09 f&dm timeseries smooth 37
TRANSCRIPT
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Decision Making&
Forecasting
Dr. Sharad Varde
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Forecasting Methods
Basic Concepts of Forecasting
Qualitative Methods of Forecasting
Quantitative Techniques of Forecasting - Causal Models
- Time Series Models
Selection of Right Forecasting Method
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Quantitative Techniques:
Time Series ModelsI. Trend Projection Models
II. Smoothing TechniquesIII. Decomposition Model
IV. Box-Jenkins Model.
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Quantitative Techniques of
Forecasting
Time Series Models:
Smoothing Techniques
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When to Use
Smoothing Techniques5When the graph of forecast variable Yagainst time T does not clearly exhibit a
single known pattern5When, in fact, it hints at many patterns
5When the plotted points fluctuate toomuch around a known curve (be it apolynomial, exponential or modifiedexponential).
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Illustration
Y
T
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Role of Smoothing
$ As the name connotes, smoothing ironsout sharp edges and softens the data
$ It tries to suppress or eliminate random &erratic fluctuations in the historical data
$ Smoothing thus highlights the hiddenunderlying basic pattern
$ Useful to obtain quick short term forecastsof several individual component factorscomprising an aggregate macro variable.
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Illustration
Y
T
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Illustration
Y
T
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Basic Steps in Smoothing
A. To Compute smoothed values based on
historical data on the forecast variableB. To use the smoothed value computed in
A as a forecast for immediate futureperiod of time.
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Sharad Varde 11
Standard Smoothing Techniques
1. Nave Method
2. Simple Moving Average3. Simple Exponential Smoothing
4. Double Moving Average
5. Double Exponential Smoothing
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1. Nave Method
Principle: Immediate past is best predictorof immediate future (Horizontal Pattern)
Example: Leaving home early on Tuesdaybecause you faced extra traffic on Monday
Nave Model: Forecast of Y at time t+1 isthe actual observed value of Y at time t.
Statistical Model: t+1 = Yt
Simple, but, has obvious drawbacks.
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2. Simple Moving Averages
A step forward from the Nave Method
Not the last observation, but the average
of last few observations is the forecastt+1 = (Yt + Yt-1) / 2 is called the moving
average of period 2
t+1=
(Yt + Yt-
1 + Yt-
2 ) / 3 is called themoving average of period 3
Judgment: How far to go in the past.
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2. Simple Moving Averages
How to decide: How far to go in the past?
Indicator: Mean Square Error (MSE)
Method: Compute moving averages ofdifferent periods, compare with actual data& select that period which shows min MSE
A
dvantage: Computation: Little & ManualLimitation: Good only for horizontal pattern
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Sharad Varde 15
2. Simple Moving AveragesYear Yt t N=301 100
02 116
03 102
04 11405 80
06 95
07 91
08 8709 86
10 85
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2. Simple Moving AveragesYear Yt t N=301 100
02 116
03 102
04 114 10605 80 111
06 95 99
07 91 96
08 87 8909 86 91
10 85 88
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Sharad Varde 17
2. Simple Moving AveragesYear Yt t N=3 t N=5 t N=701 100
02 116
03 102
04 114 10605 80 111
06 95 99
07 91 96
08 87 8909 86 91
10 85 88
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2. Simple Moving AveragesYear Yt t N=3 t N=5 t N=7 E N=3 E N=5 E N=701 100
02 116
03 102
04 114 10605 80 111
06 95 99 102
07 91 96 101
08 87 89 96 10009 86 91 93 98
10 85 88 88 94
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Sharad Varde 19
2. Simple Moving AveragesYear Yt t N=3 t N=5 t N=7 E N=3 E N=5 E N=701 100
02 116
03 102
04 114 106 805 80 111 -31
06 95 99 102 -4 -7
07 91 96 101 -5 -10
08 87 89 96 100 -2 -9 -1309 86 91 93 98 -5 -7 -12
10 85 88 88 94 -3 -3 -9
TSE
MSE
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2. Simple Moving Averages
Total Square Error:
1104 (for N=3), 300(for N=5), 394(for N=7)
Mean Square Error:158(for N=3), 60(for N=5), 132(for N=7)
Select moving av. of period 5 for forecast
Y10 + Y09 + Y08 + Y07 + Y06Forecast 11 =--------------------------------- = 88.8
5
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3. Simple Exponential Smoothing
Basis: Most recent data are more informative,more valuable & so more useful than older data
So, recent data deserve more weightage
Exponential smoothing = weighted moving avg.
Select a smoothing constant (0 < < 1)
t+1 = Yt + (1- ) t Assumption: 1 = Y1
t+1 = Yt + (1- )Yt-1 + (1- )2Yt-2 + . . . .. . . . . . . . . . . . . . . . . . . + (1- )t-1Y1
Note , (1- ), (1- )2, ... in decreasing order.
