multiplicative winter’s smoothing method · initializing the holt-winters method 1. fit a 2×...

Multiplicative Winter’s Smoothing MethodLECTURE 6|TIME SERIES FORECASTING METHOD [email protected]. id

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?

Seasonal Data

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

Outline Time series data set with multiplicative seasonal

component

Winter’s smoothing method for multiplicative seasonal time series data

Ilustration

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

Aditive 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

𝑌𝑡 = 𝐿𝑡 + 𝑆𝑡 + 휀𝑡

Multiplicative 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

𝑌𝑡 = 𝐿𝑡 𝑆𝑡 + 휀𝑡

Additive Vs Multiplicative Holt-Winter’s Method

𝑌𝑡+ℎ 𝑡 = 𝐿𝑡 + 𝐵𝑡ℎ + 𝑆𝑡+ℎ−𝑚

Level

Trend

Seasonal

Additive:

𝑌𝑡+ℎ 𝑡 = 𝐿𝑡 + 𝐵𝑡ℎ 𝑆𝑡+ℎ−𝑚Multiplicative:

Holt-Winters Multiplicative Formulation Suppose the time series is denoted by 𝑦1, … , 𝑦𝑛 with 𝑚

seasonal period.

𝐿𝑡 = 𝛼𝑌𝑡

𝑆𝑡−𝑚+ 1 − 𝛼 𝐿𝑡−1 + 𝐵𝑡−1

𝐵𝑡 = 𝛾 𝐿𝑡 − 𝐿𝑡−1 + 1 − 𝛾 𝐵𝑡−1

𝑆𝑡 = 𝛿𝑌𝑡

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

Holt-Winters Additive Vs Multiplicative Formulation

Suppose the time series is denoted by 𝑦1, … , 𝑦𝑛 with 𝑚 seasonal period.

Additive Multiplicative

Est. of level

𝐿𝑡 = 𝛼 𝑌𝑡 − 𝑆𝑡−𝑚 + 1 − 𝛼 𝐿𝑡−1 + 𝐵𝑡−1 𝐿𝑡 = 𝛼𝑌𝑡

𝑆𝑡−𝑚+ 1 − 𝛼 𝐿𝑡−1 + 𝐵𝑡−1

Est. of trend

𝐵𝑡 = 𝛾 𝐿𝑡 − 𝐿𝑡−1 + 1 − 𝛾 𝐵𝑡−1 𝐵𝑡 = 𝛾 𝐿𝑡 − 𝐿𝑡−1 + 1 − 𝛾 𝐵𝑡−1

Est. of seasonal

𝑆𝑡 = 𝛿 𝑌𝑡 − 𝐿𝑡 + 1 − 𝛿 𝑆𝑡−𝑚 𝑆𝑡 = 𝛿𝑌𝑡

𝐿𝑡+ 1 − 𝛿 𝑆𝑡−𝑚

Forecast 𝑌𝑡+ℎ 𝑡 = 𝐿𝑡 + 𝐵𝑡ℎ + 𝑆𝑡+ℎ−𝑚 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)


Montgomery (2015):

Suppose a dataset consist of 𝑘 seasons.

𝐿0 = 𝑦𝑘− 𝑦1

𝑘−1 𝑚where 𝑦𝑖 =

1

𝑚 𝑡= 𝑖−1 𝑚+1

𝑖𝑚 𝑦𝑡

𝐵0 = 𝑦1 −𝑚

2 𝐿0

𝑆𝑗−𝑚 = 𝑚 𝑆𝑗∗

𝑖=1𝑚 𝑆𝑗

∗ , for 1 ≤ 𝑗 ≤ 𝑠, where 𝑆𝑗∗ =

1

𝑘 𝑡=1

𝑘 𝑦 𝑡−1 𝑚+𝑗

𝑦𝑡−𝑠+1

2−𝑗 𝛽0


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.

Hansun (2017)

Procedures of Multiplicative Holt-Winters Method

Use the Sports Drink example as an illustration


0

50

100

150

200

250

0 5 10 15 20 25 30 35

Time

Sp

ort

s D

rin

k (y

)

Procedures of Multiplicative Holt-Winters MethodObservations: ◦ 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


Step 1: Obtain initial values for the level ℓ0, the growth rate b0, and the seasonal factors s-3, s-2, s-1, and s0, by fitting a least squares trend line to at least four or five years of the historical data. ◦ y-intercept = ℓ0; slope = b0


Example ◦ Fit a least squares trend line to the first 16 observations

◦ Trend line

◦ ℓ0 = 95.2500; b0 = 2.4706

ˆ 95.2500 2.4706ty t


Step 2: Find the initial seasonal factors1. Compute for the in-sample observations used for

fitting the regression. In this example, t = 1, 2, …, 16. ˆ

ty

1

2

16

ˆ 95.2500 2.4706(1) 97.7206

ˆ 95.2500 2.4706(2) 100.1912

......

ˆ 95.2500 2.4706(16) 134.7794

y

y

y


Step 2: Find the initial seasonal factors2. Detrend the data by computing for each time

period that is used in finding the least squares regression equation. In this example, t = 1, 2, …, 16.

ˆ/t t tS y y

1 1 1

2 2 2

16 16 16

ˆ/ 72 / 97.7206 0.7368

ˆ/ 116 /100.1912 1.1578

......

