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Smoothing methodsSmoothing methods
Marzena NarodzonekMarzena Narodzonek--KarpowskaKarpowska
Prof. Dr. W.Prof. Dr. W. ToporowskiToporowskiInstitut fr Marketing & HandelInstitut fr Marketing & HandelAbteilung HandelAbteilung Handel
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What Is Forecasting?What Is Forecasting?
Process of predicting a future eventProcess of predicting a future event
Underlying basis of all business decisionsUnderlying basis of all business decisions
Youll never get it right. But, you can always get it lessYoull never get it right. But, you can always get it less
wrong.wrong.
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Realities of ForecastingRealities of Forecasting
Forecasts are seldom perfectForecasts are seldom perfect
Most forecasting methods assume that there is someMost forecasting methods assume that there is some
underlying stability in the system and future will be likeunderlying stability in the system and future will be like
the past (causal factors will be the same).the past (causal factors will be the same).
Accuracy decreases with length of forecastAccuracy decreases with length of forecast
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Forecasting MethodsForecasting Methods
Qualitative MethodsQualitative Methods are subjective in nature since they relyare subjective in nature since they relyon human judgment and opinion.on human judgment and opinion.
Used when situation is vague & little data existUsed when situation is vague & little data existNew productsNew products
New technologyNew technology
Involve intuition, experienceInvolve intuition, experience
Quantitative MethodsQuantitative Methods use mathematical or simulationuse mathematical or simulationmodels based on historical demand or relationshipsmodels based on historical demand or relationships
between variables.between variables. Used when situation is stable & historical data existUsed when situation is stable & historical data exist
Existing productsExisting products
Current technologyCurrent technology
Involve mathematical techniquesInvolve mathematical techniques
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Forecasting MethodsForecasting Methods
ForecastingMethods
ForecastingForecastingMethodsMethods
QuantitativeQuantitativeQuantitative QualitativeQualitativeQualitative
CausalCausalCausal Time SeriesTime SeriesTime Series
SmoothingSmoothingSmoothing TrendProjection
TrendTrendProjectionProjection
Trend ProjectionAdjusted for
Seasonal Influence
Trend ProjectionTrend ProjectionAdjusted forAdjusted for
Seasonal InfluenceSeasonal Influence
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What is a Time Series?What is a Time Series?
Set of evenly spaced numerical dataSet of evenly spaced numerical data
Obtained by observing response variable at regular timeObtained by observing response variable at regular time
periodsperiods
Forecast based only on past valuesForecast based only on past values
Assumes that factors influencing past, present, & futureAssumes that factors influencing past, present, & future
will continuewill continue
ExampleExample
Year:Year: 20022002 20032003 20042004 20052005 20062006
Sales:Sales: 78.778.7 63.563.5 89.789.7 93.293.2 92.192.1
By reviewing historical data over time, we can better understandBy reviewing historical data over time, we can better understand
the pattern of past behavior of a variable and better predict ththe pattern of past behavior of a variable and better predict thee
future behavior.future behavior.
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Time Series ForecastingTime Series Forecasting
Linear
TimeSeries
Trend?Smoothing
Methods
Trend
Models
YesNo
ExponentialSmoothing
Quadratic ExponentialAuto-
Regressive
MovingAverage
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Components of a Time SeriesComponents of a Time Series
The pattern or behavior of the data in a time series hasThe pattern or behavior of the data in a time series hasseveral components.several components.
TrendTrendTrend
CyclicalCyclicalCyclical
SeasonalSeasonalSeasonal
IrregularIrregularIrregular
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Trend ComponentTrend Component
The trend component accounts for the gradual shifting ofThe trend component accounts for the gradual shifting ofthe time series to relatively higher or lower values over athe time series to relatively higher or lower values over a
long period of time.long period of time.
Trend is usually the result of longTrend is usually the result of long--term factors such asterm factors such as
changes in the population, demographics, technology, orchanges in the population, demographics, technology, or
consumer preferencesconsumer preferences
Upwardtren
dSales
Time
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Cyclical ComponentCyclical Component AnyAny regular pattern of sequences of values above and belowregular pattern of sequences of values above and below
the trend line lasting more than one year can be attributed tothe trend line lasting more than one year can be attributed to
the cyclical componentthe cyclical component..
