Multiplicative Winter s Smoothing Method

Transcription

1 Multiplicative Winter s Smoothing Method LECTURE 6 TIME SERIES FORECASTING METHOD rahmaanisa@apps.ipb.ac.id

2 Review What is the difference between additive and multiplicative seasonal pattern in time series data? What is the basic idea of additive Winter s smoothing method? What are the issues of additive Winter s smoothing procedure?

3 Seasonal Data

4 Seasonal Data Aditif Multiplikatif Additive Multiplicative

5 Outline Time series data set with multiplicative seasonal component Winter s smoothing method for multiplicative seasonal time series data Ilustration

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8 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

9 Aditive Model Y t = L t + S t + ε t level or linear trend component can in turn be represented by β 0 + β 1 t the seasonal adjustment S t = S t+m = S t+2m = for t = 1,, m 1 length of the season (period) of the cycles

10 Multiplicative Model Y t = L t S t + ε t level or linear trend component can in turn be represented by β 0 + β 1 t the seasonal adjustment S t = S t+m = S t+2m = for t = 1,, m 1 length of the season (period) of the cycles

11 Additive Vs Multiplicative Holt- Winter s Method Additive: Y t+h t = L t + B t h + S t+h m Level Trend Seasonal Multiplicative: Y t+h t = L t + B t h S t+h m

12 Holt-Winters Multiplicative Formulation Suppose the time series is denoted by y 1,, y n with m seasonal period. Estimate of the level: L t = α Estimate of the trend: Y t S t m + 1 α L t 1 + B t 1 B t = γ L t L t γ B t 1 Estimate of the seasonal factor: S t = δ Y t L t + 1 δ S t m

13 h-step-ahead forecast Let Y t+h t be the h-step forecast made using data to time t Y t+h t = L t + B t h S t+h m

14 Holt-Winters Additive Vs Multiplicative Formulation Suppose the time series is denoted by y 1,, y n with m seasonal period. Est. of level Additive L t = α Y t S t m + 1 α L t 1 + B t 1 L t = α Multiplicative Y t S t m + 1 α L t 1 + B t 1 Est. of trend B t = γ L t L t γ B t 1 B t = γ L t L t γ B t 1 Est. of seasonal S t = δ Y t L t + 1 δ S t m S t = δ Y t L t + 1 δ S t m Forecast Y t+h t = L t + B t h + S t+h m Y t+h t = L t + B t h S t+h m

15 The Procedure Step 1: Initialize the value of L t, B t, and S t Step 2: Update the estimate of L t Step 3: Update the estimate of B t Step 4: Update the estimate of S t Step 5: Conduct the h-step-ahead forecast

16 Initializing the Holt-Winters method Hyndman (2010) 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 detrended 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 L 0 (the intercept) and the initial slope B 0.

17 Initializing the Holt-Winters method Montgomery (2015): Suppose a dataset consist of k seasons. L 0 = y k y 1 k 1 m where y i = 1 m im t= i 1 m+1 y t B 0 = y 1 m 2 L 0 S j m = m S j m i=1 S, for 1 j s, where S j = 1 j k t=1 k y t 1 m+j y t s+1 2 j β 0

18 Initializing the Holt-Winters method Hansun (2017) Used the basic principle of weighted moving average to give more weight to more recent data and estimate the initial values for the overall smoothing and the trend smoothing components. The initial values for the seasonal indices can be computed by calculating the average level for each observed season.

19 Procedures of Multiplicative Holt-Winters Method Use the Sports Drink example as an illustration Slide 19

20 Procedures of Multiplicative Holt-Winters Method Sports Drink (y) Time Slide 20

21 Procedures of Multiplicative Holt-Winters Method Observations: Linear upward trend over the 8-year period Magnitude of the seasonal span increases as the level of the time series increases Multiplicative Holt-Winters method can be applied to forecast future sales Slide 21

22 Procedures of Multiplicative Holt-Winters Method Step 1: Obtain initial values for the level l 0, the growth rate b 0, and the seasonal factors s -3, s -2, s -1, and s 0, by fitting a least squares trend line to at least four or five years of the historical data. y-intercept = l 0 ; slope = b 0 Slide 22

23 Procedures of Multiplicative Holt-Winters Method Example Fit a least squares trend line to the first 16 observations Trend line yˆ t t l 0 = ; b 0 = Slide 23

24 Procedures of Multiplicative Holt-Winters Method Step 2: Find the initial seasonal factors 1. Compute for the in-sample observations used for fitting the regression. In this example, t = 1, 2,, 16. y ˆt yˆ (1) yˆ (2) yˆ (16) Slide 24