ˆ/ 120 /134.7794 0.8903

S y y

S y y

S y y

𝑆0,1

𝑆0,2

……

𝑆0,16

𝑆0,𝑡


Step 2: Find the initial seasonal factors3. 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,

1 5 9 13[1]

4

0.7368 0.7156 0.6894 0.68310.7062

4

S S S SS


Step 2: Find the initial seasonal factors4. Multiply the average seasonal values by the normalizing

constant

such that the average of the seasonal factors is 1. The initial seasonal factors are

[ ]( ) ( 1,2,..., )i L isn S CF i L 𝑆𝑖−𝑚

𝐶𝐹 =𝑚

𝑖=1𝑚 𝑆[𝑖]


Step 2: Find the initial seasonal factors4. Multiply the average seasonal values by the normalizing

constant such that the average of the seasonal factors is 1. ◦ Example

CF = 4/3.9999 = 1.0000

3 1 4 [1]

2 2 4 [2]

1 3 4 [3]

0 4 4 [1]

( ) 0.7062(1) 0.7062

( ) 1.1114(1) 1.1114

( ) 1.2937(1) 1.2937

( ) 0.8886(1) 0.8886

sn sn S CF

sn sn S CF

sn sn S CF

sn sn S CF

𝑆−3 = 𝑆1−4

𝑆−2 = 𝑆2−4

𝑆−1 = 𝑆3−4

𝑆0 = 𝑆4−4


Step 3: Calculate a point forecast of y1 from time 0 using the initial values

𝑦𝑡+ℎ 𝑡 = 𝐿𝑡 + 𝐵𝑡ℎ 𝑆𝑡+ℎ−𝑚

𝑦1 0 = 𝐿0 + 𝐵0 𝑆1−4 = 𝐿0 + 𝐵0 𝑆−3

𝑦1 0 = 95.25 + 2.4706 0.7062

𝑦1 0 = 69.0103

𝑡 = 1 , ℎ = 0


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

1 1 1 4 0 0( / ) (1 )( )

0.2(72 / 0.7062) 0.8(95.2500 2.4706) 98.5673

y sn b

l l

1 1 0 0( ) (1 )

0.1(98.5673 95.2500) 0.9(2.4706) 2.5553

b b

l l

2 1 1 2 4ˆ (1) ( )

(98.5673 2.5553)(1.1114) 112.3876

y b sn

l

1 1 1 1 4( / ) (1 )

0.1(72 / 98.5673) 0.9(0.7062) 0.7086

sn y sn

l

𝑠1−4

𝑠1−4

𝑠2−4

053.135

2937.162031.27727.101

2ˆ

114239.1

1114.19.07727.1011161.0

1

62031.2

5553.29.05673.987727.1011.0

1

7727.101

5553.25673.988.01114.11162.0

1

43223

42222

1122

114222

snby

snysn

bb

bsny

𝑠2−4

𝑠2−4

𝑠3−4

945.77

7086.065212.23464.107

4ˆ

889170.0

8886.09.03464.107961.0

1

65212.2

6349.29.05393.1043464.1071.0

1

3464.107

6349.25393.1048.08886.0962.0

1

45445

44444

3344

334444

snby

snysn

bb

bsny

𝑠4−4

𝑠4−4

𝑠5−4

…… ……


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

…… ……

Multiplicative Holt-Winters Method

p-step-ahead forecast made at time T

Example

33 32 32 33 4ˆ (32) ( ) (168.1213 2.3028)(0.7044) 120.0467y b sn l

34 32 32 34 4ˆ (32) ( 2 ) [168.1213 2(2.3028)](1.1038) 190.6560y b sn l

35 32 32 35 4ˆ (32) ( 3 ) [(168.1213 3(2.3028)](1.2934) 226.3834y b sn l

36 32 32 36 4ˆ (32) ( 4 ) [(168.1213 4(2.3028)](0.8908) 157.9678y b sn l

𝑦𝑡+ℎ 𝑡 = 𝐿𝑡 + 𝐵𝑡ℎ 𝑆𝑡+ℎ−𝑚 ℎ = 1, 2, 3, … .

𝑠33−4

𝑠34−4

𝑠35−4

𝑠36−4

Multiplicative Holt-Winters Method

Example

Forecast Plot for Sports Drink Sales

0

50

100

150

200

250

0 5 10 15 20 25 30 35 40

Time

Fo

recasts

Observed values

Forecasts

Chapter Summary Additive Vs multiplicative Holt-Winter’s

smoothing ?

Basic idea of multiplicative Holt-Winter’s smoothing?

Procedure in multiplicative Holt-Winter’s smoothing?

Another Example

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

Exercise1. Montgomery (2015) exercise 4.27 , 4.28, 4.29

2. Montgomery (2015) exercise 4.30 part (c)

3. Montgomery (2015) exercise 4.32 part (c)

Next Topic…

Regression for Time Series Data Set (part 1)

ReferencesHyndman, 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 andpractice. https://www.otexts.org/ fpp/6/2/ [March 7th, 2018]

Montgomery, D.C., Jennings, C.L., Kulahci, M. 2015. Introduction to TimeSeries Analysis and Forecasting, 2nd ed. New Jersey: John Wiley &Sons.

Hansun, S. 2017. New Estimation Rules for Unknown Parameters onHolt-Winters Multiplicative Method. J.Math. Fund. Sci. , Vol. 49 (2):127-135. DOI: 10.5614/j.math.fund.sci.2017.49.2.3.

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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https://robjhyndman.com/hyndsight/hw-initialization/

https://www.otexts.org/fpp/6/2/

http://personal.cb.cityu.edu.hk/msawan/teaching/ms6215/Exponential Smoothing Methods.ppt

The handouts are available on the following site:

stat.ipb.ac.id/en

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multiplicative winter’s smoothing method · initializing the holt-winters method 1. fit a 2×...

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