
#Akurasi Data Training
ssedes.train1<-des.1$SSE
msedes.train1<-ssedes.train1/length(train.ts)
sisaandes1<-ramalandes1$residuals
head(sisaandes1)Time Series:
Start = 1
End = 6
Frequency = 1
[1] NA NA 3.50000 5.36000 4.63360 15.86714mapedes.train1 <- sum(abs(sisaandes1[3:length(train.ts)]/train.ts[3:length(train.ts)])
*100)/length(train.ts)
akurasides.1 <- matrix(c(ssedes.train1,msedes.train1,mapedes.train1))
row.names(akurasides.1)<- c("SSE", "MSE", "MAPE")
colnames(akurasides.1) <- c("Akurasi lamda=0.2 dan gamma=0.2")
akurasides.1 Akurasi lamda=0.2 dan gamma=0.2
SSE 989.65686
MSE 24.74142
MAPE 7.75642ssedes.train2<-des.2$SSE
msedes.train2<-ssedes.train2/length(train.ts)
sisaandes2<-ramalandes2$residuals
head(sisaandes2)Time Series:
Start = 1
End = 6
Frequency = 1
[1] NA NA 3.50000 3.47000 0.83340 11.62875mapedes.train2 <- sum(abs(sisaandes2[3:length(train.ts)]/train.ts[3:length(train.ts)])
*100)/length(train.ts)
akurasides.2 <- matrix(c(ssedes.train2,msedes.train2,mapedes.train2))
row.names(akurasides.2)<- c("SSE", "MSE", "MAPE")
colnames(akurasides.2) <- c("Akurasi lamda=0.6 dan gamma=0.3")
akurasides.2 Akurasi lamda=0.6 dan gamma=0.3
SSE 632.851720
MSE 15.821293
MAPE 6.232456Hasil akurasi dari data latih didapatkan skenario 2 dengan lamda=0.6 dan gamma=0.3 memiliki hasil yang lebih baik. Namun untuk kedua skenario dapat dikategorikan peramalan sangat baik berdasarkan nilai MAPE-nya.
7.0.2 Akurasi Data Uji
#Akurasi Data Testing
selisihdes1<-ramalandes1$mean-testing$Yt
selisihdes1Time Series:
Start = 41
End = 50
Frequency = 1
[1] 5.03448806 3.73736622 9.54024438 0.04312254 1.64600071 2.44887887
[7] 4.15175703 6.35463519 3.95751335 7.96039151SSEtestingdes1<-sum(selisihdes1^2)
MSEtestingdes1<-SSEtestingdes1/length(testing$Yt)
MAPEtestingdes1<-sum(abs(selisihdes1/testing$Yt)*100)/length(testing$Yt)
selisihdes2<-ramalandes2$mean-testing$Yt
selisihdes2Time Series:
Start = 41
End = 50
Frequency = 1
[1] 3.45922576 1.74032361 7.12142147 -2.79748068 -1.61638283 -1.23528497
[7] 0.04581288 1.82691074 -0.99199141 2.58910645SSEtestingdes2<-sum(selisihdes2^2)
MSEtestingdes2<-SSEtestingdes2/length(testing$Yt)
MAPEtestingdes2<-sum(abs(selisihdes2/testing$Yt)*100)/length(testing$Yt)
selisihdesopt<-ramalandesopt$mean-testing$Yt
selisihdesoptTime Series:
Start = 41
End = 50
Frequency = 1
[1] 3.87605603 2.23999765 7.70393928 -2.13211910 -0.86817747 -0.40423584
[7] 0.95970578 2.82364741 0.08758903 3.75153066SSEtestingdesopt<-sum(selisihdesopt^2)
MSEtestingdesopt<-SSEtestingdesopt/length(testing$Yt)
MAPEtestingdesopt<-sum(abs(selisihdesopt/testing$Yt)*100)/length(testing$Yt)
akurasitestingdes <-
matrix(c(SSEtestingdes1,MSEtestingdes1,MAPEtestingdes1,SSEtestingdes2,MSEtestingdes2,
MAPEtestingdes2,SSEtestingdesopt,MSEtestingdesopt,MAPEtestingdesopt),
nrow=3,ncol=3)
row.names(akurasitestingdes)<- c("SSE", "MSE", "MAPE")
colnames(akurasitestingdes) <- c("des ske1","des ske2","des opt")
akurasitestingdes des ske1 des ske2 des opt
SSE 275.686644 88.701354 107.830825
MSE 27.568664 8.870135 10.783082
MAPE 9.330928 4.877651 5.2250227.0.3 Perbandingan SES dan DES
MSEfull <-
matrix(c(MSEtesting1,MSEtesting2,MSEtestingopt,MSEtestingdes1,MSEtestingdes2,
MSEtestingdesopt),nrow=3,ncol=2)
row.names(MSEfull)<- c("ske 1", "ske 2", "ske opt")
colnames(MSEfull) <- c("ses","des")
MSEfull ses des
ske 1 9.528706 27.568664
ske 2 10.802806 8.870135
ske opt 8.847392 10.783082Kedua metode dapat dibandingkan dengan menggunakan ukuran akurasi yang sama. Contoh di atas adalah perbandingan kedua metode dengan ukuran akurasi MSE. Hasilnya didapatkan metode DES lebih baik dibandingkan metode SES dilihat dari MSE yang lebih kecil nilainya.
8 Pemulusan Data Musiman
Pertama impor kembali data baru untuk latihan data musiman.
#Import data
library(rio)
data3 <- import("https://raw.githubusercontent.com/rizkynurhambali/praktikum-sta1341/main/Pertemuan%201/Electric_Production.csv")
data3.ts <- ts(data3$Yt)Selanjutnya melakukan pembagian data dan mengubahnya menjadi data deret waktu.
#membagi data menjadi training dan testing
training<-data3[1:192,2]
testing<-data3[193:241,2]
training.ts<-ts(training, frequency = 13)
testing.ts<-ts(testing, frequency = 13)Kemudian akan dilakukan eskplorasi dengan plot data deret waktu sebagai berikut.
#Membuat plot time series
plot(data3.ts, col="red",main="Plot semua data")
points(data3.ts)
plot(training.ts, col="blue",main="Plot data latih")
points(training.ts)
plot(testing.ts, col="green",main="Plot data uji")
points(testing.ts)
Metode Holt-Winter untuk peramalan data musiman menggunakan tiga persamaan pemulusan yang terdiri atas persamaan untuk level \((L_t)\), trend \((B_t)\), dan komponen seasonal / musiman \((S_t)\) dengan parameter pemulusan berupa \(\alpha\), \(\beta\), dan \(\gamma\). Metode Holt-Winter musiman terbagi menjadi dua, yaitu metode aditif dan metode multiplikatif. Perbedaan persamaan dan contoh datanya adalah sebagai berikut.
Pemulusan data musiman dengan metode Winter dilakukan menggunakan fungsi HoltWinters() dengan memasukkan argumen tambahan, yaitu gamma() dan seasonal() . Arguman seasonal() diinisialisasi menyesuaikan jenis musiman, aditif atau multiplikatif.
8.1 Winter Aditif
Perhitungan dengan model aditif dilakukan jika plot data asli menunjukkan fluktuasi musiman yang relatif stabil (konstan).
