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Metode PemulusanWinter (Aditif)KULIAH 5|METODE PERAMALAN DERET WAKTU
Review Untuk apa metode pemulusan (smoothing)
dilakukan terhadap data deret waktu?
Kapan metode pemulusan eksponensial tunggaldigunakan?
Kapan metode pemulusan eksponensial gandadigunakan?
Review: The Process of Smoothing Data Set
Outline Data deret waktu yang mengandung komponen
musiman aditif
Pemulusan metode Winter untuk data deretwaktu musiman aditif
Contoh aplikasi pada data
Seasonal Data
Seasonal Data
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Aditif
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1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58
Multiplikatif
Additive Multiplicative
Ilustrasi: US Clothing Sales
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
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
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:
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:
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
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
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)
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
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)
Procedures of Additive Holt-Winters Method
Consider the Mountain Bike example,
Slide 17
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
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
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
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
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
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
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.
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
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
Slide 27
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
Slide 29
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
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
Another Example
See example 4.8 on Montgomery (2015) , chapter 4 page 309.
Chapter Summary Exponential smoothing Vs Holt-Winter’s
smoothing ?
Basic idea of additive Holt-Winter’s smoothing?
Procedure in additive Holt-Winter’s smoothing?
Exercise1. Montgomery (2015) exercise 4.30 part (a) and (b)
2. Montgomery (2015) exercise 4.32 part (a) and (b)
Next Topic…
“Multiplicative Holt-Winter’s Method”
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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