1

Time Series Decomposition & Exponential Smoothing

2

Readings• Multiplicative Time Series Decomposition: Read “Time

Series Forecasting”, Notes Abridged from Operations Management by K N Dervitsiotis, McGraw Hill, 1981

• Additive Time Series Decomposition: Notes on PPT Slides

• Exponential Smoothing: – Chapter 3, Business Forecasting, 5th Ed, Wilson & Keating,

Tata-McGrawHill;

• “Marriot Rooms Forecasting” Case

3

Three Systems of Techniques for Business Forecasting

• First forecasting model is cause-and-effect.• This model assumes a cause determines an

outcome. • Cause may be an investment in information

technology, and the effect is sales. • This model requires historical data not only of

effect (say, sales), but also the “cause” (say, information technology expenditure).

4

Three Systems of Techniques for Business Forecasting

• Second is the time-series model• Data are projected forward based on an

established method like -- moving average, simple average, exponential smoothing, decomposition, and Box-Jenkins.

• This model assumes data patterns from the recent past will remain stable in future.

5

Three Systems of Techniques for Business Forecasting

• Third is the judgmental model. • To produce a forecast without useful historical

data (while projecting sales for a brand new product or when market conditions change making past data obsolete).

• In absence of historical data, alternative data collected from experts in the field (Delphi method), prospective customers (Conjoint Analysis), trade groups, business partners, or other relevant source of information.

Time Series Decomposition

• Multiplicative Decomposition: Y=T*S*C*R

• Additive Decompostion: Y=T+S+C+R

6

7

WBSEDCL Energy Sales Data

Apr 2004 – Mar 2008

8

WBSEDCL Energy Sales (MU) - April 2004 to Nov 2007

700750

800850900950

100010501100

11501200

Apr-04

Jul-0

4

Oct-04

Jan-0

5

Apr-05

Jul-0

5

Oct-05

Jan-0

6

Apr-06

Jul-0

6

Oct-06

Jan-0

7

Apr-07

Jul-0

7

Oct-07

End of Nov 2007: How to Predict Future Sales?? (for Dec 2007, …)

9Multiplicative Model: Sales = T*S*C*R

Additive Model: Sales = T+S+C+R

WBSEDCL Energy Sales (MU) - April 2004 to Nov 2007

700750

800850900950

100010501100

11501200

Apr-04

Jul-0

4

Oct-04

Jan-0

5

Apr-05

Jul-0

5

Oct-05

Jan-0

6

Apr-06

Jul-0

6

Oct-06

Jan-0

7

Apr-07

Jul-0

7

Oct-07

10

WBSEDCL Energy Sales (MU) - April 2004 to Nov 2007

700750

800850900950

100010501100

11501200

Apr-04

Jul-0

4

Oct-04

Jan-0

5

Apr-05

Jul-0

5

Oct-05

Jan-0

6

Apr-06

Jul-0

6

Oct-06

Jan-0

7

Apr-07

Jul-0

7

Oct-07

11

Seasonal Index

0.85

0.90

0.95

1.00

1.05

1.10

Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar

12

Deseasonalized Data Apr 2004 - Nov 2007

800.00

850.00

900.00

950.00

1000.00

1050.00

1100.00

1150.00

Apr-04

Jul-0

4

Oct-04

Jan-0

5

Apr-05

Jul-0

5

Oct-05

Jan-0

6

Apr-06

Jul-0

6

Oct-06

Jan-0

7

Apr-07

Jul-0

7

Oct-07

13

Cyclical Component (May 2004 - Oct 2007)

0.920

0.940

0.960

0.980

1.000

1.020

1.040

1.060

1.080

May-04

Aug-04

Nov-04

Feb-05

May-05

Aug-05

Nov-05

Feb-06

May-06

Aug-06

Nov-06

Feb-07

May-07

Aug-07

14

Multiplicative Model: Sales = T*S*C*R

APE = Absolute Percentage Error

MAPE= Mean Absolute Percentage Error

15

Additive Model: Sales = T+S+C+R

Sales in current period = a1*time + (b1*Jan+ b2*Feb + … b12*Dec)+ (c1*Sales last period) + Error

16

17

Additive Model: Sales = T+S+C+R

Sales = 2.83*Time + [258.14(If Month is Jan) + 238.02*(If Month is Feb) + 290.90*(If Month is Mar) + 161.15*(If Month is Apr) + 309.87*(If Month is May) + 271.00*(If Month is Jun) + 335.06*(If Month is Jul) + 291.76*(If Month is Aug) + 309.07(If Month is Sep) + 311.58*(If Month is Oct)+269.76*(If Month is Nov) + 319.74*(If Month is Dec)]+ 0.64*(Prev Month Sale)

18

Additive Model: Sales = T+S+C+R

19

Moving Averages & Exponential Smoothing

• Exponential Smoothing• Holt’s Exponential Smoothing• Holt-Winters Exponential Smoothing

