This document discusses models for predicting customer lifetime value that incorporate regularity in customer purchase patterns. It introduces the Pareto/GGG and (M)BG/CNBD-k models, which extend existing models by allowing for heterogeneity in customer purchase regularity. A simulation study shows the Pareto/GGG model improves predictive accuracy when customers exhibit regular patterns. Empirical analysis of various datasets finds purchase regularity varies both across and within categories, and accounting for regularity leads to different predictions of next purchase timing. The document advocates leveraging regularity to improve predictions of customer lifetime value.

1. Leveraging Regularity in Predicting Customer
Lifetime Value
Michael Platzer & Thomas Reutterer
Seminar Riezlern 2016

2. Warm Up
PAGE 2
Customer A
Customer B
1-Jan-16, 09:00 21-Jun-16, 10:28
1) Which customer would you prefer? The regular one, or the clumpy one?
2) Which type of customers are more prevalent? The regular ones, or the clumpy ones?
Two customers – Same Recency, Same Frequency

3. PAGE 3
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

4. PAGE 4
dead?
non-contractual setting
Customer purchases, until she stops purchasing.
However, dropout event is not observed.
Buy-Till-You-Die
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
alive!

5. Key Issues in the Management
of Customer Relationships
PAGE 5
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
?
?
Given: Purchase history of customer
cohort in non-contractual setting.
Example:
CD Sales
Broadening the context:
• purchase ≈ transaction ≈ event …
• customer relationship ≈ channel activity ≈
service activity …
Questions:
How valuable is that cohort?
How many purchases to expect?
Who will still be active?
Who will be most active?
When will next purchase take place?

6. PAGE 6
BTYD “Gold Standard”
Pareto/NBD
Schmittlein, Morrison and Colombo, 1987
Assumptions
1. Purchase process (while ‘alive’)
• Purchases follow Poisson process, i.e. exponentially-distributed
inter-transaction times, itti,j ~ Exponential(λi)
• λi are Gamma (r, α) distributed across customers
Pareto
NBD
(Ehrenberg 1959)
à parameter estimation of (r, α, s, β) via Maximum Likelihood
à closed-form solutions for key expressions P(alive), # of future purchases
à require only recency/frequency summary statistics (x, tx, T) per customer
2. Dropout (‘death’) process
• (Unobserved) customer’s lifetime is exponentially distributed,
lifetime τi ~ Exponential(μi)
• μi are Gamma (s, β) distributed across customers
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
3. λ and μ vary independently

7. PAGE 7
BTYD Models
• BG/NBD (Fader, Hardie, and Lee 2005)
Discrete time defection process (after any transaction) instead of continuous
• MBG/NBD (Batislam et al. 2007), CBG/NBD (Hoppe and Wagner 2007)
Customers can drop out at time zero (immediately after first purchase)
• PDO/NBD (Jerath et al. 2011)
Defection opportunities tied to calendar time (indep. of transaction timing)
• GG/NBD (Bemmaor and Glady 2012)
Flexible lifetime model, departing from exponential (Gamma-Gompertz)
• Pareto/NBD variant (Abe 2009)
Hierarchical Bayes extension of Pareto/NBD (dependencies of λi and μi)
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
à All modify dropout process, but not purchase process

8. PAGE 8
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

9. Regularity improves
Predictability
PAGE 9
futurepast
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

10. PAGE 10
next event?
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
Regularity improves
Predictability

11. PAGE 11
next event?
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
Regularity improves
Predictability

12. PAGE 12
next event?
Well, so what?
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
Regularity improves
Predictability

13. PAGE 13
still alive?
A
B
Buy-Till-You-Die Setting
Customer A and B exhibit same Recency and Frequency,
yet we come to different assessments regarding P(alive).
Regularity improves
Predictability
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

14. PAGE 14
• Erlang-k Herniter (1971)
• Gamma Wheat & Morrison (1990)
• CNBD Chatfield and Goodhardt (1973)
Schmittlein and Morrison (1983)
Morrison and Schmittlein (1988)
• CNBD Models Gupta (1991)
Wu and Chen (2000)
Schweidel and Fader (2009)
Regularity in Purchase
Timings
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

15. PAGE 15
• RFMC Zhang, Bradlow and Small (2015)
Irregularity in
Purchase Timings
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

