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Review Untuk apa metode pemulusan (smoothing)
dilakukan terhadap data deret waktu?
Kapan metode pemulusan eksponensial tunggaldigunakan?
Kapan metode pemulusan eksponensial gandadigunakan?
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Review: The Process of Smoothing Data Set
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Outline Data deret waktu yang mengandung komponen
musiman aditif
Pemulusan metode Winter untuk data deretwaktu musiman aditif
Contoh aplikasi pada data
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Seasonal Data
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Seasonal Data
50.00
55.00
60.00
65.00
70.00
75.00
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58
Aditif
30.00
40.00
50.00
60.00
70.00
80.00
90.00
100.00
110.00
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58
Multiplikatif
Additive Multiplicative
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Ilustrasi: US Clothing Sales
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EXPONENTIAL SMOOTHING FOR SEASONAL DATA Originally introduced by Holt (1957) and Winters (1960)
Generally known as Winters’ method
Basic idea:
seasonal adjustment linear trend model
Two types of adjustments are suggested: Additive
Multiplicative
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Additive Model
level or linear trend component the seasonal adjustment
St = St+m = St+2m =… for t = 1,…, m − 1
length of the season (period) of the cycles
can in turn be represented by 𝛽0 + 𝛽1t
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Double Exponential Vs Additive Holt-Winter’s Method
𝑦𝑡+ℎ 𝑡 = 𝐿𝑡 + 𝑇𝑡ℎ
𝑦𝑡+ℎ 𝑡 = 𝐿𝑡 + 𝐵𝑡ℎ + 𝑆𝑡+ℎ−𝑚
Level Trend
Level
Trend
Seasonal
Holt Winter≈Triple Exponential Smoothing
Double Exponential:
Holt-Winter:
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Holt-Winters Additive Formulation Suppose the time series is denoted by 𝑦1, … , 𝑦𝑛 with 𝑚
seasonal period.
𝑙𝑡 = 𝛼 𝑦𝑡 − 𝑠𝑡−𝑚 + 1 − 𝛼 𝑙𝑡−1 + 𝑏𝑡−1
𝑏𝑡 = 𝛾 𝑙𝑡 − 𝑙𝑡−1 + 1 − 𝛾 𝑏𝑡−1
𝑠𝑡 = 𝛿 𝑦𝑡 − 𝑙𝑡 + 1 − 𝛿 𝑠𝑡−𝑚
Estimate of the level:
Estimate of the trend:
Estimate of the seasonal factor:
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ℎ-step-ahead forecast Let 𝑦𝑡+ℎ 𝑡 be the ℎ-step forecast made using data to
time 𝑡
𝑦𝑡+ℎ 𝑡 = 𝑙𝑡 + 𝑏𝑡ℎ + 𝑠𝑡+ℎ−𝑚
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The Procedure
Step 1: Initialize the value of 𝑙𝑡 , 𝑏𝑡, and 𝑠𝑡
Step 2: Update the estimate of 𝑙𝑡
Step 3: Update the estimate of 𝑏𝑡
Step 4: Update the estimate of 𝑠𝑡
Step 5: Conduct the ℎ-step-ahead forecast
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Initializing the Holt-Winters method
1. Fit a 2×𝑚 moving average smoother to the first 2 or 3 years of data.
2. Subtract smooth trend from the original data to get de-trended data. The initial seasonal values are then obtained from the averaged de-trended data.
3. Subtract the seasonal values from the original data to get seasonally adjusted data.
4. Fit a linear trend to the seasonally adjusted data to get the initial level ℓ0 (the intercept) and the initial slope b0.
Hyndman (2010)
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Initializing the Holt-Winters method
use the least squares estimates of the following model:
Montgomery (2015):
𝑙0 𝑏0
𝑠𝑗−𝑠 = 𝑦𝑗 for 1 ≤ 𝑗 ≤ 𝑚 − 1 , and 𝑠0 = − 𝑗=1𝑚−1 𝑦𝑗
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Initializing the Holt-Winters method
• fitting a regression with linear trend to the first few years of data (usually 3 or 4 years are used)
• the initial level ℓ0 is set to the intercept
• the initial slope 𝑏0 is set to the regression slope
• the initial seasonal values 𝑠−𝑚+1, … , 𝑠0 are computed from the detrended data.
Bowerman, O’Connell & Koehler (2005)
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Procedures of Additive Holt-Winters Method
Consider the Mountain Bike example,
Slide 17
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Slide 18
Procedures of Additive Holt-Winters Method
0
10
20
30
40
50
60
0 2 4 6 8 10 12 14 16 18
Time
Bik
e sa
les
(y)
Observations:
Linear upward trend over the 4-year period
Magnitude of seasonal span is almost constant as the level of the time series increases
Additive Holt-Winters method can be applied to forecast future sales
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Slide 19
Procedures of Additive Holt-Winters Method
Step 1: Obtain initial values for the level ℓ0, the growth rate b0, and the seasonal factors sn-3, sn-2, sn-1, and sn0, by fitting a least squares trend line to at least four or five years of the historical data. ◦ y-intercept = ℓ0; slope = b0
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Slide 20
Procedures of Additive Holt-Winters Method
Example ◦ Fit a least squares trend line to all 16 observations
◦ Trend line
ℓ0 = 20.85; b0 = 0.9809tyt 980882.085.20ˆ
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Slide 21
Procedures of Additive Holt-Winters Method
Step 2: Find the initial seasonal factors1. Compute for each time period that is used in finding
the least squares regression equation. In this example, t = 1, 2, …, 16.