8.1.1 Pemulusan
#Pemulusan dengan winter aditif
winter1 <- HoltWinters(training.ts,alpha=0.2,beta=0.1,gamma=0.1,seasonal = "additive")
winter1$fittedTime Series:
Start = c(2, 1)
End = c(15, 10)
Frequency = 13
xhat level trend season
2.000000 88.54720 87.09424 0.182710904 1.2702538462
2.076923 89.35006 87.80409 0.235424883 1.3105461538
2.153846 80.20116 88.31712 0.263185721 -8.3791461538
2.230769 79.24275 88.64359 0.269514524 -9.6703615385
2.307692 86.99289 88.94230 0.272433583 -2.2218384615
2.384615 96.28834 89.38477 0.289437697 6.6141307692
2.461538 96.00536 89.91460 0.313476850 5.7772846154
2.538462 96.01483 90.31015 0.321683558 5.3830000000
2.615385 88.18864 88.87382 0.145882946 -0.8310692308
2.692308 79.68943 87.51016 -0.005071818 -7.8156615385
2.769231 81.37692 88.00770 0.045189654 -6.6759769231
2.846154 93.80604 90.67977 0.307877345 2.8183923077
2.923077 105.56926 92.67246 0.476358567 12.4204461538
3.000000 92.62681 90.89474 0.250951296 1.4811097582
3.076923 91.82073 90.23887 0.160269181 1.4215895083
3.153846 79.92237 88.32350 -0.047295489 -8.3538309434
3.230769 79.56744 89.18277 0.043361094 -9.6586853012
3.307692 89.63976 91.52076 0.272824225 -2.1538220045
3.384615 100.17500 93.06477 0.399942979 6.7102873806
3.461538 99.49776 93.30379 0.383850936 5.8101114488
3.538462 96.86370 91.97165 0.212251801 4.6797975513
3.615385 87.94223 89.43933 -0.062206265 -1.4348882870
3.692308 81.72203 89.39687 -0.060230876 -7.6146156534
3.769231 88.21262 93.48340 0.354444600 -5.6252261566
3.846154 102.28161 98.01694 0.772354299 3.4923171948
3.923077 109.95628 97.76731 0.670156106 11.5188170705
4.000000 96.45785 95.01187 0.327596431 1.1183812995
4.076923 93.27897 92.63090 0.056739478 0.5913308268
4.153846 82.39203 90.54114 -0.157909864 -7.9912046133
4.230769 83.24525 91.98393 0.002159568 -8.7408327760
4.307692 93.16522 94.55184 0.258734485 -1.6453469861
4.384615 103.12062 96.08821 0.386498014 6.6459192062
4.461538 98.72020 93.50678 0.089705553 5.1237149089
4.538462 93.74063 90.38964 -0.230978441 3.5819652871
4.615385 86.12278 87.99692 -0.447153075 -1.4269867301
4.692308 82.66807 88.93283 -0.308846681 -5.9559137485
4.769231 88.83358 92.68947 0.097701909 -3.9535873604
4.846154 97.37066 94.06196 0.225180215 3.0835244197
4.923077 103.75558 93.46412 0.142879019 10.1485783719
5.000000 89.83351 90.01489 -0.216332596 0.0349534879
5.076923 88.30330 88.87887 -0.308300739 -0.2672665429
5.153846 81.98633 89.54783 -0.210574838 -7.3509268847
5.230769 85.90162 93.41859 0.197558586 -7.7145331056
5.307692 96.13950 96.76169 0.512112279 -1.1342928718
5.384615 102.62193 96.70768 0.455500182 5.4587493611
5.461538 98.00088 94.01883 0.141065657 3.8409789325
5.538462 94.94451 92.27470 -0.047453868 2.7172667510
5.615385 92.59431 93.39841 0.069661833 -0.8737611515
5.692308 93.07702 96.98535 0.421389715 -4.3297193897
5.769231 96.75259 99.55959 0.636675391 -3.4436741371
5.846154 103.10340 99.75639 0.592687520 2.7543196387
5.923077 105.68439 96.74080 0.231859596 8.7117319094
6.000000 92.42340 92.92884 -0.172522180 -0.3329190837
6.076923 92.41103 92.48686 -0.199468137 0.1236370614
6.153846 88.52750 94.24918 -0.003288663 -5.7183931869
6.230769 91.29324 97.43403 0.315525294 -6.4563183365
6.307692 97.06080 98.07361 0.347930457 -1.3607412575
6.384615 100.64814 96.31032 0.136808437 4.2010112593
6.461538 97.01442 94.03220 -0.104684399 3.0869008327
6.538462 98.44481 95.23319 0.025883230 3.1857295552
6.615385 99.17923 98.31464 0.331439093 0.5331503774
6.692308 97.04449 100.04203 0.471034586 -3.4685766847
6.769231 97.00332 100.18475 0.438202829 -3.6196256246
6.846154 100.11056 98.56694 0.232602346 1.3110079451
6.923077 103.68868 96.58348 0.010995242 7.0942048051
7.000000 94.36425 94.95764 -0.152688284 -0.4407029133
7.076923 97.19534 96.29106 -0.004077185 0.9083549571
7.153846 92.50134 96.88841 0.056066081 -4.4431373574
7.230769 91.00883 97.24901 0.086519218 -6.3266976845
7.307692 94.45064 96.63900 0.016866551 -2.2052293388
7.384615 98.84380 95.68862 -0.079858282 3.2350399181
7.461538 100.65223 96.98526 0.057791633 3.6091713458
7.538462 103.82082 99.14489 0.267975103 4.4079530094
7.615385 100.32328 99.00460 0.227148744 1.0915323476
7.692308 95.55830 98.95839 0.199813096 -3.5999037105
7.769231 92.91824 97.34207 0.018199036 -4.4420275564
7.846154 96.55018 96.22130 -0.095697716 0.4245795264
7.923077 103.12001 96.71708 -0.036549304 6.4394707021
8.000000 97.94835 97.72655 0.068052587 0.1537414845
8.076923 101.54991 100.10218 0.298809633 1.1489280204
8.153846 95.84290 99.91410 0.250121347 -4.3213248080
8.230769 92.47007 98.94698 0.128397347 -6.6053083563
8.307692 96.16915 98.67311 0.088169871 -2.5921286679
8.384615 105.28753 101.17259 0.329300899 3.7856395767
8.461538 106.10177 101.33886 0.312998339 4.4499052278
8.538462 105.20143 100.73543 0.221355011 4.2446475737
8.615385 100.66439 99.59684 0.085360402 0.9821897524
8.692308 92.54708 97.05118 -0.177741338 -4.3263599496
8.769231 91.24261 96.36846 -0.228238908 -4.8976145629
8.846154 98.61515 97.99658 -0.042603094 0.6611731718
8.923077 107.48299 100.42101 0.204099864 6.8578782687
9.000000 102.79394 101.43235 0.284824131 1.0767696682
9.076923 101.40941 100.31103 0.144209235 0.9541748739
9.153846 93.36339 98.24811 -0.076502971 -4.8082208067
9.230769 91.10740 97.97025 -0.096638763 -6.7662182592
9.307692 98.78026 100.26533 0.142533328 -1.6276045550
9.384615 107.19184 103.06334 0.408080060 3.7204293355
9.461538 108.98540 104.40103 0.501041166 4.0833319157
9.538462 106.94934 102.94349 0.305183177 3.7006691355
9.615385 99.93468 100.02234 -0.017449623 -0.0702172047
9.692308 93.72638 98.42970 -0.174969143 -4.5283502321
9.769231 95.51776 99.70297 -0.030144716 -4.1550713072
9.846154 103.94185 102.08300 0.210872141 1.6479850052
9.923077 111.92110 104.32622 0.414107058 7.1807753385
10.000000 103.38305 102.66243 0.206317044 0.5143100816
10.076923 100.86361 100.79349 -0.001208003 0.0713260494
10.153846 94.37309 99.40208 -0.140228201 -4.8887639742
10.230769 96.03563 101.73780 0.107366006 -5.8095298959
10.307692 105.52255 105.60466 0.483315370 -0.5654176269
10.384615 111.26036 106.63052 0.537570308 4.0922737620
