1. 1
Time Series Decomposition &
Exponential Smoothing

2. 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. 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. 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. 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.

6. Time Series Decomposition
• Multiplicative Decomposition: Y=T*S*C*R
• Additive Decompostion: Y=T+S+C+R
6

7. 7
WBSEDCL Energy
Sales Data
Apr 2004 – Mar
2008

8. 8
WBSEDCL Energy Sales (MU) - April 2004 to Nov 2007
700
750
800
850
900
950
1000
1050
1100
1150
1200
Apr-04
Jul-04O
ct-04
Jan-05Apr-05
Jul-05O
ct-05
Jan-06Apr-06
Jul-06O
ct-06Jan-07Apr-07
Jul-07O
ct-07
End of Nov 2007: How to Predict Future Sales?? (for Dec 2007, …)

9. 9
Multiplicative Model: Sales = T*S*C*R
Additive Model: Sales = T+S+C+R
WBSEDCL Energy Sales (MU) - April 2004 to Nov 2007
700
750
800
850
900
950
1000
1050
1100
1150
1200
Apr-04
Jul-04O
ct-04
Jan-05Apr-05
Jul-05O
ct-05
Jan-06Apr-06
Jul-06O
ct-06Jan-07Apr-07
Jul-07O
ct-07

10. 10
WBSEDCL Energy Sales (MU) - April 2004 to Nov 2007
700
750
800
850
900
950
1000
1050
1100
1150
1200Apr-04
Jul-04O
ct-04
Jan-05Apr-05
Jul-05O
ct-05
Jan-06Apr-06
Jul-06O
ct-06Jan-07Apr-07
Jul-07O
ct-07

11. 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. 12
Deseasonalized Data Apr 2004 - Nov 2007
800.00
850.00
900.00
950.00
1000.00
1050.00
1100.00
1150.00Apr-04
Jul-04O
ct-04Jan-05Apr-05
Jul-05O
ct-05Jan-06Apr-06
Jul-06O
ct-06Jan-07Apr-07
Jul-07O
ct-07

13. 13
Cyclical Component (May 2004 - Oct 2007)
0.920
0.940
0.960
0.980
1.000
1.020
1.040
1.060
1.080M
ay-04Aug-04Nov-04Feb-05M
ay-05Aug-05Nov-05Feb-06M
ay-06Aug-06Nov-06Feb-07M
ay-07Aug-07

14. 14
Multiplicative Model: Sales = T*S*C*R
APE = Absolute Percentage Error
MAPE= Mean Absolute Percentage Error

15. 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. 16

17. 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. 18
Additive Model: Sales = T+S+C+R

19. 19
Moving Averages &
Exponential Smoothing
• Exponential Smoothing
• Holt’s Exponential Smoothing
• Holt-Winters Exponential Smoothing

20. 20
Moving Averages for Forecasts

21. 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 EWS
EWS Holt
EWS Winters

22. 22
In-sample Prediction Error
using MA & EWS Methods

23. 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
Sales

24. 24
Forecasting with Various Averages:
Exponential Smoothing
0.8
Month Sales
All Prev.
Period
average
Last
Period
Moving
Average
(3
month)
Exponen
tial
Moving
Average
(w= )
Jan 21
Feb 23 21.00 21 21.00
Mar 21 22.00 23 22.60
Apr 20 21.67 21 21.67 21.32
May 21 21.25 20 21.33 20.26
June 19 21.20 21 20.67 20.85
Jul 28 20.83 19 20.00 19.37
Aug 32 21.86 28 22.67 26.27
Sep 26 23.13 32 26.33 30.85
Oct ?? 23.44 26 28.67 26.97

25. 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. 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. 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. 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 0.900
0.090 0.160 0.250 0.160 0.090
0.081 0.128 0.125 0.032 0.009
0.073 0.102 0.063 0.006 0.001
0.066 0.082 0.031 0.001 0.000
0.059 0.066 0.016 0.000 0.000
0.053 0.052 0.008 0.000 0.000
0.048 0.042 0.004 0.000 0.000
0.043 0.034 0.002 0.000 0.000
0.039 0.027 0.001 0.000 0.000
0.035 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.
Weight

29. 29

30. 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 10
Time Period
Sales
Sales Smoothed

31. 31
Exponential Smoothing for Trent Data

32. 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. 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. 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

35. Calculation
35

36. 36
Forecast by Exponential Smoothing

37. 37
Comparing Forecasts by Various Methods

38. Exponential Moving Average
(Special Type of EWS)
38

39. 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)

40. Stock Market Data
40

41. Stock Market Data
41

42. Stock Market Data
42

43. Stock Market Data
43