Presentation of a newly derived stochastic prediction model for customer lifetime values, which is able to incorporate regularity within the transaction patterns.

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