09 F&DM TimeSeries Smooth 37

8/7/2019 09 F&DM TimeSeries Smooth 37

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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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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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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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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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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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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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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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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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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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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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t Yt1 450

2 500

3 518

4 455

5 502

6 545

7 557

8 586

9 612


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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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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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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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Forecast

10 = 609.22 + 24.22 = 633.44 MT


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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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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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Forecast

11 = 373.84 + 7.3788 = 381.22 cms

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09 F&DM TimeSeries Smooth 37