Tugas 1 Praktikum MPDW
Fadly Syabani
2026-09-08
Packages
library(forecast)## Warning: package 'forecast' was built under R version 4.5.3library(graphics)
library(TTR)## Warning: package 'TTR' was built under R version 4.5.3library(TSA)## Warning: package 'TSA' was built under R version 4.5.3## Registered 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, arima## The following object is masked from 'package:utils':
##
## tarlibrary(rio)## Warning: package 'rio' was built under R version 4.5.3library(ggplot2)## Warning: package 'ggplot2' was built under R version 4.5.2Impor Data
data <- read.csv("C:/Users/hp/Downloads/dataset_mpdw.csv")
data$tanggal <- as.Date(data$tanggal, format = "%m/%d/%Y")
head(data)## tanggal pm25
## 1 2024-09-18 76
## 2 2024-09-19 83
## 3 2024-09-20 88
## 4 2024-09-21 94
## 5 2024-09-22 101
## 6 2024-09-23 87Eksplorasi Data
View() : menampilkan data dalam bentuk tabel,
str() : menampilkan struktur data, dim() :
menampilkan dimensi data
View(data)
str(data)## 'data.frame': 161 obs. of 2 variables:
## $ tanggal: Date, format: "2024-09-18" "2024-09-19" ...
## $ pm25 : int 76 83 88 94 101 87 74 89 71 70 ...dim(data)## [1] 161 2Mengubah data agar terbaca sebagai data deret waktu dengan fungsi
ts() dan menampilkan ringkasan data.
data.ts <- ts(data$pm25)
summary(data.ts)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 10.00 51.00 64.00 64.24 79.00 115.00Membuat plot data deret waktu
ts.plot(data.ts, xlab="Periode", ylab="Konsentrasi PM2.5 (µg/m³)",
main = "Time Series Plot Konsentrasi PM2.5")
points(data.ts)
Moving Average
Pembagian Data
Pembagian data latih dan data uji dilakukan dengan perbandingan 80% data latih dan 20% data uji.
# hitung jumlah data
n <- nrow(data)
# tentukan batas 80%
n_train <- floor(0.8 * n)
# bagi data
training_ma <- data[1:n_train, ]
testing_ma <- data[(n_train+1):n, ]
# ubah ke time series
train_ma.ts <- ts(training_ma$pm25)
test_ma.ts <- ts(testing_ma$pm25)
n_train## [1] 128nrow(testing_ma)## [1] 33Eksplorasi data dilakukan pada keseluruhan data, data latih, serta data uji menggunakan plot data deret waktu.
#eksplorasi keseluruhan data
plot(
data.ts,
col = "red",
main = "Plot Semua Data",
xlab = "Periode",
ylab = "Konsentrasi PM2.5 (µg/m³)"
)
points(data.ts)
#eksplorasi data latih
plot(
train_ma.ts,
col = "blue",
main = "Plot Data Latih",
xlab = "Periode",
ylab = "Konsentrasi PM2.5 (µg/m³)"
)
points(train_ma.ts)
#eksplorasi data uji
plot(
test_ma.ts,
col = "blue",
main = "Plot Data Uji",
xlab = "Periode",
ylab = "Konsentrasi PM2.5 (µg/m³)"
)
points(test_ma.ts)
Eksplorasi data juga dapat dilakukan menggunakan package
ggplot2
ggplot() +
geom_line(data = training_ma, aes(x = tanggal, y = pm25, col = "Data Latih")) +
geom_line(data = testing_ma, aes(x = tanggal, y = pm25, col = "Data Uji")) +
labs(x = "Periode", y = "Konsentrasi PM2.5 (µg/m³)", 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))
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).
\[S_t = \frac{1}{m} \sum_{i=t-m+1}^{t} X_i \]\[S_t = \frac{1}{m} \sum_{i=t-m+1}^{t} X_i\]
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.sma## Time Series:
## Start = 1
## End = 128
## Frequency = 1
## [1] NA NA NA 85.25 91.50 92.50 89.00 87.75 80.25 76.00
## [11] 78.75 78.75 78.50 83.75 83.75 84.00 87.00 84.25 81.25 85.75
## [21] 84.25 83.25 86.50 80.25 80.50 84.50 84.50 81.50 85.75 80.25
## [31] 77.00 79.75 76.75 79.75 85.00 85.50 84.50 82.25 77.75 76.25
## [41] 77.50 78.00 80.50 80.50 81.50 84.50 80.50 76.00 70.50 63.25
## [51] 66.50 72.25 80.75 84.00 81.00 73.50 67.25 73.50 82.25 96.25
## [61] 106.50 99.50 88.25 76.50 63.75 58.75 57.00 54.25 49.25 51.00
## [71] 48.00 43.25 42.25 44.75 47.25 47.00 48.25 38.50 37.25 55.25
## [81] 60.75 71.50 73.75 54.50 47.75 43.50 39.50 43.00 45.00 41.25
## [91] 48.00 46.75 46.25 50.25 53.50 64.25 66.50 73.00 68.50 69.75
## [101] 81.25 66.75 53.75 35.75 26.25 30.50 42.25 54.75 53.00 58.00
## [111] 55.25 51.50 44.00 44.00 46.50 49.50 52.25 51.25 50.00 49.75
## [121] 54.25 49.25 49.00 44.50 43.75 48.00 50.25 58.75Data 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## [1] NA NA NA NA 85.25 91.50 92.50 89.00 87.75 80.25
## [11] 76.00 78.75 78.75 78.50 83.75 83.75 84.00 87.00 84.25 81.25
## [21] 85.75 84.25 83.25 86.50 80.25 80.50 84.50 84.50 81.50 85.75
## [31] 80.25 77.00 79.75 76.75 79.75 85.00 85.50 84.50 82.25 77.75
## [41] 76.25 77.50 78.00 80.50 80.50 81.50 84.50 80.50 76.00 70.50
## [51] 63.25 66.50 72.25 80.75 84.00 81.00 73.50 67.25 73.50 82.25
## [61] 96.25 106.50 99.50 88.25 76.50 63.75 58.75 57.00 54.25 49.25
## [71] 51.00 48.00 43.25 42.25 44.75 47.25 47.00 48.25 38.50 37.25
## [81] 55.25 60.75 71.50 73.75 54.50 47.75 43.50 39.50 43.00 45.00
## [91] 41.25 48.00 46.75 46.25 50.25 53.50 64.25 66.50 73.00 68.50
## [101] 69.75 81.25 66.75 53.75 35.75 26.25 30.50 42.25 54.75 53.00
## [111] 58.00 55.25 51.50 44.00 44.00 46.50 49.50 52.25 51.25 50.00
## [121] 49.75 54.25 49.25 49.00 44.50 43.75 48.00 50.25 58.75Selanjutnya dilakukan peramalan sebanyak 33 periode sesuai dengan jumlah data uji. Pada metode SMA, seluruh hasil peramalan untuk 33 periode ke depan akan memiliki nilai yang sama dengan hasil ramalan satu periode ke depan.
n_test <- nrow(testing_ma)
data.gab <- cbind(
aktual = c(data.ts),
pemulusan = c(data.sma, rep(NA, n_test)),
ramalan = c(data.ramal,
rep(data.ramal[length(data.ramal)], n_test - 1))
)
data.gab ## aktual pemulusan ramalan
## [1,] 76 NA NA
## [2,] 83 NA NA
## [3,] 88 NA NA
## [4,] 94 85.25 NA
## [5,] 101 91.50 85.25
## [6,] 87 92.50 91.50
## [7,] 74 89.00 92.50
## [8,] 89 87.75 89.00
## [9,] 71 80.25 87.75
## [10,] 70 76.00 80.25
## [11,] 85 78.75 76.00
## [12,] 89 78.75 78.75
## [13,] 70 78.50 78.75
## [14,] 91 83.75 78.50
## [15,] 85 83.75 83.75
## [16,] 90 84.00 83.75
## [17,] 82 87.00 84.00
## [18,] 80 84.25 87.00
## [19,] 73 81.25 84.25
## [20,] 108 85.75 81.25
## [21,] 76 84.25 85.75
## [22,] 76 83.25 84.25
## [23,] 86 86.50 83.25
## [24,] 83 80.25 86.50
## [25,] 77 80.50 80.25
## [26,] 92 84.50 80.50
## [27,] 86 84.50 84.50
## [28,] 71 81.50 84.50
## [29,] 94 85.75 81.50
## [30,] 70 80.25 85.75
## [31,] 73 77.00 80.25
## [32,] 82 79.75 77.00
## [33,] 82 76.75 79.75
## [34,] 82 79.75 76.75
## [35,] 94 85.00 79.75
## [36,] 84 85.50 85.00
## [37,] 78 84.50 85.50
## [38,] 73 82.25 84.50
## [39,] 76 77.75 82.25
## [40,] 78 76.25 77.75
## [41,] 83 77.50 76.25
## [42,] 75 78.00 77.50
## [43,] 86 80.50 78.00
## [44,] 78 80.50 80.50
## [45,] 87 81.50 80.50
## [46,] 87 84.50 81.50
## [47,] 70 80.50 84.50
## [48,] 60 76.00 80.50
## [49,] 65 70.50 76.00
## [50,] 58 63.25 70.50
## [51,] 83 66.50 63.25
## [52,] 83 72.25 66.50
## [53,] 99 80.75 72.25
## [54,] 71 84.00 80.75
## [55,] 71 81.00 84.00
## [56,] 53 73.50 81.00
## [57,] 74 67.25 73.50
## [58,] 96 73.50 67.25
## [59,] 106 82.25 73.50
## [60,] 109 96.25 82.25
## [61,] 115 106.50 96.25
## [62,] 68 99.50 106.50
## [63,] 61 88.25 99.50
## [64,] 62 76.50 88.25
## [65,] 64 63.75 76.50
## [66,] 48 58.75 63.75
## [67,] 54 57.00 58.75
## [68,] 51 54.25 57.00
## [69,] 44 49.25 54.25
## [70,] 55 51.00 49.25
## [71,] 42 48.00 51.00
## [72,] 32 43.25 48.00
## [73,] 40 42.25 43.25
## [74,] 65 44.75 42.25
## [75,] 52 47.25 44.75
## [76,] 31 47.00 47.25
## [77,] 45 48.25 47.00
## [78,] 26 38.50 48.25
## [79,] 47 37.25 38.50
## [80,] 103 55.25 37.25
## [81,] 67 60.75 55.25
## [82,] 69 71.50 60.75
## [83,] 56 73.75 71.50
## [84,] 26 54.50 73.75
## [85,] 40 47.75 54.50
## [86,] 52 43.50 47.75
## [87,] 40 39.50 43.50
## [88,] 40 43.00 39.50
## [89,] 48 45.00 43.00
## [90,] 37 41.25 45.00
## [91,] 67 48.00 41.25
## [92,] 35 46.75 48.00
## [93,] 46 46.25 46.75
## [94,] 53 50.25 46.25
## [95,] 80 53.50 50.25
## [96,] 78 64.25 53.50
## [97,] 55 66.50 64.25
## [98,] 79 73.00 66.50
## [99,] 62 68.50 73.00
## [100,] 83 69.75 68.50
## [101,] 101 81.25 69.75
## [102,] 21 66.75 81.25
## [103,] 10 53.75 66.75
## [104,] 11 35.75 53.75
## [105,] 63 26.25 35.75
## [106,] 38 30.50 26.25
## [107,] 57 42.25 30.50
## [108,] 61 54.75 42.25
## [109,] 56 53.00 54.75
## [110,] 58 58.00 53.00
## [111,] 46 55.25 58.00
## [112,] 46 51.50 55.25
## [113,] 26 44.00 51.50
## [114,] 58 44.00 44.00
## [115,] 56 46.50 44.00
## [116,] 58 49.50 46.50
## [117,] 37 52.25 49.50
## [118,] 54 51.25 52.25
## [119,] 51 50.00 51.25
## [120,] 57 49.75 50.00
## [121,] 55 54.25 49.75
## [122,] 34 49.25 54.25
## [123,] 50 49.00 49.25
## [124,] 39 44.50 49.00
## [125,] 52 43.75 44.50
## [126,] 51 48.00 43.75
## [127,] 59 50.25 48.00
## [128,] 73 58.75 50.25
## [129,] 55 NA 58.75
## [130,] 59 NA 58.75
## [131,] 50 NA 58.75
## [132,] 36 NA 58.75
## [133,] 32 NA 58.75
## [134,] 30 NA 58.75
## [135,] 51 NA 58.75
## [136,] 51 NA 58.75
## [137,] 47 NA 58.75
## [138,] 45 NA 58.75
## [139,] 73 NA 58.75
## [140,] 47 NA 58.75
## [141,] 53 NA 58.75
## [142,] 50 NA 58.75
## [143,] 48 NA 58.75
## [144,] 52 NA 58.75
## [145,] 38 NA 58.75
## [146,] 51 NA 58.75
## [147,] 74 NA 58.75
## [148,] 74 NA 58.75
## [149,] 68 NA 58.75
## [150,] 54 NA 58.75
## [151,] 67 NA 58.75
## [152,] 73 NA 58.75
## [153,] 50 NA 58.75
## [154,] 72 NA 58.75
## [155,] 63 NA 58.75
## [156,] 70 NA 58.75
## [157,] 55 NA 58.75
## [158,] 66 NA 58.75
## [159,] 57 NA 58.75
## [160,] 85 NA 58.75
## [161,] 57 NA 58.75Adapun plot data deret waktu dari hasil peramalan yang dilakukan adalah sebagai berikut.
ts.plot(data.ts, xlab="Periode", ylab="Konsentrasi PM2.5 (µg/m³)", main= "SMA N=4 Data Konsentrasi PM2.5")
points(data.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=1, col=c("black","green","red"), cex=0.7)
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.
\[SSE = \sum_{t=1}^{n} (y_t - \hat{y}_t)^2\]
\[MSE = \frac{1}{n} \sum_{t=1}^{n} (y_t - \hat{y}_t)^2\]
\[MAPE = \frac{1}{n} \sum_{t=1}^{n} \left|\frac{y_t - \hat{y}_t}{y_t}\right| * 100\]
Akurasi 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 41070.93750
## MSE 331.21724
## MAPE 30.69689Berdasarkan hasil perhitungan akurasi data latih, metode SMA dengan m = 4 menghasilkan nilai SSE sebesar 41.070,94, MSE sebesar 331,22, dan MAPE sebesar 30,70%. Nilai MAPE sebesar 30,70% menunjukkan bahwa rata-rata kesalahan hasil pemulusan terhadap data aktual adalah sebesar 30,70%. Berdasarkan nilai tersebut, metode SMA dengan m = 4 memiliki tingkat akurasi yang cukup pada data latih, meskipun masih terdapat kesalahan pemulusan yang relatif cukup besar.