16. PAGE 16
Empirical Findings
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
Data Sets
Grocery kwheat = 2.5
Donations kwheat = 2.2
Health Supplements kwheat = 2.1
Office Supply kwheat = 1.8
CD Sales kwheat = 1.0
Fashion & Accessoires kwheat = 0.6
Grocery Categories
Coffee pads kwheat = 3.1
Detergents kwheat = 2.8
Toilet Paper kwheat = 2.8
Cat food kwheat = 2.8
…
Light bulbs kwheat = 1.9
Cosmetics & perfumes kwheat = 1.6
Sparkling Wine kwheat = 1.6

17. PAGE 17
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

18. Pareto/GGG
Platzer and Reutterer, forthcoming
PAGE 18
Customer Level
• Purchase Process: While alive, customer purchases with Gamma
distributed waiting times; i.e. itti,j ~ Gamma(ki, ki λi)
• Dropout Process: Each customer remains alive for an exponentially
distributed lifetime with death rate μi; i.e. lifetime τi ~ Exponential(μi)
Heterogeneity across Customers
• λi ~ Gamma(r, α)
• μi ~ Gamma(s, β)
• ki ~ Gamma(t, γ)
• λi, μi, ki vary independently
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
Pareto/GGG =
Pareto/NBD + Varying Regularity

19. Gamma Distributed
Interpurchase Times
PAGE 19
k=0.3 k=1
Exponential
k=8
Erlang-8
regularrandomclumpy
Coefficient of Variation = 1 / sqrt(k)
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

20. Pareto/GGG
Estimation via MCMC
Component-wise Slice Sampling
within Gibbs with Data Augmentation
SEITE 20
L Significantly Increased Computational Costs
(2mins for drawing 1’000 customers)
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

21. Pareto/GGG
Estimation via MCMC
Component-wise Slice Sampling
within Gibbs with Data Augmentation
SEITE 21
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
L Significantly Increased Computational Costs
(2mins for drawing 1’000 customers)
J but…
• Posterior Distributions instead of Point Estimates
• Also for Individual Level Parameters
• Direct Simulation of Key Metrics of Managerial Interest
• And only one additional summary statistic required

22. Simulation Study
Design
160 scenarios covering a wide range of parameter settings
(similar to simulation design from BG/BB paper)
• N = {1000, 4000}
• r = {0.25, 0.75}, α = {5, 15}
• s = {0.25, 0.75}, β = {5, 15}
• (t, γ) = {(1.6, 0.4), (5, 2.5), (6, 4), (8, 8), (17, 20)}
=> Total of 400’000 simulated customers
=> Total of 64 billion individual-level parameter draws (via slice sampling)
Compare individual-level forecast accuracy of Pareto/GGG vs. Pareto/NBD
in terms of mean absolute error (MAE). Study relative improvement in terms
in MAE.
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

23. Simulation Study
Regularity improves Predictability
Regularity improves Predictability
• bigger lift for bigger regularity
• even for mildly regular patterns
we see lift
• no lift for random and clumpy
customers
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

24. Simulation Study
Lift in Predictive Accuracy
by Segment
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

25. Simulation Study
Interplay of Recency,
Frequency and Regularity
Assumptions: mean(itt) = 6 weeks, mean(lifetime) = 52 weeks
A
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

26. Simulation Study
Interplay of Recency,
Frequency and Regularity
Same RF, but different P(alive)
for different k! Particularly when
customer is already “overdue”.
Regular customers are less
likely and clumpy customers are
more likely to be still alive,
when compared to the
randomly purchasing customer.
Assumptions: mean(itt) = 6 weeks, mean(lifetime) = 52 weeks
A
B
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

27. Simulation Study
Interplay of Recency,
Frequency and Regularity
Same RF, but different P(alive)
for different k! Particularly when
customer is already “overdue”.
Regular customers are less
likely and clumpy customers are
more likely to be still alive,
when compared to the
randomly purchasing customer.
Assumptions: mean(itt) = 6 weeks, mean(lifetime) = 52 weeks
A
B
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

28. Empirical Findings
regular
Poisson
clumpy
à regularity varies across but also within datasets
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

29. à improved predictive accuracy for datasets with regular patterns
median(k) rel. Lift in MAE
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
Empirical Findings

30. à estimates for next transaction timings differ, when
regularity is taking into consideration
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
Empirical Findings

31. PAGE 31
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

32. (M)BG/CNBD-k
Platzer and Reutterer, forthcoming
PAGE 32
Customer Level
• Purchase Process: While alive, customer purchases with Erlang-k
distributed waiting times; i.e. itti,j ~ Erlang-k(λi)
• Dropout Process: A customer drops out at a (re-)purchaseevent with
probability pi
Heterogeneity across Customers
• λi ~ Gamma(r, α)
• pi ~ Beta(a, b)
• λi, pi vary independently
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
BG/CNBD-k =
BG/NBD + Fixed Regularity
MBG/CNBD-k =
MBG/NBD + Fixed Regularity

