Analisis Deret Waktu: Materi

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Model ARIMA Box-Jenkins untuk data time series yang TIDAK STASIONER

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Identification of NONSTATIONARY TIME SERIES

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Estimation of ARIMA model Diagnostic Check of ARIMA model Forecasting

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Studi Kasus : Model ARIMAX (Analisis Intervensi, Fungsi Transfer dan Neural Networks)

Example : Weekly sales of Ultra Shine toothpaste (in units of 1000 tubes) [Bowerman and O’Connell, pg. 478] t 1.

2.

3.

4.

5.

… … 26.

27.

28.

29.

30.

Y t 235.000

239.000

244.090

252.731

264.377

… … 517.237

524.349

532.104

538.097

544.948

t 31.

32.

33.

34.

35.

… … 56.

57.

58.

59.

60.

Y t 551.925

557.929

564.285

572.164

582.926

… … 805.844

815.122

822.905

930.663

839.600

t 61.

62.

63.

64.

65.

… ...

86.

87.

88.

89.

90.

Y t 846.962

853.830

860.840

871.075

877.792

… … 996.291

1003.100

1010.320

1018.420

1029.480

Example

: IDENTIFICATION step [stationarity and ACF]

ACF

Dying down extremely slowly Nonstationary time series

Example : IDENTIFICATION step … Difference [W t = Y t – Y t-1 ] Stationary time series

ACF

Y W t t   AR(1) or ARI(1,1)

PACF

Dies down Cuts off after lag 1

Example

: ESTIMATION and DIAGNOSTIC CHECK step Y t = 3.0232 + 1.6591 Y t-1 – 0.6591 Y t-2 + a t Estimation and Testing parameter Diagnostic Check ( white noise residual)

Example: DIAGNOSTIC CHECK step … [ Normality test of residuals ]

Example

: FORECASTING step [MINITAB output]

Comparison

: ARIMA versus Trend Analysis ARIMA(1,1,0) MSE = 7.647

Plot comparison

: ARIMA versus Trend Analysis ARIMA(1,1,0) MSE = 7.647

Trend Analysis MSE = 598.212

Forecast comparison

Plot RESIDUAL comparison : ARIMA versus Trend Analysis ARIMA(1,1,0) MSE = 7.647

Trend Analysis MSE = 598.212

MINITAB command:

IDENTIFICATION

Step  Plot Data stationarity data To make stationarity data ACF & PACF data  to find tentative ARIMA model