Non Linear Modelling by Y4rv8i2

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									Non Linear Modelling

     An example
                  Background
 Ace Snackfoods, Inc. has developed a new snack product
called Krunchy Bits. Before deciding whether or not to ―go
national‖ with the new product, the marketing manager for
Krunchy Bits has decided to commission a year-long test
market using IRI’s BehaviorScan service, with a view to
getting a clearer picture of the product’s potential.

The product has now been under test for 24 weeks. On
hand is a dataset documenting the number of households
that 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 a
forecast of the product’s year-end performance in the test
market. First, she wants a forecast of the percentage of
households that will have made a trial purchase by week 52.
Data
Approaches to Forecasting Trial
• French curve
• ―Curve fitting‖—specify a flexible functional
   form
• fit it to the data, and project into the future.
• Inspect the data (see Non Linear
   Modelling .xls)
       Proposed Model for this
              example
              Y = p0 (1 – e–bx)


Decreasing returns and saturation.

Here: p0 = saturation proportion

     b = decreasing returns parameter

Widely used in marketing.
                                                                                 Data
Week        # HHs         Propn. of Households
        1             8                   0.005                                         Cumulative Trial vs Week
        2            14                   0.009
        3            16                   0.011
                                                                     8.00%
        4            32                   0.021
        5            40                   0.027
        6            47                   0.031                      7.00%
        7            50                   0.033
        8            52                   0.035                      6.00%
        9            57                   0.038
                                                  Cumulative Trial


       10            60                   0.040                      5.00%
       11            65                   0.043
       12            67                   0.045                      4.00%
       13            68                   0.045
       14            72                   0.048
                                                                     3.00%
       15            75                   0.050
       16            81                   0.054
       17            90                   0.060                      2.00%
       18            94                   0.063
       19            96                   0.064                      1.00%
       20            96                   0.064
       21            96                   0.064                      0.00%
       22            97                   0.065                              0     5           10             15   20   25
       23            97                   0.065
                                                                                                     Week
       24           101                   0.067
                     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-10
        9            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-05
       19            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
                   How well does the model do?
                                            "R^2"                 0.985



                                         Cumulative Trial vs Week

                   8.00%

                   7.00%       Propn. of Households
                               Modelled Proportion
                   6.00%
Cumulative Trial




                   5.00%

                   4.00%

                   3.00%

                   2.00%

                   1.00%

                   0.00%
                           0       5                  10              15   20   25
                                                           Week
How well does the model do –
        forecasting?
                                         Cumulative Trial vs Week
                                                          forecast region-->
                   10.00%
                   9.00%        Propn. of Households
                   8.00%        Modelled Proportion

                   7.00%
Cumulative Trial




                   6.00%
                   5.00%
                   4.00%

                   3.00%
                   2.00%
                   1.00%
                   0.00%
                            0     10              20           30              40   50
                                                        Week
     Doing the same thing in R
NLeg.df=read.csv(file.choose(),header=T)
attach(NLeg.df)

fit.nls<- nls( propHH ~ p0*(1-exp(-
   beta*Week )),
                 data = NLeg.df,
                 start = list( p0=.05,beta=.1),
                  trace = TRUE )
          Doing the same thing in R
> fit.nls<- nls( propHH ~ p0*(1-exp(-beta*Week )),
+                  data = NLeg.df,
+                  start = list( p0=.05,beta=.1),
+                   trace = TRUE )
0.00450657 : 0.05 0.10
0.003392708 : 0.08067993 0.04530598
0.0002417058 : 0.07951665 0.06836061
0.0001274101 : 0.08637640 0.06347303
0.0001261767 : 0.08661768 0.06369747
0.0001261767 : 0.08661641 0.06369891
>
> 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
Different types of Models

               &
     Their Interpretations
                   A Simple Model



Y (Sales Level)
                                  }b   (slope of the
                    a



                           }
                                       salesline)
     (sales level when      1
      advertising = 0)




                          X (Advertising)
                      Phenomena
     P1: Through Origin           P2: Linear



Y                            Y




                 X                        X


    P3: Decreasing Returns
                                 P4: Saturation
            (concave)
                             —
                             Q

Y                            Y



                 X                        X
                    Phenomena
    P5: Increasing Returns
                                    P6: S-shape
             (convex)


Y                            Y




              X                              X



    P7: Threshold                P8: Super-saturation



Y                            Y



              X                              X
Aggregate Response Models:
       Linear Model


          Y = a + bX


• Linear/through origin

• Saturation and threshold (in
  ranges)
Aggregate Response Models:
   Fractional Root Model

                         c
           Y = a + bX


c can be interpreted as elasticity
when a = 0.

Linear, increasing or decreasing
returns (depends on c).
Aggregate Response Models:
    Exponential Model


       Y = aebx; x > 0


     Increasing or
     decreasing returns
     (depends on b).
Aggregate Response Models:
     Adbudg Function

                      Xc
     Y = b + (a–b)
                     d + Xc



 S-shaped and concave;
 saturation effect.

 Widely used. Amenable to
 judgmental calibration.
  Aggregate Response Models:
     Multiple Instruments

• Additive model for handling
  multiple marketing instruments

       Y = af (X1) + bg (X2)

  Easy to estimate using linear
  regression.
   Aggregate Response Models:
    Multiple Instruments cont’d
• Multiplicative model for handling multiple
  marketing instruments

                   Y = aXb1 Xc
                             2




  b and c are elasticities.

  Widely used in marketing.

  Can be estimated by linear regression.

								
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