33. (M)BG/CNBD-k
Platzer and Reutterer, forthcoming
PAGE 33
Closed-Form Expressions
• Likelihood à 100-1000x faster parameter estimation via MLE than MCMC
• P(X(t)=x | r, α, a, b) à approximate Unconditional Expectation
• P(alive | r, α, a, b, x, tx, T) à key component for Conditional Expectation
• Conditional Expected Transactions à “pretty good” approximation possible
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
Erlang-k = Poisson with every kth event counted

34. PAGE 34
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
Simulation Study
Design
324 scenarios covering a wide range of parameter settings – 5 repeats each
(similar to simulation design from BG/NBD paper)
• N = 4000, T.cal = 52, T.star = {4, 16, 52}
• r = {0.25, 0.50, 0.75}, α = {5, 10, 15}
• s = {0.50, 0.75, 1.00}, β = {2.5, 5, 10}
• k = {1, 2, 3, 4}
=> total of 1’300’000 simulated customers
Compare individual-level forecast accuracy of Pareto/GGG vs. Pareto/NBD
in terms of mean absolute error (MAE). Study relative improvement in terms
in MAE.

35. PAGE 35
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
Simulation Study
Example

36. PAGE 36
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
Simulation Study
Results
Regularity improves Predictability
• bigger lift for bigger regularity
• even for mildly regular patterns we see lift

37. PAGE 37
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
Empirical Findings
Results
Findings
1. MBG/NBD either on par
or better than BG/NBD
2. MBG/CNBD-k sees lift in
forecast accuracy, if
regularity present
3. MBG/CNBD-k comes
close to P/GGG

38. PAGE 38
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
Empirical Findings
Results
Yet to come: Study Lift by Retail Category

39. PAGE 39
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package

40. PAGE 40
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
BTYDplus
• https://github.com/mplatzer/BTYDplus
• GPL-3 license
• Implementations of
• MBG/NBD – Batislam et al. (2007)
• GammaGompertz/NBD – Bemmaor & Glady (2012)
• (M)BG/CNBD-k – Platzer and Reutterer (forthcoming)
• Pareto/NBD (MCMC) - Shao-Hui and Liu (2007)
• Pareto/NBD variant (MCMC) – Abe (2009)
• Pareto/GGG (MCMC) – Platzer and Reutterer (forthcoming)
• Fully tested and documented, incl. demos
• Vignette will be coming
…
Users

41. PAGE 41
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
BTYDplus
demo
> elog
cust date
1: 4 1997-01-18
2: 4 1997-08-02
3: 4 1997-12-12
4: 18 1997-01-04
5: 21 1997-01-01
---
6914: 23556 1997-07-26
6915: 23556 1997-09-27
6916: 23556 1998-01-03
6917: 23556 1998-06-07
6918: 23569 1997-03-25
> (cbs <- elog2cbs(elog, per="week",
T.cal=as.Date("1997-09-30"), T.tot=as.Date("1997-09-30")))
cust x t.x litt T.cal T.star x.star
1: 4 1 28.000000 3.3322045 36.42857 39 1
2: 18 0 0.000000 0.0000000 38.42857 39 0
3: 21 1 1.714286 0.5389965 38.85714 39 0
4: 50 0 0.000000 0.0000000 38.85714 39 0
5: 60 0 0.000000 0.0000000 34.42857 39 0
---
2353: 23537 0 0.000000 0.0000000 27.00000 39 2
2354: 23551 5 24.285714 5.5243721 27.00000 39 0
2355: 23554 0 0.000000 0.0000000 27.00000 39 1
2356: 23556 4 26.571429 6.3127713 27.00000 39 2
2357: 23569 0 0.000000 0.0000000 27.00000 39 0
calibration summary stats
x = Frequency
t.x = Recency
litt = Sum Over Logarithmic
Intertransaction Times
holdout summary stats
Transform event-log to summary stats
(optionally one can split data into calibration and holdout)
customer ID