10.461538 108.62477 105.00374 0.321135025 3.2998999571
10.538462 104.17746 101.79880 -0.031472426 2.4101379369
10.615385 98.18817 99.17879 -0.290325698 -0.7002952864
10.692308 95.91847 100.04245 -0.174927137 -3.9490525238
10.769231 99.76741 102.83645 0.121965399 -3.1910038772
10.846154 107.25711 104.51823 0.277947147 2.4609246729
10.923077 109.32521 102.88842 0.087171032 6.3496152814
11.000000 98.71496 99.31013 -0.279375094 -0.3157901071
11.076923 97.18659 98.04890 -0.377560384 -0.4847547387
11.153846 96.08721 100.11844 -0.132850116 -3.8983871467
11.230769 100.11078 104.13447 0.282037749 -4.3057324407
11.307692 105.29822 105.27839 0.368226191 -0.3483978739
11.384615 107.60341 104.15756 0.219319730 3.2265326286
11.461538 102.21632 100.49565 -0.168802443 1.8894701528
11.538462 98.30324 97.39091 -0.462396883 1.3747248492
11.615385 95.99321 96.71560 -0.483687592 -0.2387010429
11.692308 94.72813 97.81499 -0.325379889 -2.7614823791
11.769231 97.02166 99.69371 -0.104970502 -2.5670768851
11.846154 100.71093 99.16086 -0.147757693 1.6978202108
11.923077 101.23787 96.73046 -0.376022233 4.8834307805
12.000000 92.65642 93.97857 -0.613609632 -0.7085312692
12.076923 97.21159 96.97055 -0.253050118 0.4940863328
12.153846 99.02272 101.07850 0.183050144 -2.2388356875
12.230769 100.01589 103.56361 0.413255791 -3.9609786697
12.307692 103.06139 103.62713 0.378282052 -0.9440237177
12.384615 102.37023 100.65313 0.043054321 1.6740439349
12.461538 98.91689 98.38942 -0.187622303 0.7150923914
12.538462 100.49986 99.28918 -0.078884125 1.2895620150
12.615385 102.23343 101.67164 0.167250707 0.3945297684
12.692308 102.47326 103.97249 0.380610203 -1.8798448320
12.769231 101.47797 103.88263 0.333563088 -2.7382256467
12.846154 102.57289 101.70562 0.082505751 0.7847620493
12.923077 103.64818 99.82855 -0.113451988 3.9330811843
13.000000 102.68460 101.85078 0.100116478 0.7337067884
13.076923 107.98625 105.31158 0.436184409 2.2384873802
13.153846 104.73152 105.62557 0.423965454 -1.3180130996
13.230769 101.13431 104.92379 0.311390991 -4.1008736240
13.307692 100.67092 102.87998 0.075870810 -2.2849346415
13.384615 101.45081 100.83563 -0.136151530 0.7513374379
13.461538 102.01468 100.97339 -0.108759813 1.1500451015
13.538462 105.82224 103.40306 0.145082604 2.2741013432
13.615385 106.65137 105.10285 0.300553760 1.2479677542
13.692308 102.05629 103.96737 0.156950282 -2.0680332916
13.769231 97.96489 101.78438 -0.077043485 -3.7424549963
13.846154 100.71704 100.87626 -0.160151196 0.0009310938
13.923077 106.79843 102.03894 -0.027868062 4.7873550482
14.000000 105.19774 103.04431 0.075455328 2.0779785129
14.076923 104.98227 102.75380 0.038858463 2.1896115597
14.153846 98.55409 100.51164 -0.189242858 -1.7683109539
14.230769 92.67075 98.12290 -0.409192608 -5.0429543447
14.307692 94.19119 97.73162 -0.407401688 -3.1330240016
14.384615 99.30646 98.71398 -0.268425525 0.8609043059
14.461538 103.36297 101.19140 0.006159350 2.1654147678
14.538462 105.52907 102.49699 0.136101853 2.8959859686
14.615385 101.50983 100.87589 -0.039617605 0.6735538408
14.692308 95.34420 98.61041 -0.262204177 -3.0040083612
14.769231 93.79057 98.14771 -0.282254107 -4.0748858368
14.846154 100.63922 100.16178 -0.052621419 0.5300636299
14.923077 108.52678 103.08151 0.244614170 5.2006486056
15.000000 105.10792 102.96757 0.208758658 1.9315910555
15.076923 104.19741 102.75371 0.166496227 1.2772062730
15.153846 97.53718 100.28256 -0.097267947 -2.6481099535
15.230769 93.60600 98.87054 -0.228743603 -5.0357906646
15.307692 97.39566 100.05973 -0.086949641 -2.5771193498
15.384615 104.65059 102.52325 0.168097095 1.9592438048
15.461538 106.32013 103.39635 0.238597278 2.6851847820
15.538462 104.51802 102.22710 0.097812660 2.1931081356
15.615385 99.40513 99.77873 -0.156805755 -0.2167924469
15.692308 95.91455 99.19796 -0.199202360 -3.0842080810xhat1 <- winter1$fitted[,2]
winter1.opt<- HoltWinters(training.ts, alpha= NULL, beta = NULL, gamma = NULL, seasonal = "additive")
winter1.optHolt-Winters exponential smoothing with trend and additive seasonal component.
Call:
HoltWinters(x = training.ts, alpha = NULL, beta = NULL, gamma = NULL, seasonal = "additive")
Smoothing parameters:
alpha: 0.4047538
beta : 0
gamma: 1
Coefficients:
[,1]
a 109.2446603
b 0.1827109
s1 -0.8266228
s2 5.6964456
s3 0.3679335
s4 -1.1981680
s5 -9.2800809
s6 -8.1509015
s7 -0.8688158
s8 4.3878031
s9 3.5559231
s10 -1.7827396
s11 -7.2290284
s12 -4.4226157
s13 4.2285397winter1.opt$fittedTime Series:
Start = c(2, 1)
End = c(15, 10)
Frequency = 13
xhat level trend season
2.000000 88.54720 87.09424 0.1827109 1.27025385
2.076923 89.83701 88.34376 0.1827109 1.31054615
2.153846 80.69475 88.89119 0.1827109 -8.37914615
2.230769 79.51454 89.00219 0.1827109 -9.67036154
2.307692 87.09484 89.13397 0.1827109 -2.22183846
2.384615 96.41638 89.61954 0.1827109 6.61413077
2.461538 96.19692 90.23692 0.1827109 5.77728462
2.538462 96.07390 90.50819 0.1827109 5.38300000
2.615385 86.46083 87.10919 0.1827109 -0.83106923
2.692308 77.30331 84.93626 0.1827109 -7.81566154
2.769231 80.60867 87.10194 0.1827109 -6.67597692
2.846154 95.91289 92.91179 0.1827109 2.81839231
2.923077 108.25457 95.65141 0.1827109 12.42044615
3.000000 93.20737 90.18551 0.1827109 2.83914360
3.076923 90.32767 88.29804 0.1827109 1.84691449
3.153846 76.58257 84.88445 0.1827109 -8.48459417
3.230769 78.69108 88.25364 0.1827109 -9.74526943
3.307692 91.84113 93.43486 0.1827109 -1.77644065
3.384615 102.73524 95.29915 0.1827109 7.25337618
3.461538 100.21016 94.11993 0.1827109 5.90751297
3.538462 90.83985 90.54153 0.1827109 0.11560808
3.615385 83.49537 87.60802 0.1827109 -4.29536258
3.692308 84.91386 89.63059 0.1827109 -4.89943847
3.769231 98.69569 96.91347 0.1827109 1.59950930
3.846154 108.07204 101.31064 0.1827109 6.57868522
3.923077 101.37750 97.08140 0.1827109 4.11338553
4.000000 93.78116 93.80379 0.1827109 -0.20534367
4.076923 86.32915 89.58838 0.1827109 -3.44194863
4.153846 84.62433 88.24005 0.1827109 -3.79843832
4.230769 88.54712 90.75867 0.1827109 -2.39426445
4.307692 94.86717 93.98792 0.1827109 0.69654855
4.384615 101.50057 96.06739 0.1827109 5.25046883
