Quantitative Analysis Forecasting

8/14/2019 Quantitative Analysis Forecasting

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Ronald E. TioReporter



Forecasting ModelsForecasting

Techniques

Causal Methods QualitativeMethods

Time SeriesMethods

RegressionAnalysis

MultipleRegression

MovingAverage

ExponentialSmoothing

TrendProjections

DelphiMethods

Jury of ExecutiveOpinion

Sales ForceComposite



Measures of Forecast Accuracy

∑= errors forecast Bias _

Mean Absolute Deviation (MAD) –a technique for determining theaccuracy of a forecasting model by taking the average of the absolutedeviations.

n

errors forecast MAD

∑=_

Bias – a technique for determining the accuracy of a forecasting

model by measuring the average error and its direction.



Mean Absolute Percent Error (MAPE) - a technique for determining the

accuracy of a forecasting model by taking the average of the squarederrors.

1001

xactual

forecast actual

nMAPE ∑ −

=

100 _ 1

xactual

errors forecast

nMAPE ∑=

Mean Square Error (MSE) - a technique for determining the accuracy of

a forecasting model by taking the average of the squared errors.

n

errors forecast MSE

∑=2) _ (

Measures of Forecast Accuracy



Decomposition of a Time Series

•Random Variations – are “blips” in the data caused by chance or unusualsituations; they follow no discernible pattern and does not reflect the typicalbehavior; their inclusion in the data series can distort the overall picture;whenever possible, these should be identified and removed from the data.

•Trend – is the gradual upward or downward movement of the data over

time. Population shifts, changing incomes, and cultural changes oftenaccount for such movement.

•Seasonality – is a pattern of the demand fluctuation above or below thetrend line that occurs every year. Also refers to short-term, fairly regular

variations generally related to factors such as the calendar or time of theday.

•Cycle – are wavelike variations or patterns in the data that occur everyseveral years. They are usually tied into the business cycle, political andeconomic conditions.



Moving Averages

n

periodsn previousindemand MA

∑=_ _ _ _

Month Actual Sales Forecast

January 10

February 12

March 13

April 16 (10 + 12 + 13) / 3 = 11.667

May 19 (12 + 13 + 16) / 3 = 13.667

June 23 (13 + 16 + 19) / 3 = 16

July 26 (16 + 19 + 23) /3 = 19.333

August 30 (19 + 23 + 26) /3 = 22.667September 28 (23 + 26 + 30) /3 = 26.333

October 18 (26 + 30 + 28) /3 = 28

November 16 (30 + 28 + 18) /3 = 25.333

December 14 (28 + 18 + 16) /3 = 20.667

Forecast (18 + 16 + 14) /3 = 16



Weighted Moving Averages( )

∑

∑=weights

n period indemand xn period for weight WMA

_ _ _ () _ _ _

WeightsApplied

Period

3 Last Month

2 2 MonthsAgo1 3 Months

Ago6 Sum of

Weights

Month ActualSales

Forecast

January 10February 12

March 13April 16 [(10x1)+(12x2)+(13x3)]/6 = 12.167May 19 [(12x1)+(13x2)+(16x3)]/6 = 14.333June 23 [(13x1)+(16x2)+(19x3)]/6 = 17July 26 [(16x1)+(19x2)+(23x3)]/6 = 20.5

August 30 [(19x1)+(23x2)+(26x3)]/6 = 23.833September 28 [(23x1)+(26x2)+(30x3)]/6 = 27.5October 18 [(26x1)+(30x2)+(28x3)]/6 = 28.333November 16 [(30x1)+(28x2)+(18x3)]/6 = 23.333December 14 [(28x1)+(18x2)+(16x3)]/6 = 18.667Forecast [(18x1)+(16x2)+(14x3)]/6 = 15.333



Exponential SmoothingNew Forecast = Previous Forecast + α(Previous Actual - Previous Forecast)

Ft = Ft-1 + α(At-1 – Ft-1 )