Usually, this component is due to multiyear cyclicalUsually, this component is due to multiyear cyclicalmovements in the economy.movements in the economy.
Sales1 Cycle
Year
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Seasonal ComponentSeasonal Component TheThe seasonal component accounts for regular patterns ofseasonal component accounts for regular patterns of
variability within certain time periods, such as a yearvariability within certain time periods, such as a year..
The variability does not always correspond with theThe variability does not always correspond with the
seasons of the year (i.e. winter, spring, summer, fall).seasons of the year (i.e. winter, spring, summer, fall).
There can be, for example, withinThere can be, for example, within--week or withinweek or within--daydayseasonal behavior.seasonal behavior.
Sales
Time (Quarterly)
Winter
Spring
Summer
Fall
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Irregular ComponentIrregular Component
TheThe irregular component is caused by shortirregular component is caused by short--term,term,
unanticipated and nonunanticipated and non--recurring factors that affect the valuesrecurring factors that affect the values
of the time seriesof the time series..
This component is the residual, or catchThis component is the residual, or catch--all, factor thatall, factor that
accounts for unexpected data values.accounts for unexpected data values.
It is unpredictable.It is unpredictable.
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Components of Time Series DataComponents of Time Series Data
Trend Seasonal
Cyclical Irregular
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Components of Time Series DataComponents of Time Series Data
1 2 3 4 5 6 7 8 9 10 11 12 13
Year
Seasonal
Cyclical
Trend
Irregular
fluctuations
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Smoothing MethodsSmoothing Methods
In cases in which the time series is fairly stable and has noIn cases in which the time series is fairly stable and has no
significant trend, seasonal, or cyclical effects, one can usesignificant trend, seasonal, or cyclical effects, one can use
smoothing methods to average out the irregular componentsmoothing methods to average out the irregular component
of the time series.of the time series.
Common smoothing methods are:Common smoothing methods are:
MovingMovingAveragesAverages
Weighted Moving AveragesWeighted Moving Averages
Exponential SmoothingExponential Smoothing
Centered Moving AverageCentered Moving Average
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Moving Averages MethodMoving Averages Method
TheThe moving averages method consists of computing an averagemoving averages method consists of computing an average
of the most recentof the most recent nn data values for the series and using thisdata values for the series and using this
average for forecasting the value of the time series for the nexaverage for forecasting the value of the time series for the nextt
period.period.
(most recent data values)Moving Average =
n
n
Moving averages are useful if one can assume item to be forecasMoving averages are useful if one can assume item to be forecastt
will stay fairly steady over time.will stay fairly steady over time.
Series of arithmetic meansSeries of arithmetic means -- used onlyused only for smoothing,for smoothing,providesprovides
overall impression of data over timeoverall impression of data over time
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Moving AveragesMoving Averages
Let usLet us forecast salesforecast sales forfor20072007 usingusing aa 33--period movingperiod movingaverage.average.
20022002 4420032003 66
20042004 55
20052005 33
20062006 77
2007 ?2007 ?
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Moving AveragesMoving Averages
Time ResponseYi
MovingTotal(n=3)
MovingAverage
(n=3)
1995 4 NA NA1996 6 NA NA
1997 5 NA NA1998 3 4+6+5=15 15/3=5.0
1999 7 6+5+3=14 14/3=4.7
2000 NA 5+3+7=15 15/3=5.0
2002
0032003
2004
2005
2006
2007
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Moving Averages GraphMoving Averages Graph
95 96 97 98 99 00
Year
Sales
2
4
6
8Actual
Forecast
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MovingMoving AveragesAverages GraphGraph
Time
Demand
Actual
Small nSmall n
Large nLarge n
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Centered Moving AveragesCentered Moving Averages
MethodMethod
The centered moving average method consists ofThe centered moving average method consists ofcomputing an average ofcomputing an average ofnn periods' data and associating itperiods' data and associating it
with the midpoint of the periods. For example, the averagewith the midpoint of the periods. For example, the average
for periods 5, 6, and 7 is associated with period 6. Thisfor periods 5, 6, and 7 is associated with period 6. Thismethodology is useful in the process of computing seasonmethodology is useful in the process of computing season
indexes.indexes.
