Kuliah 2 | Metode Peramalan Deret Waktu

rahmaanisa@apps.ipb.ac.id

REVIEW Tentukan pola dari data deret waktu berikut:

Gambar (1) Gambar (2)

Gambar (3) Gambar (4)

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Kriteria kebaikan peramalan data deret waktu

MAD

MAPE

MSE

AIC

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Data deret waktu stasioner (tanpa tren)

Pemulusan rataan bergerak sederhana (RBS)

Peramalan melalui RBS

Data deret waktu tak-stasioner (ada tren)

Pemulusan rataan bergerak ganda (RBG)

Peramalan melalui RBG

Contoh aplikasi pada data

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“A time series is said to be strictly stationary if its properties are not affected by a change in the time origin.”

Montgomerry (2015)

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Plotting smoothed dataOverlay a smoothed version of the original data

help reveal patterns in the original data

The simplest approach: moving average

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Bagaimana akurasi dari

peramalannya?

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Single Moving Average

Double Moving Average

…

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Note that the smoothed data will have less variance*:

*assuming independence between observations.

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𝐹𝑡 = 𝑀𝑡−1

Sedangkan untuk periode ke-n:

𝐹𝑛,ℎ = 𝑀𝑛

Artinya, peramalan untuk periode selanjutnya

adalah konstan.18

Single moving average of order three:

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Monthly Time

Periods

Sales

(units)MA(3) Forecast Error

Jan 10

Feb 9

Mar 810+9+8

3= 9.00

Apr 79+8+7

3= 8.00 9.00 -2.00

May 3 6.00 8.00 -5.00

Jun 2 4.00 6.00 -4.00

Jul 1 2.00 4.00 -3.00

Aug 0 1.00 2.00 -2.00

Sep 1 0.67 1.00 0.00

Oct 5 2.00 0.67 4.33

Nov 12 6.00 2.00 10.00

Dec 14 10.33 6.00 8.00

Forecast 10.3320

0

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4

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Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Sales (units) Forecast21

Monthly Time

PeriodsSales (units) MA(5) Forecast

Jan 10

Feb 9

Mar 8

Apr 7

May 3 7.4

Jun 2 5.8 7.4

Jul 1 4.2 5.8

Aug 0 2.6 4.2

Sep 1 1.4 2.6

Oct 5 1.8 1.4

Nov 12 3.8 1.8

Dec 14 6.4 3.8

Forecast 6.4 23

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Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Sales (units) Forecast24

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an outlier will dominate the moving averages that contain that observation

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100

150

200

250

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Actual Forecast 28

Moving Median

Centered Moving Average

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Odd-span moving medians (also called running medians) are an alternative to moving averages that are effective data smoothers when the time series may be contaminated with unusual values or outliers.

The moving median of span N is defined as

where N = 2u + 1. The median is the middle observation in rank order (or order of value). The moving median of span 3 is a very popular and effective data smoother, where

SUPPLEMENTARY TOPICS

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This is common for even numbers of observations.

Monthly Time PeriodsSales

(units)MA(3) Forecast

Jan 10

Feb 910+9+8

3= 9.00

Mar 89+8+7

3= 8.00 9.00

Apr 7 6.00 8.00

May 3 4.00 6.00

Jun 2 2.00 4.00

Jul 1 1.00 2.00

Aug 0 0.67 1.00

Sep 1 2.00 0.67

Oct 5 6.00 2.00

Nov 12 10.33 6.00

Dec 14 10.33

Forecast 10.33

SUPPLEMENTARY TOPICS

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0

2

4

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10

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Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Sales (units) Forecase (centered) Forecast

SUPPLEMENTARY TOPICS

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SUPPLEMENTARY TOPICS

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A time series that exhibits a trend is a nonstationary time series.

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A double moving average may be used for additional smoothing of a single moving average.

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1. Compute a single moving average (𝑆1) of order 𝑇

2. A second moving average (𝑆2) series is calculated from the first moving average, is of order 𝑁

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dengan:

𝑎𝑡 = 2 𝑆1,𝑡 − 𝑆2,𝑡

𝑏𝑡 =2

𝑁−1𝑆1,𝑡 − 𝑆2,𝑡

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Period Data Series

1 34

2 36

3 38

4 40

5 42

6 44

7 46

8 48

9 50

10 52

Langkah 1:

Lakukan pemulusan single moving

average (misal, T=3):

𝑆1,𝑡 =1

3𝑦𝑡−2 + 𝑦𝑡−3 + 𝑦𝑡

misal:

𝑆1,3 =1

3𝑦1 + 𝑦2 + 𝑦3

𝑆1,3 =1

334 + 36 + 38

𝑆1,3 = 36

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Period Data Series 𝑺𝟏1 34

2 36

3 38 36

4 40 38

5 42 40

6 44 42

7 46 44

8 48 46

9 50 48

10 52 50

Langkah 2:

