Metode PemulusanWinter (Aditif)KULIAH 5|METODE PERAMALAN DERET WAKTU

rahmaanisa@apps.ipb.ac. id

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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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

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40

50

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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

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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

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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 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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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

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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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Materi perkuliahan dapatdiakses pada:

stat.ipb.ac.id/en

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