3. FORECASTING_Regression_TimeSeriesDecomposition.ppt

*TREND ANALYSISLinear trend:At = + t Exponential trend:At = tNote that: ln At = ln lnt Gompertz trend:At = ADM 3301 ~ Rim Jaber

ADM 3301 ~ Rim Jaber

*NON LINEAR TREND

AsymptoticTrendS-Curve TrendADM 3301 ~ Rim Jaber


*EXAMPLE 4(Blitz Beer Sales)ADM 3301 ~ Rim Jaber


Period

t

Sales

At

Period

t

Sales

At

1

2

3

4

5

6

7

8

9

10

4,890

4,910

4,970

5,010

5,060

5,100

5,050

5,170

5,180

5,240

11

12

13

14

15

16

17

18

19

20

5,220

5,280

5,330

5,380

5,440

5,460

5,520

5,490

5,550

5,600

*BLITZ BEER SALES

*

CAUSAL (ASSOCIATIVE) MODELS: Linear Regression Model For Linear TrendTechnique for fitting a straight line to a set of points to describe the relationship between two variables:Quantity being forecasted ( y )Variable that influence the quantity being forecasted (x)

Estimated by least squares methodMinimizes sum of squared errorsADM 3301 ~ Rim Jaber^


Least Squares MethodFigure 4.4* 2011 Pearson Education, Inc. publishing as Prentice Hall

2011 Pearson Education, Inc. publishing as Prentice Hall

Least Squares MethodFigure 4.4Least squares method minimizes the sum of the squared errors (deviations)* 2011 Pearson Education, Inc. publishing as Prentice Hall


*CAUSAL METHODS: LINEAR REGRESSION MODEL FOR LINEAR TRENDY = + T + T is the independent (explanatory, stimulus, exogenous, predictor) variable (in this case, time);Y is the dependent (explained, response, endogenous, predicted) variable; is the y-intercept of the line Y = + T (value of Y when T = 0); is the slope of the regression line Y = + T (increment in Y when T increases by 1); is a random error term (deviation)=Y ( + T)= Y-- T



*ESTIMATING THE MODEL PARAMETERSTo find the line that best fits the data, we minimize sum of squared errors 2 = (yi - - ti)2The value b of which minimizes the sum of the errors squared is the slope of the regression lineThe value a of which minimizes the sum of the errors squared is the y-intercept of the regression linea + bT is the forecast at time TADM 3301 ~ Rim Jaber


*b > 0b < 0aaFTime, TLinear Trend Projection ModelADM 3301 ~ Rim Jaber


*How Good Is the RegressionWe need a relative measure of the degree of variation of data about the regression lineRelative measure: compares the variation of Y about the regression line with the variation of Y without the regression line. This measure is called coefficient of determination, R2R2 is a descriptive measure of strength of the regression relationship, a measure of how well the regression line fits the data



*

Total variation: variation in Y or AT from its mean (Y or AT) before using the regression (Y Y or AT AT)Total Variation = Explained Variation + Residual Explained Variation/Regression Deviation: What has been eliminated from the total variation by using the regression It is FT - ATResidual/Deviation: what is left after using the regression. It is = AT FTThe coefficients of determination and of correlation:R2 = Explained Variation / Total VariationCoefficient of correlationThe three sources of variation

*Coefficient of Determination: R2 0

How strong is the linear relationship between the variables?Correlation does not necessarily imply causality!Coefficient of correlation, r, measures degree of associationValues range from -1 to +1Correlation*ADM 3301 ~ Rim Jaber


*COEFFICIENT OF CORRELATION VALUES, r-1.0+1.00Increasing degree of negative correlation-.5+.5Increasing degree of positive correlationADM 3301 ~ Rim Jaber


*EXCEL OUTPUT(Blitz Beer Sales)SUMMARY OUTPUTRegression StatisticsMultiple R0.9937799R Square0.9875985Adjusted R Square0.9869096Standard Error25.426602Observations20ADM 3301 ~ Rim Jaber n|r| R2


*EXCEL OUTPUT(Blitz Beer Sales)ANOVA dfSSMSFRegression1926737.78926737.781433.4423Residual1811637.218646.51211Total19938375CoefficientsStandard Error t StatP-valueIntercept4850.526311.811457 410.662843.331E-37X Variable 137.3308270.9860014 37.8608271.294E-37ADM 3301 ~ Rim Jaberabt Stat >= 2P-value

*MINITAB OUTPUT FOR BLITZ BEER SALESMTB > Regress C2 1 C1; The regression equation isSALES = 4851 + 37.3 PERIOD Predictor Coef Stdev t-ratio pConstant 4850.53 11.81 410.66 0.000PERIOD 37.33 0.99 37.86 0.000 s = 25.43 R-sq = 98.8% R-sq(adj) = 98.7%ADM 3301 ~ Rim Jaber


*MINITAB OUTPUT FOR BLITZ BEER SALESAnalysis of Variance SOURCE DF SS MS F pRegression 1 926738 926738 1433.44 0.000Error 18 11637 647Total 19 938375 Unusual ObservationsObs. PERIOD SALES Fit Stdev.Fit Residual 7 7.0 5050.00 5111.84 6.65 -61.84 R R denotes an obs. with a large st. resid.

