Non Linear Modelling Internet

8/3/2019 Non Linear Modelling Internet

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Non Linear Modelling

An example


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BackgroundAce Snackfoods, Inc. has developed a new snack productcalled Krunchy Bits. Before deciding whether or not to gonational with the new product, the marketing manager forKrunchy Bits has decided to commission a year-long testmarket using IRIs BehaviorScan service, with a view togetting a clearer picture of the products potential.

The product has now been under test for 24 weeks. Onhand is a dataset documenting the number of householdsthat have made a trial purchase by the end of each week.(The total size of the panel is 1499 households.)

The marketing manager for Krunchy Bits would like aforecast of the products year-end performance in the testmarket. First, she wants a forecast of the percentage ofhouseholds that will have made a trial purchase by week 52.


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Data


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Approaches to Forecasting Trial

French curve

Curve fittingspecify a flexible functionalform

fit it to the data, and project into the future.

Inspect the data (see Non Linear

Modelling .xls)


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Proposed Model for thisexample

Y = p0(1ebx)

Decreasing returns and saturation.

Here: p0 = saturation proportion

b = decreasing returns parameter

Widel used in marketin .


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Data

Cumulative Trial vs Week

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

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0 5 10 15 20 25

Week

CumulativeTrial

Week # HHs Propn. of Households

1 8 0.005

2 14 0.009

3 16 0.011

4 32 0.021

5 40 0.027

6 47 0.031

7 50 0.033

8 52 0.0359 57 0.038

10 60 0.040

11 65 0.043

12 67 0.045

13 68 0.045

14 72 0.048

15 75 0.050

16 81 0.054

17 90 0.060

18 94 0.06319 96 0.064

20 96 0.064

21 96 0.064

22 97 0.065

23 97 0.065

24 101 0.067


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Modelled data

Week # HHs Propn. of Households Modelled Proportion diff p0 beta

1 8 0.005 0.005 9.08E-10 0.0862 0.064285

2 14 0.009 0.010 1.12E-06 LS 0.000128

3 16 0.011 0.015 1.98E-05

4 32 0.021 0.020 3.25E-06

5 40 0.027 0.024 8.94E-06

6 47 0.031 0.028 1.42E-05

7 50 0.033 0.031 4.49E-06

8 52 0.035 0.035 9.82E-109 57 0.038 0.038 2.49E-08

10 60 0.040 0.041 7.23E-07

11 65 0.043 0.044 1.13E-07

12 67 0.045 0.046 2.72E-06

13 68 0.045 0.049 1.2E-05

14 72 0.048 0.051 9.74E-06

15 75 0.050 0.053 1.09E-05

16 81 0.054 0.055 1.81E-06

17 90 0.060 0.057 7.5E-06

18 94 0.063 0.059 1.3E-0519 96 0.064 0.061 1.06E-05

20 96 0.064 0.062 2.8E-06

21 96 0.064 0.064 3.58E-08

22 97 0.065 0.065 2.86E-07

23 97 0.065 0.067 3.38E-06

24 101 0.067 0.068 1.56E-07


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How well does the model do?


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2.00%

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4.00%

5.00%

6.00%

7.00%

8.00%

0 5 10 15 20 25

Week

CumulativeTrial

Propn. of Households

Modelled Proportion

"R^2" 0.985


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How well does the model doforecasting?


0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

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9.00%

10.00%

0 10 20 30 40 50

Week

CumulativeTrial

Propn. of Households

Modelled Proportion

forecast region-->


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Doing the same thing in R

NLeg.df=read.csv(file.choose(),header=T)

attach(NLeg.df)

fit.nls


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Doing the same thing in R> fit.nls> summary(fit.nls)Formula: propHH ~ p0 * (1 - exp(-beta * Week))Parameters:

Estimate Std. Error t value Pr(>|t|)p0 0.086616 0.004462 19.41 2.49e-15 ***

beta 0.063699 0.005721 11.13 1.65e-10 ***---Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 .' 0.1 ` ' 1

Residual standard error: 0.002395 on 22 degrees of freedom

Correlation of Parameter Estimates:p0

beta -0.9798


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Different types of Models

&

Their Interpretations


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A Simple Model

Y (Sales Level)

} b (slope of thesalesline)

}

1

X (Advertising)

a

(sales level when

advertising = 0)


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Phenomena

P1: Through Origin

P4: SaturationP3: Decreasing Returns

(concave)

P2: Linear

Y

X

Y

X

Y

X

Q

Y

X


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Phenomena

P5: Increasing Returns

(convex)

P8: Super-saturationP7: Threshold

P6: S-shape

Y

X

Y

X

Y

X

Y

X


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Aggregate Response Models:Linear Model

Y = a + bX

Linear/through origin

Saturation and threshold (inranges)


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Aggregate Response Models:Fractional Root Model

Y = a + bXc

ccan be interpreted as elasticitywhen a= 0.

Linear, increasing or decreasingreturns (depends on c).


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Aggregate Response Models:Exponential Model

Y = aebx; x > 0

Increasing or

decreasing returns(depends on b).


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Aggregate Response Models:Adbudg Function

Y = b + (ab)

S-shaped and concave;saturation effect.

Widely used. Amenable tojudgmental calibration.

Xc

d + Xc


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Aggregate Response Models:Multiple Instruments

Additive model for handlingmultiple marketing instruments

Y = af(X1) + bg(X2)

Easy to estimate using linearregression.


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Aggregate Response Models:Multiple Instruments contd

Multiplicative model for handling multiplemarketing instruments

Y = aXbXc

band care elasticities.

Widely used in marketing.

Can be estimated by linear regression

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Non Linear Modelling Internet