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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
                     Lifetime Value


                                                  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
   Advantages
       Minimal data requirements
       Closed form expressions for likelihood
   Disadvantages
       Transaction: Constant hazard rate
       Death process : Customer defection hazard
        independent of previous number of
        transactions
                                                    10
Assumptions of BG-NBD Model

   Disadvantages contd.
       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
   Advantages
       More flexible models
       Different combinations of the three facets yield
        an array of new models
            CWDW Model – Correlated Discrete Weibull -
             Weibull
   Disadvantage
       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
       People buy reports online
   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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posted:10/18/2011
language:English
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