Where:Ft = New ForecastFt-1 = Previous ForecastAt-1 = Actual of Previous Periodα = smoothing constant (value between 0 and 1)



Trend ProjectionsGeneral Regression Equation: Ŷ=a + bX

where:

Ŷ = computed value of the variable to be predicted(Dependent Variable)

a = Y – axis intercept

X = Independent Variableb = slope of the line



Trend Projections

∑∑ −

−= 22 X n X

XY n XY b

b = slope of the line

X = values of the independent variable Y = values of the dependent variable= average of the values of X’s= average of the values of Y’s

n = number of data points or observations

X Y

X bY a −=



Trend Projections – Time SeriesLet us consider the case of Midwestern Manufacturing Company; that firm’sdemand for electrical generators over the period 1996 – 2002 is shown below:

Year ElectricalGeneratorsSold

1996 74

1997 791998 80

1999 90

2000 105

2001 1422002 122

Year TimePeriod (X)

GenDemand (Y)

X2 XY

1996 1 74 1 74

1997 2 79 4 158

1998 3 80 9 2401999 4 90 16 360

2000 5 105 25 525

2001 6 142 36 852

2002 7 122 49 854ΣX = 28 ΣY = 692 ΣX2 = 140 ΣXY = 3,063



Trend Projections – Time Series4

7

28 === ∑n

X X 86.98

7

692 === ∑n

Y Y

54.1028

295)4)(7(140

)86.98)(4)(7(063,3222

==−

−=−

−=

∑∑

X n X

XY n XY b

70.56)4)(54.10(86.98 =−=−= X bY a

Y = a + bX = 56.70 + 10.54X

(Sales in 2003) = 56.70 + 10.54(8) = 141.02 or 141 Gen

(Sales in 2004) = 56.70 + 10.54(9) = 151.56 or 152 Gen



Trend Projections – Time Series

Generator Deman d and C omp uted Trend

0

2 0

4 0

6 0

8 0

100

120

140

160

1995 1996 1997 199 8 1999 2000 2001 2002 200 3 2004

G e n e r a

t o r D e m a n d

Act

F or



Trend Projections – Seasonal Variations

Month SalesYear 1

SalesYear 2

Ave Sales Ave MonthlyDemand

SeasonalIndex

Next Year Forecast

Jan 80 100 90 94 0.957 1,200 / 12 x 0.957 = 96Feb 85 75 80 94 0.851 1,200 / 12 x 0.851 = 85Mar 80 90 85 94 0.904 1,200 / 12 x 0.904 = 90Apr 110 90 100 94 1.064 1,200 / 12 x 1.064 = 106May 115 131 123 94 1.309 1,200 / 12 x 1.309 = 131June 120 110 115 94 1.223 1,200 / 12 x 1.223 = 122July 100 110 105 94 1.117 1,200 / 12 x 1.117 = 112

Aug 110 90 100 94 1.064 1,200 / 12 x 1.064 = 106Sep 85 95 90 94 0.957 1,200 / 12 x 0.957 = 96Oct 75 85 80 94 0.851 1,200 / 12 x 0.851 = 85Nov 85 75 80 94 0.851 1,200 / 12 x 0.851 = 85Dec 80 80 80 94 0.851 1,200 / 12 x 0.851 = 85

1,128

n Demand Ave

Demand Monthly Ave∑=

_ _ _ Demand Monthly Ave

Demand Sales Ave IndexSeasonal

_ _

_ _ _ =



General Regression Equation: Ŷ=a + bX

where:

Ŷ = computed value of the variable to be predicted(Dependent Variable)

a = Y – axis intercept

X = Independent Variableb = slope of the line

Causal Forecasting Method



∑∑ −

−= 22 X n X

XY n XY b

b = slope of the line

X = values of the independent variable Y = values of the dependent variable= average of the values of X’s= average of the values of Y’s

n = number of data points or observations

X Y

X bY a −=




Causal Forecasting Method Y

Triple A’s Sales

($100,000)

