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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 4th Annual Hawaii International Conference on Statistics, Mathematics, and Related Fields 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 2 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 3 Cumulative Sales (Units Sold) Training Data – Cumulative Sales for the First 39 Weeks 180 160 140 120 100 80 60 40 20 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 Week T = 39 4 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). 5 Time Series • Autoregressive Moving Average Model: ARMA(1,1) – generally recognized to be a good approximation for many observed time series y t y t 1 t t 1 or 1 B yt 1 B t 6 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 7 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 8 Neural Networks (cont’d) Given the rule on the left, we deduce the pattern on the right: X1X2X3 X AND X1X2X3 X X1 X2 X3 X1 X2 X3 X1 X2 X3 X1 X2 X3 X1 X2 X3 X1 X2 X3 X1 X2 X3 X1 X2 X3 X X X X X X X X or X or X or X or X 9 Neural Networks Structure of Neural Net Models: 10 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 11 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 12 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) 1 e 0 t r r 1e r d 13 Probability Modeling • After solving the integral, the cumulative probability function becomes: • F(t) = • LL = 1 t r F t F t 1 Sales t ln F T t 1 T • Estimation uses Excel Solver 14 15 Results • All three models do a relatively good job predicting future sales, but Exponential Gamma is the best Mean Absolute Percentage Error (MAPE) T MAPE t 1 Exp. Gamma 2.7% Neural Nets 9.0% Time Series 5.5% Actual Salest P redicted Salest Actual Salest T Where T is the total number of time periods (weeks). Here, t=1 is the first validation week (week 40) 16 New Product Sales – Results Cumulative Sales (Units Sold) 200 Actual Exp. Gamma Neural Nets Time Series 180 160 140 120 100 80 60 40 20 T=39 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 Week 17 Time Series - Results • Captures “jumps” in the training data • Implies no additional sales (the product is “dead”), extreme case of forecast 180 160 140 120 100 80 60 40 20 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 Forecast Actual 18 Neural Nets - Results • Can sometimes be over-responsive to “jumps” in training data 180 160 140 120 100 80 60 40 20 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 Actual Forecast 19 Probability Model - Results • Overall, the best method • Furthermore, allows the analyst to make statements about the consumers in the market 160 140 120 100 80 60 40 20 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 Actual Forecast 20 Next Steps • Include covariates • Different training periods • Perform comparative analysis for other areas of forecasting – Customer Lifetime Value 21 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: 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/repo rt.html 22