25 Procedures of Multiplicative Holt-Winters Method Step 2: Find the initial seasonal factors 2. Detrend the data by computing SS 0,t ˆ t yt / yt for each time period that is used in finding the least squares regression equation. In this example, t = 1, 2,, 16. S 0,1 S 0,2 S y / yˆ 72 / S y / yˆ 116 / S 0, S y / yˆ 120 / Slide 25

26 Procedures of Multiplicative Holt-Winters Method Step 2: Find the initial seasonal factors 3. Compute the average seasonal values for each of the k seasons. The k averages are found by computing the average of the detrended values for the corresponding season. For example, for quarter 1, S [1] S1 S5 S9 S Slide 26

27 Procedures of Multiplicative Holt-Winters Method Step 2: Find the initial seasonal factors 4. Multiply the average seasonal values by the normalizing constant CF = m m i=1 S [i] such that the average of the seasonal factors is 1. The initial seasonal factors are sn S ( ) ( 1,2,..., ) i m i L S[] CF i L i Slide 27

28 Procedures of Multiplicative Holt-Winters Method Step 2: Find the initial seasonal factors 4. Multiply the average seasonal values by the normalizing constant such that the average of the seasonal factors is 1. Example CF = 4/ = sn S 3 = sn S 1 4 S ( CF) (1) [1] sn S 2 = sn S 2 4 S ( CF) (1) [2] sn S 1 = sn S 3 4 S ( CF) (1) [3] sn S 0 = sn S S[1] ( CF) (1) Slide 28

29 Procedures of Multiplicative Holt-Winters Method Step 3: Calculate a point forecast of y 1 from time 0 using the initial values y t+h t = L t + B t h S t+h m t = 1, h = 0 y 1 0 = L 0 + B 0 S 1 4 = L 0 + B 0 S 3 y 1 0 = y 1 0 = Slide 29

30 Procedures of Multiplicative Holt-Winters Method Step 4: Update the estimates l T, b T, and sn T by using some predetermined values of smoothing constants. Example: let = 0.2, = 0.1, and δ = 0.1 l b ( y / sn ) (1 )( l b ) s (72 / ) 0.8( ) ( l l ) (1 ) b ( ) 0.9(2.4706) sn1 ( y1 / l 1) (1 ) sn s (72 / ) 0.9(0.7062) yˆ 2(1) ( l 1 b1 ) sn s ( )(1.1114) Slide 30

31 yˆ 3 b sn y2 sn s b b y2 2 1 sn s b2 sn s Slide 31

32 yˆ 5 b sn y4 sn s b b y4 4 1 sn s b4 sn s Slide 32

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34 Procedures of Multiplicative 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 illustration SSE alpha gamma delta Slide 34

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36 Multiplicative Holt-Winters Method p-step-ahead forecast made at time T y t+h t = L t + B t h S t+h m h = 1, 2, 3,. Example yˆ (32) ( l b ) sn s 33 4 ( )(0.7044) yˆ (32) ( l 2 b ) sn [ (2.3028)](1.1038) s yˆ (32) ( l 3 b ) sn [( (2.3028)](1.2934) s yˆ (32) ( l 4 b ) sn [( (2.3028)](0.8908) s Slide 36

37 Multiplicative Holt-Winters Method Example Forecast Plot for Sports Drink Sales Observed values Forecasts Forecasts Time Slide 37

38 Chapter Summary Additive Vs multiplicative Holt-Winter s smoothing? Basic idea of multiplicative Holt-Winter s smoothing? Procedure in multiplicative Holt-Winter s smoothing?

39 Another Example See example 4.8 on Montgomery (2015), chapter 4 page 309.

40 Exercise 1. Montgomery (2015) exercise 4.27, 4.28, Montgomery (2015) exercise 4.30 part (c) 3. Montgomery (2015) exercise 4.32 part (c)

41 Next Topic Regression for Time Series Data Set (part 1)

42 References Hyndman, R.J Initializing the Holt-Winters method. [March 7 th, 2018] Hyndman, R.J and Athanasopoulos, G Forecasting: principles and practice. fpp/6/2/ [March 7 th, 2018] Montgomery, D.C., Jennings, C.L., Kulahci, M Introduction to Time Series Analysis and Forecasting, 2 nd ed. New Jersey: John Wiley & Sons. Hansun, S New Estimation Rules for Unknown Parameters on Holt-Winters Multiplicative Method. J.Math. Fund. Sci., Vol. 49 (2): DOI: /j.math.fund.sci Wan, A Exponential Smoothing. cityu.edu.hk/msawan/teaching/ms6215/exponential%20smoothin g%20methods.ppt [March 7 th, 2018] 42

43 The handouts are available on the following site: stat.ipb.ac.id/en 43

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