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 4th Annual Hawaii International

Conference on Statistics, Mathematics, and Related Fields

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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

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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

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Training Data – Cumulative Sales for the First 39 Weeks

T = 39

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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).

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Time Series

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

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tt ByB 11or

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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

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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

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Neural Networks (cont’d)

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X1X2X3 X

AND

X1X2X3 XX1X2X3 X or XX1X2X3 XX1X2X3 X or XX1X2X3 X or XX1X2X3 XX1X2X3 X or XX1X2X3 X

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

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Neural Networks

Structure of Neural Net Models:

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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

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Probability Modeling

• 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

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Probability Modeling

• 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)

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Probability Modeling

• After solving the integral, the cumulative probability function becomes:

• F(t) =

• LL =

• Estimation uses Excel Solver

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Results

Exp. Gamma

Neural Nets

Time Series

Mean Absolute Percentage Error (MAPE)

2.7% 9.0% 5.5%

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SalesActual

SalesredictedPSalesActual

MAPE

T

t t

tt

1

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

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Neural Nets Time Series

New Product Sales – Results

T=39

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Time Series - Results• Captures “jumps” in the training data• Implies no additional sales (the product is “dead”),

extreme case of forecast

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Forecast Actual

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Neural Nets - Results

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

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Actual Forecast

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Probability Model - Results

• Overall, the best method• Furthermore, allows the analyst to make statements

about the consumers in the market

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Actual Forecast

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Next Steps

• Include covariates• Different training periods• Perform comparative analysis for other areas of

forecasting – Customer Lifetime Value

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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/report.html