Smoothing Methods

8/2/2019 Smoothing Methods

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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


SmoothingSmoothing

DoubleDouble

(Holts)(Holts)


SmoothingSmoothing

TripleTriple

(Winters)(Winters)


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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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 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



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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


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


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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