20

Moving Averages for Forecasts

21

Exponential Smoothing for Forecasts

700.00

800.00

900.00

1000.00

1100.00

1200.00

1300.00

Apr

-05

Jun-

05

Aug

-05

Oct

-05

Dec

-05

Feb-

06

Apr

-06

Jun-

06

Aug

-06

Oct

-06

Dec

-06

Feb-

07

Apr

-07

Jun-

07

Aug

-07

Oct

-07

Energy Sales (MU)Simple EWSEWS HoltEWS Winters

22

In-sample Prediction Error using MA & EWS Methods

23

Forecasting with Various Averages: Exponential Smoothing

9-month Sales

17

19

21

23

25

27

29

31

33

Jan Feb Mar Apr May June Jul Aug Sep

Month

Sale

s

24

Forecasting with Various Averages: Exponential Smoothing

0.8

Month Sales

All Prev. Period average

Last Period

Moving Average (3 month)

Exponential Moving Average (w= )

Jan 21Feb 23 21.00 21 21.00Mar 21 22.00 23 22.60Apr 20 21.67 21 21.67 21.32May 21 21.25 20 21.33 20.26June 19 21.20 21 20.67 20.85Jul 28 20.83 19 20.00 19.37Aug 32 21.86 28 22.67 26.27Sep 26 23.13 32 26.33 30.85Oct ?? 23.44 26 28.67 26.97

25

Exponential Smoothing

• A weighted moving average– Weights decline exponentially

– Most recent observation weighted most

• Used for smoothing and forecasting (one period into the future)

26

Exponential Smoothing

• Weight (smoothing coefficient) is W– Range from 0 to 1– Smaller W gives better smoothing

(smoothing out unwanted cyclical and noise components),

– Larger W forecasts better

(continued)

27

Exponential Smoothing: Method

11 YE =

,)1( 1−−+= iii EWWYE for i = 2, 3, 4, …

Ei = weighted average of actual obs Yi and its forecast Ei-1

= forecast for next period (i+1)

Weights: w, w*(1-w), w*(1-w)^2, w*(1-w)^3, w*(1-w)^4, …

Yn, Yn-1, Yn-2, Yn-3, Yn-4, …

28

EWS or EMA Weights decline fast: w, w*(1-w), w*(1-w)^2, w*(1-w)^3, w*(1-w)^4, …

0.1 0.2 0.5 0.8 0.9

Weight = W

Weight = W

Weight = W

Weight = W

Weight = W

0.100 0.200 0.500 0.800 0.9000.090 0.160 0.250 0.160 0.0900.081 0.128 0.125 0.032 0.0090.073 0.102 0.063 0.006 0.0010.066 0.082 0.031 0.001 0.0000.059 0.066 0.016 0.000 0.0000.053 0.052 0.008 0.000 0.0000.048 0.042 0.004 0.000 0.0000.043 0.034 0.002 0.000 0.0000.039 0.027 0.001 0.000 0.0000.035 0.021 0.000 0.000 0.000

… … … … …

Weight W = 0.5

0.000

0.100

0.200

0.300

0.400

0.500

0.600

1 2 3 4 5 6 7 8 9 10 11

Observation No.

Wei

ght

29

30

Sales vs. Smoothed Sales

• Fluctuations have been smoothed

• NOTE: the smoothed value in this case is generally a little low, since the trend is upward sloping and the weighting factor is only .2

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 10Time Period

Sal

es

Sales Smoothed

31

Exponential Smoothing for Trent Data

32

Exponential Smoothing: Holt’s Method

Initial Values: L1 = Y1, T1 = 0

******************Preliminary forecast of Y for next period (t+1): Lt = a*Yt + (1-a)*(Lt-1+Tt-1) for t = 2, 3, 4, …

Correction Factor of “slope”:Tt = b*(Lt - Lt-1) + (1-b)*Tt-1 for t = 2, 3, 4, …

Modified forecast of Y for next period (t+1): Ft = (Lt + Tt)

33

Exponential Smoothing: Holt Winters Method

Initial Values:

St = Yt/Average(Y1:Ys), t=1,2,…,s,

Ls = Ys/Ss,

Ts = [Average(Ys+1:Y2s)– Average(Y1:Ys) ] /s

34

Exponential Smoothing: Holt Winters Method

1. Preliminary forecast of deseasonalized Y for (t+1)Lt = a*(Yt /St-s) + (1-a)*(Lt-1+Tt-1) for t = s+1, …

2. Correction Factor of “slope” to add to preliminary forecast of deseasonalized Y for (t+1) : Tt = b*(Lt - Lt-1) + (1-b)*Tt-1 for t = s+1, …

3. Modified Forecast of deseasonalized Y for (t+1): (Lt + Tt)

4. Correction Factor of “seasonality” (will be used s periods later) : St = c*(Yt /Lt) + (1-c)*St-s, t=s+1, …

5. Final forecast of seasonal Y for (t+1):Ft = (Lt + Tt)*St+1-s

Calculation

35

36

Forecast by Exponential Smoothing

37

Comparing Forecasts by Various Methods

Exponential Moving Average (Special Type of EWS)

38

Exponential Moving Average (special type of EWS)

39

for 20-Period EMA, 0.0952(approx) of current period value is considered and for 50-Period EMA, 0.0392(approx) of the current value is considered.

Formula:EMA(current) = Price(current)x Multiplier +(1-Multiplier) x EMA(previous)

Exact Weight or Multiplier= 2/(n+1)

Stock Market Data

40

Stock Market Data

41

Stock Market Data

42

Stock Market Data

43