42. PAGE 42
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
BTYDplus
demo MBG/CNBD-k
> params <- mbgcnbd.EstimateParameters(cbs)
> round(params, 2)
k r alpha a b
1.00 0.52 6.17 0.89 1.62
> cbs$xstar_est <- mbgnbd.ConditionalExpectedTransactions(params, cbs$T.star, cbs$x, cbs$t.x,
cbs$T.cal)
> cbs$palive_est <- mbgnbd.PAlive(params, cbs$x, cbs$t.x, cbs$T.cal)
> cbs
cust x t.x litt T.cal T.star x.star palive_est xstar_est
1: 4 1 28.000000 3.3322045 36.42857 39 1 0.6771113 0.7838636
2: 18 0 0.000000 0.0000000 38.42857 39 0 0.3919457 0.1558104
3: 21 1 1.714286 0.5389965 38.85714 39 0 0.1711458 0.1890291
4: 50 0 0.000000 0.0000000 38.85714 39 0 0.3907532 0.1540336
5: 60 0 0.000000 0.0000000 34.42857 39 0 0.4037292 0.1742668
---
2353: 23537 0 0.000000 0.0000000 27.00000 39 2 0.4294331 0.2206554
2354: 23551 5 24.285714 5.5243721 27.00000 39 0 0.8222069 3.9501015
2355: 23554 0 0.000000 0.0000000 27.00000 39 1 0.4294331 0.2206554
2356: 23556 4 26.571429 6.3127713 27.00000 39 2 0.8557381 3.4019351
2357: 23569 0 0.000000 0.0000000 27.00000 39 0 0.4294331 0.2206554
E(X(T+T.star))P(alive)

43. PAGE 43
1. Intro to BTYD models
2. On the Subject of Regularity
3. Our Pareto/GGG model
4. Our (M)BG/CNBD-k model
5. Our BTYDplus R package
BTYDplus
demo Pareto/GGG
> params_draws <- pggg.mcmc.DrawParameters(cbs)
> round(summary(params_draws$level_2)$quantiles[, "50%"], 2)
t gamma r alpha s beta
45.31 43.36 0.55 10.74 0.66 12.51
> est_draws <- mcmc.DrawFutureTransactions(cbs, params_draws, cbs$T.star)
> cbs$palive_est <- sapply(params_draws$level_1, function(draws) mean(as.matrix(draws)[, 'z']))
> cbs$xstar_est <- apply(est_draws, 2, mean)
> cbs
cust x t.x litt T.cal T.star x.star palive_est xstar_est
1: 4 1 28.000000 3.3322045 36.42857 39 1 0.92 0.77
2: 18 0 0.000000 0.0000000 38.42857 39 0 0.26 0.08
3: 21 1 1.714286 0.5389965 38.85714 39 0 0.17 0.11
4: 50 0 0.000000 0.0000000 38.85714 39 0 0.33 0.05
5: 60 0 0.000000 0.0000000 34.42857 39 0 0.34 0.27
---
2353: 23537 0 0.000000 0.0000000 27.00000 39 2 0.38 0.15
2354: 23551 5 24.285714 5.5243721 27.00000 39 0 0.95 4.55
2355: 23554 0 0.000000 0.0000000 27.00000 39 1 0.36 0.17
2356: 23556 4 26.571429 6.3127713 27.00000 39 2 1.00 3.41
2357: 23569 0 0.000000 0.0000000 27.00000 39 0 0.51 0.31
E(X(T+T.star))P(alive)

44. PAGE 44
Questions?
michael.platzer@gmail.com
thomas.reutterer@wu.ac.at
Try BTYDplus !!!

45. Appendix
• C Measure by Zhang, Bradlow, Small 2015
• MCMC Sampling Scheme

46. ZBS: Clumpiness Measure C
a metric-based approach
Predicting Customer Value Using Clumpiness: From RFM to RFMC
Zhang, Bradlow, Small
• Introduce metric C which captures the “non-randomness” in timing patterns
• Straightforward calculation at individual-level;
• Useful for descriptive analysis and segmentation;

47. ZBS: Clumpiness Measure C
a metric-based approach
Main Empirical Findings
• Capturing timing patterns adds
predictive power
• When controlling for R and F, then
clumpy customers tend to be more
active in the future
both findings are supported
and can be explained by our
model-based approach

48. ZBS: Clumpiness Measure C
a metric-based approach
Shortcomings
• Requires many transactions at
individual-level
• Metric C will be skewed when
dealing with different acquisition
dates and churn settings
both are appropriately handled
by our model-based approach

49. ZBS: Clumpiness Measure C
a metric-based approach
à sparse individual-level data mandates a model-based approach

50. Parameter Estimation via MCMC
Component-wise Slice Sampling within Gibbs with Data Augmentation
SEITE 50