*SIGNIFICANCE(Is the model useful?)The regression will only be useful if there is a linear relationship between T and Y (that is, if 0)We must test the hypotheses:H0: =0H1: 0The t-ratio must be larger than t, where is the desired significance level (generally, a t-ratio which is greater than 2 in absolute value is significant)The P-value must be small (smaller than 0.05) ADM 3301 ~ Rim Jaber


*CAVEATSVariation () around the regression line must be random, with mean equal to 0 and standard deviation s (typically normally distributed).

Predictions outside the range of observed values are not very accurate. Unfortunately, in forecasting, you are almost always going out of the observed range of T values.ADM 3301 ~ Rim Jaber


*Pattern of Forecast ErrorADM 3301 ~ Rim Jaber


*BLITZ BEER SALESADM 3301 ~ Rim Jaber


Multiple Regression AnalysisIf more than one independent variable is to be used in the model, linear regression can be extended to multiple regression to accommodate several independent variablesComputationally, this is quite complex and generally done on the computer*ADM 3301 ~ Rim Jaber


*TECHNIQUES FOR SEASONALITYNave MethodTime Series Decomposition ModelsADM 3301 ~ Rim Jaber


*Time Series ComponentsADM 3301 ~ Rim Jaber


Components of DemandAverage demand over 4 yearsTrend componentRandom variationFigure 4.1* 2011 Pearson Education, Inc. publishing as Prentice Hall


*Any observed value in a time series is the product (or sum) of time series componentsMultiplicative modelAt = Tt St Ct RtAdditive modelAt = Tt + St + Ct + Rt ObjectiveIsolate the four components of the model and determine their effect on the time series in order to be able to forecast the future

Time Series Decomposition ModelsADM 3301 ~ Rim Jaber


*QuantityQuantity months months Additive Model Multiplicative model ADM 3301 ~ Rim Jaber


*EXAMPLE 5(Tackey Toys)

Month

Demand

Month

Demand

Month

Demand

Month

Demand

1

2

3

4

5

6

7

8

9

10

11

12

54525

58142

18362

25429

22322

14617

15534

15108

15408

53918

83188

72913

13

14

15

16

17

18

19

20

21

22

23

24

52978

58145

19756

25975

23720

13376

16609

18359

18124

56279

83298

74194

25

26

27

28

29

30

31

32

33

34

35

36

52066

61921

23249

27083

25072

15598

14807

18969

20202

56149

82176

75539

37

38

39

40

41

42

43

44

45

46

47

48

51141

62647

23278

26150

27445

16579

18261

18627

22084

56868

84064

76531

14

*DEMAND

*The Multiplicative ModelAt = Tt St Ct Rt

Estimating seasonal indices, St ,from the history of the series (At):The seasonal indices are used:To include seasonality in the forecasts Or to remove such effects from the observed values (Deseasonalize the data) in order to get a clearer picture of the non seasonal components.



ISOLATING THE FOUR COMPONENTS OF THE MODEL: ESTIMATING StAt = Tt St Ct RtEliminate St Rt: centered moving average = Tt Ct.The number of periods needed in a moving average (MA) is equal to the number of seasons , N, involvedIf the number of period, N, is odd then MA(N) = Centered Moving average (CMA)MA(N) will be centered at the period: t + (N-1)/2 If the number of period, N, is even then use MA(2) on the MA(N) which will correspond to the Centered Moving Average (CMA)monthly data 12-month moving average MA(12) then MA(2) on the MA(12) (MA(2) = CMA)quarterly data 4-quarter moving average MA (4) then MA(2) on the MA(4) (MA(2) = CMA)

*

ISOLATING THE FOUR COMPONENTS OF THE MODEL: ESTIMATING StAt = Tt St Ct RtEliminate St Rt: centered moving average = Tt Ct.Calculate St Rt = (Tt St Ct Rt) / (Tt Ct) = At / (centered moving average).Calculate the seasonal indices St by averaging.Example: to compute the seasonal index for the month of January, we average out all the value of St Rt that corresponding to the month of January. (1.41 + 1.33 + 1.29)/3 = 1.34Adjust the seasonal indices such as the total sum of the indices over one year is equal to the number of period, N.

*SEASONAL INDICES

*ISOLATING THE FOUR COMPONENTS OF THE MODEL: ESTIMATING Tt At = Tt St Ct Rt Eliminate St Rt : centered moving average = Tt Ct Calculate St Rt = (Tt St Ct Rt ) / (Tt Ct ) = At / (centered moving average).Calculate the seasonal indices St by averaging.Deseasonalize the data: deseasonalized demand = At / St Determine a trend curve using regression: y = deseasonalized demand x or t = period.ADM 3301 ~ Rim Jaber


*DESEASONALIZED DEMAND WITH TRENDADM 3301 ~ Rim Jaber


*ISOLATING THE FOUR COMPONENTS OF THE MODEL: ESTIMATING Ct At = Tt St Ct Rt Eliminate St Rt : centered moving average = Tt Ct Calculate St Rt = (Tt St Ct Rt ) / (Tt Ct ) = At / (centered moving average).Calculate the seasonal indices St by averaging.Deseasonalize the data: deseasonalized demand = At / St Determine a trend curve using regression: y = deseasonalized demand and t = period.Calculate Ct Rt = At / (Tt St )

*CYCLE AND RANDOM VARIATIONSADM 3301 ~ Rim Jaber


*USING THE MULTIPLICATIVE MODELFt = Tt St Ct Here, Ct 1, so we can neglect it).

****************The formula for the correlation coefficient.*Slide 48 (Figure 4.10) presents a nice graphical demonstration of the concept of correlation.************************

3. FORECASTING_Regression_TimeSeriesDecomposition.ppt

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