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```					Generalized Customer Base Models for
Non-Contractual Settings

Raghu Iyengar - The Wharton School
Asim Ansari - Columbia University
Peter Fader   - The Wharton School

1
Customer Base Analysis

   Managers are interested in predicting

Future                    Retention
Transactions                  Rates

Customer

2
Examples of Customer Bases

Web transactions           Telephone service
Purchase of a consumer     Cable TV service
package good              Bank accounts
Doctor or dentist visits
Mail catalog sales

Church attendance          Renewable service contracts
TV program viewing         Magazine subscriptions
HMO memberships

3
A Framework for Classifying Customer Bases
This is the most
Time At Which Customers Become Inactive
interesting and challenging
case                           Unobserved              Observed

Web transactions           Telephone service
Continuous     Purchase of a consumer     Cable TV service
package good              Bank accounts
Doctor or dentist visits
Opportunities
Mail catalog sales
for Transactions

Renewable service contracts
Discrete      Church attendance
Magazine subscriptions
TV program viewing
HMO memberships

4
Non-Contractual Customer Base Models

   Customer base models use simple stochastic
processes to model
   Transaction process and
   Defection process

5
Non-Contractual Models

   Previous Models
   Pareto-NBD Model (Schmittlein, Morrison and
Colombo 1987)
   BG-NBD Model (Fader, Hardie and Lee 2005)
   We generalize these models to improve
predictions

6
Transaction and Death Process

τ1            τ2                          τxi       τ*xi+1

X             X       ...    X           X
0             1            2              xi-1          xi          Ti

Time interval: 0 - Ti
Is the customer alive or dead?
Total number of observations: xi

7
Pareto-NBD Model

   Transaction Process: Poisson
   Death Process: Exponential
    A person can die anytime

τ1       τ2                          τxi       τ*xi+1

X        X    ...         X           X
0          1        2               xi-1          xi      Ti

Customer can potentially die anytime
in this interval             8
BG-NBD Model

   Transaction Process: Poisson
   Death Process: Geometric
   A person can die only after a purchase – discrete

τ1       τ2                          τxi       τ*xi+1

X        X     ...        X           X
0          1        2               xi-1          xi      Ti

Customer can only die at this discrete point
9
Assumptions of BG-NBD Model
   Minimal data requirements
   Closed form expressions for likelihood
   Transaction: Constant hazard rate
   Death process : Customer defection hazard
independent of previous number of
transactions
10
Assumptions of BG-NBD Model

   Transaction and death process assumed
independent of each other
   Cannot uncover any relationship between
frequency of purchase and death process

11
Our Models

   General Transaction process
   Exponential distribution  More general
distributions
   Two examples
 Weibull distribution
 Loglogistic distribution

12
Hazard Rate – Weibull Distribution

Increasing Hazard
Hazard

Decreasing Hazard      Exponential

13
Our Models

   General Death process
   Discrete death process  More general
distribution
   Discrete-Weibull distribution
   Nagakawa and Osaki (1975), Fader and Hardie (2006)
   Allows for hazard of death to change with the number
of transactions

14
Discrete Weibull Distribution

   Two parameters – pi and θ
   Hazard function after k’th purchase:

θ
k θ (k 1)
h(k)  1  (1  p i )

   Note that if θ =1 then it reduces to a standard
geometric distribution

15
Hazard Function for Discrete Weibull
Hazard

p = 0.5

Number of Observations
16
Individual-Level Likelihood Expression

τ1                 τ2                                     τxi         τ*xi+1

X                  X           ...           X             X
0                1                 2                        xi-1             xi     Ti

L( y i |λ i , α, pi , θ) 

f(τ i1|λ i , α)(1  h(1|pi , θ))f(τ i1|λ i , α)(1  h(2|pi , θ))                          ...