4.461538 91.45842 90.89943 0.1827109 0.37627394
4.538462 83.24697 87.53147 0.1827109 -4.46721922
4.615385 86.17978 87.58666 0.1827109 -1.58959214
4.692308 96.27035 90.54530 0.1827109 5.54233193
4.769231 101.43021 93.45005 0.1827109 7.79745781
4.846154 91.38710 91.11409 0.1827109 0.09030026
4.923077 91.26030 92.05308 0.1827109 -0.97549101
5.000000 83.53302 90.02369 0.1827109 -6.67338307
5.076923 85.38448 90.89533 0.1827109 -5.69355390
5.153846 94.05673 94.23719 0.1827109 -0.36316922
5.230769 100.06283 97.79404 0.1827109 2.08608484
5.307692 102.27950 98.61078 0.1827109 3.48600641
5.384615 92.72689 95.16261 0.1827109 -2.61843196
5.461538 88.32418 92.98695 0.1827109 -4.84547383
5.538462 88.79909 93.27114 0.1827109 -4.65476077
5.615385 100.98689 98.31138 0.1827109 2.49279706
5.692308 111.94349 102.21532 0.1827109 9.54545443
5.769231 103.39476 99.11864 0.1827109 4.09341078
5.846154 97.10793 95.72270 0.1827109 1.20251756
5.923077 86.98379 91.02977 0.1827109 -4.22869146
6.000000 85.12035 90.59787 0.1827109 -5.66022600
6.076923 92.32631 93.19119 0.1827109 -1.04758767
6.153846 102.16008 97.37841 0.1827109 4.59896421
6.230769 101.69657 98.49534 0.1827109 3.01851924
6.307692 93.45207 95.12307 0.1827109 -1.85370801
6.384615 86.58977 92.49380 0.1827109 -6.08674523
6.461538 88.96591 93.47944 0.1827109 -4.69623507
6.538462 102.23383 99.56220 0.1827109 2.48891673
6.615385 112.54312 104.39503 0.1827109 7.96537679
6.692308 106.89912 101.99375 0.1827109 4.72265749
6.769231 96.53654 97.52332 0.1827109 -1.16949671
6.846154 87.94903 93.73409 0.1827109 -5.96777363
6.923077 89.40455 94.35441 0.1827109 -5.13256869
7.000000 95.07372 97.00610 0.1827109 -2.11509058
7.076923 104.93349 99.90919 0.1827109 4.84159144
7.153846 104.33259 98.17701 0.1827109 5.97286168
7.230769 92.16043 94.18729 0.1827109 -2.20956889
7.307692 86.68788 92.49428 0.1827109 -5.98910466
7.384615 89.13828 93.86151 0.1827109 -4.90593768
7.461538 104.92159 100.75828 0.1827109 3.98060122
7.538462 112.97685 103.46658 0.1827109 9.32756555
7.615385 103.46508 99.11712 0.1827109 4.16525234
7.692308 95.53725 97.47496 0.1827109 -2.12042143
7.769231 87.16267 93.99075 0.1827109 -7.01078793
7.846154 89.05654 94.19804 0.1827109 -5.32420844
7.923077 97.29197 98.61085 0.1827109 -1.50159702
8.000000 105.33771 103.26939 0.1827109 1.88560833
8.076923 107.33941 105.13122 0.1827109 2.02548523
8.153846 102.00469 101.98527 0.1827109 -0.16328577
8.230769 92.42519 97.21056 0.1827109 -4.96807537
8.307692 92.53293 96.59732 0.1827109 -4.24710571
8.384615 108.28247 103.13174 0.1827109 4.96801882
8.461538 109.64985 101.77231 0.1827109 7.69482244
8.538462 101.50937 98.66428 0.1827109 2.66238312
8.615385 99.25339 97.58915 0.1827109 1.48153716
8.692308 85.68796 93.01839 0.1827109 -7.51314505
8.769231 88.16348 94.95540 0.1827109 -6.97463745
8.846154 101.22070 100.14124 0.1827109 0.89674536
8.923077 109.52547 104.26205 0.1827109 5.08071372
9.000000 109.78942 105.25173 0.1827109 4.35498391
9.076923 97.07022 99.75727 0.1827109 -2.86976727
9.153846 89.95844 97.22958 0.1827109 -7.45385671
9.230769 92.42705 98.38296 0.1827109 -6.13862240
9.307692 108.14849 102.87183 0.1827109 5.09395743
9.384615 107.51956 104.63676 0.1827109 2.70008776
9.461538 109.60618 106.56815 0.1827109 2.85532466
9.538462 103.53115 102.53588 0.1827109 0.81255474
9.615385 92.24640 97.57278 0.1827109 -5.50908977
9.692308 92.92902 97.67952 0.1827109 -4.93320255
9.769231 101.68174 101.11587 0.1827109 0.38315559
9.846154 110.55227 103.68131 0.1827109 6.68825239
9.923077 111.75162 105.30143 0.1827109 6.26747259
10.000000 97.53619 101.34755 0.1827109 -3.99407157
10.076923 93.02390 99.69698 0.1827109 -6.85578297
10.153846 94.39574 100.23939 0.1827109 -6.02635940
10.230769 105.80055 105.42367 0.1827109 0.19417341
10.307692 116.86588 109.26234 0.1827109 7.42083737
10.384615 111.40625 105.95179 0.1827109 5.27175296
10.461538 98.53463 101.69530 0.1827109 -3.34338034
10.538462 92.25371 98.82607 0.1827109 -6.75507567
10.615385 93.15827 98.59638 0.1827109 -5.62081558
10.692308 103.18480 103.15036 0.1827109 -0.14827069
10.769231 110.47042 106.40042 0.1827109 3.88728831
10.846154 114.39263 105.40776 0.1827109 8.80216511
10.923077 99.20822 98.84147 0.1827109 0.18404548
11.000000 89.19355 95.70102 0.1827109 -6.69017605
11.076923 91.60643 97.75051 0.1827109 -6.32679004
11.153846 106.65603 105.14418 0.1827109 1.32913269
11.230769 115.19897 109.44550 0.1827109 5.57076350
11.307692 107.73168 105.26546 0.1827109 2.28351458
11.384615 100.37571 101.44969 0.1827109 -1.25669860
11.461538 89.05419 96.70315 0.1827109 -7.83167310
11.538462 89.09274 96.27161 0.1827109 -7.36157698
11.615385 100.74188 99.75143 0.1827109 0.80774081
11.692308 105.76128 101.21588 0.1827109 4.36268394
11.769231 103.73492 101.39346 0.1827109 2.15874378
11.846154 97.05259 97.99304 0.1827109 -1.12316779
11.923077 90.51653 95.03693 0.1827109 -4.70311879
12.000000 90.98883 94.75093 0.1827109 -3.94481345
12.076923 107.36613 102.90550 0.1827109 4.27791726
12.153846 115.37260 107.80378 0.1827109 7.38610424
12.230769 105.36510 106.02765 0.1827109 -0.84525848
12.307692 99.92337 103.33746 0.1827109 -3.59679403
12.384615 89.68292 98.00606 0.1827109 -8.50584869
12.461538 90.10334 98.65564 0.1827109 -8.73501070
12.538462 102.27626 104.60628 0.1827109 -2.51272800
12.615385 111.92661 109.05119 0.1827109 2.69271903
12.692308 114.16629 109.62844 0.1827109 4.35513871
12.769231 101.19820 104.12623 0.1827109 -3.11074224
12.846154 93.78483 99.34136 0.1827109 -5.73923383
12.923077 93.90562 99.11533 0.1827109 -5.39242933
13.000000 115.52513 107.56352 0.1827109 7.77890166
13.076923 120.74574 109.35021 0.1827109 11.21281734
13.153846 108.80925 104.12119 0.1827109 4.50535204
13.230769 95.48763 100.37518 0.1827109 -5.07025671
13.307692 86.55367 98.07702 0.1827109 -11.70605539
13.384615 92.04636 99.68289 0.1827109 -7.81924268
13.461538 104.15667 104.22644 0.1827109 -0.25248025
13.538462 112.61748 108.67935 0.1827109 3.75541499
13.615385 112.71371 109.25804 0.1827109 3.27295613
13.692308 100.25818 104.08080 0.1827109 -4.00532839
13.769231 90.02226 100.25581 0.1827109 -10.41625662