Akurasi Data Uji
error_test.sma <- test_ma.ts -
data.gab[(n_train + 1):n, 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 5703.06250
## MSE 172.82008
## MAPE 22.43497Berdasarkan hasil perhitungan akurasi data uji, metode SMA dengan m = 4 menghasilkan nilai SSE sebesar 5.703,06, MSE sebesar 172,82, dan MAPE sebesar 22,43%. Nilai MAPE sebesar 22,43% menunjukkan bahwa rata-rata kesalahan peramalan terhadap data aktual adalah sekitar 22,43%, sehingga metode SMA dengan m = 4 memiliki tingkat akurasi yang cukup pada data uji.
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 (\(B_t\)):
\[A_t = 2S_{1,t} - S_{2,t} \qquad\text{dan}\qquad B_t = \frac{2}{m-1}\,\big(S_{1,t} - S_{2,t}\big)\]
# Pemulusan Pertama dan Kedua
dma <- SMA(data.sma, n = 4)
At <- 2 * data.sma - dma
Bt <- 2/(4-1) * (data.sma - dma)
data.dma <- At + Bt
# Forecast
data.ramal2 <- c(NA, data.dma)
t <- 1:nrow(testing_ma)
f <- c()
for (i in t) {
f[i] <-
At[length(At)] +
Bt[length(Bt)] * i
}
# Gabungkan hasil
data.gab2 <- cbind(
aktual = c(data.ts),
pemulusan1 = c(
data.sma,
rep(NA, nrow(testing_ma))
),
pemulusan2 = c(
dma,
rep(NA, nrow(testing_ma))
),
At = c(
At,
rep(NA, nrow(testing_ma))
),
Bt = c(
Bt,
rep(NA, nrow(testing_ma))
),
ramalan = c(
rep(
NA,
n_train
),
f
)
)
data.gab2## aktual pemulusan1 pemulusan2 At Bt ramalan
## [1,] 76 NA NA NA NA NA
## [2,] 83 NA NA NA NA NA
## [3,] 88 NA NA NA NA NA
## [4,] 94 85.25 NA NA NA NA
## [5,] 101 91.50 NA NA NA NA
## [6,] 87 92.50 NA NA NA NA
## [7,] 74 89.00 89.5625 88.4375 -0.3750000 NA
## [8,] 89 87.75 90.1875 85.3125 -1.6250000 NA
## [9,] 71 80.25 87.3750 73.1250 -4.7500000 NA
## [10,] 70 76.00 83.2500 68.7500 -4.8333333 NA
## [11,] 85 78.75 80.6875 76.8125 -1.2916667 NA
## [12,] 89 78.75 78.4375 79.0625 0.2083333 NA
## [13,] 70 78.50 78.0000 79.0000 0.3333333 NA
## [14,] 91 83.75 79.9375 87.5625 2.5416667 NA
## [15,] 85 83.75 81.1875 86.3125 1.7083333 NA
## [16,] 90 84.00 82.5000 85.5000 1.0000000 NA
## [17,] 82 87.00 84.6250 89.3750 1.5833333 NA
## [18,] 80 84.25 84.7500 83.7500 -0.3333333 NA
## [19,] 73 81.25 84.1250 78.3750 -1.9166667 NA
## [20,] 108 85.75 84.5625 86.9375 0.7916667 NA
## [21,] 76 84.25 83.8750 84.6250 0.2500000 NA
## [22,] 76 83.25 83.6250 82.8750 -0.2500000 NA
## [23,] 86 86.50 84.9375 88.0625 1.0416667 NA
## [24,] 83 80.25 83.5625 76.9375 -2.2083333 NA
## [25,] 77 80.50 82.6250 78.3750 -1.4166667 NA
## [26,] 92 84.50 82.9375 86.0625 1.0416667 NA
## [27,] 86 84.50 82.4375 86.5625 1.3750000 NA
## [28,] 71 81.50 82.7500 80.2500 -0.8333333 NA
## [29,] 94 85.75 84.0625 87.4375 1.1250000 NA
## [30,] 70 80.25 83.0000 77.5000 -1.8333333 NA
## [31,] 73 77.00 81.1250 72.8750 -2.7500000 NA
## [32,] 82 79.75 80.6875 78.8125 -0.6250000 NA
## [33,] 82 76.75 78.4375 75.0625 -1.1250000 NA
## [34,] 82 79.75 78.3125 81.1875 0.9583333 NA
## [35,] 94 85.00 80.3125 89.6875 3.1250000 NA
## [36,] 84 85.50 81.7500 89.2500 2.5000000 NA
## [37,] 78 84.50 83.6875 85.3125 0.5416667 NA
## [38,] 73 82.25 84.3125 80.1875 -1.3750000 NA
## [39,] 76 77.75 82.5000 73.0000 -3.1666667 NA
## [40,] 78 76.25 80.1875 72.3125 -2.6250000 NA
## [41,] 83 77.50 78.4375 76.5625 -0.6250000 NA
## [42,] 75 78.00 77.3750 78.6250 0.4166667 NA
## [43,] 86 80.50 78.0625 82.9375 1.6250000 NA
## [44,] 78 80.50 79.1250 81.8750 0.9166667 NA
## [45,] 87 81.50 80.1250 82.8750 0.9166667 NA
## [46,] 87 84.50 81.7500 87.2500 1.8333333 NA
## [47,] 70 80.50 81.7500 79.2500 -0.8333333 NA
## [48,] 60 76.00 80.6250 71.3750 -3.0833333 NA
## [49,] 65 70.50 77.8750 63.1250 -4.9166667 NA
## [50,] 58 63.25 72.5625 53.9375 -6.2083333 NA
## [51,] 83 66.50 69.0625 63.9375 -1.7083333 NA
## [52,] 83 72.25 68.1250 76.3750 2.7500000 NA
## [53,] 99 80.75 70.6875 90.8125 6.7083333 NA
## [54,] 71 84.00 75.8750 92.1250 5.4166667 NA
## [55,] 71 81.00 79.5000 82.5000 1.0000000 NA
## [56,] 53 73.50 79.8125 67.1875 -4.2083333 NA
## [57,] 74 67.25 76.4375 58.0625 -6.1250000 NA
## [58,] 96 73.50 73.8125 73.1875 -0.2083333 NA
## [59,] 106 82.25 74.1250 90.3750 5.4166667 NA
## [60,] 109 96.25 79.8125 112.6875 10.9583333 NA
## [61,] 115 106.50 89.6250 123.3750 11.2500000 NA
## [62,] 68 99.50 96.1250 102.8750 2.2500000 NA
## [63,] 61 88.25 97.6250 78.8750 -6.2500000 NA
## [64,] 62 76.50 92.6875 60.3125 -10.7916667 NA
## [65,] 64 63.75 82.0000 45.5000 -12.1666667 NA
## [66,] 48 58.75 71.8125 45.6875 -8.7083333 NA
## [67,] 54 57.00 64.0000 50.0000 -4.6666667 NA
## [68,] 51 54.25 58.4375 50.0625 -2.7916667 NA
## [69,] 44 49.25 54.8125 43.6875 -3.7083333 NA
## [70,] 55 51.00 52.8750 49.1250 -1.2500000 NA
## [71,] 42 48.00 50.6250 45.3750 -1.7500000 NA
## [72,] 32 43.25 47.8750 38.6250 -3.0833333 NA
## [73,] 40 42.25 46.1250 38.3750 -2.5833333 NA
## [74,] 65 44.75 44.5625 44.9375 0.1250000 NA
## [75,] 52 47.25 44.3750 50.1250 1.9166667 NA
## [76,] 31 47.00 45.3125 48.6875 1.1250000 NA
## [77,] 45 48.25 46.8125 49.6875 0.9583333 NA
## [78,] 26 38.50 45.2500 31.7500 -4.5000000 NA
## [79,] 47 37.25 42.7500 31.7500 -3.6666667 NA
## [80,] 103 55.25 44.8125 65.6875 6.9583333 NA
## [81,] 67 60.75 47.9375 73.5625 8.5416667 NA
## [82,] 69 71.50 56.1875 86.8125 10.2083333 NA
## [83,] 56 73.75 65.3125 82.1875 5.6250000 NA
## [84,] 26 54.50 65.1250 43.8750 -7.0833333 NA
## [85,] 40 47.75 61.8750 33.6250 -9.4166667 NA
## [86,] 52 43.50 54.8750 32.1250 -7.5833333 NA
## [87,] 40 39.50 46.3125 32.6875 -4.5416667 NA
## [88,] 40 43.00 43.4375 42.5625 -0.2916667 NA
## [89,] 48 45.00 42.7500 47.2500 1.5000000 NA
## [90,] 37 41.25 42.1875 40.3125 -0.6250000 NA
## [91,] 67 48.00 44.3125 51.6875 2.4583333 NA
## [92,] 35 46.75 45.2500 48.2500 1.0000000 NA
## [93,] 46 46.25 45.5625 46.9375 0.4583333 NA
## [94,] 53 50.25 47.8125 52.6875 1.6250000 NA
## [95,] 80 53.50 49.1875 57.8125 2.8750000 NA
## [96,] 78 64.25 53.5625 74.9375 7.1250000 NA
## [97,] 55 66.50 58.6250 74.3750 5.2500000 NA
## [98,] 79 73.00 64.3125 81.6875 5.7916667 NA
## [99,] 62 68.50 68.0625 68.9375 0.2916667 NA
## [100,] 83 69.75 69.4375 70.0625 0.2083333 NA
## [101,] 101 81.25 73.1250 89.3750 5.4166667 NA
## [102,] 21 66.75 71.5625 61.9375 -3.2083333 NA
## [103,] 10 53.75 67.8750 39.6250 -9.4166667 NA
## [104,] 11 35.75 59.3750 12.1250 -15.7500000 NA
## [105,] 63 26.25 45.6250 6.8750 -12.9166667 NA
## [106,] 38 30.50 36.5625 24.4375 -4.0416667 NA
## [107,] 57 42.25 33.6875 50.8125 5.7083333 NA
## [108,] 61 54.75 38.4375 71.0625 10.8750000 NA
## [109,] 56 53.00 45.1250 60.8750 5.2500000 NA
## [110,] 58 58.00 52.0000 64.0000 4.0000000 NA
## [111,] 46 55.25 55.2500 55.2500 0.0000000 NA
## [112,] 46 51.50 54.4375 48.5625 -1.9583333 NA
## [113,] 26 44.00 52.1875 35.8125 -5.4583333 NA
## [114,] 58 44.00 48.6875 39.3125 -3.1250000 NA
## [115,] 56 46.50 46.5000 46.5000 0.0000000 NA
## [116,] 58 49.50 46.0000 53.0000 2.3333333 NA
## [117,] 37 52.25 48.0625 56.4375 2.7916667 NA
## [118,] 54 51.25 49.8750 52.6250 0.9166667 NA
## [119,] 51 50.00 50.7500 49.2500 -0.5000000 NA
## [120,] 57 49.75 50.8125 48.6875 -0.7083333 NA
## [121,] 55 54.25 51.3125 57.1875 1.9583333 NA
## [122,] 34 49.25 50.8125 47.6875 -1.0416667 NA
## [123,] 50 49.00 50.5625 47.4375 -1.0416667 NA
## [124,] 39 44.50 49.2500 39.7500 -3.1666667 NA
## [125,] 52 43.75 46.6250 40.8750 -1.9166667 NA
## [126,] 51 48.00 46.3125 49.6875 1.1250000 NA
## [127,] 59 50.25 46.6250 53.8750 2.4166667 NA
## [128,] 73 58.75 50.1875 67.3125 5.7083333 NA
## [129,] 55 NA NA NA NA 73.02083
## [130,] 59 NA NA NA NA 78.72917
## [131,] 50 NA NA NA NA 84.43750
## [132,] 36 NA NA NA NA 90.14583
## [133,] 32 NA NA NA NA 95.85417
## [134,] 30 NA NA NA NA 101.56250
## [135,] 51 NA NA NA NA 107.27083
## [136,] 51 NA NA NA NA 112.97917
## [137,] 47 NA NA NA NA 118.68750
## [138,] 45 NA NA NA NA 124.39583
## [139,] 73 NA NA NA NA 130.10417
## [140,] 47 NA NA NA NA 135.81250
## [141,] 53 NA NA NA NA 141.52083
## [142,] 50 NA NA NA NA 147.22917
## [143,] 48 NA NA NA NA 152.93750
## [144,] 52 NA NA NA NA 158.64583
## [145,] 38 NA NA NA NA 164.35417
## [146,] 51 NA NA NA NA 170.06250
## [147,] 74 NA NA NA NA 175.77083
## [148,] 74 NA NA NA NA 181.47917
## [149,] 68 NA NA NA NA 187.18750
## [150,] 54 NA NA NA NA 192.89583
## [151,] 67 NA NA NA NA 198.60417
## [152,] 73 NA NA NA NA 204.31250
## [153,] 50 NA NA NA NA 210.02083
## [154,] 72 NA NA NA NA 215.72917
## [155,] 63 NA NA NA NA 221.43750
## [156,] 70 NA NA NA NA 227.14583
## [157,] 55 NA NA NA NA 232.85417
## [158,] 66 NA NA NA NA 238.56250
## [159,] 57 NA NA NA NA 244.27083
## [160,] 85 NA NA NA NA 249.97917
## [161,] 57 NA NA NA NA 255.68750Hasil pemulusan menggunakan metode DMA divisualisasikan sebagai berikut
ts.plot(data.ts,
xlab = "Periode",
ylab = "Konsentrasi PM2.5 (µg/m³)",
main = "DMA N=4 Data Konsentrasi PM2.5")
points(data.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.
