Incorporating Regularity into Models of Noncontractual Customer-Firm Relationships

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

Incorporating Regularity into Models ofNoncontractual Customer-Firm Relationships

M. Platzer T. Reutterer

Marketing DepartmentVienna University of Economics

and Business Administration

May, 2009

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

Outline

1 Motivation

2 Regularity

3 Model Development

4 Empirical Application

5 Summary

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

A Simple ExampleNoncontractual SettingsStochastic ModelsNBD Assumptions

A Simple Example: Aunt Betty

Aunt Betty buys cookies for her favorite nephews at the end ofevery month at Mr. Baker’s local store. She adheres to thiscustom as long as Mr. Baker can recall back in time.

But recently Mr. Baker noticed that Aunt Betty has not been tohis shop since 35 days!

Mr. Baker immediately concluded that something terrible musthave happened...

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

A Simple ExampleNoncontractual SettingsStochastic ModelsNBD Assumptions

A Simple Example: Aunt Betty

Aunt Betty buys cookies for her favorite nephews at the end ofevery month at Mr. Baker’s local store. She adheres to thiscustom as long as Mr. Baker can recall back in time.

But recently Mr. Baker noticed that Aunt Betty has not been tohis shop since 35 days!

Mr. Baker immediately concluded that something terrible musthave happened...

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

A Simple ExampleNoncontractual SettingsStochastic ModelsNBD Assumptions

A Simple Example: Aunt Betty

Aunt Betty buys cookies for her favorite nephews at the end ofevery month at Mr. Baker’s local store. She adheres to thiscustom as long as Mr. Baker can recall back in time.

But recently Mr. Baker noticed that Aunt Betty has not been tohis shop since 35 days!

Mr. Baker immediately concluded that something terrible musthave happened...

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

A Simple ExampleNoncontractual SettingsStochastic ModelsNBD Assumptions

A Simple Example: Aunt Betty

Aunt Betty must have changed her buying behavior !!!

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

A Simple ExampleNoncontractual SettingsStochastic ModelsNBD Assumptions

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

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

A Simple ExampleNoncontractual SettingsStochastic ModelsNBD Assumptions

Noncontractual SettingsIn noncontractual customer relationships organizations can notobserve directly whether a customer is still active. Hence, thestatus is a latent variable and other indicators need to be usedto assess activity.

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

A Simple ExampleNoncontractual SettingsStochastic ModelsNBD Assumptions

Stochastic Models for Noncontractual Settings

Pareto/NBDby Schmittlein, Morrison, and Colombo, 1957BG/NBDby Fader, Hardie, and Lee, 2005CBG/NBDby Hoppe and Wagner, 2007

All of these models share Ehrenberg’s well-known andwidely-accepted NBD assumptions.

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

A Simple ExampleNoncontractual SettingsStochastic ModelsNBD Assumptions

NBD Assumptions1 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

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

A Simple ExampleNoncontractual SettingsStochastic ModelsNBD Assumptions

NBD Assumptions

Concerns regarding Exponential Distribution

Mode zero: The most likely time of purchase is immediatelyafter a purchase. No dead period.

Memoryless Property: No regularity within timing patterns.Succeeding interpurchase times are assumed to beuncorrelated.

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

A Simple ExampleNoncontractual SettingsStochastic ModelsNBD Assumptions

NBD Assumptions

Concerns regarding Exponential Distribution

Mode zero: The most likely time of purchase is immediatelyafter a purchase. No dead period.

Memoryless Property: No regularity within timing patterns.Succeeding interpurchase times are assumed to beuncorrelated.

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

A Simple ExampleNoncontractual SettingsStochastic ModelsNBD Assumptions

NBD Assumptions

ImplicationsNBD-based models only consider recency and frequencywhen assessing the activity status of a customer.Thus, these models know nothing about regularity andsubsequently they all (mis)interpret Aunt Betty’s 35-dayinactivity simply as a ‘longer than average’ but still unsuspiciousintertransaction period.

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

A Simple ExampleNoncontractual SettingsStochastic ModelsNBD Assumptions

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 andfrequency.

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

A Simple ExampleNoncontractual SettingsStochastic ModelsNBD Assumptions

Regularity

Thus, regularity is crucial!

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

MeasuresErlang-k

Regularity

But what is regularity, and how can it be measured?

The observed timings can fall anywhere between totallyrandom patterns and ‘clockwork-like’, deterministic patterns.

A regularity measure for a given timing pattern should thereforeindicate the location between these two extremes.

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

MeasuresErlang-k

Regularity

MeasuresVariability Ratio (=variance/mean) of the IPTsShape parameter of a fitted Gamma distribution toindividual IPTsShape parameter of a fitted Gamma distribution to all IPTs

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

MeasuresErlang-k

Erlang-k

A relatively easy-to-handle alternative to the exponentialdistribution for modeling regularity within the IPTs is the familyof Erlang-k distributions.

Erlang-k is equivalent to the Gamma distribution with its shapeparameter being fixed to some specified integer k , whichdetermines the assumed degree of regularity.

The exponential distribution equals the Erlang-1 distribution.

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

MeasuresErlang-k

Erlang-k

Figure: Erlang-k Distributions with Sampled Timing Patterns

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

IdeaReplace the exponential distribution from the stochasticmodels for noncontractual settings with the more generalErlang-k distribution.

The Gamma mixture of Erlang-k distributions will result in theCondensed Negative Binomial Distribution (cf. Chatfield andGoodhardt, 1973).

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

The CBG/CNBD-k Model1 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

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

The CBG/CNBD-k Model1 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

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

The CBG/CNBD-k Model1 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

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

The CBG/CNBD-k Model1 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

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

The CBG/CNBD-k Model1 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

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

The CBG/CNBD-k Model1 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

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

Empirical Application

DMEF Contest: Data21,166 donors53,998 donations4.7 years of observation

DMEF Contest: TaskPredict the donations for theupcoming 2 years on andisaggregated level.

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

Empirical Application

Figure: Worst Estimates of a ‘Classic’ Model

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

Empirical Application

Figure: Observed Regularities

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

Empirical Application

Thus, CBG/CNBD-2 seems to be the better choice!

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

Empirical Application

Results

LogLik MSLE RMSE Corr SUMRegression 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 errorRMSE = root mean squared errorCorr = CorrelationSUM = Error on Aggregated Level

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

Summary

ConclusionIncorporating regularity improves predictability on adisaggregated level in noncontractual settings.

This finding can be possibly generalized to all kind of predictivemodels that condense past transaction records to recency andfrequency.

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

For Further Reading I

M. Platzer.Stochastic Models of Noncontractual ConsumerRelationships.Master Thesis, 2008.

Malthouse, E.The Results from the Lifetime Value and Customer EquityModeling Competition.Journal of Interactive Marketing, 23(3):272-275, 2009.

M. Platzer, T. Reutterer Regularity within Purchase Timings

MotivationRegularity

Model DevelopmentEmpirical Application

Summary

For Further Reading II

C. Chatfield and G.J. Goodhardt.A Consumer Purchasing Model with Erlang Inter-PurchaseTime.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 ofthe Betageometric/NBD Model.Marketing - Journal of Research and Management,2:75-90, 2007.

M. Platzer, T. Reutterer Regularity within Purchase Timings