13.846154 95.81379 101.97142 0.1827109 -6.34033350
13.923077 113.76164 106.81584 0.1827109 6.76308435
14.000000 116.59169 106.27119 0.1827109 10.13778603
14.076923 104.53835 101.10153 0.1827109 3.25411242
14.153846 95.75801 96.84767 0.1827109 -1.27237575
14.230769 85.17482 93.71083 0.1827109 -8.71871707
14.307692 87.53341 96.96379 0.1827109 -9.61309391
14.384615 101.43050 102.65382 0.1827109 -1.40603758
14.461538 113.74394 107.53378 0.1827109 6.02744559
14.538462 110.66496 106.14450 0.1827109 4.33775607
14.615385 96.26539 100.69228 0.1827109 -4.60959820
14.692308 88.77656 98.49306 0.1827109 -9.89920668
14.769231 92.94907 100.92828 0.1827109 -8.16191698
14.846154 106.79689 106.09882 0.1827109 0.51535843
14.923077 115.68067 109.80455 0.1827109 5.69340501
15.000000 108.81515 106.36607 0.1827109 2.26637414
15.076923 101.10521 104.19297 0.1827109 -3.27046998
15.153846 94.31776 100.28928 0.1827109 -6.15423099
15.230769 95.09352 99.11430 0.1827109 -4.20349180
15.307692 100.23340 101.56452 0.1827109 -1.51382220
15.384615 111.44482 105.76020 0.1827109 5.50191628
15.461538 108.51799 104.61968 0.1827109 3.71560446
15.538462 97.29717 101.06364 0.1827109 -3.94917968
15.615385 91.08628 99.01613 0.1827109 -8.11255803
15.692308 95.30405 101.70792 0.1827109 -6.58657959xhat1.opt <- winter1.opt$fitted[,2]8.1.2 Peramalan
#Forecast
forecast1 <- predict(winter1, n.ahead = 49)
forecast1.opt <- predict(winter1.opt, n.ahead = 49)8.1.3 Plot Deret Waktu
#Plot time series
plot(training.ts,main="Winter 0.2;0.1;0.1",type="l",col="black",
xlim=c(1,25),pch=12)
lines(xhat1,type="l",col="red")
lines(xhat1.opt,type="l",col="blue")
lines(forecast1,type="l",col="red")
lines(forecast1.opt,type="l",col="blue")
legend("topleft",c("Actual Data",expression(paste(winter1)),
expression(paste(winter1.opt))),cex=0.5,
col=c("black","red","blue"),lty=1)
8.1.4 Akurasi Data Latih
#Akurasi data training
SSE1<-winter1$SSE
MSE1<-winter1$SSE/length(training.ts)
RMSE1<-sqrt(MSE1)
akurasi1 <- matrix(c(SSE1,MSE1,RMSE1))
row.names(akurasi1)<- c("SSE", "MSE", "RMSE")
colnames(akurasi1) <- c("Akurasi")
akurasi1 Akurasi
SSE 19249.76741
MSE 100.25921
RMSE 10.01295SSE1.opt<-winter1.opt$SSE
MSE1.opt<-winter1.opt$SSE/length(training.ts)
RMSE1.opt<-sqrt(MSE1.opt)
akurasi1.opt <- matrix(c(SSE1.opt,MSE1.opt,RMSE1.opt))
row.names(akurasi1.opt)<- c("SSE1.opt", "MSE1.opt", "RMSE1.opt")
colnames(akurasi1.opt) <- c("Akurasi")
akurasi1.opt Akurasi
SSE1.opt 13896.693461
MSE1.opt 72.378612
RMSE1.opt 8.507562akurasi1.train = data.frame(Model_Winter = c("Winter 1","Winter1 optimal"),
Nilai_SSE=c(SSE1,SSE1.opt),
Nilai_MSE=c(MSE1,MSE1.opt),Nilai_RMSE=c(RMSE1,RMSE1.opt))
akurasi1.train8.1.5 Akurasi Data Uji
#Akurasi Data Testing
forecast1<-data.frame(forecast1)
testing.ts<-data.frame(testing.ts)
selisih1<-forecast1-testing.ts
SSEtesting1<-sum(selisih1^2)
MSEtesting1<-SSEtesting1/length(testing.ts)
forecast1.opt<-data.frame(forecast1.opt)
selisih1.opt<-forecast1.opt-testing.ts
SSEtesting1.opt<-sum(selisih1.opt^2)
MSEtesting1.opt<-SSEtesting1.opt/length(testing.ts)akurasi_testing <- data.frame(
Model = c("Model1", "Model Optimum"),
SSE = c(SSEtesting1, SSEtesting1.opt),
MSE = c(MSEtesting1, MSEtesting1.opt)
)
akurasi_testing8.2 Winter Multiplikatif
Model multiplikatif digunakan cocok digunakan jika plot data asli menunjukkan fluktuasi musiman yang bervariasi.
8.2.1 Pemulusan
#Pemulusan dengan winter multiplikatif
winter2 <- HoltWinters(training.ts,alpha=0.2,beta=0.1,gamma=0.3,seasonal = "multiplicative")
winter2$fittedTime Series:
Start = c(2, 1)
End = c(15, 10)
Frequency = 13
xhat level trend season
2.000000 88.53261 87.09424 0.1827109045 1.0143871
2.076923 89.35114 87.79949 0.2349649788 1.0149565
2.153846 80.24923 88.30776 0.2622954117 0.9060538
2.230769 79.27077 88.62929 0.2682192600 0.8917097
2.307692 87.01558 88.92396 0.2708642461 0.9755676
2.384615 96.19416 89.36447 0.2878291241 1.0729692
2.461538 96.06461 89.89390 0.3119889327 1.0649483
2.538462 96.16187 90.27182 0.3185824660 1.0615017
2.615385 88.25357 88.90655 0.1501970777 0.9909813
2.692308 79.77034 87.52036 -0.0034420368 0.9114848
2.769231 81.45889 88.05059 0.0499250122 0.9246131
2.846154 94.18287 90.92384 0.3322574326 1.0320721
2.923077 106.25007 92.81553 0.4882006614 1.1387548
3.000000 93.46197 91.20474 0.2783017380 1.0216317
3.076923 92.30178 90.43192 0.1731900926 1.0187259
3.153846 80.18948 88.47318 -0.0400032385 0.9067805
3.230769 79.77237 89.37403 0.0540816599 0.8920280
3.307692 90.21214 91.95454 0.3067245472 0.9777900
3.384615 101.04367 93.44425 0.4250231729 1.0764297
3.461538 100.14205 93.55838 0.3939339204 1.0658817
3.538462 95.89750 92.22149 0.2208517869 1.0373764
3.615385 87.26861 89.98293 -0.0250898510 0.9701057
3.692308 82.72934 90.11707 -0.0091659726 0.9181141
3.769231 90.94402 94.40508 0.4205508742 0.9590659
3.846154 104.66983 98.61350 0.7993377080 1.0528804
3.923077 109.23068 97.98853 0.6569069904 1.1073060
4.000000 96.75250 95.68286 0.3606495383 1.0073820
4.076923 92.38005 93.29629 0.0859275109 0.9892681
4.153846 83.81772 91.39417 -0.1128770171 0.9182354
4.230769 85.45819 92.71399 0.0303930079 0.9214379
4.307692 94.60832 95.04857 0.2608113224 0.9926444
4.384615 103.63941 96.30573 0.3604457716 1.0721373
4.461538 97.80625 93.80116 0.0739449194 1.0418763
4.538462 91.06052 90.97260 -0.2163054328 1.0033521
4.615385 86.28690 89.13601 -0.3783346868 0.9721627
4.692308 87.05379 90.14658 -0.2394443535 0.9682634
4.769231 93.59512 93.19998 0.0898402441 1.0032726
4.846154 96.96847 93.61124 0.1219827348 1.0345155
4.923077 99.22324 93.01543 0.0502027675 1.0661641
5.000000 87.79593 90.54665 -0.2016953336 0.9717857
5.076923 86.28955 89.81792 -0.2543989997 0.9634453
5.153846 85.00109 90.99589 -0.1111620516 0.9352627
5.230769 89.95472 94.60387 0.2607528223 0.9482431
5.307692 98.32000 97.32698 0.5069887906 1.0049678
5.384615 100.43759 96.83671 0.4072623072 1.0328413
5.461538 94.95598 94.62258 0.1451235701 1.0019867
5.538462 91.77859 93.49402 0.0177549954 0.9814656
5.615385 94.60933 95.35019 0.2015964981 0.9901366
5.692308 100.13739 98.69709 0.5161262797 1.0093151