Akurasi 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 61507.82075
## MSE 508.32910
## MAPE 35.26034Berdasarkan hasil perhitungan akurasi data latih, metode Double Moving Average (DMA) dengan m = 4 menghasilkan nilai SSE sebesar 61.507,82, MSE sebesar 508,33, dan MAPE sebesar 35,26%. Nilai MAPE sebesar 35,26% menunjukkan bahwa rata-rata kesalahan hasil pemulusan terhadap data aktual adalah sekitar 35,26%. Berdasarkan nilai tersebut, metode DMA dengan m = 4 memiliki tingkat akurasi yang cukup pada data latih, meskipun masih terdapat kesalahan pemulusan yang relatif cukup besar.
Akurasi Data Uji
error_test.dma <-
test_ma.ts -
data.gab2[
(n_train + 1):n,
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) <-
"Akurasi m = 4"
akurasi_test.dma## Akurasi m = 4
## SSE 462651.8754
## MSE 14019.7538
## MAPE 195.3331Berdasarkan hasil perhitungan akurasi data uji, metode Double Moving Average (DMA) dengan m = 4 menghasilkan nilai SSE sebesar 462.651,88, MSE sebesar 14.019,75, dan MAPE sebesar 195,33%. Nilai MAPE sebesar 195,33% menunjukkan bahwa rata-rata kesalahan hasil peramalan terhadap data aktual sangat besar, yaitu sekitar 195,33%. Berdasarkan nilai tersebut, metode DMA dengan m = 4 memiliki tingkat akurasi yang sangat rendah pada data uji.
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.
Pembagian Data
Pembagian data latih dan data uji dilakukan dengan perbandingan 80% data latih dan 20% data uji.
n <- nrow(data)
n_train <- floor(
0.8 * n
)
training <- data[
1:n_train,
]
testing <- data[
(n_train + 1):n,
]
train.ts <- ts(
training$pm25
)
test.ts <- ts(
testing$pm25
)
h_test <- nrow(testing)
n_train## [1] 128h_test## [1] 33Eksplorasi
Eksplorasi dilakukan dengan membuat plot data deret waktu untuk keseluruhan data, data latih, dan data uji.
# eksplorasi data keseluruhan
plot(
data.ts,
col = "black",
main = "Plot Semua Data",
xlab = "Periode",
ylab = "Konsentrasi PM2.5 (µg/m³)"
)
points(data.ts)
# eksplorasi data latih
plot(
train.ts,
col = "red",
main = "Plot Data Latih",
xlab = "Periode",
ylab = "Konsentrasi PM2.5 (µg/m³)"
)
points(train.ts)
# eksplorasi data uji
plot(
test.ts,
col = "blue",
main = "Plot Data Uji",
xlab = "Periode",
ylab = "Konsentrasi PM2.5 (µg/m³)"
)
points(test.ts)
ggplot() +
geom_line(data = training,
aes(x = tanggal, y = pm25, color = "Data Latih")) +
geom_line(data = testing,
aes(x = tanggal, y = pm25, color = "Data Uji")) +
labs(
x = "Periode",
y = "Konsentrasi PM2.5 (µg/m³)",
color = "Keterangan"
) +
scale_colour_manual(
values = c(
"Data Latih" = "blue",
"Data Uji" = "red"
)
) +
theme_bw() +
theme(
legend.position = "bottom"
)
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 .
# SES alpha = 0.2
ses.1 <- ses(train.ts, h = h_test, alpha = 0.2)
# SES alpha = 0.7
ses.2 <- ses(train.ts, h = h_test, alpha = 0.7)
# SES dengan alpha optimum
ses.opt <- ses(train.ts, h = h_test, alpha = NULL)
ses.1## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 129 55.05754 33.68469 76.43039 22.370583 87.74449
## 130 55.05754 33.26142 76.85365 21.723254 88.39182
## 131 55.05754 32.84622 77.26885 21.088258 89.02681
## 132 55.05754 32.43864 77.67643 20.464917 89.65016
## 133 55.05754 32.03827 78.07680 19.852610 90.26246
## 134 55.05754 31.64475 78.47032 19.250773 90.86430
## 135 55.05754 31.25774 78.85733 18.658886 91.45619
## 136 55.05754 30.87692 79.23816 18.076470 92.03860
## 137 55.05754 30.50200 79.61307 17.503086 92.61199
## 138 55.05754 30.13272 79.98235 16.938326 93.17675
## 139 55.05754 29.76884 80.34623 16.381812 93.73326
## 140 55.05754 29.41012 80.70496 15.833193 94.28188
## 141 55.05754 29.05634 81.05873 15.292142 94.82293
## 142 55.05754 28.70732 81.40776 14.758354 95.35672
## 143 55.05754 28.36286 81.75222 14.231545 95.88353
## 144 55.05754 28.02278 82.09229 13.711448 96.40362
## 145 55.05754 27.68693 82.42814 13.197812 96.91726
## 146 55.05754 27.35516 82.75992 12.690403 97.42467
## 147 55.05754 27.02731 83.08777 12.188999 97.92607
## 148 55.05754 26.70325 83.41183 11.693393 98.42168
## 149 55.05754 26.38285 83.73222 11.203387 98.91169
## 150 55.05754 26.06599 84.04908 10.718796 99.39628
## 151 55.05754 25.75256 84.36251 10.239444 99.87563
## 152 55.05754 25.44245 84.67263 9.765165 100.34991
## 153 55.05754 25.13555 84.97953 9.295802 100.81927
## 154 55.05754 24.83176 85.28331 8.831204 101.28387
## 155 55.05754 24.53100 85.58407 8.371229 101.74384
## 156 55.05754 24.23317 85.88190 7.915742 102.19933
## 157 55.05754 23.93820 86.17688 7.464614 102.65046
## 158 55.05754 23.64599 86.46908 7.017722 103.09735
## 159 55.05754 23.35648 86.75860 6.574950 103.54012
## 160 55.05754 23.06958 87.04549 6.136184 103.97889
## 161 55.05754 22.78524 87.32983 5.701319 104.41375ses.2## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 129 68.02137 46.30830990 89.73443 34.814110 101.2286
## 130 68.02137 41.51720244 94.52554 27.486744 108.5560
## 131 68.02137 37.46838622 98.57435 21.294614 114.7481
## 132 68.02137 33.89661744 102.14612 15.832065 120.2107
## 133 68.02137 30.66480891 105.37793 10.889440 125.1533
## 134 68.02137 27.69115086 108.35159 6.341622 129.7011
## 135 68.02137 24.92217632 111.12056 2.106841 133.9359
## 136 68.02137 22.32066525 113.72207 -1.871826 137.9146
## 137 68.02137 19.85947288 116.18327 -5.635895 141.6786
## 138 68.02137 17.51808047 118.52466 -9.216746 145.2595
## 139 68.02137 15.28053029 120.76221 -12.638784 148.6815
## 140 68.02137 13.13412109 122.90862 -15.921433 151.9642
## 141 68.02137 11.06854731 124.97419 -19.080456 155.1232
## 142 68.02137 9.07531049 126.96743 -22.128849 158.1716
## 143 68.02137 7.14730450 128.89544 -25.077479 161.1202
## 144 68.02137 5.27851566 130.76422 -27.935545 163.9783
## 145 68.02137 3.46380120 132.57894 -30.710911 166.7537
## 146 68.02137 1.69872221 134.34402 -33.410367 169.4531
## 147 68.02137 -0.02058418 136.06332 -36.039819 172.0826
## 148 68.02137 -1.69750440 137.74024 -38.604447 174.6472
## 149 68.02137 -3.33502682 139.37777 -41.108822 177.1516
## 150 68.02137 -4.93580433 140.97854 -43.556999 179.5997
## 151 68.02137 -6.50220484 142.54494 -45.952602 181.9953
## 152 68.02137 -8.03635233 144.07909 -48.298878 184.3416
## 153 68.02137 -9.54016070 145.58290 -50.598755 186.6415
## 154 68.02137 -11.01536174 147.05810 -52.854880 188.8976
## 155 68.02137 -12.46352848 148.50627 -55.069660 191.1124
## 156 68.02137 -13.88609488 149.92883 -57.245288 193.2880
## 157 68.02137 -15.28437244 151.32711 -59.383769 195.4265
## 158 68.02137 -16.65956435 152.70230 -61.486943 197.5297
## 159 68.02137 -18.01277765 154.05552 -63.556504 199.5992
## 160 68.02137 -19.34503362 155.38777 -65.594014 201.6368
## 161 68.02137 -20.65727681 156.70002 -67.600917 203.6437ses.opt## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 129 59.3812 38.09987 80.66254 26.8342083 91.92820
## 130 59.3812 36.86161 81.90079 24.9404642 93.82194
## 131 59.3812 35.68799 83.07442 23.1455560 95.61685
## 132 59.3812 34.56981 84.19259 21.4354557 97.32695
## 133 59.3812 33.49990 85.26250 19.7991697 98.96324
## 134 59.3812 32.47250 86.28991 18.2278922 100.53451
## 135 59.3812 31.48291 87.27950 16.7144406 102.04796
## 136 59.3812 30.52723 88.23517 15.2528647 103.50954
## 137 59.3812 29.60221 89.16019 13.8381699 104.92424
## 138 59.3812 28.70508 90.05733 12.4661150 106.29629
## 139 59.3812 27.83344 90.92897 11.1330620 107.62934
## 140 59.3812 26.98525 91.77716 9.8358628 108.92654
## 141 59.3812 26.15870 92.60370 8.5717712 110.19063
## 142 59.3812 25.35223 93.41018 7.3383746 111.42403
## 143 59.3812 24.56443 94.19798 6.1335400 112.62887
## 144 59.3812 23.79406 94.96834 4.9553706 113.80703
## 145 59.3812 23.04003 95.72238 3.8021705 114.96023
## 146 59.3812 22.30132 96.46108 2.6724163 116.08999
## 147 59.3812 21.57705 97.18536 1.5647338 117.19767
## 148 59.3812 20.86639 97.89602 0.4778777 118.28453
## 149 59.3812 20.16861 98.59380 -0.5892844 119.35169
## 150 59.3812 19.48303 99.27937 -1.6377857 120.40019
## 151 59.3812 18.80904 99.95337 -2.6685722 121.43098
## 152 59.3812 18.14606 100.61635 -3.6825126 122.44492
## 153 59.3812 17.49357 101.26884 -4.6804068 123.44281
## 154 59.3812 16.85109 101.91132 -5.6629934 124.42540
## 155 59.3812 16.21817 102.54423 -6.6309558 125.39336
## 156 59.3812 15.59440 103.16800 -7.5849282 126.34733
## 157 59.3812 14.97940 103.78301 -8.5255003 127.28791
## 158 59.3812 14.37279 104.38961 -9.4532214 128.21563
## 159 59.3812 13.77426 104.98815 -10.3686042 129.13101
## 160 59.3812 13.18347 105.57893 -11.2721283 130.03453
## 161 59.3812 12.60015 106.16225 -12.1642430 130.92665Pada 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).
Untuk mendapatkan gambar hasil pemulusan pada data latih dengan
fungsi ses() , perlu digunakan fungsi
autoplot() dan autolayer() .
autoplot(ses.1) +
autolayer(fitted(ses.1), series="Fitted") +
ylab("Konsentrasi PM2.5 (µg/m³)") + xlab("Periode")
Selanjutnya akan digunakan fungsi HoltWinters() dengan
nilai inisialisasi parameter dan panjang periode peramalan yang sama
dengan fungsi ses() .
