Analisis dan Peramalan Penjualan Menggunakan Model SARIMA
Kanaya
2026-03-28
# Input data
penjualan <- c(120,135,150,160,170,180,200,210,190,220,240,260,
130,145,160,175,185,200,220,230,210,240,260,280,
140,155,170,180,190,210,230,240,220,250,270,300,
150,165,180,195,205,225,250,260,240,270,290,320,
160,175,190,205,220,240,260,275,255,290,310,340)
# Ubah ke time series
ts_data <- ts(penjualan, start=c(2018,1), frequency=12)
# Plot data
plot(ts_data, main="Penjualan Bulanan 2018-2022",
ylab="Penjualan", xlab="Tahun")
# Decomposition (trend & musiman)
dekomposisi <- decompose(ts_data)
plot(dekomposisi)
Uji Stasioneritas
library(tseries)## Warning: package 'tseries' was built under R version 4.5.3## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zooadf.test(ts_data)##
## Augmented Dickey-Fuller Test
##
## data: ts_data
## Dickey-Fuller = -3.3542, Lag order = 3, p-value = 0.07161
## alternative hypothesis: stationaryp-value = 0,07161 > 0,05 artinya data tidak stasioner.
Differencing
diff1 <- diff(ts_data, differences=1)
adf.test(diff1)## Warning in adf.test(diff1): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: diff1
## Dickey-Fuller = -4.1858, Lag order = 3, p-value = 0.01
## alternative hypothesis: stationarydiff_final <- diff(diff1, lag=12)
adf.test(diff_final)##
## Augmented Dickey-Fuller Test
##
## data: diff_final
## Dickey-Fuller = -3.1571, Lag order = 3, p-value = 0.113
## alternative hypothesis: stationaryp-value = 0,01 < 0,05 artinya data sudah stasioner. Oleh karena itu, proses differencing dihentikan pada orde pertama (d = 1).
Mengidentifikasi Model (Plot ACF & PACF)
library(forecast)## Warning: package 'forecast' was built under R version 4.5.3# Plot ACF
acf(diff1, main="ACF")
# Plot PACF
pacf(diff1, main="PACF")
Berdasarkan plot ACF, terdapat spike signifikan pada lag 1 dan lag 12 yang menunjukkan adanya komponen MA dan pola musiman. Sementara itu, plot PACF menunjukkan spike pada lag 1 dan 2 yang mengindikasikan adanya komponen AR. Oleh karena itu, kandidat model yang dipertimbangkan adalah ARIMA(1,1,1), ARIMA(2,1,1), serta model musiman SARIMA(1,1,1)(1,0,1)[12] dan SARIMA(2,1,1)(1,0,1)[12].
Estimasi Model ARIMA
library(forecast)
model_arima1 <- Arima(ts_data, order=c(1,1,1))
model_arima2 <- Arima(ts_data, order=c(2,1,1))
summary(model_arima1)## Series: ts_data
## ARIMA(1,1,1)
##
## Coefficients:
## ar1 ma1
## 0.5784 -0.9043
## s.e. 0.1366 0.0675
##
## sigma^2 = 1610: log likelihood = -300.84
## AIC=607.68 AICc=608.12 BIC=613.91
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 9.15566 39.10743 26.84525 1.473338 13.37917 1.881127 -0.007709712summary(model_arima2)## Series: ts_data
## ARIMA(2,1,1)
##
## Coefficients:
## ar1 ar2 ma1
## 0.6086 -0.0855 -0.8949
## s.e. 0.1452 0.1380 0.0721
##
## sigma^2 = 1626: log likelihood = -300.65
## AIC=609.3 AICc=610.04 BIC=617.61
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 9.358504 38.95469 26.30155 1.607976 13.03462 1.843028 -0.06100009Berdasarkan hasil estimasi model ARIMA, diperoleh bahwa model ARIMA(1,1,1) memiliki nilai AIC (607.68) dan BIC (613.91) yang lebih kecil dibandingkan ARIMA(2,1,1). Oleh karena itu, model ARIMA(1,1,1) dipilih sebagai model terbaik untuk pendekatan non-musiman.