XLocal Payroll

(100,000,000)

X2 XY

2.0 1 1 2.0

3.0 3 9 9.0

2.5 4 16 10.0

2.0 2 4 4.0

2.0 1 1 2.0

3.5 7 49 24.5

Σ Y = 15.0 Σ X = 18 Σ X2 = 80 Σ XY = 51.5

36

18 === ∑n

X X

5.26

15 === ∑n

Y Y

∑∑

−

−=

22 X n X

XY n XY

b

)9)(6(80)5.2)(3)(6(5.51

−

−

b = 0.2575.1)3)(25.0(5.2 =−=−= X bY a

Ŷ = 1.75 + 0.25X sales = 1.75 + 0.25 (payroll)

sales = 1.75 + 0.25 (6) = 3.25




Tr ip le A C o ns truc tion C om pa

0

0.5

1

1 .5

2

2 .53

3 .5

4

0 1 2 3 4 5 6 7 8

S a l e

s ( $ 1 0 0

, 0 0 0 )

A



Standard Error of the EstimateStandard Deviation of the Regression (S Y,X ) is used to measure theaccuracy of the regression estimates.

22

)ˆ( 22

, −=

−

−= ∑∑

n

Error

n

Y Y S X Y

2

2

, −

−−= ∑ ∑∑

n

XY bY aY S X Y

306.009375.04

375.0

26

375.0,

===−

= X Y

S

The standard error of estimate is $30,600 in sales.



Measures the strength and direction of relationship between two variables.

It refers to any kind of association or interdependence between two sets of data or variables. Correlation can range from -1.00 to + 1.00. A correlation of +1.00 indicates that changes of one variable are always matched bychanges in the other; a correlation of -1.00 indicates that increases in onevariable are matched by decreases in the other; a correlation close to zeroindicates little linear relationship between the two variables.

∑ ∑ ∑ ∑∑ ∑∑

−−

−=

])(][)([ 2222 Y Y n X X n

Y X XY nr

Correlation of Coefficient



Correlation of Coefficient



Monitoring & Controlling Forecast

Tracking Signal – is a measurement of how well the forecast is predictingactual values. The intent is to detect any bias in errors over time – values canbe positive or negative. A value of zero would be ideal; control limits of ± 4 or ± 5 are often used for a range of acceptable values.

t

t t MAD

Errors Forecast of Sum Running Signal Tracking

_ _ _ _ MADRSFE

_ t

t

==

t MAD

t period indemand forecast t period indemand actual ∑ − ) _ _ _ _ _ _ _ _ (



Monitoring & Controlling ForecastControl Chart – tells you how a forecast is behaving. It has a forecast average

and upper and lower control limits, which represents the amount of variationthat can be expected.



Control Chart – Out of BoundsOne or more points above

the Upper Control Limit

One or more points belowthe Lower Control Limit

A run of 6 points in a row

above the process average

A run of 6 points in a rowbelow the process average



Monitoring & Controlling ForecastQtr Actual Forecast Error

eRSFE |e| Cumulative

eMADt TSt

1 90 100 -10 -10 10 10 (10) / 1 = 10.0 (-10) / 10 = -1.02 95 100 -5 -15 5 15 (15) / 2 = 7.5 (-15) / 7.5 = -2.0

3 115 100 15 0 15 30 (30) / 3 = 10.0 (0) / 10 = 0.0

4 100 110 -10 -10 10 40 (40) / 4 = 10.0 (-10) / 10 = -1.0

5 125 110 15 5 15 55 (55) / 5 = 11.0 (5) / 11 = 0.5

6 140 110 30 35 30 85 (85) / 6 = 14.2 (35) / 14.2 = 2.5

Control Chart

-2.0-1.5-1.0-0.50.00.51.01.52.0

2.53.0

1 2 3 4 5 6 M A D

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Quantitative Analysis Forecasting