# Holt-Winters SES
ses1<- HoltWinters(train.ts, gamma = FALSE, beta = FALSE, alpha = 0.2)
plot(ses1)
ses2<- HoltWinters(train.ts, gamma = FALSE, beta = FALSE, alpha = 0.7)
plot(ses2)
HWopt <- HoltWinters(
train.ts,
gamma = FALSE,
beta = FALSE,
alpha = NULL
)
plot(HWopt)
# Forecast
ramalan1<- forecast(ses1, h=h_test)
ramalan1## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 129 55.05754 33.65353 76.46154 22.322935 87.79214
## 130 55.05754 33.22965 76.88542 21.674662 88.44041
## 131 55.05754 32.81384 77.30123 21.038740 89.07633
## 132 55.05754 32.40567 77.70941 20.414490 89.70058
## 133 55.05754 32.00472 78.11036 19.801291 90.31378
## 134 55.05754 31.61062 78.50445 19.198577 90.91650
## 135 55.05754 31.22304 78.89203 18.605827 91.50925
## 136 55.05754 30.84167 79.27340 18.022562 92.09251
## 137 55.05754 30.46621 79.64887 17.448342 92.66673
## 138 55.05754 30.09639 80.01868 16.882759 93.23231
## 139 55.05754 29.73198 80.38310 16.325433 93.78964
## 140 55.05754 29.37273 80.74234 15.776014 94.33906
## 141 55.05754 29.01844 81.09663 15.234175 94.88090
## 142 55.05754 28.66891 81.44617 14.699609 95.41546
## 143 55.05754 28.32394 81.79113 14.172032 95.94304
## 144 55.05754 27.98337 82.13170 13.651176 96.46390
## 145 55.05754 27.64703 82.46804 13.136792 96.97828
## 146 55.05754 27.31477 82.80030 12.628643 97.48643
## 147 55.05754 26.98645 83.12863 12.126509 97.98856
## 148 55.05754 26.66191 83.45316 11.630180 98.48489
## 149 55.05754 26.34105 83.77402 11.139460 98.97561
## 150 55.05754 26.02373 84.09134 10.654162 99.46091
## 151 55.05754 25.70984 84.40523 10.174112 99.94096
## 152 55.05754 25.39928 84.71580 9.699142 100.41593
## 153 55.05754 25.09193 85.02314 9.229094 100.88598
## 154 55.05754 24.78770 85.32737 8.763819 101.35125
## 155 55.05754 24.48650 85.62857 8.303173 101.81190
## 156 55.05754 24.18824 85.92683 7.847022 102.26805
## 157 55.05754 23.89283 86.22224 7.395237 102.71984
## 158 55.05754 23.60020 86.51487 6.947694 103.16738
## 159 55.05754 23.31027 86.80481 6.504276 103.61080
## 160 55.05754 23.02295 87.09212 6.064871 104.05020
## 161 55.05754 22.73820 87.37688 5.629372 104.48570ramalan2<- forecast(ses2, h=h_test)
ramalan2## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 129 68.02137 46.30616357 89.73658 34.810828 101.2319
## 130 68.02137 41.51458250 94.52816 27.482737 108.5600
## 131 68.02137 37.46536606 98.57737 21.289995 114.7527
## 132 68.02137 33.89324421 102.14950 15.826906 120.2158
## 133 68.02137 30.66111622 105.38162 10.883793 125.1589
## 134 68.02137 27.68716422 108.35558 6.335525 129.7072
## 135 68.02137 24.91791597 111.12482 2.100326 133.9424
## 136 68.02137 22.31614774 113.72659 -1.878735 137.9215
## 137 68.02137 19.85471208 116.18803 -5.643176 141.6859
## 138 68.02137 17.51308822 118.52965 -9.224381 145.2671
## 139 68.02137 15.27531686 120.76742 -12.646757 148.6895
## 140 68.02137 13.12869549 122.91404 -15.929731 151.9725
## 141 68.02137 11.06291753 124.97982 -19.089066 155.1318
## 142 68.02137 9.06948368 126.97326 -22.137760 158.1805
## 143 68.02137 7.14128710 128.90145 -25.086682 161.1294
## 144 68.02137 5.27231354 130.77043 -27.945030 163.9878
## 145 68.02137 3.45741969 132.58532 -30.720671 166.7634
## 146 68.02137 1.69216623 134.35057 -33.420393 169.4631
## 147 68.02137 -0.02731012 136.07005 -36.050105 172.0928
## 148 68.02137 -1.70439611 137.74714 -38.614987 174.6577
## 149 68.02137 -3.34208039 139.38482 -41.119609 177.1623
## 150 68.02137 -4.94301614 140.98576 -43.568029 179.6108
## 151 68.02137 -6.50957148 142.55231 -45.963869 182.0066
## 152 68.02137 -8.04387063 144.08661 -48.310377 184.3531
## 153 68.02137 -9.54782765 145.59057 -50.610480 186.6532
## 154 68.02137 -11.02317451 147.06591 -52.866829 188.9096
## 155 68.02137 -12.47148441 148.51422 -55.081828 191.1246
## 156 68.02137 -13.89419143 149.93693 -57.257670 193.3004
## 157 68.02137 -15.29260720 151.33535 -59.396363 195.4391
## 158 68.02137 -16.66793505 152.71068 -61.499745 197.5425
## 159 68.02137 -18.02128211 154.06402 -63.569511 199.6123
## 160 68.02137 -19.35366978 155.39641 -65.607222 201.6500
## 161 68.02137 -20.66604269 156.70878 -67.614323 203.6571ramalanopt <- forecast(
HWopt,
h = h_test
)
ramalanopt## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 129 59.97166 38.66771 81.27561 27.3900797 92.55324
## 130 59.97166 37.28131 82.66200 25.2697689 94.67355
## 131 59.97166 35.97488 83.96844 23.2717538 96.67156
## 132 59.97166 34.73599 85.20733 21.3770362 98.56628
## 133 59.97166 33.55514 86.38818 19.5710801 100.37224
## 134 59.97166 32.42486 87.51845 17.8424691 102.10085
## 135 59.97166 31.33917 88.60415 16.1820425 103.76127
## 136 59.97166 30.29316 89.65015 14.5823169 105.36100
## 137 59.97166 29.28279 90.66053 13.0370849 106.90623
## 138 59.97166 28.30464 91.63868 11.5411303 108.40219
## 139 59.97166 27.35581 92.58751 10.0900193 109.85330
## 140 59.97166 26.43381 93.50951 8.6799458 111.26337
## 141 59.97166 25.53649 94.40683 7.3076133 112.63570
## 142 59.97166 24.66197 95.28135 5.9701444 113.97317
## 143 59.97166 23.80858 96.13473 4.6650098 115.27831
## 144 59.97166 22.97488 96.96843 3.3899719 116.55335
## 145 59.97166 22.15956 97.78376 2.1430400 117.80028
## 146 59.97166 21.36145 98.58187 0.9224334 119.02088
## 147 59.97166 20.57950 99.36382 -0.2734479 120.21676
## 148 59.97166 19.81278 100.13054 -1.4460484 121.38937
## 149 59.97166 19.06042 100.88289 -2.5966768 122.53999
## 150 59.97166 18.32166 101.62166 -3.7265239 123.66984
## 151 59.97166 17.59577 102.34755 -4.8366767 124.77999
## 152 59.97166 16.88210 103.06121 -5.9281304 125.87145
## 153 59.97166 16.18007 103.76325 -7.0017992 126.94512
## 154 59.97166 15.48911 104.45420 -8.0585253 128.00184
## 155 59.97166 14.80873 105.13459 -9.0990862 129.04240
## 156 59.97166 14.13844 105.80488 -10.1242018 130.06752
## 157 59.97166 13.47781 106.46550 -11.1345403 131.07786
## 158 59.97166 12.82645 107.11687 -12.1307226 132.07404
## 159 59.97166 12.18395 107.75936 -13.1133279 133.05664
## 160 59.97166 11.54999 108.39333 -14.0828964 134.02621
## 161 59.97166 10.92421 109.01910 -15.0399337 134.98325Fungsi HoltWinters memiliki argumen yang sama dengan
fungsi ses() . Nilai parameter \(\alpha\) dari kedua fungsi dapat
dioptimalkan menyesuaikan dari error-nya paling minimumnya.
Caranya adalah dengan membuat parameter \(\alpha =\) NULL .
Akurasi Data Latih
Perhitungan akurasi data dapat dilakukan dengan cara langsung maupun manual. Dalam hal ini, kita akan menggunakan cara langsung dengan mengakses nilai-nilai dari objek hasil pemulusan.
# Alpha = 0.2
SSE1 <- ses1$SSE
MSE1 <- SSE1 / length(train.ts)
RMSE1 <- sqrt(MSE1)
# Alpha = 0.7
SSE2 <- ses2$SSE
MSE2 <- SSE2 / length(train.ts)
RMSE2 <- sqrt(MSE2)
# Alpha optimum
SSEopt <- HWopt$SSE
MSEopt <- SSEopt / length(train.ts)
RMSEopt <- sqrt(MSEopt)
# Tabel ringkas
akurasi_ses_train <- data.frame(
Model = c(
"SES Alpha 0.2",
"SES Alpha 0.7",
"SES Optimal"
),
SSE = c(
SSE1,
SSE2,
SSEopt
),
MSE = c(
MSE1,
MSE2,
MSEopt
),
RMSE = c(
RMSE1,
RMSE2,
RMSEopt
)
)
akurasi_ses_train## Model SSE MSE RMSE
## 1 SES Alpha 0.2 35233.35 275.2606 16.59098
## 2 SES Alpha 0.7 36177.50 282.6367 16.81180
## 3 SES Optimal 34834.24 272.1425 16.49674Berdasarkan hasil akurasi data latih, SES optimal merupakan model terbaik karena menghasilkan nilai SSE, MSE, dan RMSE paling kecil, yaitu masing-masing 34.834,24; 272,14; dan 16,50. Hal ini menunjukkan bahwa SES optimal memberikan hasil pemulusan yang paling mendekati data aktual dibandingkan SES dengan alpha 0,2 dan 0,7.
Akurasi Data Uji
# error (ramalan - aktual), samakan panjang dan tipe numeric
e1 <- as.numeric(ramalan1$mean)[1:n_test] - as.numeric(testing$pm25)
e2 <- as.numeric(ramalan2$mean)[1:n_test] - as.numeric(testing$pm25)
eopt <- as.numeric(ramalanopt$mean)[1:n_test] - as.numeric(testing$pm25)
# 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
akurasi_ses_test <- data.frame(
Model = c(
"SES Alpha 0.2",
"SES Alpha 0.7",
"SES Optimal"
),
SSE = c(
SSEtesting1,
SSEtesting2,
SSEtestingopt
),
MSE = c(
MSEtesting1,
MSEtesting2,
MSEtestingopt
),
RMSE = c(
RMSEtesting1,
RMSEtesting2,
RMSEtestingopt
)
)
akurasi_ses_test## Model SSE MSE RMSE
## 1 SES Alpha 0.2 5519.736 167.2647 12.93309
## 2 SES Alpha 0.7 10129.726 306.9614 17.52031
## 3 SES Optimal 5961.828 180.6614 13.44104Berdasarkan hasil akurasi data uji, SES Alpha 0,2 menghasilkan SSE sebesar 5.519,74, MSE sebesar 167,26, dan RMSE sebesar 12,93. Model SES Alpha 0,2 memiliki nilai error paling kecil dibandingkan model lainnya, sehingga memberikan hasil peramalan terbaik pada data uji.
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 .
# Model 1
des.1 <- HoltWinters(
train.ts,
gamma = FALSE,
beta = 0.2,
alpha = 0.2
)
ramalandes1 <- forecast(
des.1,
h = h_test
)
# Model 2
des.2 <- HoltWinters(
train.ts,
gamma = FALSE,
beta = 0.3,
alpha = 0.6
)
ramalandes2 <- forecast(
des.2,
h = h_test
)Selanjutnya jika ingin membandingkan plot data latih dan data uji adalah sebagai berikut.
plot(
data.ts,
main = "Double Exponential Smoothing",
xlab = "Periode",
ylab = "Konsentrasi PM2.5 (µg/m³)"
)
lines(
des.1$fitted[, 1],
lty = 2,
col = "blue"
)
lines(
ramalandes1$mean,
col = "red"
)
Untuk mendapatkan nilai parameter optimum dari DES, argumen
alpha dan beta dapat dibuat NULL
seperti berikut.
des.opt <- HoltWinters(
train.ts,
gamma = FALSE
)
plot(des.opt)
ramalandesopt <- forecast(
des.opt,
h = h_test
)Akurasi Data Latih
ssedes.train1 <- des.1$SSE
msedes.train1 <- ssedes.train1 / length(train.ts)
sisaandes1 <- ramalandes1$residuals
mapedes.train1 <-
mean(
abs(
sisaandes1[3:length(train.ts)] /
train.ts[3:length(train.ts)]
) * 100,
na.rm = TRUE
)
ssedes.train2 <- des.2$SSE
msedes.train2 <- ssedes.train2 / length(train.ts)
sisaandes2 <- ramalandes2$residuals
mapedes.train2 <-
mean(
abs(
sisaandes2[3:length(train.ts)] /
train.ts[3:length(train.ts)]
) * 100,
na.rm = TRUE
)
ssedes.trainopt <- des.opt$SSE
msedes.trainopt <-
ssedes.trainopt / length(train.ts)
sisaandesopt <- ramalandesopt$residuals
mapedes.trainopt <-
mean(
abs(
sisaandesopt[3:length(train.ts)] /
train.ts[3:length(train.ts)]
) * 100,
na.rm = TRUE
)
# Tabel ringkas
akurasi_des_train <- data.frame(
Model = c(
"DES Skenario 1",
"DES Skenario 2",
"DES Optimal"
),
SSE = c(
ssedes.train1,
ssedes.train2,
ssedes.trainopt
),
MSE = c(
msedes.train1,
msedes.train2,
msedes.trainopt
),
MAPE = c(
mapedes.train1,
mapedes.train2,
mapedes.trainopt
)
)
akurasi_des_train## Model SSE MSE MAPE
## 1 DES Skenario 1 45271.06 353.6802 31.35653
## 2 DES Skenario 2 44899.28 350.7756 27.06199
## 3 DES Optimal 38414.89 300.1163 26.66994Berdasarkan hasil akurasi data latih, DES Optimal menghasilkan SSE sebesar 38.414,89, MSE sebesar 300,12, dan MAPE sebesar 26,67%. Model DES Optimal memiliki nilai error paling kecil dibandingkan kedua skenario lainnya, sehingga memberikan hasil pemulusan terbaik pada data latih.