(1  h(x i - 1| p i , θ))f(τ ixi | λ i , α)[h(x i | p i , θ)  (1  h(x i | p i , θ))S(τ * i1 | λ i , α)]
ix

17
Our Models

   Correlation between transaction and death
process.
   Allows us to uncover if there is any relationship
between time between purchases and customer
death

18
Correlation and Heterogeneity

   Let γi  {log(λ i ), log(pi/(1  pi ))}
   We specify heterogeneity across people as
follows:

γi  N( Ziμ, Λ )

19
Our Models
   More flexible models
   Different combinations of the three facets yield
an array of new models
   CWDW Model – Correlated Discrete Weibull -
Weibull
   No closed form expressions for likelihood
20
Bayesian Estimation Procedures

   MCMC methods
   Circumvent the need for any closed form
expressions
   Allow researchers to use flexible distributions
that capture the underlying process better

21
Simulation Studies

   Compare the performance of BG-NBD and
CWDW
   Data generation
   Simulation I
   Use Weibull-Geometric with normal heterogeneity
   Simulation II
   Use BG-NBD model with no correlation across the
two processes
22
Predictive performance

   Aggregate number of purchases in calibration
and holdout dataset
   Correlation between the individual-level
predicted number of purchases and actual
number of purchases

23
Simulation - I

   Data generation - 3 x 3 design = 9 cells
   3 levels of a to account for different hazard
shapes ( 0.5, 1.0, 1.5 )
   3 levels of r , the correlation between death and
transaction parameters ( -0.5, 0.0, 0.5 )
   Data Estimation – using both BG-NBD and
CWDW

24
Simulation 1: Key Results

   When data is exponential, and process are
uncorrelated (i.e., a = 1.0 and r = 0.0 )
   BG-NBD and CWDW predict equally well

25
Simulation 1: Key Results

   Non-Exponential cells
   a= 0.5 : BG-NBD model systematically
underpredicts the aggregate number of purchases in
holdout
   a = 1.5 : BG-NBD model systematically
overpredicts the aggregate number of purchases in
holdout
   No systematic bias with the CWDW model
26
Simulation 1: Key Results
   Correlated cells
   r= 0.5 : BG-NBD model systematically
underpredicts the aggregate number of purchases in
holdout
   r= -0.5 : BG-NBD model systematically
overpredicts the aggregate number of purchases in
holdout
   No systematic bias with the CWDW model
27
Simulation - II

   Data generation – BG-NBD model
   2 cells – Low / High probability of death
after each transaction
   Low – On average, 0.1 chance of death
   High – One average, 0.5 chance of death
   CWDW model does as well even when the
data is generated through a BG-NBD model
28
Simulation: Overall Results

   CWDW Model does better than the BG-
NBD model when the data is generated from
neither of the two models
   CWDW does as well as the BG-NBD model
when the data is from a BG-NBD model

29
Application
   Consumer transactions from a German
website like Consumer Reports
   Randomly sampled 1000 customers
   1 year of data: March 2001 – March 2002

30
Application
   Calibration
   First 6 months of data yielding 1927 transactions
   Average interpurchase time ~ 42 days
   Holdout
   Second 6 months yielding 1263 transactions

31
Estimated Models

   BG-NBD
   UEG – Uncorrelated Exponential-Geometric
   UWG – Uncorrelated Weibull-Geometric
   CEG – Correlated Exponential-Geometric
   CWG – Correlated Weibull-Geometric
   CLG – Correlated Loglogistic - Geometric
   CWDW
32
Application Results

   Calibration – 1927 transactions
   Holdout – 1263 transactions
Calibration    Holdout
Model     Predicted     Predicted
BGNBD       1974           784
UWG         1960           816
UWG         1942           879
CEG         1924          1095
CWG         1927          1116
CLG         1917          1031
CWDW        1932          1157

33
Application Results

   Models without correlated processes
underpredict in the holdout period
   Correlation is important
   CWDW shows a correlation of 0.89
   Allowing for non-exponential interpurchase
times is also important but less so
   a= 0.85 with 95% posterior interval of (0.80,
0.91)
34
Conclusions

   We developed new models with
   flexible transaction distributions
   flexible death time distributions
   correlation between death-transaction process
   More general models are better at predictions
and should be used in the future

35

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 views: 4 posted: 10/18/2011 language: English pages: 35