5 105 10
6 13 10+13+11: 3= 11.336 13 10+13+11: 3= 11.33
7 117 11
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WeightedWeighted Moving AveragesMoving Averages
Used when trend is presentUsed when trend is present
Older data usually less importantOlder data usually less important The more recent observations are typicallyThe more recent observations are typically
given more weight than older observationsgiven more weight than older observations
Weights based on intuitionWeights based on intuition Often lay between 0 & 1, & sum to 1.0Often lay between 0 & 1, & sum to 1.0
WMA =WMA =
(Weight for period n) ((Weight for period n) (Value inValue in period n)period n)
WeightsWeights
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2x 12
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Weighted Moving Average GraphWeighted Moving Average Graph
Time
Demand
Actual
Large weight onLarge weight on
recent datarecent data
Small weight onSmall weight on
recent datarecent data
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Disadvantages of M.A. MethodsDisadvantages of M.A. Methods
Increasing n makesIncreasing n makes
forecast less sensitive toforecast less sensitive to
changeschanges
Do not forecast trendsDo not forecast trendswellwell
Require sufficientRequire sufficienthistorical datahistorical data
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Responsiveness of M.A. MethodsResponsiveness of M.A. Methods
The problem with M.A. Methods :The problem with M.A. Methods :
Forecast lags with increasing demandForecast lags with increasing demand
Forecast leads with decreasing demandForecast leads with decreasing demand
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Actual Demand, Moving Average,
Weighted Moving Average
0
5
10
15
20
2530
35
Jan
Feb
Mar Ap
rM
ay Jun
Jul
Aug
Sep
Oct
Nov
Dec
Month
Sale
sDemand
All forecasting methods lag ahead of or behind actual demandAll forecasting methods lag ahead of or behind actual demand
Actual sales
Weighted moving average
Moving average
M A E i lM A E ti l
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M.A. versus ExponentialM.A. versus Exponential
SmoothingSmoothing Moving averages and weighted moving averages areMoving averages and weighted moving averages are
effective in smoothing out sudden fluctuations in demandeffective in smoothing out sudden fluctuations in demand
pattern in order to provide stable estimates.pattern in order to provide stable estimates.
Requires maintaining extensive records of past data.Requires maintaining extensive records of past data.
Exponential smoothing requires little record keeping ofExponential smoothing requires little record keeping of
past data.past data. Form of weighted moving averageForm of weighted moving average
Weights decline exponentiallyWeights decline exponentially
Most recent data weighted mostMost recent data weighted most
Requires smoothing constant (Requires smoothing constant ())
Ranges from 0 to 1Ranges from 0 to 1
Subjectively chosenSubjectively chosen
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Forecasting UsingForecasting Using SmoothingSmoothing MethodsMethods
ExponentialExponential
SmoothingSmoothingMethodsMethods
SingleSingle
ExponentialExponential
SmoothingSmoothing
DoubleDouble
(Holts)(Holts)
ExponentialExponential
SmoothingSmoothing
TripleTriple
(Winters)(Winters)
ExponentialExponential
SmoothingSmoothing
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Exponential Smoothing MethodsExponential Smoothing Methods
Single Exponential SmoothingSingle Exponential Smoothing
Similar to single MASimilar to single MA
Double (Holts) Exponential SmoothingDouble (Holts) Exponential Smoothing
Similar to double MASimilar to double MA
Estimates trendEstimates trend
Triple (Winters) Exponential SmoothingTriple (Winters) Exponential Smoothing
Estimates trend and seasonalityEstimates trend and seasonality
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Exponential Smoothing ModelExponential Smoothing Model Single exponential smoothing modelSingle exponential smoothing model
)Fy(FF ttt1t +=+
tt1t F)1(yF +=+
where:where: FFt+1t+1= forecast value for period t + 1= forecast value for period t + 1
yytt = actual value for period t= actual value for period t
