CHAPTER 4 SIMPLE SMOOTHING METHODS

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CHAPTER 4 SIMPLE SMOOTHING METHODS
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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
F
(At-1 At-2 At-3 At-4)
/4
t

SMA4(Jun)(Feb Mar Apr May)/4

(124122123125)/4
123.50
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Ft1 (At At-1 At-2 At-3)/4
et ?(At - Ft) (4-1)
n et2 ?(At - Ft)2
(4-2) RSE ? et2 (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 RSEA 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
Averages For 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 EXPONENTIAL
SMOOTHING 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. period?
Smoothing constant, called alphaFt ?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 "
0.300 One period old "
(1- ") 0.210 Two periods old
" (1- " )(1- ") 0.147 Three 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 ALPHAAlpha 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 ACTUALSThe basic exponential smoothing
modelFt "At-1 (1-")Ft-1 (4-9)Thus, the
following equations are also trueFt-1 "At-2
(1-")Ft-2 (4-10)Ft-2 "At-3
(1-")Ft-3 (4-11)Ft-3 "At-4
(1-")Ft-4 (4-12)Ft-4 "At-5
(1-")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- ")4 " At-5 (1- ")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 "At-s
(1-")Ft-s (4-18)where s length of the
seasonal cycleQuarterly Marriages Ft
"At-4 (1-")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)Mod
el ResultsR2 0.9879 RSE 12823.98 MAPE
1.79R-2 1-(12,823.98)2/(116,739.84)2 .9879
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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
-1.8-----------------------------------------Mea
n 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) TSTt SADt
(4-20) MADt SADt ?(At - Ft)
(1 - ?)SADt-1 (4-21)MADt ?/At - Ft/ (1 -
?)MADt-1 (4-22)where TSTt Tracking
signal in t used for alpha
in forecasting period t 1 ? Beta,
a smoothing constant often 0.2
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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 Ft Ft-1 TSTt-1(At-1 - Ft-1)
(4-23) 0 lt TSTt-1 gt 1
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FORECASTING LOW-VALUE
OR ERRATIC SERIESWith high Cv and no seasonal
or trend, patternsdifficult 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.