# 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
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

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:

Xc
Y = b + (a–b)
d + Xc

S-shaped and concave;
saturation effect.

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

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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