FFtt = forecast value for period t= forecast value for period t
= alpha (smoothing constant)= alpha (smoothing constant)
oror:
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Single Exponential SmoothingSingle Exponential Smoothing AA weighted movingweighted moving averageaverage
WeightsWeights declinedecline exponentially, mostexponentially, most recent observationrecent observation
weightedweighted mostmost The weighting factor isThe weighting factor is
Subjectively chosenSubjectively chosen
Range from 0 to 1Range from 0 to 1 SmallerSmaller gives more smoothing, largergives more smoothing, larger gives lessgives less
smoothingsmoothing
The weight is:The weight is:
Close to 0 for smoothing out unwanted cyclical andClose to 0 for smoothing out unwanted cyclical andirregular componentsirregular components
Close to 1 for forecastingClose to 1 for forecasting
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3000
2500
2000
1500
1000
1 2 3 4 5 6 7 8 9 10 11 12
Actual demand
alpha = 0.1
alpha = 0.5
alpha = 0.9
ResponsivenessResponsiveness to Different Values ofto Different Values of
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Exponential Smoothing ExampleExponential Smoothing Example Suppose we use weightSuppose we use weight == 0.0.22
NA
23
26.426.12
26.296
27.437
31.549
31.840
32.872
33.697etc
Forecast
from prior
period
23
40
2527
32
48
33
37
37
50etc
Sales
(yt)
23
(.2)(40)+(.8)(23)=26.4
(.2)(25)+(.8)(26.4)=26.12(.2)(27)+(.8)(26.12)=26.296
(.2)(32)+(.8)(26.296)=27.437
(.2)(48)+(.8)(27.437)=31.549
(.2)(48)+(.8)(31.549)=31.840
(.2)(33)+(.8)(31.840)=32.872
(.2)(37)+(.8)(32.872)=33.697
(.2)(50)+(.8)(33.697)=36.958etc
Forecast for next period
(Ft+1)
1
2
34
5
6
7
8
9
10etc
Quarter
(t)
tt
1t
F)1(y
F
+=
+
F1 = y1since no
prior
informatio
n exists
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Sales vs. Smoothed SalesSales vs. Smoothed Sales SeasonalSeasonalfluctuations havefluctuations have
beenbeen smoothedsmoothed
TheThe smoothedsmoothed
value in this casevalue in this case
is generallyis generally a littlea littlelow, since thelow, since the
trend is upwardtrend is upward
sloping and thesloping and the
weighting factor isweighting factor is
onlyonly 0.0.22
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10Quarter
Sale
s
Sales Smoothed
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Double Exponential SmoothingDouble Exponential Smoothing
Double exponential smoothing is sometimes calledDouble exponential smoothing is sometimes called
exponential smoothing with trendexponential smoothing with trend
If trend exists, single exponential smoothing may needIf trend exists, single exponential smoothing may need
adjustmentadjustment
There is a need to addThere is a need to add a second smoothing constant toa second smoothing constant to
account for trendaccount for trend
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Double Exponential SmoothingDouble Exponential Smoothing
ModelModel)TC)(1(yC 1t1ttt ++=
1t1ttt T)1()CC(T +=tt1t TCF +=+
yytt = actual value in time t= actual value in time t
= constant= constant--process smoothing constantprocess smoothing constant
= trend= trend--smoothing constantsmoothing constant
CCtt = smoothed constant= smoothed constant--process value for period tprocess value for period t
TTtt = smoothed trend value for period t= smoothed trend value for period t
FFt+1t+1= forecast value for period t + 1= forecast value for period t + 1
t = current time periodt = current time period
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Double Exponential SmoothingDouble Exponential Smoothing
Double exponential smoothing is generally done byDouble exponential smoothing is generally done by
computercomputer
One uses larger smoothing constantsOne uses larger smoothing constants andand when lesswhen less
smoothing is desiredsmoothing is desired
One uses smaller smoothing constantsOne uses smaller smoothing constants andand when morewhen moresmoothing is desiredsmoothing is desired
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Exponential Smoothing MethodExponential Smoothing Method
You want to forecast sales forYou want to forecast sales for20072007 using exponentialusing exponentialsmoothing (smoothing ( = 0.10= 0.10). The 2001 forecast was). The 2001 forecast was 175175..