ˆty
1
2
16
ˆ 20.85 0.980882(1) 21.8309
ˆ 20.85 0.980882(2) 22.8118
......
ˆ 20.85 0.980882(16) 36.5441
y
y
y
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Slide 22
Procedures of Additive Holt-Winters Method
Step 2: Find the initial seasonal factors2. Detrend the data by computing for each
observation used in the least squares fit. In this example, t = 1, 2, …, 16.
ttt yyS ˆ
5441.115441.3625ˆ
......
1882.88112.2231ˆ
8309.118309.2110ˆ
161616
222
111
yyS
yyS
yyS
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Slide 23
Procedures of Additive Holt-Winters Method
Step 2: Find the initial seasonal factors3. Compute the average seasonal values for each of the L
seasons. The L averages are found by computing the average of the detrended values for the corresponding season. For example, for quarter 1,
2162.144
)6015.14()6779.15()7544.14()8309.11(
4
13951
]1[
SSSSS
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Slide 24
Procedures of Additive Holt-Winters Method
Step 2: Find the initial seasonal factors4. Compute the average of the L seasonal factors. The
average should be 0.
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Slide 25
Procedures of Additive Holt-Winters Method
Step 3: Calculate a point forecast of y1 from time 0 using the initial values
7.6147(-14.2162)0.980920.85
)0(ˆ
1),0( )(ˆ
30041001
snbsnby
pTsnpbTy LpTTTpT
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Slide 26
Procedures of Additive Holt-Winters Method
Step 4: Update the estimates ℓT, bT, and snT by using some predetermined values of smoothing constants.
Example: let = 0.2, = 0.1, and δ = 0.1
3079.22)9808.085.20(8.0))2162.14(10(2.0
))(1()( 004111
bsny
0286.1)9809.0(9.0)85.203079.22(1.0
)1()( 0011
bb
0254.14)2162.14(9.0)3079.2210(1.0
)1()( 41111
snysn
8895.295529.60286.13079.22
)1(ˆ21142112
snbsnby
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Slide 27
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Slide 28
Procedures of Additive Holt-Winters Method
Step 5: Find the most suitable combination of , , and δ that minimizes SSE (or MSE)
Example: Use Solver in Excel as an illustrationSSE
alpha
gamma
delta
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Slide 29
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Slide 30
Additive Holt-Winters Methodp-step-ahead forecast made at time T
Example
3,...) 2, 1,( )(ˆ psnpbTy LpTTTpT
1073.232162.149809.03426.36)16(ˆ417161617 snby
8573.445529.6)9809.0(23426.362)16(ˆ418161618 snby
8573.575721.18)9809.0(33426.363)16(ˆ419161619 snby
3573.299088.10)9809.0(43426.364)16(ˆ420161620 snby
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Slide 31
Additive Holt-Winters MethodExample
Forecast Plot for Mountain Bike Sales
0
10
20
30
40
50
60
70
0 2 4 6 8 10 12 14 16 18 20
Time
Fo
recasts
Observed values
Forecasts
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Another Example
See example 4.8 on Montgomery (2015) , chapter 4 page 309.
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Chapter Summary Exponential smoothing Vs Holt-Winter’s
smoothing ?
Basic idea of additive Holt-Winter’s smoothing?
Procedure in additive Holt-Winter’s smoothing?
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Exercise1. Montgomery (2015) exercise 4.30 part (a) and (b)
2. Montgomery (2015) exercise 4.32 part (a) and (b)
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Next Topic…
“Multiplicative Holt-Winter’s Method”
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ReferensiHyndman, R.J. 2010. Initializing the Holt-Winters method.
https://robjhyndman.com/hyndsight/hw-initialization/[March 7th, 2018]
Hyndman, R.J and Athanasopoulos, G. 2013. Forecasting:principles and practice. https://www.otexts.org/ fpp/6/2/[March 7th, 2018]
Montgomery, D.C., Jennings, C.L., Kulahci, M. 2015. Introductionto Time Series Analysis and Forecasting, 2nd ed. New Jersey:John Wiley & Sons.
Wan, A. 2017. Exponential Smoothing. http://personal.cb.cityu.edu.hk/msawan/teaching/ms6215/Exponential%20Smoothing%20Methods.ppt [March 7th, 2018]
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Materi perkuliahan dapatdiakses pada:
stat.ipb.ac.id/en
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