business forecasting decomposition & exponential smoothing - bhawani nandan prasad - it...
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Time Series Decomposition & Exponential Smoothing
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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
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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).
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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.
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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.
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Time Series Decomposition
• Multiplicative Decomposition: Y=T*S*C*R
• Additive Decompostion: Y=T+S+C+R
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WBSEDCL Energy Sales Data
Apr 2004 – Mar 2008
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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, …)
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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
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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
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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
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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
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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
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Multiplicative Model: Sales = T*S*C*R
APE = Absolute Percentage Error
MAPE= Mean Absolute Percentage Error
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Additive Model: Sales = T+S+C+R
Sales in current period = a1*time + (b1*Jan+ b2*Feb + … b12*Dec)+ (c1*Sales last period) + Error
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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)
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Additive Model: Sales = T+S+C+R
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Moving Averages & Exponential Smoothing
• Exponential Smoothing• Holt’s Exponential Smoothing• Holt-Winters Exponential Smoothing
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Moving Averages for Forecasts
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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
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In-sample Prediction Error using MA & EWS Methods
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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
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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
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Exponential Smoothing
• A weighted moving average– Weights decline exponentially
– Most recent observation weighted most
• Used for smoothing and forecasting (one period into the future)
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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)
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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, …
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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
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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
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Exponential Smoothing for Trent Data
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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)
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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
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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
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Calculation
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Forecast by Exponential Smoothing
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Comparing Forecasts by Various Methods
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Exponential Moving Average (Special Type of EWS)
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Exponential Moving Average (special type of EWS)
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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)
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Stock Market Data
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Stock Market Data
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Stock Market Data
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Stock Market Data
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