Akurasi Data Uji
selisihdes1 <-
as.numeric(ramalandes1$mean) -
as.numeric(testing$pm25)
selisihdes2 <-
as.numeric(ramalandes2$mean) -
as.numeric(testing$pm25)
selisihdesopt <-
as.numeric(ramalandesopt$mean) -
as.numeric(testing$pm25)
SSEtestingdes1 <-
sum(selisihdes1^2)
MSEtestingdes1 <-
mean(selisihdes1^2)
MAPEtestingdes1 <-
mean(
abs(
selisihdes1 /
testing$pm25
) * 100
)
SSEtestingdes2 <-
sum(selisihdes2^2)
MSEtestingdes2 <-
mean(selisihdes2^2)
MAPEtestingdes2 <-
mean(
abs(
selisihdes2 /
testing$pm25
) * 100
)
SSEtestingdesopt <-
sum(selisihdesopt^2)
MSEtestingdesopt <-
mean(selisihdesopt^2)
MAPEtestingdesopt <-
mean(
abs(
selisihdesopt /
testing$pm25
) * 100
)
# Tabel ringkas
akurasi_des_test <- data.frame(
Model = c(
"DES Skenario 1",
"DES Skenario 2",
"DES Optimal"
),
SSE = c(
SSEtestingdes1,
SSEtestingdes2,
SSEtestingdesopt
),
MSE = c(
MSEtestingdes1,
MSEtestingdes2,
MSEtestingdesopt
),
MAPE = c(
MAPEtestingdes1,
MAPEtestingdes2,
MAPEtestingdesopt
)
)
akurasi_des_test## Model SSE MSE MAPE
## 1 DES Skenario 1 20183.53 611.6220 43.98177
## 2 DES Skenario 2 377976.96 11453.8474 177.77658
## 3 DES Optimal 19066.96 577.7867 44.94684Berdasarkan hasil akurasi data uji, DES Optimal menghasilkan MSE terkecil sebesar 577,79, sedangkan DES Skenario 1 menghasilkan MAPE terkecil sebesar 43,98%. Dengan demikian, DES Optimal memiliki kinerja terbaik berdasarkan MSE, sementara DES Skenario 1 lebih baik berdasarkan MAPE. DES Skenario 2 memiliki tingkat kesalahan paling tinggi berdasarkan kedua ukuran tersebut.
Perbandingan SES dan DES
MSEfull <- data.frame(
Model = c(
"SES Alpha 0.2",
"SES Alpha 0.7",
"SES Optimal",
"DES Skenario 1",
"DES Skenario 2",
"DES Optimal"
),
MSE = c(
MSEtesting1,
MSEtesting2,
MSEtestingopt,
MSEtestingdes1,
MSEtestingdes2,
MSEtestingdesopt
)
)
MSEfull## Model MSE
## 1 SES Alpha 0.2 167.2647
## 2 SES Alpha 0.7 306.9614
## 3 SES Optimal 180.6614
## 4 DES Skenario 1 611.6220
## 5 DES Skenario 2 11453.8474
## 6 DES Optimal 577.7867# Menentukan model terbaik berdasarkan MSE terkecil
MSEfull[
which.min(MSEfull$MSE),
]## Model MSE
## 1 SES Alpha 0.2 167.2647Berdasarkan hasil perbandingan MSE pada data uji, metode SES Alpha 0,2 menghasilkan nilai MSE paling kecil, yaitu 167,26. Oleh karena itu, SES Alpha 0,2 dipilih sebagai metode terbaik dalam melakukan peramalan data PM2.5 karena menghasilkan error peramalan paling rendah dibandingkan metode lainnya.
Seasonal Smoothing
Karena data yang digunakan merupakan data harian, frekuensi 7 digunakan untuk mengakomodasi pola musiman mingguan. Selanjutnya, dilakukan pemodelan menggunakan metode Holt-Winters aditif dan multiplikatif.
# Pembagian Data
n <- nrow(data)
n_train <- floor(
0.8 * n
)
training_hw <- data[
1:n_train,
]
testing_hw <- data[
(n_train + 1):n,
]
train_hw.ts <- ts(
training_hw$pm25,
frequency = 7
)
test_hw.ts <- ts(
testing_hw$pm25,
frequency = 7
)
n_train## [1] 128nrow(testing_hw)## [1] 33Kemudian akan dilakukan ekSplorasi dengan plot data deret waktu sebagai berikut.
## Eksplorasi Data
plot(data$tanggal, data$pm25,
type = "o",
xlab = "Periode",
ylab = "Konsentrasi PM2.5 (µg/m³)",
main = "Plot Data PM2.5")
# Data Latih
plot(training_hw$tanggal, training_hw$pm25,
type = "o",
col = "blue",
xlab = "Periode",
ylab = "Konsentrasi PM2.5 (µg/m³)",
main = "Data Latih PM2.5")
# Data Uji
plot(testing_hw$tanggal, testing_hw$pm25,
type = "o",
col = "red",
xlab = "Periode",
ylab = "Konsentrasi PM2.5 (µg/m³)",
main = "Data Uji PM2.5")
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.
\[L_t = \alpha(Y_t - S_{t-s}) + (1-\alpha)(L_{t-1} + B_{t-1})\]
\[B_t = \beta(L_t - L_{t-1}) + (1-\beta)B_{t-1}\] \[S_t = \gamma(Y_t - L_t) + (1-\gamma)S_{t-s}\]
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.
Winter Aditif
# Model Aditif
winter1 <- HoltWinters(
train_hw.ts,
alpha = 0.2,
beta = 0.1,
gamma = 0.1,
seasonal = "additive"
)
winter1## Holt-Winters exponential smoothing with trend and additive seasonal component.
##
## Call:
## HoltWinters(x = train_hw.ts, alpha = 0.2, beta = 0.1, gamma = 0.1, seasonal = "additive")
##
## Smoothing parameters:
## alpha: 0.2
## beta : 0.1
## gamma: 0.1
##
## Coefficients:
## [,1]
## a 54.4727422
## b 0.5928781
## s1 3.2838657
## s2 4.1250190
## s3 2.0746832
## s4 -3.1749200
## s5 -0.6912022
## s6 -1.8129926
## s7 3.5466512winter1$fitted## Time Series:
## Start = c(2, 1)
## End = c(19, 2)
## Frequency = 7
## xhat level trend season
## 2.000000 94.40816 88.88776 -1.244897959 6.765306122
## 2.142857 75.68776 86.56122 -1.353061224 -9.520408163
## 2.285714 74.73196 84.27061 -1.446816327 -8.091836735
## 2.428571 86.60125 81.87740 -1.541455510 6.265306122
## 2.571429 91.63609 80.01570 -1.573480604 13.193877551
## 2.714286 77.19696 77.91500 -1.626202496 0.908163265
## 2.857143 63.55885 74.84940 -1.770141673 -9.520408163
## 3.000000 83.67883 78.56749 -1.221318753 6.332653061
## 3.142857 66.52008 77.61041 -1.194895267 -9.895428571
## 3.285714 71.91581 81.11150 -0.725296940 -8.470393469
## 3.428571 88.01663 82.40304 -0.523613041 6.137205747
## 3.571429 92.57514 80.27610 -0.683945646 12.982989985
## 3.714286 74.93408 75.67712 -1.075448501 0.332406555
## 3.857143 73.47561 81.21486 -0.414130147 -7.325116484
## 4.000000 87.38031 81.30561 -0.363642400 6.438347007
## 4.142857 70.05762 78.66590 -0.591248624 -8.017035263
## 4.285714 73.32707 81.26313 -0.272400986 -7.663657875
## 4.428571 88.34225 82.92531 -0.078942403 5.495875328
## 4.571429 91.68911 80.57792 -0.305787353 11.416978562
## 4.714286 83.01242 80.33431 -0.299569630 2.977679971
## 4.857143 73.26927 80.63226 -0.239818088 -7.123165495
## 5.000000 85.18130 79.93859 -0.285203583 5.527922111
## 5.142857 74.56665 81.41712 -0.108829659 -6.741644708
## 5.285714 73.30498 80.39496 -0.200162591 -6.889823544
## 5.428571 84.51604 80.13380 -0.206262107 4.588495529
## 5.571429 90.60960 79.42433 -0.256582860 11.441849453
## 5.714286 80.23374 77.44583 -0.428774884 3.216686141
## 5.857143 69.67215 77.37031 -0.393449738 -7.304707475
## 6.000000 88.16895 81.84243 0.093107245 6.233417805
## 6.142857 74.00450 81.10174 0.009728181 -7.106976435
## 6.285714 75.08599 81.91057 0.089638251 -6.914221611
## 6.428571 86.01814 81.58301 0.047918445 4.387212518
## 6.571429 90.22794 79.62730 -0.152444450 10.753081359
## 6.714286 79.99025 77.02927 -0.397003255 3.357986722
## 6.857143 71.53893 77.23422 -0.336808341 -5.358479542
## 7.000000 83.22194 77.58962 -0.267586917 5.899901549
## 7.142857 70.87829 77.87765 -0.212025662 -6.787336155
## 7.285714 71.93927 79.08997 -0.069591390 -7.081100833
## 7.428571 85.84990 82.03252 0.231623148 3.585760935
## 7.571429 92.52363 82.49416 0.254625081 9.774846141
## 7.714286 81.64698 78.24406 -0.195847579 3.598766379
## 7.857143 68.00844 73.71882 -0.628787160 -5.081593848
## 8.000000 77.92153 72.48834 -0.688955877 6.122146570
## 8.142857 60.51009 67.81508 -1.087386541 -6.217599068
## 8.285714 64.71184 71.22567 -0.637588429 -5.876242678
## 8.428571 77.65166 74.24572 -0.271825299 3.677768665
## 8.571429 86.37166 78.24356 0.155141485 7.972955502
## 8.714286 77.03909 75.32437 -0.152291655 1.867008055
## 8.857143 68.36892 73.96426 -0.273073385 -5.322268716
## 9.000000 74.56538 70.61740 -0.580451765 4.528423913
## 9.142857 64.91371 69.92388 -0.591759287 -4.418406620
## 9.285714 71.16615 75.54938 0.029966492 -4.413190157
## 9.428571 88.65839 82.54611 0.726643456 5.385635799
## 9.571429 95.21778 87.34108 1.133475639 6.743222943
## 9.714286 95.34400 92.43100 1.529120130 1.383881136
## 9.857143 82.92178 88.49132 0.982240156 -6.551782239
## 10.000000 90.11620 85.08920 0.543804641 4.483193826
## 10.142857 78.05974 80.00977 -0.018519385 -1.931503504
## 10.285714 75.25310 77.17930 -0.299714271 -1.626482300
## 10.428571 77.59715 71.42896 -0.844776319 7.012964531
## 10.571429 72.87384 65.86476 -1.316719367 8.325800906
## 10.714286 57.61544 60.17327 -1.754196146 -0.803638759
## 10.857143 45.36396 55.69599 -2.026504853 -8.305524298
## 11.000000 55.99680 55.59669 -1.833784011 2.233897722
## 11.142857 45.79354 50.96355 -2.113720097 -3.056283050
## 11.285714 39.89480 46.09112 -2.389590949 -3.806730492
## 11.428571 46.46027 43.72257 -2.387486862 5.125192337
## 11.571429 49.60223 45.04303 -2.016692313 6.575893791
## 11.714286 39.64428 43.50589 -1.968736856 -1.892873587
## 11.857143 30.13203 39.80830 -2.141622405 -7.534640928
## 12.000000 39.91016 40.64027 -1.844263050 1.114153377
## 12.142857 29.73174 36.01397 -2.122466191 -4.159766457
## 12.285714 31.76974 37.34516 -1.777100983 -3.798314146
## 12.428571 56.06998 49.81411 -0.352495844 6.608370536
## 12.571429 58.28144 51.64762 -0.133895509 6.767715619
## 12.714286 51.15349 53.65743 0.080475768 -2.584415786
## 12.857143 48.53941 54.70721 0.177405902 -6.345203508
## 13.000000 50.10469 50.37673 -0.273382355 0.001340813
## 13.142857 44.82863 48.08241 -0.475476199 -2.778305626
## 13.285714 50.60927 49.04121 -0.332048822 1.900106413
## 13.428571 53.52585 46.58731 -0.544234185 7.482771876
## 13.571429 50.14835 43.33790 -0.814751100 7.625200725
## 13.714286 39.03907 42.09348 -0.857718188 -2.196695250
## 13.857143 31.78109 40.82795 -0.898499575 -8.148356534
## 14.000000 45.97208 46.97323 -0.194121467 -0.807034565
## 14.142857 41.96654 44.58470 -0.413562991 -2.204596117
## 14.285714 45.69630 44.97783 -0.332893719 1.051364962
## 14.428571 52.31956 46.10567 -0.186819649 6.400704214
## 14.571429 59.27506 51.45494 0.366789216 7.453332375
## 14.714286 53.94819 55.56672 0.741287959 -2.359820797
## 14.857143 51.94985 56.51837 0.762324259 -5.330844102
## 15.000000 62.30925 62.69072 1.303327279 -1.684800660
## 15.142857 63.34742 63.93220 1.297142281 -1.881919030
## 15.285714 72.48571 69.15986 1.690193804 1.635661243
## 15.428571 87.42853 76.55291 2.260479541 8.615139670
## 15.571429 75.41092 65.52768 0.931908971 8.951327350
## 15.714286 50.72542 53.37741 -0.376309417 -2.275675600
## 15.857143 40.71836 45.05601 -1.170817881 -3.166832021
## 16.000000 45.90680 48.34152 -0.725185166 -1.709540653
## 16.142857 44.84194 46.03498 -0.883321118 -0.309712937
## 16.285714 50.85991 47.58327 -0.640160011 3.916804192
## 16.428571 51.83463 48.97113 -0.437358269 3.300857391
## 16.571429 52.73125 49.36684 -0.354050773 3.718453795
## 16.714286 44.36416 50.06654 -0.248675689 -5.453709455
## 16.857143 48.54478 50.14504 -0.215958843 -1.384301161
## 17.000000 46.81118 49.42012 -0.266854356 -2.342084461
## 17.142857 44.97088 44.99103 -0.683078012 0.662931489
## 17.285714 51.21929 46.91378 -0.422495696 4.728011163
## 17.428571 50.75463 47.44742 -0.326881523 3.634087374
## 17.571429 52.52759 48.56961 -0.181974078 4.139954130
## 17.714286 39.46675 45.28212 -0.492525976 -5.322842071
## 17.857143 45.90650 47.69625 -0.201861051 -1.587883210
## 18.000000 44.40611 48.51308 -0.099991067 -4.006979088
## 18.142857 52.78902 50.93187 0.151886659 1.705260753
## 18.285714 56.83253 51.52595 0.196106310 5.110467856
## 18.428571 51.10873 47.15555 -0.260544237 4.213717153
## 18.571429 49.28829 46.67326 -0.282718776 2.897746539
## 18.714286 39.68422 44.33289 -0.488484620 -4.160182370
## 18.857143 44.88499 46.30756 -0.242169024 -1.180403274
## 19.000000 44.16906 47.28839 -0.119868749 -2.999468183
## 19.142857 52.19360 50.13471 0.176750144 1.882139356# Model Aditif Optimal
winter1.opt <- HoltWinters(
train_hw.ts,
alpha = NULL,
beta = NULL,
gamma = NULL,
seasonal = "additive"
)
winter1.opt## Holt-Winters exponential smoothing with trend and additive seasonal component.