Estimasi Model SARIMA
library(forecast)
model_sarima1 <- Arima(ts_data,
order=c(1,1,1),
seasonal=c(0,1,1))
model_sarima2 <- Arima(ts_data,
order=c(2,1,1),
seasonal=c(0,1,1))
summary(model_sarima1)## Series: ts_data
## ARIMA(1,1,1)(0,1,1)[12]
##
## Coefficients:
## ar1 ma1 sma1
## -0.0408 -0.1318 0.1949
## s.e. 0.6212 0.6093 0.1384
##
## sigma^2 = 13.51: log likelihood = -126.55
## AIC=261.1 AICc=262.05 BIC=268.5
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.1644189 3.147447 1.73685 -0.05359323 0.8730661 0.1217063
## ACF1
## Training set -0.006879982summary(model_sarima2)## Series: ts_data
## ARIMA(2,1,1)(0,1,1)[12]
##
## Coefficients:
## ar1 ar2 ma1 sma1
## -0.4819 -0.0997 0.3072 0.1973
## s.e. 1.2812 0.2418 1.2799 0.1385
##
## sigma^2 = 13.79: log likelihood = -126.51
## AIC=263.02 AICc=264.48 BIC=272.27
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.1666443 3.144215 1.749437 -0.05339637 0.8790533 0.1225883
## ACF1
## Training set -0.003612474Menentukan Model Terbaik berdasarkan AIC/BIC
AIC(model_arima1, model_arima2,
model_sarima1, model_sarima2)## Warning in AIC.default(model_arima1, model_arima2, model_sarima1,
## model_sarima2): models are not all fitted to the same number of observations## df AIC
## model_arima1 3 607.6804
## model_arima2 4 609.3021
## model_sarima1 4 261.0961
## model_sarima2 5 263.0152BIC(model_arima1, model_arima2,
model_sarima1, model_sarima2)## Warning in BIC.default(model_arima1, model_arima2, model_sarima1,
## model_sarima2): models are not all fitted to the same number of observations## df BIC
## model_arima1 3 613.9130
## model_arima2 4 617.6123
## model_sarima1 4 268.4967
## model_sarima2 5 272.2659Berdasarkan nilai AIC dan BIC model terbaik adalah SARIMA(1,1,1)(0,0,1)[12] memiliki nilai AIC terkecil sebesar 261.0961 dan BIC terkecil sebesar 268.4967.
Uji Diagnostik Model
checkresiduals(model_sarima1)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(1,1,1)(0,1,1)[12]
## Q* = 9.1664, df = 9, p-value = 0.4221
##
## Model df: 3. Total lags used: 12# Uji Ljung-Box
Box.test(residuals(model_sarima1),
lag=20,
type="Ljung-Box")##
## Box-Ljung test
##
## data: residuals(model_sarima1)
## X-squared = 20.398, df = 20, p-value = 0.4333Hasil uji diagnostik residual menggunakan uji Ljung-Box menunjukkan bahwa p-value > 0.05, sehingga residual bersifat white noise. Hal ini menunjukkan bahwa model SARIMA(1,1,1)(0,1,1)[12] telah memenuhi asumsi diagnostik dan layak digunakan untuk forecasting.
Forecasting
library(forecast)
forecast_final <- forecast(model_sarima1, h=12)
# Lihat hasil angka
forecast_final## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 2023 178.2786 173.5684 182.9889 171.0749 185.4823
## Feb 2023 193.2819 187.1686 199.3953 183.9324 202.6315
## Mar 2023 208.2818 201.0141 215.5495 197.1669 219.3967
## Apr 2023 223.0480 214.7864 231.3097 210.4130 235.6831
## May 2023 239.0224 229.8742 248.1707 225.0314 253.0135
## Jun 2023 259.0524 249.0962 269.0087 243.8257 274.2792
## Jul 2023 277.8882 267.1848 288.5915 261.5187 294.2576
## Aug 2023 293.8626 282.4609 305.2642 276.4252 311.2999
## Sep 2023 273.8626 261.8030 285.9222 255.4190 292.3061
## Oct 2023 309.8370 297.1535 322.5204 290.4393 329.2346
## Nov 2023 329.8370 316.5589 343.1150 309.5300 350.1440
## Dec 2023 359.9109 346.0638 373.7580 338.7336 381.0882#plot
plot(forecast_final)
Kesimpulan
Berdasarkan model SARIMA(1,1,1)(0,1,1)[12], hasil forecasting menunjukkan bahwa penjualan pada tahun 2023 cenderung meningkat secara bertahap.
Terdapat penurunan sementara pada bulan September, yang mengindikasikan adanya pola musiman, sebelum kembali meningkat hingga mencapai puncak pada bulan Desember dengan penjualan sebesar 359.9109.