Chapter 4 SIMPLE SMOOTHING METHODS

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Chapter 4SIMPLE SMOOTHING METHODS

• MOVING AVERAGES
• SIMPLE MOVING AVERAGES (SMA)
• WEIGHTED MOVING AVERAGES (WMA)
• SINGLE EXPONENTIAL
• SMOOTHING DERIVATION OF EXPONENTIAL WEIGHTS
• SEASONAL EXPOS - FORECASTING U.S. MARRIAGES
• ADAPTIVE RESPONSE-RATE EXPOS (ARRES)
• FORECASTING LOW-VOLUME AND ERRATIC SERIES
• I know of no way of judging the future but by the
  past. Patrick Henry, Virginia Convention, 1775

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INTRODUCTION

• Simple Smoothing are not General Models
• Simple Smoothing for Simple Series.
• Do Not Include Trends or Seasonality.
• May Work Well c Deseasonalized Data.

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SIMPLE MOVING AVERAGES (SMA)

• SMA4(May) (JanFebMarApr)/4
• (120124122123)/4
• 122.25
• Ft (At-1 At-2 At-3 At-4)
• SMA4(Jun) (Feb Mar Apr May)/4
• (124122123125)/4
• 123.50

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• Ft1 (At At-1 At-2 At-3)/4 S
  (At - Ft) et (4-1)
• n et2 S(At - Ft)2
  (4-2) S et2 RSE
  (4-3)
• n - 1 where
  At Actual demand
  Ft Forecasted demand
  n Number of errors

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Choosing the Best Forecasting Model - Min
RSE

• A probability statementActual Jan
  (t25)(NovDec)/2 /-1.96RSE
  (137138)/2 /-1.962.12 137.5-4.16
  to 137.54.16 133.34 to
  141.66Figure 4-1.Two, Four, and Eight -
  Period Moving Average for Data of Table 4-1.

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Optimal Number of Periods in a Moving Average

• is that number minimizing the RSE.
• When to Use Simple Moving AveragesFor
  patternless series W/O trend or seasonality.
  Patternless-erratic use a longer-period.
• Smooth (highly autocorrelated) a shorter-period
  average.

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WEIGHTED MOVING AVERAGES (WMA)

• WMA(May) .1Jan.2Feb.3Mar.4Apr
  .1120.2124.3122.4123
  122.6 Ft .1At-4 .2At-3 .3At -2
  .4At-1
• Limitations of the SMA and the WMA
• Do not model seasonality or trend.
• Expo. Smo. is more efficient.
• Difficult to determine the optimal No. of
  periods.

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SINGLE EXPONENTIALSMOOTHING

• Ft ?At-1 (1-?)Ft-1 (4-4)whereFt
  Exp. smoothed F. for period tAt-1 Actual
  demand in the prior periodFt-1 Exp. smoothed
  F. of the pri. perioda Smoothing constant,
  called alpha
• Ft ?At-1 (1-?)Ft-1 .301,000
  (1-.3)900 300 630 930 unitsAlpha(?)
  yields weights for each term

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• Ft ?At-1 (1-?)Ft-1 .301,000
  (1-.3)900 300 630 930 At
  980, then Ft1 ?At (1-?)Ft
  .30980 (1-.3)930 945
• A Re-expression of the SES equation is Ft
  Ft-1 ?(At-1 - Ft-1) (4-5)

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The Smoothing Constant

• Actuals Weight
  Most recent " 0.300One period
  old " (1- ") 0.210 Two periods
  old " (1- " )(1- ") 0.147Three
  periods old "(1- ")(1-")(1-") 0.1029
  Alpha 1.0
  -gt zero smoothing Ft "At-1 (1-")Ft-1
  1At-1 (1-1)Ft-1 At-1 (4-6)

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CHOOSING THE BEST ALPHA

• Alpha Based on AutocorrelationsAlpha Based on
  Desired Simple Moving Average ? 2/(n1)
  or n 2/? - 1 (4-8)Consider the use of the
  following alphasFor alpha of .1 n 2/.1 - 1
  19.00For alpha of .3 n 2/.3 - 1
  5.67For alpha of .6 n 2/.6 - 1 2.33For
  alpha of .9 n 2/.9 - 1 1.22Alpha Based
  on Minimum Residual Standard Error