20022002 18018020032003 168168
20042004 159159
20052005 175175
20062006 190190
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Ft= Ft-1 + (At-1 - Ft-1)
Time ActualForecast, Ft
( =0.10)
20022002 180180 175.00 (Given)175.00 (Given)
20032003 168168 175.00 + .10(180175.00 + .10(180 -- 175.00) = 175.50175.00) = 175.50
20042004 159159 175.50 + .10(168175.50 + .10(168 -- 175.50) = 174.75175.50) = 174.75
20052005 175175 174.75 + .10(159174.75 + .10(159 -- 174.75) = 173.18174.75) = 173.18
20062006 190190 173.18 + .10(175173.18 + .10(175 -- 173.18) = 173.36173.18) = 173.36
20072007 NANA 173.36 + .10(190173.36 + .10(190 -- 173.36) = 175.02173.36) = 175.02
Exponential SmoothingExponential Smoothing MethodMethod
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Exponential Smoothing GraphExponential Smoothing Graph
Sales
140150
160170
180
190
02 03 04 05 06 07
Actual
Forecast
Year
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Exponential Smoothing GraphExponential Smoothing Graph
Time
Demand
Actual
Large Large
Small Small
Comparing Smoothing TechniquesComparing Smoothing Techniques
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Let us determine the smoothing technique that is best forLet us determine the smoothing technique that is best for
forecasting these sales data : A two period movingforecasting these sales data : A two period moving
average, a three period moving average, exponentialaverage, a three period moving average, exponential
smoothing (smoothing (=0.1), or exponential smoothing (=0.1), or exponential smoothing (=0.2)=0.2)
WeekWeek SalesSales WeekWeek SalesSales
1 110 6 1201 110 6 120
2 115 7 1302 115 7 130
3 125 8 1153 125 8 115
4 120 9 1104 120 9 1105 125 10 1305 125 10 130
Comparing Smoothing TechniquesComparing Smoothing Techniques
M f F t AM f F t A
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Mean Squared Error (MSE)Mean Squared Error (MSE)
The average of the squared forecast errors for the historicalThe average of the squared forecast errors for the historical
data is calculated. The forecasting method or parameter(s)data is calculated. The forecasting method or parameter(s)which minimize this mean squared error is then selected.which minimize this mean squared error is then selected.
Mean Absolute Deviation (MAD)Mean Absolute Deviation (MAD)
The mean of the absolute values of all forecast errors isThe mean of the absolute values of all forecast errors iscalculated, and the forecasting method or parameter(s) whichcalculated, and the forecasting method or parameter(s) which
minimize this measure is selected. The mean absoluteminimize this measure is selected. The mean absolute
deviation measure is less sensitive to individual large forecastdeviation measure is less sensitive to individual large forecast
errors than the mean squared error measure.errors than the mean squared error measure.
You may choose either of the above criteria for evaluating theYou may choose either of the above criteria for evaluating the
accuracy of a method (or parameter).accuracy of a method (or parameter).
Measures of Forecast AccuracyMeasures of Forecast Accuracy
2 period moving average2 period moving average
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Sales n=2 Error
Week (t
) Yt Ft (Yt - Ft) (Yt - Ft)
2
1 110
2 115 #NV
3 125 112,5 12,5 156,25
4 120 120 0 05 125 122,5 2,5 6,25
6 120 122,5 -2,5 6,25
7 130 122,5 7,5 56,25
8 115 125 -10 1009 110 122,5 -12,5 156,25
10 130 112,5 17,5 306,25
11 120
MSE 98,4375
2 period moving average2 period moving average
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3 period moving average3 period moving average