##
## Call:
## HoltWinters(x = train_hw.ts, alpha = NULL, beta = NULL, gamma = NULL, seasonal = "additive")
##
## Smoothing parameters:
## alpha: 0.3745335
## beta : 0.007144506
## gamma: 0.2173608
##
## Coefficients:
## [,1]
## a 55.5464985
## b -0.6370214
## s1 8.4907279
## s2 6.2634731
## s3 1.7091289
## s4 -0.3426872
## s5 5.2566858
## s6 -0.5371225
## s7 10.0440882winter1.opt$fitted## Time Series:
## Start = c(2, 1)
## End = c(19, 2)
## Frequency = 7
## xhat level trend season
## 2.000000 94.40816 88.88776 -1.2448980 6.76530612
## 2.142857 74.83754 85.61732 -1.2593694 -9.52040816
## 2.285714 73.55919 82.92066 -1.2696381 -8.09183673
## 2.428571 85.30413 80.31799 -1.2791620 6.26530612
## 2.571429 90.83882 78.92492 -1.2799758 13.19387755
## 2.714286 76.57951 76.95624 -1.2848963 0.90816327
## 2.857143 62.38419 73.20710 -1.3025021 -9.52040816
## 3.000000 87.42630 82.62218 -1.2259303 6.03005622
## 3.142857 69.21296 80.48752 -1.2324227 -10.04212906
## 3.285714 77.28802 87.04053 -1.1767996 -8.57571481
## 3.428571 92.68830 87.62853 -1.1641910 6.22395870
## 3.571429 93.45789 81.71215 -1.1981431 12.94388647
## 3.714286 71.61262 72.85184 -1.2528854 0.01366658
## 3.857143 78.44169 85.22725 -1.1555180 -5.63003523
## 4.000000 87.69538 83.15723 -1.1620516 5.70019569
## 4.142857 69.20543 77.61487 -1.1933468 -7.21609261
## 4.285714 73.62813 82.71165 -1.1484069 -7.93511234
## 4.428571 88.44896 85.07332 -1.1233292 4.49896100
## 4.571429 88.67061 79.66198 -1.1539649 10.16259818
## 4.714286 83.57052 79.75498 -1.1450559 4.96059908
## 4.857143 72.41930 79.51984 -1.1385550 -5.96198793
## 5.000000 80.81755 77.84971 -1.1423529 4.11018742
## 5.142857 75.60471 81.64463 -1.1070785 -4.93283953
## 5.285714 70.65533 78.43840 -1.1220759 -6.66098953
## 5.428571 80.02113 78.19448 -1.1158019 2.94245408
## 5.571429 87.32456 77.81983 -1.1105068 10.61523494
## 5.714286 78.88123 74.71510 -1.1247545 5.29089098
## 5.857143 67.48707 74.75843 -1.1164091 -6.15494465
## 6.000000 88.42890 83.57200 -1.0454643 5.90236640
## 6.142857 74.11564 80.86776 -1.0573154 -5.69481070
## 6.285714 73.87612 81.26527 -1.0469214 -6.34222781
## 6.428571 82.05243 79.89021 -1.0492658 3.21148492
## 6.571429 85.40000 76.57411 -1.0654613 9.89135128
## 6.714286 77.36673 72.73710 -1.0852626 5.71489301
## 6.857143 70.14103 73.76169 -1.0701888 -2.55046249
## 7.000000 78.75440 74.51134 -1.0571869 5.30024934
## 7.142857 69.96335 76.16787 -1.0377987 -5.16672404
## 7.285714 70.66244 78.14007 -1.0162938 -6.46133816
## 7.428571 84.65881 83.24274 -0.9725768 2.38864554
## 7.571429 91.06601 83.14702 -0.9663121 8.88530770
## 7.714286 79.74884 74.29078 -1.0226817 6.48074660
## 7.857143 62.90609 65.87149 -1.0755268 -1.88987700
## 8.000000 70.79558 65.58021 -1.0699238 6.28530157
## 8.142857 54.53962 59.71791 -1.1041630 -4.07412657
## 8.285714 64.00489 69.27311 -1.0280070 -4.24021599
## 8.428571 77.08917 75.35941 -0.9771788 2.70693482
## 8.571429 87.69137 82.58857 -0.9185486 6.02134377
## 8.714286 78.25119 75.41855 -0.9632123 3.79585453
## 8.857143 69.15170 71.73952 -0.9826154 -1.60520585
## 9.000000 68.22744 64.70755 -1.0258351 4.54571808
## 9.142857 64.62846 65.84374 -1.0103885 -0.20488437
## 9.285714 73.99880 76.58304 -0.9264428 -1.65779502
## 9.428571 92.48706 87.64212 -0.8408122 5.68575365
## 9.571429 95.94145 92.98596 -0.7966259 3.75212107
## 9.714286 101.39181 99.32740 -0.7456280 2.81004172
## 9.857143 81.43938 86.07542 -0.8349797 -3.80105971
## 10.000000 82.02604 77.58520 -0.8896725 5.33050893
## 10.142857 72.31198 69.19511 -0.9432593 4.06013463
## 10.285714 66.86606 65.13873 -0.9655010 2.69282766
## 10.428571 64.02200 57.10726 -1.0159839 7.93071874
## 10.571429 57.63807 52.33770 -1.0428013 6.34316645
## 10.714286 46.01852 48.80872 -1.0605638 -1.72963713
## 10.857143 39.34636 46.99215 -1.0659651 -6.57983112
## 11.000000 53.37286 51.78900 -1.0240782 2.60793123
## 11.142857 48.38100 46.50541 -1.0545103 2.93010489
## 11.285714 38.34527 39.31566 -1.0983435 0.12795162
## 11.428571 44.31137 38.83707 -1.0939157 6.56820979
## 11.571429 49.89390 45.49174 -1.0385559 5.44070865
## 11.714286 42.20501 45.24199 -1.0329203 -2.00405896
## 11.857143 34.49783 40.01242 -1.0629033 -4.45168933
## 12.000000 42.90990 42.88293 -1.0348010 1.06177017
## 12.142857 35.13783 35.51481 -1.0800495 0.70307682
## 12.285714 38.18214 38.87754 -1.0483080 0.35291499
## 12.428571 70.61169 62.10569 -0.8748647 9.38086832
## 12.571429 64.72063 59.87812 -0.8845291 5.72703749
## 12.714286 56.19588 60.59636 -0.8730781 -3.52740160
## 12.857143 55.75242 59.64992 -0.8736022 -3.02389931
## 13.000000 45.44266 47.63304 -0.9532154 -1.23716255
## 13.142857 45.98935 44.64136 -0.9677792 2.31576048
## 13.285714 54.13811 45.92478 -0.9516956 9.16502385
## 13.428571 47.57821 39.67789 -0.9895271 8.88985212
## 13.571429 41.14909 35.85007 -1.0098053 6.30882557
## 13.714286 32.86065 37.40616 -0.9914733 -3.55403207
## 13.857143 29.91582 37.96501 -0.9803970 -7.06879608
## 14.000000 48.01561 50.87388 -0.8811650 -1.97710252
## 14.142857 47.33486 45.11793 -0.9159929 3.13292013
## 14.285714 50.02535 43.70199 -0.9195648 7.24292202
## 14.428571 50.84451 43.89653 -0.9116051 7.85958002
## 14.571429 60.31127 53.90464 -0.8335891 7.24022016
## 14.714286 55.91853 59.69607 -0.7862566 -2.99127996
## 14.857143 55.74995 58.56579 -0.7887145 -2.02713188
## 15.000000 62.01190 66.48500 -0.7265007 -3.74659959
## 15.142857 67.97895 65.75404 -0.7265325 2.95144340
## 15.285714 77.61439 70.65340 -0.6863383 7.64733161
## 15.428571 89.92531 78.72575 -0.6237618 11.82332374
## 15.571429 61.12399 52.28715 -0.8081961 9.64503640
## 15.714286 28.27015 32.33131 -0.9449966 -3.11615637
## 15.857143 25.06061 24.91806 -0.9912090 1.13375645
## 16.000000 33.49852 38.13642 -0.8896886 -3.74821768
## 16.142857 43.04863 38.93269 -0.8776433 4.99358258
## 16.285714 53.26664 43.28030 -0.8403114 10.82664907
## 16.428571 46.96957 45.33639 -0.8196181 2.45279835
## 16.571429 49.79815 47.89897 -0.7954540 2.69463427
## 16.714286 43.93781 50.17539 -0.7735070 -5.46406602
## 16.857143 55.69794 50.17424 -0.7679889 6.29168799
## 17.000000 41.84387 45.77405 -0.7939392 -3.13623277
## 17.142857 45.10001 39.04604 -0.8363351 6.89029731
## 17.285714 54.11739 43.04119 -0.8018166 11.87801366
## 17.428571 45.82820 42.94447 -0.7967790 3.68050159
## 17.571429 49.75193 46.70644 -0.7642090 3.80969064
## 17.714286 35.18417 41.16621 -0.7983313 -5.18370779
## 17.857143 51.64029 47.41504 -0.7479828 4.97323546
## 18.000000 40.38731 46.42724 -0.7496961 -5.29023706
## 18.142857 59.83839 51.89956 -0.7052430 8.64407545
## 18.285714 60.79794 49.38218 -0.7181898 12.13395847
## 18.428571 43.17264 38.62726 -0.7898973 5.33528091
## 18.571429 41.69885 40.39443 -0.7716282 2.07604253
## 18.714286 35.20749 38.61200 -0.7788500 -2.62566084
## 18.857143 48.27478 44.12251 -0.7339156 4.88618672
## 19.000000 40.65094 44.40928 -0.7266233 -3.03171104
## 19.142857 57.86375 50.55499 -0.6775239 7.98628731# Forecast
h_test <- nrow(testing_hw)
forecast1 <- predict(
winter1,
n.ahead = h_test
)
forecast1.opt <- predict(
winter1.opt,
n.ahead = h_test
)# Plot
xhat1 <- winter1$fitted[, 2]
xhat1.opt <- winter1.opt$fitted[, 2]
plot(
train_hw.ts,
main = "Winter Aditif PM2.5",
type = "l",
xlab = "Periode",
ylab = "Konsentrasi PM2.5 (µg/m³)"
)
lines(
xhat1,
col = "red",
lwd = 2
)
lines(
xhat1.opt,
col = "blue",
lwd = 2
)
lines(
forecast1,
col = "red",
lwd = 2
)
lines(
forecast1.opt,
col = "blue",
lwd = 2
)
legend(
"topleft",
c(
"Data Aktual",
"Winter Aditif",
"Winter Aditif Optimal"
),
col = c(
"black",
"red",
"blue"
),
lty = 1,
cex = 0.7
)
# Akurasi Data Latih
SSE_hw1 <- winter1$SSE
MSE_hw1 <- SSE_hw1 / length(train_hw.ts)
RMSE_hw1 <- sqrt(MSE_hw1)
SSE_hw1.opt <- winter1.opt$SSE
MSE_hw1.opt <- SSE_hw1.opt / length(train_hw.ts)
RMSE_hw1.opt <- sqrt(MSE_hw1.opt)
akurasi_hw_train <- data.frame(
Model = c("Winter Aditif", "Winter Aditif Optimal"),
SSE = c(SSE_hw1, SSE_hw1.opt),
MSE = c(MSE_hw1, MSE_hw1.opt),
RMSE = c(RMSE_hw1, RMSE_hw1.opt)
)
akurasi_hw_train## Model SSE MSE RMSE
## 1 Winter Aditif 41747.55 326.1527 18.05970
## 2 Winter Aditif Optimal 37820.10 295.4695 17.18923best_hw1_train <-
akurasi_hw_train$Model[
which.min(
akurasi_hw_train$RMSE
)
]
cat(
"Berdasarkan hasil akurasi data latih,",
best_hw1_train,
"memiliki nilai RMSE paling kecil. ",
"Dengan demikian, model tersebut memberikan",
"hasil pemulusan yang lebih baik pada data latih."
)## Berdasarkan hasil akurasi data latih, Winter Aditif Optimal memiliki nilai RMSE paling kecil. Dengan demikian, model tersebut memberikan hasil pemulusan yang lebih baik pada data latih.# Akurasi Data Uji
error_hw1 <-
as.numeric(forecast1) -
as.numeric(test_hw.ts)
error_hw1.opt <-
as.numeric(forecast1.opt) -
as.numeric(test_hw.ts)
SSE_hw1.test <-
sum(error_hw1^2)
MSE_hw1.test <-
mean(error_hw1^2)
RMSE_hw1.test <-
sqrt(MSE_hw1.test)
SSE_hw1.opt.test <-
sum(error_hw1.opt^2)
MSE_hw1.opt.test <-
mean(error_hw1.opt^2)
RMSE_hw1.opt.test <-
sqrt(MSE_hw1.opt.test)
akurasi_hw_test <- data.frame(
Model = c(
"Winter Aditif",
"Winter Aditif Optimal"
),
SSE = c(
SSE_hw1.test,
SSE_hw1.opt.test
),
MSE = c(
MSE_hw1.test,
MSE_hw1.opt.test
),
RMSE = c(
RMSE_hw1.test,
RMSE_hw1.opt.test
)
)
akurasi_hw_test## Model SSE MSE RMSE
## 1 Winter Aditif 6956.005 210.7880 14.51854
## 2 Winter Aditif Optimal 11779.928 356.9675 18.89358best_hw1_test <-
akurasi_hw_test$Model[
which.min(
akurasi_hw_test$RMSE
)
]
cat(
"Berdasarkan hasil akurasi data uji,",
best_hw1_test,
"memiliki nilai RMSE paling kecil. ",
"Dengan demikian, model tersebut memberikan",
"hasil peramalan yang lebih baik pada data uji."