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DERIVATION OF EXPONENTIAL WEIGHTS FOR
PAST ACTUALS

• The basic exponential smoothing model Ft
  aAt-1 (1-a)Ft-1 (4-9)
• Thus, the following equations are also
  true Ft-1 aAt-2 (1-a)Ft-2 (4-10) Ft-2
  aAt-3 (1-a)Ft-3 (4-11) Ft-3 aAt-4
  (1-a)Ft-4 (4-12) Ft-4 aAt-5
  (1-a)Ft-5 (4-13)

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• Through SubstitutionFt "At-1(1-")"At-2(1
  -")Ft-2 (4-14)Ft "At-1(1-")"At-2
  (1-")("At-3
• (1-")Ft-3) (4-15) Ft
  "At-1(1-")1"At-2(1-")2"At-3
• (1-")3Ft-3 (4-16)Ft
  weighted moving average of At-k's
• and one Ft-kF(t) " At-1 (1- ")1
  " At-2 (1- ")2 " At-3 (1-
  ")3At-4 (1- a)4 a At-5
• (1- a)5 " At- 6 (1- ")6 " At-7
  ...
• (1 - ")nFt-n (4 -17)where
  Ft-n Initial forecast in period t-n

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SEASONAL EXPOS EXAMPLE FORECASTING MARRIAGES IN
THE UNITED STATES

• Ft aAt-s (1-a)Ft-s (4-18)where s
  length of the seasonal cycle
• Quarterly Marriages Ft aAt-4 (1-
  a)Ft-4 (4-19) Optimal alpha .435

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• Statistics of the original series At are Mean
  599,909.44 Standard Deviation
  116,739.84 The ACFS of At 1 2 3
  4 -0.12496 -0.71134 -0.10884 0.87531
  5 6 7 8 -0.10276 -0.61304
  -0.10676 0.74338 2SeACF .35

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• Ft .435At-4 (1-.435)Ft-4 (4-20)Model
  ResultsR2 0.9879 RSE 12823.98 MAPE
  1.79R-2 1-(12,823.98)2/(116,739.84)2 .9879
  chapter

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• Table 4-5. Fitted and Residuals of Marriages
  using Eq. 4-10.DATE
  MARRIAGES FITTED RESIDUAL ERROR198501
  420240 NA NA
  NA198502 703900 NA
  NA NA198503 709010
  NA NA NA198504
  579475 NA NA
  NA198601 416040 420240.0
  -4200.0 -1.0198602 701072
  703900.0 -2828.0 -0.4 ?199201
  423000 414657.0 8343.0
  2.0199202 662000 687064.0 -25064.0
  -3.8199203 697000
  709045.5 -12045.5 -1.7199204
  579000 589175.3 -10175.3
  -1.8Mean
  599,909 601,878 -2,432.2 -0.39
  Std.Dev. 116,740 118,211 12,824
  2.13

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• The ets have patternless ACFs 1
  2 3 4 -0.02162 0.28913 0.01340 0.11240
  5 6 7 8 -0.08242 -0.15865
  -0.18035 -0.22533

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• Most EXPSMO models do not yield white noise
  residuals. However, if a model has
• low ACFs
• high R2 and
• low RSE
• EXPOS is Versatile.

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ADAPTIVE RESPONSE-RATE EXPONENTIAL SMOOTHING
(ARRES)

• SADt
• TSTt (4-20)
• MADt
• SADt b(At - Ft) (1 - b)SADt-1 (4-21)
• MADt bAt - Ft/ (1 - b)MADt-1 (4-22)

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• where TSTt Tracking signal in t used for
  alpha in forecasting period t
  1 b Beta, a smoothing constant often
  0.2
• SADt An exponentially weighted average
  deviation (mean forecast error) in
  
• period t MADt An
  exponentially weighted mean
• absolute forecast error in period t
  Denotes absolute values

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• Ft Ft-1 TSTt-1(At-1 - Ft-1) (4-23) 0 ?
  TSTt-1 ? 1

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FORECASTING LOW-VALUE OR ERRATIC SERIES

• With high Cv and no seasonal or trend, patterns
  difficult to forecast. Then (1) forecast the
  series as well as possible (2) group
  the demands to improve forecast accuracy.

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• Patterns in Low-Value SeriesLow volume series
  can possess patterns and are more easily
  forecasted.
• Low-Volume and Erratic DemandsWith these simple
  smoothing methods are the best.

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• Group Patterns in Low-Volume or Erratic
  SeriesIndividual f (20 from pattern 80
  from random) Group f (60 from pattern
  40 from random) Extremely
  Low-Volume ValuesWith extremely low means,
  sometimes a fractional unit per period, Errors
  should be modeled with different distributions.