### Probability Model -Results

- Overall, the best method
- Furthermore, allows the analyst to make statements about the consumers in the market

Eugene Brusilovskiy

Ka Lok Lee

* These slides are based on the authors' presentation at the 4th Annual Hawaii International Conference on Statistics, Mathematics, and Related Fields

* 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

* 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

* 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).

* Autoregressive Moving Average Model: ARMA (1,1) - generally recognized to be a good approximation for many observed time series

* 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

* 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

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

Structure of Neural Net Models:

* 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

* Probability models:

- Are representations of individual buying behavior

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

* Specific assumptions of purchase process and latent propensity (Bayesian flavor)

* Explicit consideration of unobserved heterogeneity

* 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 (Î»)

* Î» ~ Gamma (r, a)

* After solving the integral, the cumulative probability function becomes: * Estimation uses Excel Solver

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

- Captures jumps in the training data
- Implies no additional sales (the product is dead), extreme case of forecast

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

- Overall, the best method
- Furthermore, allows the analyst to make statements about the consumers in the market

- Include covariates
- Different training periods
- Perform comparative analysis for other areas of forecasting
- Customer Lifetime Value

Hardie B. G.S., Zeithammer R., and Fader P. (2003), Forecasting New Product Trial in a Controlled Test Market Environment, Journal of Forecasting, 22: 391410

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/report.html