)## Berdasarkan hasil akurasi data uji, Winter Aditif memiliki nilai RMSE paling kecil. Dengan demikian, model tersebut memberikan hasil peramalan yang lebih baik pada data uji.Winter Multiplikatif
Model multiplikatif digunakan cocok digunakan jika plot data asli menunjukkan fluktuasi musiman yang bervariasi.
# Model Multiplikatif
winter2 <- HoltWinters(
train_hw.ts,
alpha = 0.2,
beta = 0.1,
gamma = 0.3,
seasonal = "multiplicative"
)
winter2## Holt-Winters exponential smoothing with trend and multiplicative seasonal component.
##
## Call:
## HoltWinters(x = train_hw.ts, alpha = 0.2, beta = 0.1, gamma = 0.3, seasonal = "multiplicative")
##
## Smoothing parameters:
## alpha: 0.2
## beta : 0.1
## gamma: 0.3
##
## Coefficients:
## [,1]
## a 56.9238132
## b 0.5862737
## s1 1.0531988
## s2 1.0556284
## s3 0.8846290
## s4 0.8973818
## s5 1.0385799
## s6 0.9058656
## s7 1.1476404winter2$fitted## Time Series:
## Start = c(2, 1)
## End = c(19, 2)
## Frequency = 7
## xhat level trend season
## 2.000000 95.09667 88.88776 -1.24489796 1.0850475
## 2.142857 75.28170 86.51910 -1.35727398 0.8839841
## 2.285714 74.34676 84.19310 -1.45414677 0.8985703
## 2.428571 86.43297 81.77146 -1.55089514 1.0774415
## 2.571429 90.39914 79.95457 -1.57749463 1.1533875
## 2.714286 77.54682 78.13447 -1.60175608 1.0132506
## 2.857143 65.10696 75.04308 -1.75071856 0.8883184
## 3.000000 83.26576 79.12204 -1.16775103 1.0681357
## 3.142857 67.25227 78.27901 -1.13527871 0.8717788
## 3.285714 72.41430 82.36243 -0.61340937 0.8858125
## 3.428571 89.62470 83.91329 -0.39698215 1.0731401
## 3.571429 93.24428 81.72256 -0.57635673 1.1490898
## 3.714286 75.85913 77.62267 -0.92871005 0.9891147
## 3.857143 80.16625 83.19288 -0.27881828 0.9668595
## 4.000000 87.68741 82.05225 -0.36499935 1.0734528
## 4.142857 74.03858 79.50972 -0.58275290 0.9380645
## 4.285714 74.10801 81.47720 -0.32772959 0.9132285
## 4.428571 86.68682 83.09684 -0.13299207 1.0448746
## 4.571429 87.77948 81.10969 -0.31840807 1.0864969
## 4.714286 87.98304 81.56819 -0.24071768 1.0818367
## 4.857143 77.02637 80.96086 -0.27737829 0.9546734
## 5.000000 82.03379 79.42098 -0.40362825 1.0381743
## 5.142857 78.98265 81.32260 -0.17310420 0.9732982
## 5.285714 74.12321 79.30368 -0.35768594 0.9389103
## 5.428571 79.59491 78.70673 -0.38161165 1.0162118
## 5.571429 86.22548 78.79847 -0.33427720 1.0989151
## 5.714286 83.15433 77.69516 -0.41117994 1.0759582
## 5.857143 71.76746 77.06941 -0.43263674 0.9364624
## 6.000000 87.41117 81.38497 0.04218290 1.0734891
## 6.142857 76.41784 80.79163 -0.02136998 0.9461136
## 6.285714 75.88357 81.10471 0.01207552 0.9354853
## 6.428571 82.34430 80.50030 -0.04957303 1.0235371
## 6.571429 85.82387 79.21105 -0.17354121 1.0858626
## 6.714286 82.87098 77.59646 -0.31764546 1.0723635
## 6.857143 77.14354 77.30288 -0.31523924 1.0020250
## 7.000000 81.02961 76.55980 -0.35802345 1.0633559
## 7.142857 73.08963 77.13663 -0.26453842 0.9507954
## 7.285714 72.05977 77.90498 -0.16124868 0.9268884
## 7.428571 81.47864 80.96747 0.16112524 1.0043146
## 7.571429 87.58643 82.22813 0.27107813 1.0616639
## 7.714286 84.88352 79.18621 -0.06022132 1.0727641
## 7.857143 73.61549 74.48685 -0.52413543 0.9953054
## 8.000000 77.17260 72.23149 -0.69725801 1.0788206
## 8.142857 64.64648 67.97987 -1.05269434 0.9659227
## 8.285714 68.03528 70.72738 -0.67267377 0.9711736
## 8.428571 74.25871 73.13649 -0.36449562 1.0204298
## 8.571429 78.39171 77.62118 0.12042335 1.0083624
## 8.714286 75.68419 76.27552 -0.02618493 0.9925882
## 8.857143 72.67971 75.30550 -0.12056835 0.9666792
## 9.000000 71.37139 71.11332 -0.52772949 1.0111326
## 9.142857 72.62169 71.10553 -0.47573612 1.0282019
## 9.285714 76.68044 75.17721 -0.02099442 1.0202808
## 9.428571 89.35283 80.90356 0.55374062 1.0969284
## 9.571429 84.67119 85.03952 0.91196216 0.9851045
## 9.714286 91.54479 92.10896 1.52771019 0.9776596
## 9.857143 80.90311 88.82011 1.04605394 0.9002622
## 10.000000 87.76981 85.44454 0.60389140 1.0200048
## 10.142857 89.43356 80.99555 0.09860336 1.1028362
## 10.285714 84.28344 76.48176 -0.36263587 1.1072571
## 10.428571 78.99240 69.56537 -1.01801107 1.1523770
## 10.571429 66.78270 64.20982 -1.45176499 1.0641295
## 10.714286 53.05391 59.79174 -1.74839622 0.9140395
## 10.857143 45.69306 56.06227 -1.94650387 0.8443576
## 11.000000 51.51759 56.32027 -1.72605363 0.9436456
## 11.142857 51.81547 52.57702 -1.92777328 1.0230256
## 11.285714 43.66344 46.77535 -2.31516291 0.9820795
## 11.428571 43.76092 43.71413 -2.38976870 1.0589617
## 11.571429 43.38079 45.33566 -1.98863839 1.0007788
## 11.714286 37.85861 45.06953 -1.81638828 0.8752801
## 11.857143 35.10686 41.68596 -1.97310637 0.8840176
## 12.000000 36.18966 41.95107 -1.74928417 0.9002003
## 12.142857 33.13397 37.93792 -1.97567077 0.9213542
## 12.285714 35.87894 38.97218 -1.67467844 0.9619665
## 12.428571 59.71001 51.25247 -0.27918158 1.1713982
## 12.571429 54.49339 52.21795 -0.15471513 1.0466771
## 12.714286 45.93320 54.83517 0.12247842 0.8357928
## 12.857143 54.30169 57.36657 0.36337057 0.9406158
## 13.000000 43.01871 51.71225 -0.23839890 0.8357393
## 13.142857 50.78100 50.75144 -0.31063948 1.0067445
## 13.285714 64.31982 50.68297 -0.28642281 1.2762743
## 13.428571 55.32672 46.58548 -0.66752923 1.2049038
## 13.571429 47.12885 43.37390 -0.92193485 1.1101689
## 13.714286 36.61112 42.60890 -0.90624078 0.8779084
## 13.857143 33.09401 41.79125 -0.89738147 0.8092657
## 14.000000 40.42743 49.27332 -0.05943692 0.8214640
## 14.142857 48.29797 47.89248 -0.19157735 1.0125169
## 14.285714 54.10773 47.24699 -0.23696850 1.1509830
## 14.428571 52.15318 46.81753 -0.25621698 1.1200968
## 14.571429 57.73253 51.53353 0.24100467 1.1150758
## 14.714286 49.30046 55.40971 0.60452195 0.8801417
## 14.857143 56.55835 57.30937 0.73403602 0.9744147
## 15.000000 50.70931 62.64959 1.19465403 0.7942659
## 15.142857 68.22378 66.68730 1.47895917 1.0008439
## 15.285714 83.48495 71.11901 1.77423434 1.1453045
## 15.428571 97.52303 75.95182 2.08009268 1.2497839
## 15.571429 80.16066 65.78611 0.85551244 1.2028617
## 15.714286 49.41768 54.97600 -0.31104994 0.9040102
## 15.857143 47.72217 46.16556 -1.16098891 1.0603849
## 16.000000 39.25140 47.88614 -0.87283259 0.8348998
## 16.142857 48.13370 46.71353 -0.90280984 1.0507081
## 16.285714 56.14764 47.49840 -0.73404173 1.2006502
## 16.428571 45.54065 47.57265 -0.65321288 0.9706136
## 16.571429 43.60655 49.07464 -0.43769244 0.8965725
## 16.714286 36.43365 51.84772 -0.11661519 0.7042890
## 16.857143 62.08090 54.44770 0.15504461 1.1369557
## 17.000000 42.78731 51.77398 -0.12783194 0.8284705
## 17.142857 51.55510 47.59355 -0.53309233 1.0955078
## 17.285714 58.58771 48.23706 -0.41543175 1.2251300
## 17.428571 47.96321 47.39919 -0.45767562 1.0217652
## 17.571429 46.85471 48.90611 -0.26121575 0.9631989
## 17.714286 34.43614 46.59865 -0.46584040 0.7464565
## 17.857143 54.64298 51.37461 0.05833917 1.0624121
## 18.000000 37.73898 50.74715 -0.01024033 0.7438171
## 18.142857 63.62171 55.91588 0.50765625 1.1275739
## 18.285714 66.96332 54.89428 0.35473136 1.2120274
## 18.428571 53.14445 49.80965 -0.18920553 1.0710194
## 18.571429 44.51386 49.03325 -0.24792443 0.9124435
## 18.714286 39.55320 47.57674 -0.36878356 0.8378504
## 18.857143 52.37142 50.17908 -0.07167096 1.0451832
## 19.000000 41.11538 49.84498 -0.09791367 0.8264885
## 19.142857 59.30012 54.07492 0.33487201 1.0898795# Model Multiplikatif Optimal
winter2.opt <- HoltWinters(
train_hw.ts,
alpha = NULL,
beta = NULL,
gamma = NULL,
seasonal = "multiplicative"
)
winter2.opt## Holt-Winters exponential smoothing with trend and multiplicative seasonal component.
