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- 1. Motivation Regularity Model Development Empirical Application Summary Incorporating Regularity into Models of Noncontractual Customer-Firm Relationships M. Platzer T. Reutterer Marketing Department Vienna University of Economics and Business Administration May, 2009 M. Platzer, T. Reutterer Regularity within Purchase Timings
- 2. Motivation Regularity Model Development Empirical Application Summary Outline 1 Motivation 2 Regularity 3 Model Development 4 Empirical Application 5 Summary M. Platzer, T. Reutterer Regularity within Purchase Timings
- 3. Motivation A Simple Example Regularity Noncontractual Settings Model Development Stochastic Models Empirical Application NBD Assumptions Summary A Simple Example: Aunt Betty Aunt Betty buys cookies for her favorite nephews at the end of every month at Mr. Baker’s local store. She adheres to this custom as long as Mr. Baker can recall back in time. But recently Mr. Baker noticed that Aunt Betty has not been to his shop since 35 days! Mr. Baker immediately concluded that something terrible must have happened... M. Platzer, T. Reutterer Regularity within Purchase Timings
- 4. Motivation A Simple Example Regularity Noncontractual Settings Model Development Stochastic Models Empirical Application NBD Assumptions Summary A Simple Example: Aunt Betty Aunt Betty buys cookies for her favorite nephews at the end of every month at Mr. Baker’s local store. She adheres to this custom as long as Mr. Baker can recall back in time. But recently Mr. Baker noticed that Aunt Betty has not been to his shop since 35 days! Mr. Baker immediately concluded that something terrible must have happened... M. Platzer, T. Reutterer Regularity within Purchase Timings
- 5. Motivation A Simple Example Regularity Noncontractual Settings Model Development Stochastic Models Empirical Application NBD Assumptions Summary A Simple Example: Aunt Betty Aunt Betty buys cookies for her favorite nephews at the end of every month at Mr. Baker’s local store. She adheres to this custom as long as Mr. Baker can recall back in time. But recently Mr. Baker noticed that Aunt Betty has not been to his shop since 35 days! Mr. Baker immediately concluded that something terrible must have happened... M. Platzer, T. Reutterer Regularity within Purchase Timings
- 6. Motivation A Simple Example Regularity Noncontractual Settings Model Development Stochastic Models Empirical Application NBD Assumptions Summary A Simple Example: Aunt Betty Aunt Betty must have changed her buying behavior !!! M. Platzer, T. Reutterer Regularity within Purchase Timings
- 7. Motivation A Simple Example Regularity Noncontractual Settings Model Development Stochastic Models Empirical Application NBD Assumptions Summary A Simple Example: Aunt Betty But if Mr. Baker knows it, why don’t our models know? M. Platzer, T. Reutterer Regularity within Purchase Timings
- 8. Motivation A Simple Example Regularity Noncontractual Settings Model Development Stochastic Models Empirical Application NBD Assumptions Summary Noncontractual Settings In noncontractual customer relationships organizations can not observe directly whether a customer is still active. Hence, the status is a latent variable and other indicators need to be used to assess activity. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 9. Motivation A Simple Example Regularity Noncontractual Settings Model Development Stochastic Models Empirical Application NBD Assumptions Summary Stochastic Models for Noncontractual Settings Pareto/NBD by Schmittlein, Morrison, and Colombo, 1957 BG/NBD by Fader, Hardie, and Lee, 2005 CBG/NBD by Hoppe and Wagner, 2007 All of these models share Ehrenberg’s well-known and widely-accepted NBD assumptions. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 10. Motivation A Simple Example Regularity Noncontractual Settings Model Development Stochastic Models Empirical Application NBD Assumptions Summary NBD Assumptions 1 Interpurchase times for an active customer follow an exponential distribution with rate parameter λ. 2 Heterogeneity in λ follows a Gamma distribution across customers. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 11. Motivation A Simple Example Regularity Noncontractual Settings Model Development Stochastic Models Empirical Application NBD Assumptions Summary NBD Assumptions Concerns regarding Exponential Distribution Mode zero: The most likely time of purchase is immediately after a purchase. No dead period. Memoryless Property: No regularity within timing patterns. Succeeding interpurchase times are assumed to be uncorrelated. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 12. Motivation A Simple Example Regularity Noncontractual Settings Model Development Stochastic Models Empirical Application NBD Assumptions Summary NBD Assumptions Concerns regarding Exponential Distribution Mode zero: The most likely time of purchase is immediately after a purchase. No dead period. Memoryless Property: No regularity within timing patterns. Succeeding interpurchase times are assumed to be uncorrelated. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 13. Motivation A Simple Example Regularity Noncontractual Settings Model Development Stochastic Models Empirical Application NBD Assumptions Summary NBD Assumptions Implications NBD-based models only consider recency and frequency when assessing the activity status of a customer. Thus, these models know nothing about regularity and subsequently they all (mis)interpret Aunt Betty’s 35-day inactivity simply as a ‘longer than average’ but still unsuspicious intertransaction period. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 14. Motivation A Simple Example Regularity Noncontractual Settings Model Development Stochastic Models Empirical Application NBD Assumptions Summary NBD Assumptions Is the customer still active at time T ? × ×× × ×× × - t0 t1 t2 t3 t4 t5 t6 T × × × × × × × - t0 t1 t2 t3 t4 t5 t6 T Figure: Regular vs. random timing pattern with identical recency and frequency. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 15. Motivation A Simple Example Regularity Noncontractual Settings Model Development Stochastic Models Empirical Application NBD Assumptions Summary Regularity Thus, regularity is crucial! M. Platzer, T. Reutterer Regularity within Purchase Timings
- 16. Motivation Regularity Measures Model Development Erlang-k Empirical Application Summary Regularity But what is regularity, and how can it be measured? The observed timings can fall anywhere between totally random patterns and ‘clockwork-like’, deterministic patterns. A regularity measure for a given timing pattern should therefore indicate the location between these two extremes. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 17. Motivation Regularity Measures Model Development Erlang-k Empirical Application Summary Regularity Measures Variability Ratio (=variance/mean) of the IPTs Shape parameter of a ﬁtted Gamma distribution to individual IPTs Shape parameter of a ﬁtted Gamma distribution to all IPTs M. Platzer, T. Reutterer Regularity within Purchase Timings
- 18. Motivation Regularity Measures Model Development Erlang-k Empirical Application Summary Erlang-k A relatively easy-to-handle alternative to the exponential distribution for modeling regularity within the IPTs is the family of Erlang-k distributions. Erlang-k is equivalent to the Gamma distribution with its shape parameter being ﬁxed to some speciﬁed integer k , which determines the assumed degree of regularity. The exponential distribution equals the Erlang-1 distribution. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 19. Motivation Regularity Measures Model Development Erlang-k Empirical Application Summary Erlang-k Figure: Erlang-k Distributions with Sampled Timing Patterns M. Platzer, T. Reutterer Regularity within Purchase Timings
- 20. Motivation Regularity Model Development Empirical Application Summary Idea Replace the exponential distribution from the stochastic models for noncontractual settings with the more general Erlang-k distribution. The Gamma mixture of Erlang-k distributions will result in the Condensed Negative Binomial Distribution (cf. Chatﬁeld and Goodhardt, 1973). M. Platzer, T. Reutterer Regularity within Purchase Timings
- 21. Motivation Regularity Model Development Empirical Application Summary The CBG/CNBD-k Model 1 Interpurchase times for an active customer follow an Erlang-k distribution with rate parameter λ. 2 Heterogeneity in λ follows a Gamma distribution across customers. 3 At time zero and directly after each transaction customers drop out with probability p. 4 Heterogeneity in p follows a Beta distribution across customers. 5 Parameters λ and p are distributed independently of each other. 6 The observation period starts out with a transaction at time zero. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 22. Motivation Regularity Model Development Empirical Application Summary The CBG/CNBD-k Model 1 Interpurchase times for an active customer follow an Erlang-k distribution with rate parameter λ. 2 Heterogeneity in λ follows a Gamma distribution across customers. 3 At time zero and directly after each transaction customers drop out with probability p. 4 Heterogeneity in p follows a Beta distribution across customers. 5 Parameters λ and p are distributed independently of each other. 6 The observation period starts out with a transaction at time zero. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 23. Motivation Regularity Model Development Empirical Application Summary The CBG/CNBD-k Model 1 Interpurchase times for an active customer follow an Erlang-k distribution with rate parameter λ. 2 Heterogeneity in λ follows a Gamma distribution across customers. 