5.769231 101.28129 99.94716 0.5895208040 1.0074064
5.846154 102.14189 99.20095 0.4559482280 1.0249355
5.923077 99.39523 96.32403 0.1226610371 1.0305717
6.000000 90.31449 93.74335 -0.1476729777 0.9649430
6.076923 91.90313 93.75353 -0.1318874069 0.9816441
6.153846 93.82687 95.72360 0.0783082468 0.9793841
6.230769 96.01347 97.97498 0.2956147751 0.9770316
6.307692 97.13890 97.63602 0.2321578999 0.9925483
6.384615 95.60013 95.72537 0.0178772024 0.9985051
6.461538 92.85689 94.33582 -0.1228657071 0.9856064
6.538462 96.87797 96.38135 0.0939735919 1.0041736
6.615385 103.06711 99.83024 0.4294659819 1.0280013
6.692308 103.19638 100.86124 0.4896193790 1.0182092
6.769231 99.26808 99.82004 0.3365373227 0.9911289
6.846154 95.98688 97.62517 0.0833960170 0.9823794
6.923077 95.74407 96.29227 -0.0582331575 0.9949086
7.000000 92.94046 96.18588 -0.0630490484 0.9668926
7.076923 98.81105 97.95434 0.1201014538 1.0075107
7.153846 99.03531 98.35065 0.1477230448 1.0054512
7.230769 94.56569 97.50155 0.0480402216 0.9694115
7.307692 92.72518 96.09727 -0.0971920884 0.9658866
7.384615 93.35030 95.35594 -0.1616050529 0.9806287
7.461538 99.00402 97.71843 0.0908044867 1.0122155
7.538462 105.03356 100.21137 0.3310176930 1.0446695
7.615385 103.73062 99.91940 0.2687192957 1.0353585
7.692308 99.38981 99.26591 0.1764976955 0.9994711
7.769231 92.93362 96.85860 -0.0818831413 0.9602890
7.846154 92.05196 95.58744 -0.2008101981 0.9650405
7.923077 96.33428 96.93178 -0.0462956801 0.9943108
8.000000 98.36221 99.30240 0.1953957420 0.9885868
8.076923 103.28299 101.74828 0.4204440891 1.0109062
8.153846 100.98221 101.34421 0.3379934370 0.9931159
8.230769 94.73900 99.42154 0.1119269153 0.9518306
8.307692 94.51815 98.63409 0.0219886463 0.9580571
8.384615 102.94833 101.51761 0.3081417194 1.0110246
8.461538 106.70059 102.12724 0.3382906693 1.0413316
8.538462 105.45757 101.47046 0.2387835794 1.0368534
8.615385 102.84323 100.34822 0.1026819546 1.0238158
8.692308 94.09543 97.45546 -0.1968626872 0.9674767
8.769231 90.93956 96.41657 -0.2810657930 0.9459519
8.846154 96.46524 98.16200 -0.0784160941 0.9835004
8.923077 103.60955 101.02919 0.2161452365 1.0233513
9.000000 104.69135 102.79117 0.3707284352 1.0148257
9.076923 101.70248 101.40236 0.1947741562 1.0010369
9.153846 95.92768 99.33374 -0.0315646779 0.9660179
9.230769 92.68928 98.56284 -0.1054986990 0.9414156
9.307692 99.81572 100.66183 0.1149506849 0.9904634
9.384615 105.12424 103.24873 0.3621457722 1.0146062
9.461538 108.49481 104.93468 0.4945254804 1.0290774
9.538462 106.01147 103.62131 0.3137361078 1.0199781
9.615385 99.56343 100.95581 0.0158126979 0.9860535
9.692308 95.07612 99.44945 -0.1364047115 0.9573376
9.769231 97.45291 100.54385 -0.0133241229 0.9693862
9.846154 104.65436 102.61756 0.1953789005 1.0179104
9.923077 109.44368 104.66953 0.3810385386 1.0418190
10.000000 103.10656 103.53167 0.2291487498 0.9936945
10.076923 99.07431 101.72805 0.0258718767 0.9736657
10.153846 96.31956 100.69366 -0.0801543902 0.9573223
10.230769 99.44740 102.79318 0.1378128134 0.9661560
10.307692 108.58852 106.11592 0.4563059634 1.0189195
10.384615 110.15718 106.50290 0.4493727539 1.0299658
10.461538 106.11909 105.06510 0.2606561821 1.0075321
10.538462 100.63282 102.32344 -0.0395761486 0.9838583
10.615385 96.93247 100.37342 -0.2306203992 0.9679425
10.692308 98.60592 101.59446 -0.0854540957 0.9714007
10.769231 103.45229 104.01203 0.1648479002 0.9930446
10.846154 109.41878 105.00548 0.2477085164 1.0395769
10.923077 105.44358 103.00218 0.0226076830 1.0234778
11.000000 96.93067 100.20193 -0.2596785244 0.9698668
11.076923 95.14966 99.29784 -0.3241197648 0.9613629
11.153846 100.04880 101.94293 -0.0271989041 0.9816817
11.230769 105.75038 105.33492 0.3147203969 1.0009535
11.307692 107.58715 105.38386 0.2881421925 1.0181235
11.384615 104.66325 103.75981 0.0969228338 1.0077657
11.461538 97.55474 100.58892 -0.2298584078 0.9720571
11.538462 94.08311 98.29783 -0.4359807662 0.9613869
11.615385 96.63056 98.51832 -0.3703340921 0.9845394
11.692308 99.34954 99.62645 -0.2224874086 0.9994525
11.769231 100.83629 100.68448 -0.0944361484 1.0024480
11.846154 100.41036 99.40215 -0.2132251807 1.0123142
11.923077 95.48671 96.99343 -0.4327747567 0.9888780
12.000000 91.19320 95.32123 -0.5567174431 0.9623139
12.076923 98.16691 98.81541 -0.1516272151 0.9949640
12.153846 105.17655 102.85483 0.2674771768 1.0199204
12.230769 104.32817 104.17268 0.3725138619 0.9979242
12.307692 103.12657 103.33047 0.2510422764 0.9956079
12.384615 96.95865 100.20136 -0.0869737416 0.9684788
12.461538 93.46784 98.85008 -0.2134040283 0.9475972
12.538462 97.82865 100.93427 0.0163552048 0.9690743
12.615385 104.58355 104.04181 0.3254739572 1.0020722
12.692308 108.08535 106.02741 0.4914869650 1.0147058
12.769231 104.02783 104.94910 0.3345064967 0.9880725
12.846154 100.70286 102.22659 0.0288055592 0.9848171
12.923077 97.84440 100.64538 -0.1321961895 0.9734484
13.000000 105.11097 103.89954 0.2064391617 1.0096535
13.076923 112.13127 106.95389 0.4912306168 1.0436143
13.153846 110.38355 106.53368 0.4000864320 1.0322610
13.230769 103.23469 104.74813 0.1815224349 0.9838467
13.307692 97.44901 102.10881 -0.1005619379 0.9553052
13.384615 95.56008 100.46335 -0.2550509841 0.9536144
13.461538 98.93321 101.73100 -0.1027815483 0.9734817
13.538462 105.47109 104.86887 0.2212839501 1.0036249
13.615385 109.33067 106.70923 0.3831912125 1.0209002
13.692308 104.98142 105.16089 0.1900387879 0.9964925
13.769231 97.46417 102.41567 -0.1034871288 0.9526154
13.846154 97.90938 101.54490 -0.1802162895 0.9659122
13.923077 104.52088 103.31554 0.0148701359 1.0115210
14.000000 109.36382 104.80220 0.1620487953 1.0419150
14.076923 107.27727 103.81331 0.0469545448 1.0329000
14.153846 101.63699 101.20752 -0.2183192677 1.0064144
14.230769 92.92857 98.19107 -0.4981321764 0.9512312
14.307692 91.10083 97.65756 -0.5016701188 0.9376768
14.384615 96.11288 99.29718 -0.2875414910 0.9707427
14.461538 103.53990 102.49621 0.0611156284 1.0095807
14.538462 106.27300 103.80937 0.1863201645 1.0218982
14.615385 101.96682 102.13055 -0.0001936158 0.9983987
14.692308 95.81515 99.80938 -0.2322916359 0.9622209
