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Kuliah 2 | Metode Peramalan Deret Waktu
[email protected]
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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
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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
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0
2
4
6
8
10
12
14
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Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Sales (units) Forecast21
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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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0
2
4
6
8
10
12
14
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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
0
50
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
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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
6
8
10
12
14
16
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
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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
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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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https://robjhyndman.com/papers/movingaverage.pdf
