# Nets and Probability Models for

AI and Robotics

Oct 19, 2013 (4 years and 8 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
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

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

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),
, 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