3 At time zero and directly after each transaction customers drop out with probability p. 4 Heterogeneity in p follows a Beta distribution across customers. 5 Parameters λ and p are distributed independently of each other. 6 The observation period starts out with a transaction at time zero. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 24. Motivation Regularity Model Development Empirical Application Summary The CBG/CNBD-k Model 1 Interpurchase times for an active customer follow an Erlang-k distribution with rate parameter λ. 2 Heterogeneity in λ follows a Gamma distribution across customers. 3 At time zero and directly after each transaction customers drop out with probability p. 4 Heterogeneity in p follows a Beta distribution across customers. 5 Parameters λ and p are distributed independently of each other. 6 The observation period starts out with a transaction at time zero. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 25. Motivation Regularity Model Development Empirical Application Summary The CBG/CNBD-k Model 1 Interpurchase times for an active customer follow an Erlang-k distribution with rate parameter λ. 2 Heterogeneity in λ follows a Gamma distribution across customers. 3 At time zero and directly after each transaction customers drop out with probability p. 4 Heterogeneity in p follows a Beta distribution across customers. 5 Parameters λ and p are distributed independently of each other. 6 The observation period starts out with a transaction at time zero. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 26. Motivation Regularity Model Development Empirical Application Summary The CBG/CNBD-k Model 1 Interpurchase times for an active customer follow an Erlang-k distribution with rate parameter λ. 2 Heterogeneity in λ follows a Gamma distribution across customers. 3 At time zero and directly after each transaction customers drop out with probability p. 4 Heterogeneity in p follows a Beta distribution across customers. 5 Parameters λ and p are distributed independently of each other. 6 The observation period starts out with a transaction at time zero. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 27. Motivation Regularity Model Development Empirical Application Summary Empirical Application DMEF Contest: Data DMEF Contest: Task 21,166 donors Predict the donations for the 53,998 donations upcoming 2 years on an 4.7 years of observation disaggregated level. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 28. Motivation Regularity Model Development Empirical Application Summary Empirical Application Figure: Worst Estimates of a ‘Classic’ Model M. Platzer, T. Reutterer Regularity within Purchase Timings
- 29. Motivation Regularity Model Development Empirical Application Summary Empirical Application Figure: Observed Regularities M. Platzer, T. Reutterer Regularity within Purchase Timings
- 30. Motivation Regularity Model Development Empirical Application Summary Empirical Application Thus, CBG/CNBD-2 seems to be the better choice! M. Platzer, T. Reutterer Regularity within Purchase Timings
- 31. Motivation Regularity Model Development Empirical Application Summary Empirical Application Results LogLik MSLE RMSE Corr SUM Regression Model - .086 .642 .644 -31% Pareto/NBD -245,674 .098 .653 .628 +22% BG/NBD -245,833 .096 .651 .640 +19% CBG/NBD -245,702 .096 .650 .639 +19% CBG/CNBD-2 -242,738 .083 .632 .660 -11% CBG/CNBD-3 -243,924 .082 .637 .663 -24% MSLE = mean squared logarithmic error RMSE = root mean squared error Corr = Correlation SUM = Error on Aggregated Level M. Platzer, T. Reutterer Regularity within Purchase Timings
- 32. Motivation Regularity Model Development Empirical Application Summary Summary Conclusion Incorporating regularity improves predictability on a disaggregated level in noncontractual settings. This ﬁnding can be possibly generalized to all kind of predictive models that condense past transaction records to recency and frequency. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 33. Motivation Regularity Model Development Empirical Application Summary For Further Reading I M. Platzer. Stochastic Models of Noncontractual Consumer Relationships. Master Thesis, 2008. Malthouse, E. The Results from the Lifetime Value and Customer Equity Modeling Competition. Journal of Interactive Marketing, 23(3):272-275, 2009. M. Platzer, T. Reutterer Regularity within Purchase Timings
- 34. Motivation Regularity Model Development Empirical Application Summary For Further Reading II C. Chatﬁeld and G.J. Goodhardt. A Consumer Purchasing Model with Erlang Inter-Purchase Time. Journal of the American Statistical Association, 68(344):828-835, 12 1973. D. Hoppe and U. Wagner. Customer Base Analysis: The Case for a Central Variant of the Betageometric/NBD Model. Marketing - Journal of Research and Management, 2:75-90, 2007. M. Platzer, T. Reutterer Regularity within Purchase Timings

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