14.769231 93.46126 99.27083 -0.2629176749 0.9439777
14.846154 100.25923 101.51029 -0.0126798894 0.9877989
14.923077 107.87566 104.58361 0.2959208571 1.0285673
15.000000 107.87494 104.65754 0.2737217585 1.0280533
15.076923 104.20359 103.98187 0.1787823748 1.0004122
15.153846 98.59750 101.52286 -0.0849966184 0.9719989
15.230769 94.72519 99.86706 -0.2420769598 0.9508176
15.307692 96.92941 100.88085 -0.1164901618 0.9619414
15.384615 104.74592 103.51268 0.1583413438 1.0103683
15.461538 107.10612 104.34992 0.2262311270 1.0241926
15.538462 103.07030 103.04807 0.0734236000 0.9995035
15.615385 97.74427 100.86374 -0.1523525163 0.9705384
15.692308 96.30204 100.61680 -0.1618105752 0.9586586xhat2 <- winter2$fitted[,2]
winter2.opt<- HoltWinters(training.ts, alpha= NULL, beta = NULL, gamma = NULL, seasonal = "multiplicative")
winter2.opt$fittedTime Series:
Start = c(2, 1)
End = c(15, 10)
Frequency = 13
xhat level trend season
2.000000 88.53261 87.09424 0.1827109 1.0143871
2.076923 89.85599 88.34915 0.1827109 1.0149565
2.153846 80.70334 88.88853 0.1827109 0.9060538
2.230769 79.51360 88.98712 0.1827109 0.8917097
2.307692 87.11336 89.11235 0.1827109 0.9755676
2.384615 96.33626 89.60203 0.1827109 1.0729692
2.461538 96.28074 90.22613 0.1827109 1.0649483
2.538462 96.21829 90.46084 0.1827109 1.0615017
2.615385 86.56156 87.16663 0.1827109 0.9909813
2.692308 77.54931 84.89750 0.1827109 0.9114848
2.769231 80.77230 87.17523 0.1827109 0.9246131
2.846154 96.64178 93.45588 0.1827109 1.0320721
2.923077 109.36987 95.86067 0.1827109 1.1387548
3.000000 93.70705 90.61214 0.1827109 1.0320744
3.076923 90.59173 88.56243 0.1827109 1.0208077
3.153846 77.13591 85.06700 0.1827109 0.9048231
3.230769 79.06775 88.56937 0.1827109 0.8908833
3.307692 92.59594 94.26733 0.1827109 0.9803696
3.384615 103.78942 95.87317 0.1827109 1.0805108
3.461538 100.78495 94.37753 0.1827109 1.0658280
3.538462 91.00444 90.76103 0.1827109 1.0006674
3.615385 83.49430 87.71880 0.1827109 0.9498619
3.692308 84.91196 89.86589 0.1827109 0.9429571
3.769231 98.97174 97.68380 0.1827109 1.0112932
3.846154 108.95070 101.97984 0.1827109 1.0664446
3.923077 101.79227 97.62984 0.1827109 1.0406872
4.000000 93.95942 94.27772 0.1827109 0.9946960
4.076923 86.24818 89.90384 0.1827109 0.9573924
4.153846 84.56313 88.49984 0.1827109 0.9535486
4.230769 88.24762 91.19265 0.1827109 0.9657705
4.307692 95.00497 94.70100 0.1827109 1.0012780
4.384615 102.05857 96.74792 0.1827109 1.0529032
4.461538 91.90183 91.56065 0.1827109 1.0017272
4.538462 83.55005 87.96788 0.1827109 0.9478105
4.615385 86.27732 87.88294 0.1827109 0.9796932
4.692308 95.52978 90.89766 0.1827109 1.0488515
4.769231 100.76753 94.00147 0.1827109 1.0698987
4.846154 91.80141 92.05151 0.1827109 0.9953074
4.923077 91.58649 92.83381 0.1827109 0.9846261
5.000000 83.72803 90.60269 0.1827109 0.9222632
5.076923 85.47422 91.45600 0.1827109 0.9327305
5.153846 94.38371 95.03332 0.1827109 0.9912586
5.230769 100.14578 98.53188 0.1827109 1.0144983
5.307692 102.38262 99.31470 0.1827109 1.0289978
5.384615 92.62050 95.87866 0.1827109 0.9641805
5.461538 88.17674 93.62665 0.1827109 0.9399567
5.538462 88.86090 93.98320 0.1827109 0.9436632
5.615385 101.88481 99.35813 0.1827109 1.0235478
5.692308 112.90948 102.86700 0.1827109 1.0956797
5.769231 103.25886 99.65327 0.1827109 1.0342850
5.846154 97.00321 96.38176 0.1827109 1.0045435
5.923077 86.99401 91.68618 0.1827109 0.9469366
6.000000 85.17274 91.20638 0.1827109 0.9319793
6.076923 92.34426 93.98854 0.1827109 0.9805992
6.153846 102.34633 98.30426 0.1827109 1.0391865
6.230769 101.82672 99.32491 0.1827109 1.0233057
6.307692 93.53970 95.93311 0.1827109 0.9731978
6.384615 86.62670 93.14927 0.1827109 0.9281567
6.461538 88.94459 94.19275 0.1827109 0.9424547
6.538462 102.37957 100.73210 0.1827109 1.0145148
6.615385 113.20001 105.50318 0.1827109 1.0710986
6.692308 107.50677 102.98819 0.1827109 1.0420260
6.769231 96.71634 98.40403 0.1827109 0.9810279
6.846154 87.75526 94.40647 0.1827109 0.9277516
6.923077 89.33483 95.15314 0.1827109 0.9370539
7.000000 95.17698 98.03785 0.1827109 0.9690128
7.076923 105.23727 101.02324 0.1827109 1.0398329
7.153846 104.54889 99.21892 0.1827109 1.0517825
7.230769 92.47249 95.29506 0.1827109 0.9685238
7.307692 86.89044 93.38193 0.1827109 0.9286675
7.384615 89.28659 94.76837 0.1827109 0.9403430
7.461538 105.16305 102.12565 0.1827109 1.0279028
7.538462 113.05719 104.70315 0.1827109 1.0779068
7.615385 103.87759 100.59221 0.1827109 1.0307881
7.692308 95.97917 98.81572 0.1827109 0.9695020
7.769231 87.41478 94.97650 0.1827109 0.9186161
7.846154 89.12684 95.07371 0.1827109 0.9356518
7.923077 97.40834 99.80948 0.1827109 0.9741595
8.000000 105.58459 104.60160 0.1827109 1.0076375
8.076923 107.61235 106.37332 0.1827109 1.0099132
8.153846 101.90838 103.10331 0.1827109 0.9866619
8.230769 92.24318 98.23178 0.1827109 0.9372927
8.307692 92.49618 97.63318 0.1827109 0.9456151
8.384615 108.52078 104.64225 0.1827109 1.0352570
8.461538 109.78091 103.22016 0.1827109 1.0616815
8.538462 101.57719 100.20955 0.1827109 1.0118030
8.615385 99.42846 99.10430 0.1827109 1.0014247
8.692308 86.12138 94.40256 0.1827109 0.9105158
8.769231 88.55589 96.34342 0.1827109 0.9174292
8.846154 101.75387 101.87985 0.1827109 0.9969755
8.923077 109.83049 105.84806 0.1827109 1.0358360
9.000000 110.01030 106.69981 0.1827109 1.0292637
9.076923 97.46338 101.20200 0.1827109 0.9613222
9.153846 90.03918 98.35822 0.1827109 0.9137237
9.230769 92.43338 99.58176 0.1827109 0.9265160
9.307692 108.24064 104.47398 0.1827109 1.0342448
9.384615 107.64421 106.17123 0.1827109 1.0121318
9.461538 109.65286 108.05514 0.1827109 1.0130731
9.538462 103.44461 104.00051 0.1827109 0.9929105
9.615385 91.90737 98.96436 0.1827109 0.9269802
9.692308 92.87522 99.21407 0.1827109 0.9343887
9.769231 101.76152 102.95094 0.1827109 0.9866956
9.846154 110.82824 105.54890 0.1827109 1.0482034
9.923077 112.03849 107.01395 0.1827109 1.0451677
10.000000 97.70480 103.07116 0.1827109 0.9462580
10.076923 93.16784 101.21640 0.1827109 0.9188230
10.153846 94.52006 101.73175 0.1827109 0.9274450
10.230769 106.06110 107.32729 0.1827109 0.9865232