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3. Simple Exponential SmoothingYear Yt t =.3 t =.5 t =.9 E =.3 E =.5 E =.901 300
02 235
03 285
04 29705 420
06 275
07 255
08 24009 320
10 380
11 340
MSE
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3. Simple Exponential SmoothingYear Yt t =.3 t =.5 t =.9 E =.3 E =.5 E =.901 300
02 235 300 300 300
03 285 281 268 242
04 297 282 276 28105 420 286 287 295
06 275 326 353 408
07 255 311 314 288
08 240 294 285 25809 320 278 262 242
10 380 291 291 312
11 340 317 336 373
MSE
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3. Simple Exponential SmoothingYear Yt t =.3 t =.5 t =.9 E =.3 E =.5 E =.901 300
02 235 300 300 300 -65 -65 -65
03 285 281 268 242 4 17 43
04 297 282 276 281 15 21 1605 420 286 287 295 134 133 125
06 275 326 353 408 -51 -78 133
07 255 311 314 288 -56 -59 -33
08 240 294 285 258 -54 -45 -1809 320 278 262 242 42 58 78
10 380 291 291 312 89 89 68
11 340 317 336 373 23 4 -33
MSE 4127 4553 5285
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3. Simple Exponential Smoothing
Mean Square Error: 4127(for = 0.3)
Fix smoothing constant = 0.3 to forecast
Forecast 12 = Y11 + (1- ) 11
= 0.3 (340) + 0.7 (317) = 324
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4. Double Moving Averages
Useful to handle upward/downward trend pattern
Moving averages of simple moving averages
Let N be the period of moving average
Let St be Simple moving average for time t:
St = (Yt + Yt-1 + Yt-2 + . . . + Yt-N+1 ) / N
Let Dt be Double moving average for time t:
Dt = (St + St-1 + St-2 + . . . + St-N+1 ) / NForecast t+1 = 2St Dt + [2/(N 1)] [St Dt].
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Sharad Varde 28
4. Double Moving Averages
Numerical Example: Forecast Steel Production
Moving average of period 3 selected by MSE
Forecast t+1 = a + b
Where a = 2St Dt and b = [2/(N 1)] [St Dt]
Data: Nine years of Steel Production in tons
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4. Double Moving Averages
t Yt1 450
2 500
3 518
4 455
5 502
6 545
7 557
8 586
9 612
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4. Double Moving Averages
t Yt St Dt1 450
2 500
3 518
4 455
5 502
6 545
7 557
8 586
9 612
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4. Double Moving Averages
t Yt St Dt a b t+11 450
2 500
3 518 489.33
4 455 491.00
5 502 491.67 490.67
6 545 500.67 494.45
7 557 534.67 509.00
8 586 562.67 532.67
9 612 585.00 560.78
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4. Double Moving Averages
t Yt St Dt a b t+11 450
2 500
3 518 489.33
4 455 491.00
5 502 491.67 490.67 492.67 1.00
6 545 500.67 494.45 506.89 6.22 493.67
7 557 534.67 509.00 560.34 25.67 513.11
8 586 562.67 532.67 592.67 30.00 586.01
9 612 585.00 560.78 609.22 24.22 622.67
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4. Double Moving Averages
Forecast
10 = 609.22 + 24.22 = 633.44 MT
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5. Double Exponential Smoothing
Trend pattern and gradually reducing weights
Let St be Simple exponential average for time t:
St = Yt + (1- )St-1
Let Dt be Double exponential average for time t:
Dt = St + (1- )Dt-1
Forecast t+1 = 2St Dt + [ /(1 )] [St Dt]
The best value of is determined using theMean Square Error method.
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5. Double Exponential Smoothing
Numerical Example: Forecast Annual Rainfall
Smoothing constant = 0.3 selected by MSE
Forecast t+1 = a + b
Where a = 2St Dt and b = [ /(1 )] [St Dt]
Ten years data on Annual Rainfall in cms.
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t Yt St Dt a b t+11 320 320 320
2 329 322.7 319.19 326.21 1.5093
3 325 323.39 320.45 326.33 1.2642 327.72
4 304 317.57 319.59 315.55 -0.8686 327.59
5 328 320.70 319.92 321.48 0.3354 314.68
6 325 321.99 320.54 323.44 0.6235 321.82
7 347 329.49 323.23 335.75 2.6918 324.06
8 349 335.34 326.86 343.82 3.6464 338.44
9 366 344.54 332.16 356.92 5.3234 347.47
10 385 356.68 339.52 373.84 7.3788 362.24
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5. Double Exponential Smoothing
Forecast
11 = 373.84 + 7.3788 = 381.22 cms