##
## Call:
## HoltWinters(x = train_hw.ts, alpha = NULL, beta = NULL, gamma = NULL, seasonal = "multiplicative")
##
## Smoothing parameters:
## alpha: 0.3373193
## beta : 0.006217193
## gamma: 0.1561606
##
## Coefficients:
## [,1]
## a 56.6503447
## b -0.6934027
## s1 1.0882570
## s2 1.0796591
## s3 0.9743877
## s4 0.9221282
## s5 1.0580116
## s6 0.9419818
## s7 1.1339191winter2.opt$fitted## Time Series:
## Start = c(2, 1)
## End = c(19, 2)
## Frequency = 7
## xhat level trend season
## 2.000000 95.09667 88.88776 -1.2448980 1.0850475
## 2.142857 74.68857 85.74753 -1.2566816 0.8839841
## 2.285714 73.51913 83.08333 -1.2654324 0.8985703
## 2.428571 85.35834 80.49683 -1.2736457 1.0774415
## 2.571429 89.77582 79.11099 -1.2743432 1.1533875
## 2.714286 77.34548 77.60976 -1.2757539 1.0132506
## 2.857143 64.48985 73.88863 -1.2909572 0.8883184
## 3.000000 87.76269 82.66432 -1.2283709 1.0776898
## 3.142857 69.76857 80.57122 -1.2337471 0.8793898
## 3.285714 76.80967 87.09791 -1.1854990 0.8940462
## 3.428571 93.37071 87.87070 -1.1733239 1.0769727
## 3.571429 93.69802 82.50953 -1.1993606 1.1523530
## 3.714286 74.23367 75.25140 -1.2370291 1.0029629
## 3.857143 77.59476 85.37076 -1.1664243 0.9215056
## 4.000000 88.56351 83.62057 -1.1700537 1.0741414
## 4.142857 69.84447 78.50512 -1.1945830 0.9034276
## 4.285714 73.98005 83.34264 -1.1570802 0.9001588
## 4.428571 89.51248 85.56563 -1.1360656 1.0602030
## 4.571429 89.11063 80.44854 -1.1608165 1.1238894
## 4.714286 82.46708 80.15492 -1.1554249 1.0438938
## 4.857143 72.63639 80.14111 -1.1483273 0.9195320
## 5.000000 81.68796 78.39249 -1.1520594 1.0575803
## 5.142857 73.91572 81.16741 -1.1276446 0.9234876
## 5.285714 70.58309 78.60948 -1.1365370 0.9110677
## 5.428571 80.64355 78.36779 -1.1309735 1.0441076
## 5.571429 86.31568 77.67504 -1.1282490 1.1276197
## 5.714286 77.71103 75.25579 -1.1362754 1.0484558
## 5.857143 68.22651 75.49940 -1.1276964 0.9173718
## 6.000000 88.84580 83.84867 -1.0687763 1.0732776
## 6.142857 73.63069 81.25691 -1.0782450 0.9183328
## 6.285714 73.79472 81.78359 -1.0682669 0.9142592
## 6.428571 82.99544 80.42210 -1.0700898 1.0459148
## 6.571429 85.26130 77.09590 -1.0841165 1.1216852
## 6.714286 76.68221 73.82813 -1.0976927 1.0543345
## 6.857143 69.92294 74.75172 -1.0851260 0.9491810
## 7.000000 79.38947 75.47088 -1.0739084 1.0671062
## 7.142857 69.68288 76.48660 -1.0609168 0.9238614
## 7.285714 70.70313 78.46242 -1.0420368 0.9132366
## 7.428571 85.44624 83.43991 -1.0046123 1.0365249
## 7.571429 91.07631 82.94094 -1.0014686 1.1115071
## 7.714286 79.20166 75.54325 -1.0412351 1.0630807
## 7.857143 64.37722 68.40926 -1.0791149 0.9561426
## 8.000000 71.52732 67.54986 -1.0777490 1.0760501
## 8.142857 57.14384 62.23158 -1.1041132 0.9348309
## 8.285714 64.79173 70.45726 -1.0461080 0.9334484
## 8.428571 77.87008 75.99105 -1.0051995 1.0384635
## 8.571429 87.57097 81.84939 -0.9625276 1.0826353
## 8.714286 77.27250 75.72381 -0.9946273 1.0340338
## 8.857143 68.60051 72.68298 -1.0073489 0.9570967
## 9.000000 68.62424 66.17739 -1.0415326 1.0535556
## 9.142857 64.03620 66.85702 -1.0308317 0.9728074
## 9.285714 72.77642 76.90958 -0.9619241 0.9582445
## 9.428571 92.40824 87.64296 -0.8892122 1.0651787
## 9.571429 96.61956 92.00800 -0.8565455 1.0599893
## 9.714286 98.59490 97.00064 -0.8201800 1.0251032
## 9.857143 79.49430 86.11294 -0.8827717 0.9327015
## 10.000000 82.41987 78.54155 -0.9243561 1.0618765
## 10.142857 71.27559 71.13055 -0.9646849 1.0158158
## 10.285714 66.60146 67.74987 -0.9797055 0.9974733
## 10.428571 64.44603 60.47964 -1.0188149 1.0838400
## 10.571429 59.56222 56.20975 -1.0390275 1.0795983
## 10.714286 50.83983 52.49547 -1.0556601 0.9883363
## 10.857143 43.63199 49.10537 -1.0701737 0.9083338
## 11.000000 52.86034 52.25683 -1.0439270 1.0321685
## 11.142857 46.81681 47.66367 -1.0659933 1.0047027
## 11.285714 39.13388 41.62307 -1.0969214 0.9656450
## 11.428571 42.30080 40.82871 -1.0950403 1.0646085
## 11.571429 48.75284 46.92587 -1.0503250 1.0627196
## 11.714286 44.66632 46.90623 -1.0439170 0.9739221
## 11.857143 37.28561 41.12896 -1.0733452 0.9308459
## 12.000000 42.15418 42.85115 -1.0559648 1.0085892
## 12.142857 34.16845 36.39248 -1.0895545 0.9678647
## 12.285714 37.46821 39.77496 -1.0617510 0.9678402
## 12.428571 67.58571 61.55287 -0.9197524 1.1146665
## 12.571429 63.69553 60.45587 -0.9208544 1.0698835
## 12.714286 56.65120 61.20744 -0.9104566 0.9395362
## 12.857143 56.16272 60.06319 -0.9119102 0.9494760
## 13.000000 45.68452 48.43540 -0.9785328 0.9626536
## 13.142857 44.52962 45.46498 -0.9909168 1.0012492
## 13.285714 49.60542 46.99082 -0.9752696 1.0780143
## 13.428571 46.79168 43.00994 -0.9939561 1.1136639
## 13.571429 42.02354 39.95884 -1.0067457 1.0788518
## 13.714286 37.37291 40.82072 -0.9951281 0.9384143
## 13.857143 34.24683 39.69155 -0.9959615 0.8850319
## 14.000000 47.73335 51.17907 -0.9183493 0.9497148
## 14.142857 45.58447 45.73809 -0.9464673 1.0177007
## 14.285714 46.39859 44.92936 -0.9456110 1.0549031
## 14.428571 49.50110 46.09463 -0.9324872 1.0960750
## 14.571429 58.71963 54.54825 -0.8741320 1.0940028
## 14.714286 55.10449 59.61893 -0.8371720 0.9374420
## 14.857143 55.08433 58.74416 -0.8374057 0.9512591
## 15.000000 60.41380 66.38732 -0.7846804 0.9209050
## 15.142857 66.62285 66.18365 -0.7810682 1.0186578
## 15.285714 74.96449 70.82573 -0.7473514 1.0697236
## 15.428571 89.53605 78.28824 -0.6963091 1.1539351
## 15.571429 63.96869 57.55741 -0.8208676 1.1274690
## 15.714286 37.17988 40.59004 -0.9212535 0.9372579
## 15.857143 28.93138 30.24664 -0.9798328 0.9885389
## 16.000000 36.92109 40.89205 -0.9075564 0.9233852
## 16.142857 41.15457 40.37863 -0.9051060 1.0425867
## 16.285714 48.28058 44.60016 -0.8732327 1.1041384
## 16.428571 48.19988 47.61277 -0.8490737 1.0307116
## 16.571429 47.99235 49.31643 -0.8332029 0.9898754
## 16.714286 43.30115 51.89353 -0.8120004 0.8476872
## 16.857143 55.18886 52.15548 -0.8053234 1.0747556
## 17.000000 44.12450 48.46616 -0.8232537 0.9261503
## 17.142857 43.36555 41.04167 -0.8642949 1.0793524
## 17.285714 49.70236 44.75093 -0.8358602 1.1317836
## 17.428571 47.08490 45.79204 -0.8241908 1.0470792
## 17.571429 48.15067 48.48418 -0.8023291 1.0098324
## 17.714286 36.79311 43.95713 -0.8254863 0.8530421
## 17.857143 51.86266 49.93578 -0.7831837 1.0551356
## 18.000000 42.34254 48.87681 -0.7848983 0.8804502
## 18.142857 58.95199 53.70750 -0.7499851 1.1131940
## 18.285714 58.44972 51.75999 -0.7574304 1.1460155
## 18.428571 46.03028 43.80600 -0.8021728 1.0703764
## 18.571429 42.74689 44.25485 -0.7943949 0.9835813
## 18.714286 36.76829 42.17546 -0.8023840 0.8887009
## 18.857143 48.86096 47.15449 -0.7664398 1.0533091
## 19.000000 42.08236 47.07307 -0.7621809 0.9086925
## 19.142857 57.32912 52.59096 -0.7231365 1.1052927# Forecast
forecast2 <- predict(
winter2,
n.ahead = h_test
)
forecast2.opt <- predict(
winter2.opt,
n.ahead = h_test
)# Plot
xhat2 <-
winter2$fitted[, 2]
xhat2.opt <-
winter2.opt$fitted[, 2]
plot(
train_hw.ts,
main = "Winter Multiplikatif PM2.5",
type = "l",
xlab = "Periode",
ylab = "Konsentrasi PM2.5 (µg/m³)"
)
lines(
xhat2,
col = "red",
lwd = 2
)
lines(
xhat2.opt,
col = "blue",
lwd = 2
)
lines(
forecast2,
col = "red",
lwd = 2
)
lines(
forecast2.opt,
col = "blue",
lwd = 2
)
legend(
"topleft",
c(
"Data Aktual",
"Winter Multiplikatif",
"Winter Multiplikatif Optimal"
),
col = c(
"black",
"red",
"blue"
),
lty = 1,
cex = 0.7
)
# Akurasi Data Latih
SSE_hw2 <- winter2$SSE
MSE_hw2 <-
SSE_hw2 /
length(train_hw.ts)
RMSE_hw2 <-
sqrt(MSE_hw2)
SSE_hw2.opt <-
winter2.opt$SSE
MSE_hw2.opt <-
SSE_hw2.opt /
length(train_hw.ts)
RMSE_hw2.opt <-
sqrt(MSE_hw2.opt)
akurasi_hw2_train <- data.frame(
Model = c(
"Winter Multiplikatif",
"Winter Multiplikatif Optimal"
),
SSE = c(
SSE_hw2,
SSE_hw2.opt
),
MSE = c(
MSE_hw2,
MSE_hw2.opt
),
RMSE = c(
RMSE_hw2,
RMSE_hw2.opt
)
)
akurasi_hw2_train## Model SSE MSE RMSE
## 1 Winter Multiplikatif 42954.42 335.5814 18.31888
## 2 Winter Multiplikatif Optimal 37386.30 292.0804 17.09036best_hw2_train <-
akurasi_hw2_train$Model[
which.min(
akurasi_hw2_train$RMSE
)
]
cat(
"Berdasarkan hasil akurasi data latih,",
best_hw2_train,
"memiliki nilai RMSE paling kecil. ",
"Dengan demikian, model tersebut memberikan",
"hasil pemulusan yang lebih baik pada data latih."
)## Berdasarkan hasil akurasi data latih, Winter Multiplikatif Optimal memiliki nilai RMSE paling kecil. Dengan demikian, model tersebut memberikan hasil pemulusan yang lebih baik pada data latih.# Akurasi Data Uji
error_hw2 <-
as.numeric(forecast2) -
as.numeric(test_hw.ts)
error_hw2.opt <-
as.numeric(forecast2.opt) -
as.numeric(test_hw.ts)
SSE_hw2.test <-
sum(error_hw2^2)
MSE_hw2.test <-
mean(error_hw2^2)
RMSE_hw2.test <-
sqrt(MSE_hw2.test)
SSE_hw2.opt.test <-
sum(error_hw2.opt^2)
MSE_hw2.opt.test <-
mean(error_hw2.opt^2)
RMSE_hw2.opt.test <-
sqrt(MSE_hw2.opt.test)
akurasi_hw2_test <- data.frame(
Model = c(
"Winter Multiplikatif",
"Winter Multiplikatif Optimal"
),
SSE = c(
SSE_hw2.test,
SSE_hw2.opt.test
),
MSE = c(
MSE_hw2.test,
MSE_hw2.opt.test
),
RMSE = c(
RMSE_hw2.test,
RMSE_hw2.opt.test
)
)
akurasi_hw2_test## Model SSE MSE RMSE
## 1 Winter Multiplikatif 8264.595 250.4423 15.82537
## 2 Winter Multiplikatif Optimal 13918.343 421.7680 20.53699best_hw2_test <-
akurasi_hw2_test$Model[
which.min(
akurasi_hw2_test$RMSE
)
]
cat(
"Berdasarkan hasil akurasi data uji,",
best_hw2_test,
"memiliki nilai RMSE paling kecil. ",
"Dengan demikian, model tersebut memberikan",
"hasil peramalan yang lebih baik pada data uji."
)## Berdasarkan hasil akurasi data uji, Winter Multiplikatif memiliki nilai RMSE paling kecil. Dengan demikian, model tersebut memberikan hasil peramalan yang lebih baik pada data uji.Perbandingan Semua Model
perbandingan_model <- data.frame(
Model = c(
"SMA m=4",
"DMA m=4",
"SES Alpha=0.2",
"SES Alpha=0.7",
"SES Optimal",
"DES Skenario 1",
"DES Skenario 2",
"DES Optimal",
"Winter Aditif",
"Winter Aditif Optimal",
"Winter Multiplikatif",
"Winter Multiplikatif Optimal"
),
RMSE = c(
sqrt(MSE_test.sma),
sqrt(MSE_test.dma),
RMSEtesting1,
RMSEtesting2,
RMSEtestingopt,
sqrt(MSEtestingdes1),
sqrt(MSEtestingdes2),
sqrt(MSEtestingdesopt),
RMSE_hw1.test,
RMSE_hw1.opt.test,
RMSE_hw2.test,
RMSE_hw2.opt.test
)
)
perbandingan_model## Model RMSE
## 1 SMA m=4 13.14610
## 2 DMA m=4 118.40504
## 3 SES Alpha=0.2 12.93309
## 4 SES Alpha=0.7 17.52031
## 5 SES Optimal 13.44104
## 6 DES Skenario 1 24.73099
## 7 DES Skenario 2 107.02265
## 8 DES Optimal 24.03719
## 9 Winter Aditif 14.51854
## 10 Winter Aditif Optimal 18.89358
## 11 Winter Multiplikatif 15.82537
## 12 Winter Multiplikatif Optimal 20.53699perbandingan_model <-
perbandingan_model[
order(perbandingan_model$RMSE),
]
perbandingan_model## Model RMSE
## 3 SES Alpha=0.2 12.93309
## 1 SMA m=4 13.14610
## 5 SES Optimal 13.44104
## 9 Winter Aditif 14.51854
## 11 Winter Multiplikatif 15.82537
## 4 SES Alpha=0.7 17.52031
## 10 Winter Aditif Optimal 18.89358
## 12 Winter Multiplikatif Optimal 20.53699
## 8 DES Optimal 24.03719
## 6 DES Skenario 1 24.73099
## 7 DES Skenario 2 107.02265
## 2 DMA m=4 118.40504model_terbaik <-
perbandingan_model$Model[1]
rmse_terbaik <-
perbandingan_model$RMSE[1]
cat(
"Berdasarkan hasil perbandingan pada data uji, model terbaik adalah",
model_terbaik,
"dengan nilai RMSE sebesar",
rmse_terbaik,
". Model tersebut dipilih karena memiliki nilai RMSE paling kecil dibandingkan model lainnya."
)## Berdasarkan hasil perbandingan pada data uji, model terbaik adalah SES Alpha=0.2 dengan nilai RMSE sebesar 12.93309 . Model tersebut dipilih karena memiliki nilai RMSE paling kecil dibandingkan model lainnya.