Lakukan pemulusan single moving

average (misal, N=3):

𝑆2,𝑡 =1

3𝑆𝑡−2 + 𝑆𝑡−3 + 𝑆𝑡

misal:

𝑆2,5 =1

3𝑆3 + 𝑆4 + 𝑆5

𝑆1,3 =1

336 + 38 + 40

𝑆1,3 = 38

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Period Data Series 𝑺𝟏 𝑺𝟐1 34

2 36

3 38 36

4 40 38

5 42 40 38

6 44 42 40

7 46 44 42

8 48 46 44

9 50 48 46

10 52 50 48

Menghitung Forecasts:

𝑎𝑡 = 2 𝑆1,𝑡 − 𝑆2,𝑡

𝑏𝑡 =2

𝑁−1𝑆1,𝑡 − 𝑆2,𝑡

misal utk t=5,

𝑎5 = 2 𝑆1,5 − 𝑆2,5𝑎5 = 2 40 − 38𝑎5 = 42

𝑏5 =2

3−1𝑆1,5 − 𝑆2,5

𝑏5 =2

3−140 − 38

𝑏5 = 2

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Period Data Series 𝑺𝟏,𝒕 𝑺𝟐,𝒕 𝒂𝒕 𝒃𝒕1 34

2 36

3 38 36

4 40 38

5 42 40 38 42 2

6 44 42 40 44 2

7 46 44 42 46 2

8 48 46 44 48 2

9 50 48 46 50 2

10 52 50 48 52 2

Menghitung Forecasts: 𝐹𝑡+ℎ = 𝑎𝑡 + 𝑏𝑡 ℎ

𝐹6 = 𝐹5+1 = 𝑎5 + 𝑏5 1 = 42 + 2 1 = 4442

Period Data Series 𝑺𝟏,𝒕 𝑺𝟐,𝒕 𝒂𝒕 𝒃𝒕 𝑭𝒕1 34

2 36

3 38 36

4 40 38

5 42 40 38 42 2 44

6 44 42 40 44 2 46

7 46 44 42 46 2 48

8 48 46 44 48 2 50

9 50 48 46 50 2 52

10 52 50 48 52 2 44

11 54Menghitung Forecasts: 𝐹𝑡+ℎ = 𝑎𝑡 + 𝑏𝑡 ℎ

𝐹6 = 𝐹5+1 = 𝑎5 + 𝑏5 1 = 42 + 2 1 = 4443

Two-sided Moving Average

Weighted Moving Average

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Perhatikan kembali ilustrasi-1 dan ilustrasi-2. HitungMAPE, MAD, dan MSE dari masing-masing kasustersebut.

Menurut Anda, mana yang lebih baik di antarakeduanya?

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Berikut disajikan data penjualan mobil di Carmen’s Chevrolet. Lakukan pemulusan rataan bergerak tunggal dengan periode 3 minggu.

a) Berapa nilai hasil peramalan pada minggu ke -7 menggunakan metode rataan bergerak dengan periode 3 minggu?

b) Buatlah sketsa data aktual dan data hasil pemulusan

Week 1 2 3 4 5 6 7

Auto Sales 8 10 9 11 10 13 -

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Volume ekspor karet mentah Indonesia ke Negara Asia selama 11 tahun terakhir disajikan dalam table dibawah ini:

Gambarkan plot data ekspor ini terhadap tahun.

Apa penjelasan Anda mengenai perilaku ekspor tersebut?

Berdasarkan pola data tersebut, menurut Anda, metode pemulusanmana yang lebih tepat utk digunakan pada data tsb?

(Single atau Double Moving Average)

Tahun 1998 1999 2000 2001 2002 2003

Ekspor (ribuan ton) 97.43 96.22 98.29 98.61 97.19 99.58

Tahun 2004 2005 2006 2007 2008

Ekspor (ribuan ton) 101.03 100.04 102.6 101.3 101.81

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Hyndman, R.J. 2010. Moving Averages. Contribution to the InternationalEncyclopedia of Statistical Science, ed. Miodrag Lovric, Springer.pp.866-869. https://robjhyndman.com/papers/movingaverage.pdf[diakses pada 13 Februari 2018]

Montgomery, D.C., Jennings, C.L., Kulahci, M. 2015 .Introduction to TimeSeries Analysis and Forecasting, 2nd ed. New Jersey: John Wiley &Sons.

Yaffee, R.A., McGee, M. 2000. Introduction to Time Series Analysis andForecasting with Applications of SAS and SPSS. San Diego:Academic Press.

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