10.307692 117.51478 111.15904 0.1827109 1.0554422
10.384615 111.69627 107.73366 0.1827109 1.0350262
10.461538 98.84435 103.45277 0.1827109 0.9537694
10.538462 92.17238 100.25786 0.1827109 0.9176808
10.615385 92.97723 100.02129 0.1827109 0.9278795
10.692308 103.21582 105.06061 0.1827109 0.9807351
10.769231 110.66191 108.40142 0.1827109 1.0191352
10.846154 114.64360 107.33769 0.1827109 1.0662498
10.923077 99.67739 101.00613 0.1827109 0.9850631
11.000000 89.82673 97.57292 0.1827109 0.9188906
11.076923 92.05120 99.53266 0.1827109 0.9231395
11.153846 107.04427 107.43761 0.1827109 0.9946474
11.230769 115.53779 111.65848 0.1827109 1.0330523
11.307692 108.10847 107.42471 0.1827109 1.0046563
11.384615 100.58245 103.41822 0.1827109 0.9708643
11.461538 89.44296 98.36577 0.1827109 0.9076037
11.538462 89.27178 97.68650 0.1827109 0.9121538
11.615385 100.56970 101.45356 0.1827109 0.9895060
11.692308 105.45217 103.02103 0.1827109 1.0217864
11.769231 103.72615 103.32279 0.1827109 1.0021317
11.846154 96.80933 99.88387 0.1827109 0.9674492
11.923077 90.52228 96.88022 0.1827109 0.9326143
12.000000 91.16764 96.55083 0.1827109 0.9424615
12.076923 107.36168 105.23185 0.1827109 1.0184711
12.153846 115.40337 110.11078 0.1827109 1.0463299
12.230769 105.52956 108.38328 0.1827109 0.9720315
12.307692 99.99543 105.49989 0.1827109 0.9461862
12.384615 89.59563 99.74260 0.1827109 0.8966260
12.461538 90.21541 100.49321 0.1827109 0.8960972
12.538462 102.87438 107.15081 0.1827109 0.9584553
12.615385 112.73185 111.58620 0.1827109 1.0086155
12.692308 114.65062 111.83790 0.1827109 1.0234780
12.769231 101.05064 106.19466 0.1827109 0.9499262
12.846154 93.39153 101.13896 0.1827109 0.9217330
12.923077 93.68900 101.04717 0.1827109 0.9255073
13.000000 116.29239 110.38086 0.1827109 1.0518147
13.076923 121.05116 111.81039 0.1827109 1.0808805
13.153846 109.10540 106.80074 0.1827109 1.0198344
13.230769 96.07012 102.95840 0.1827109 0.9314436
13.307692 86.83997 100.18397 0.1827109 0.8652271
13.384615 92.27190 101.89860 0.1827109 0.9039059
13.461538 104.25868 106.87043 0.1827109 0.9738965
13.538462 112.85947 111.45579 0.1827109 1.0109368
13.615385 113.18630 111.93741 0.1827109 1.0095093
13.692308 100.62319 106.54471 0.1827109 0.9428054
13.769231 90.07018 102.25860 0.1827109 0.8792369
13.846154 95.82520 104.18663 0.1827109 0.9181355
13.923077 113.61614 109.51221 0.1827109 1.0357466
14.000000 116.72322 109.04055 0.1827109 1.0686663
14.076923 104.83834 104.09465 0.1827109 1.0053798
14.153846 96.12373 99.68072 0.1827109 0.9625519
14.230769 85.97748 96.21085 0.1827109 0.8919421
14.307692 88.12381 99.51433 0.1827109 0.8839160
14.384615 101.90875 105.74016 0.1827109 0.9621033
14.461538 114.08514 110.66903 0.1827109 1.0291686
14.538462 110.96977 109.16701 0.1827109 1.0148154
14.615385 96.88938 103.59658 0.1827109 0.9336100
14.692308 89.33367 100.91822 0.1827109 0.8836088
14.769231 93.28991 103.42685 0.1827109 0.9003986
14.846154 107.07768 109.07083 0.1827109 0.9800843
14.923077 115.99251 112.78056 0.1827109 1.0268161
15.000000 108.68158 109.26298 0.1827109 0.9930184
15.076923 100.70952 107.09553 0.1827109 0.9387693
15.153846 93.93569 103.03776 0.1827109 0.9100491
15.230769 95.13584 101.88013 0.1827109 0.9321301
15.307692 100.13873 104.51063 0.1827109 0.9564956
15.384615 111.50513 108.98780 0.1827109 1.0213851
15.461538 108.70123 107.83274 0.1827109 1.0063489
15.538462 97.45285 104.17391 0.1827109 0.9338445
15.615385 91.39364 101.86678 0.1827109 0.8955816
15.692308 95.71457 104.74922 0.1827109 0.9121587xhat2.opt <- winter2.opt$fitted[,2]8.2.2 Peramalan
#Forecast
forecast2 <- predict(winter2, n.ahead = 49)
forecast2.opt <- predict(winter2.opt, n.ahead = 49)8.2.3 Plot Deret Waktu
#Plot time series
plot(training.ts,main="Winter 0.2;0.1;0.1",type="l",col="black",
xlim=c(1,25),pch=12)
lines(xhat2,type="l",col="red")
lines(xhat2.opt,type="l",col="blue")
lines(forecast2,type="l",col="red")
lines(forecast2.opt,type="l",col="blue")
legend("topleft",c("Actual Data",expression(paste(winter2)),
expression(paste(winter2.opt))),cex=0.5,
col=c("black","red","blue"),lty=1)
8.2.4 Akurasi Data Latih
#Akurasi data training
SSE2<-winter2$SSE
MSE2<-winter2$SSE/length(training.ts)
RMSE2<-sqrt(MSE2)
akurasi1 <- matrix(c(SSE2,MSE2,RMSE2))
row.names(akurasi1)<- c("SSE2", "MSE2", "RMSE2")
colnames(akurasi1) <- c("Akurasi lamda=0.2")
akurasi1 Akurasi lamda=0.2
SSE2 18378.242877
MSE2 95.720015
RMSE2 9.783661SSE2.opt<-winter2.opt$SSE
MSE2.opt<-winter2.opt$SSE/length(training.ts)
RMSE2.opt<-sqrt(MSE2.opt)
akurasi1.opt <- matrix(c(SSE2.opt,MSE2.opt,RMSE2.opt))
row.names(akurasi1.opt)<- c("SSE2.opt", "MSE2.opt", "RMSE2.opt")
colnames(akurasi1.opt) <- c("Akurasi")
akurasi1.opt Akurasi
SSE2.opt 14014.482201
MSE2.opt 72.992095
RMSE2.opt 8.543541akurasi2.train = data.frame(Model_Winter = c("Winter 1","winter2 optimal"),
Nilai_SSE=c(SSE2,SSE2.opt),
Nilai_MSE=c(MSE2,MSE2.opt),Nilai_RMSE=c(RMSE2,RMSE2.opt))
akurasi2.train8.2.5 Akurasi Data Uji
#Akurasi Data Testing
forecast2<-data.frame(forecast2)
testing.ts<-data.frame(testing.ts)
selisih2<-forecast2-testing.ts
SSEtesting2<-sum(selisih2^2)
MSEtesting2<-SSEtesting2/length(testing.ts)
forecast2.opt<-data.frame(forecast2.opt)
selisih2.opt<-forecast2.opt-testing.ts
SSEtesting2.opt<-sum(selisih2.opt^2)
MSEtesting2.opt<-SSEtesting2.opt/length(testing.ts)# Atau rapikan dalam tabel
akurasi_testing2 <- data.frame(
Model = c("Model2", "Model2 Optimal"),
SSE = c(SSEtesting2, SSEtesting2.opt),
MSE = c(MSEtesting2, MSEtesting2.opt)
)
akurasi_testing29 Tugas
Setiap kelompok memilih satu data time series dengan panjang ≥ 500 periode.
Data dibagi rata ke setiap anggota; minimal 100 observasi per orang.
Terapkan metode pemulusan yang sudah dipelajari (SMA, DMA, SES, DES, atau Holt-Winters) pada bagian data masing-masing
Bandingkan kinerja metode dan tentukan metode terbaik untuk data tersebut menggunakan metrik akurasi (mis. SSE/MSE/RMSE/MAPE), lalu jelaskan alasannya secara singkat.
Upload tugas per individu ke GitHub pada folder
Pertemuan-2dengan format HTML bernamaTugas-Pertemuan-2.html.Deadline Pengumpulan : Senin, Jam 23.59 WIB