Nets and Probability Models for

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Oct 19, 2013 (3 years and 5 months ago)

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Comparing Time Series, Neural
Nets and Probability Models for
New Product Trial Forecasting



Eugene Brusilovskiy



Ka Lok Lee



These slides are based on the authors’
presentation at the 4
th

Annual Hawaii
International Conference on Statistics,
Mathematics, and Related Fields

2

Problem Introduction


Goal: To predict future sales using sales information from
an introductory period


Product: A new (unnamed) soft beverage that was
introduced to a test market


Data: We have 52 weeks of sales data, which we split into
training (first 39 weeks) and validation (last 13 weeks)
datasets


We build the models using the training dataset and
then examine how well the models predict sales in the
last 13 weeks


The methods employed here apply to predicting the sales
of any newly introduced consumer good


3

Prediction Methods Used


Time Series


Most common technique, available in almost every
statistics software


Neural Nets


Extensive data
-
mining tool (requires expensive
software)


Probability Modeling


Not always available in standard statistical packages,
may be coded in Excel


4

Training Data


Cumulative Sales for the First 39
Weeks

T = 39


5

Time Series


A
time
-
series
(TS)

model

accounts for patterns in the
past movements of a variable and uses that information
to predict its future movements. In a sense a time
-
series
model is just a sophisticated method of extrapolation
(Pindyck and Rubinfeld, 1998).


6

Time Series


Autoregressive Moving Average Model: ARMA(1,1)


generally recognized to be a good approximation for
many observed time series

or


7

Neural Networks


A
Neural Network

(NN)

is an information processing
paradigm inspired by the way the brain processes
information (Stergiou and Siganos, 1996).


MLP (The Multi
-
Layer Perceptron) is used here


8

Neural Networks


A Neural Network consists of neuron
layers

of 3 types:


Input
layer


Hidden

layer


Output
layer


We use two models with different MLP architectures: a
model with one hidden layer and a model with a skip
layer


9

Neural Networks (cont’d)

AND

Given the rule on the left, we deduce the pattern on the right:


10

Neural Networks

Structure of Neural Net Models:


11

Neural Networks


Neural Networks are especially useful for problems
where


Prediction is more important than explanation


There are lots of training data


No mathematical formula that relates inputs to
outputs is known


Source: SAS Enterprise Miner Reference Help.
Neural Network Node: Reference


12

Probability Modeling


Probability m
odels
:


Are representations of
individual
buying
behavior



Provide structural insight into the ways in which
consumers make purchase decisions (Massy el at.,1970)


Specific a
ssumption
s

of purchase process and latent
propensity
(Bayesian flavor)


Explicit consideration of unobserved heterogeneity


13

Probability Model
ing


Individual purchase time or time
-
to
-
trial is modeled by
“Diffusion Model”.


Exponential
-
Gamma
(EG)
, also known as
the
Pareto
distribution

(Hardie et al., 2003)


Time to trial ~
Exponential

(
λ
)


λ
~ Gamma (r,
α
)



14

Probability Model
ing


After solving the integral, the cumulative probability
function becomes:



F(t) =



LL =




Estimation uses Excel Solver



15


16

Results






Exp.
Gamma

Neural
Nets

Time
Series

Mean Absolute
Percentage Error
(MAPE)

2.7%

9.0%

5.5%

Where T is the total number of time periods (weeks). Here, t=1 is the
first validation week (week 40)


All three models do a relatively good job predicting future
sales, but Exponential Gamma is the best


17

New Product Sales


Results

T=39


18

Time Series
-

Results


Captures “jumps” in the training data


Implies no additional sales (the product is “dead”),
extreme case of forecast


19

Neural Nets
-

Results


Can sometimes be over
-
responsive to “jumps” in training
data


20

Probability Model
-

Results


Overall, the best method


Furthermore, allows the analyst to make statements
about the consumers in the market



21

Next Steps


Include covariates


Different training periods


Perform comparative analysis for other areas of
forecasting


Customer Lifetime Value



22

References



Hardie B. G.S., Zeithammer R., and Fader P. (2003),
Forecasting New Product Trial in a Controlled Test
Market Environment,
Journal of Forecasting,

22: 391
-
410


Massy, W.F., Montgomery, D.B. and Morrison, D.G.
(1970),
Stochastic Models of Buying Behavior
, The M.I.T.
Press, 464 pp.


Pindyck, R.S. and Rubinfeld D.L. (1998),
Econometric
Models and Economic Forecasts
, Irwin/McGraw
-
Hill.


SAS Enterprise Miner Reference Help. Article:
Neural
Network Node: Reference


Stergiou, C., & Siganos, D. (1996), Introduction to Neural
Networks. Available online at
www.doc.ic.ac.uk/~nd/surprise_96/journal/vol4/cs11/repo
rt.html