Sales n=3 Error
Week (t) Yt Ft (Yt - Ft) (Yt - Ft)2
1 110
2 115 #NV
3 125 #NV
4 120 116,6667 3,333333 11,11111
5 125 120 5 25
6 120 123,3333 -3,33333 11,11111
7 130 121,6667 8,333333 69,44444
8 115 125 -10 100
9 110 121,6667 -11,6667 136,1111
10 130 118,3333 11,66667 136,1111
11 118,3333
MSE 69,84127
E i l hi (E ti l thi ( 0 1)0 1)
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Exponential smoothing (Exponential smoothing (=0.1)=0.1)
Sales =0.1 Error
Week (t) Yt Ft (Yt - Ft) (Yt - Ft)2
1 110 #NV
2 115 110 5 25
3 125 110,5 14,5 210,25
4 120 111,95 8,05 64,8025
5 125 112,755 12,245 149,94
6 120 113,9795 6,0205 36,24642
7 130 114,5816 15,41845 237,7286
8 115 116,1234 -1,1234 1,262016
9 110 116,0111 -6,01106 36,1327910 130 115,4099 14,59005 212,8696
11
MSE 108,248
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Exponential smoothing (Exponential smoothing (
=0.2)=0.2)
Sales =0.2 Error
Week (t) Yt Ft (Yt - Ft) (Yt - Ft)2
1 110 #NV2 115 110 5 25
3 125 111 14 196
4 120 113,8 6,2 38,44
5 125 115,04 9,96 99,20166 120 117,032 2,968 8,809024
7 130 117,6256 12,3744 153,1258
8 115 120,1005 -5,10048 26,0149
9 110 119,0804 -9,08038 82,4533710 130 117,2643 12,73569 162,1979
11
MSE 87,91584
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Comparing SmoothingComparing SmoothingTechniquesTechniques
Since the three period moving average technique (MASince the three period moving average technique (MA33))
provides to lowest MSE value, this is the best smoothingprovides to lowest MSE value, this is the best smoothing
technique to use for forecasting these Sales data in ourtechnique to use for forecasting these Sales data in ourexample.example.
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Thank You for YourThank You for YourAttentionAttention
ReferencesReferences
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wwwwww..lsvlsv.uni.uni--saarlandsaarland.de/Seminar/LMIR_WS0506/LM4IR_.de/Seminar/LMIR_WS0506/LM4IR_slidesslides//
AlejandroAlejandro__FigueraFiguera__SmoothingSmoothing__MethodsMethods__forfor_LM_in_IR._LM_in_IR.pptppt wwwwww..forecastforecast..umkcumkc..eduedu//ftppubftppub//formulaformula__pagespages/ch04lec./ch04lec.pptppt
wwwwww..angelfireangelfire..comcom/empire2/qnt531/workshop4./empire2/qnt531/workshop4.pptppt
wwwwww..swlearningswlearning..comcom//quantquant//aswasw//sbesbe_8e/_8e/powerpointpowerpoint/ch18./ch18.pptppt
wwwwww..coursesitecoursesite.cl.uh..cl.uh.eduedu/BPA//BPA/revererevere/DSCI5431//DSCI5431/gradqntchgradqntch_16&14._16&14.pptppt wwwwww..cscs..wrightwright..eduedu/~/~rhillrhill/EGR_702//EGR_702/LectureLecture%203.%203.pptppt
wwwwww..lehighlehigh..eduedu/~rhs2/ie409//~rhs2/ie409/BasicForecastingMethodsBasicForecastingMethods..pptppt
wwwwww..cobcob..sjsusjsu..eduedu//hibshohibsho_a/ballou08aconcforecast.PPT_a/ballou08aconcforecast.PPT
wwwwww..mgtclassmgtclass..mgtmgt..unmunm..eduedu/MIDS//MIDS/KrayeKraye//MgtMgt%20300/%20300/SlideSlide%20Show%2%20Show%20Week%20%232%20MGT%20300%20Forecasting.0Week%20%232%20MGT%20300%20Forecasting.pptppt
wwwwww..infohostinfohost..nmtnmt..eduedu/~/~toshitoshi/EMGT%20501_/EMGT%20501_filesfiles//LecturesLectures/EMGT50113t/EMGT50113thweek.hweek.pptppt
wwwwww..docpdocp..wrightwright..eduedu//bvibvi//classesclasses/MBA780/MBA%20/MBA780/MBA%20ForecastingForecasting--1.1.pptppt wwwwww..cbtpdccbtpdc..tamutamu--commercecommerce..eduedu/gbus302/gbus302/ba578/Chapter14./gbus302/gbus302/ba578/Chapter14.pptppt
wwwwww..mgtclassmgtclass..mgtmgt..unmunm..eduedu/Weber//Weber/MgtMgt%20720/Unit%201/4.0_%20intro%20720/Unit%201/4.0_%20intro%20to%20forecasting.%20to%20forecasting.pptppt
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