Stochastic Models of Noncontractual Consumer Relationships

Stochastic Models of
Noncontractual Consumer Relationships

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Calibration Period Validation Period

Michael Platzer
michael.platzer@gmail.com

Master Thesis at the
Vienna University of Economics
and Business Adminstration

Under the Supervision of
Dr. Thomas Reutterer

November 2008

Abstract

The primary goal of this master thesis is to evaluate several well-established
probabilistic models for forecasting customer behavior in noncontractual set-
tings on an individual level. This research has been carried out with the
particular purpose of participating in a lifetime value competition that has
been organized by the Direct Marketing Educational Foundation throughout
fall 2008.
First, an in-depth exploratory analysis of the provided contest data set
is undertaken, with its key characteristics being displayed in several in-
formative visualizations. Subsequently, the NBD (Ehrenberg, 1959), the
Pareto/NBD (Schmittlein et al., 1987), the BG/NBD (Fader et al., 2005a)
and the CBG/NBD (Hoppe and Wagner, 2007) model are applied on the
data. Since the data seems to violate the Poisson assumption, which is a
prevalent assumption regarding the random nature of the transaction timing
process, the presented models produce rather mediocre results. This becomes
apparent as we will show that a simple linear regression model outperforms
these probabilistic models for the contest data.
As a consequence a new variant based on the CBG/NBD model, namely the
CBG/CNBD-k model, is being developed. This model is able to take a certain
degree of regularity in the timing process into account by modeling Erlang-k
intertransaction times, and thereby delivers considerably better predictions
for the data set at hand. Out of 25 participating teams at the contest the
model ﬁnished at second place, only marginally behind the winning model. A
result that demonstrates that under certain conditions this newly developed
variant is able to outperform numerous other existent, in particular stochastic
models.
Keywords: marketing, consumer behavior, lifetime value, stochastic predic-
tion models, customer base analysis, Pareto/NBD, regularity

i

Contents

Abstract i

1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Problem Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Discussed Models . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Usage Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 DMEF Competition 6
2.1 Contest Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Game Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Exploratory Data Analysis 11
3.1 Key Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Distribution of Individual Donation Behavior . . . . . . . . . . 13
3.3 Trends on Aggregated Level . . . . . . . . . . . . . . . . . . . . 15
3.4 Distribution of Intertransaction Times . . . . . . . . . . . . . . 19

4 Forecast Models 21
4.1 NBD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Pareto/NBD Model . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 BG/NBD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 CBG/NBD Model . . . . . . . . . . . . . . . . . . . . . . . . . . 37

ii

CONTENTS iii

5 Model Comparison 41
5.1 Parameter Interpretation . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Data Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Forecast Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.4 Simple Forecast Benchmarks . . . . . . . . . . . . . . . . . . . 51
5.5 Error Composition . . . . . . . . . . . . . . . . . . . . . . . . . 52

6 CBG/CNBD-k Model 56
6.1 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.3 Comparison of Models . . . . . . . . . . . . . . . . . . . . . . . 64
6.4 Final Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7 Conclusion 72

A Derivation of CBG/CNBD-k 74
A.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
A.2 Erlang-k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.3 Individual Likelihood . . . . . . . . . . . . . . . . . . . . . . . . 76
A.4 Aggregate Likelihood . . . . . . . . . . . . . . . . . . . . . . . . 77
A.5 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . 79
A.6 Probability Distribution of Purchase Frequencies . . . . . . . . 79
A.7 Probability of Being Active . . . . . . . . . . . . . . . . . . . . 81
A.8 Expected Number of Transactions . . . . . . . . . . . . . . . . 83
A.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 88

Bibliography 89

Chapter 1

Introduction

1.1 Background
Over 80% of those companies that participated in a German study on the
usage of information instruments in retail controlling regarded the concept of
customer lifetime value as useful (Schr¨der et al., 1999, p. 9). But only less
o
than 10% actually had a working implementation at that time. No other con-
sumer related information, for example customer satisfaction, penetration or
sociodemographic variables, showed such a big discrepancy between assessed
usefulness and actual usage. Therefore, accurate lifetime value models can
be expected to become, despite but also because of their inherent challenging
complexity, a crucial information advance in highly competitive markets.
Typical fundamental managerial questions that arise, are (Schmittlein et al.,
1987; Morrison and Schmittlein, 1988):

• How much is my current customer base worth?
• How many purchases, and which sales volume can I expect from my
client`le in the future?
e
• How many customers are still active customers? Who has already, and
who will likely defect?
• Who will be my most, respectively my least proﬁtable customers?
• Who should we target with a speciﬁc marketing activity?
• How much of the sales volume has been attributed to such a marketing
activity?

1

CHAPTER 1. INTRODUCTION 2

And a key part for finding answers to those questions is the accurate assess-
ment of lifetime value on an aggregated as well as on an individual level.
Hardly any organization can afford to make budget plans for the upcoming
period without making careful estimations regarding the future sales. Such
estimates on the aggregate level are therefore widely common and numerous
methods exist which range from simple managerial heuristics to advanced
time series analyses. Fairly more challenging is the prediction of future sales
broken down between trial and repetitive customers. And, considering how
little information we have on an individual level, an even more demanding
task is the accurate forecasting for each single client.
Nevertheless, the increasing prevalence of computerized transaction systems
and the drop in data storage costs, which we have seen over the past decade,
provide more and more companies with customer databases coupled with
large records of transaction history (‘Who bought which product at what
price at what time?’). But the sheer data itself is no good unless models and
tools are implemented that condense the desired characteristics, trends and
forecasts out of the data. Such tools are nowadays commonly provided as
part of customer relationship management software, which enables the orga-
nizations to act and react individually to each customer. The heterogeneity
in one’s customer base is thereby taken into account and this allows a further
optimization of marketing activities and their efficiency.1 And one essential
information bit for CRM implementations is the (monetary) valuation of an
individual customer (Rosset et al., 2003, p. 321).

1.2 Problem Scope
The primary focus of this thesis is the evaluation and implementation of sev-
eral probabilistic models for forecasting customer behavior in noncontractual
settings on an individual level. This research has been carried out with the
main focus on participating in a lifetime value competition which has been
organized by the Direct Marketing Educational Foundation in fall 2008.
The limitations of the research scope in this thesis are fairly well defined by
the main task of the competition, which is the estimation of the future pur-
chase amount for an existent customer base on a disaggregated level based
1
Clustering a customer base into segments can be seen as a first step in dealing with
heterogeneity. But one-to-one marketing, as it is described here, is the consequent contin-
uation of this approach.


upon transaction history. Therefore, we will not provide a complete overview
of existing lifetime value models (see Gupta et al. (2006) for such an overview)
but will rather focus on models that can make such accurate future predic-
tions on an individual level.
Due to the large amount of one-time purchases and the long time span of
the data, we have to use models that can also incorporate the defection of
customers in addition to modeling the purchase frequency. Furthermore, we
are faced with noncontractual consumer relationships, a characteristic that is
widely common but which unfortunately adds considerably some complexity
to the forecasting task (Reinartz and Kumar, 2000). The difficulty arises
because no definite information regarding the status of a customer-firm rela-
tionship is available. Neither now nor later. This means that it is impossible
to tell whether a specific customer is still active or whether he/she has already
defected. On the contrary to that, in a contractual setting2 , such as the client
base of a telecommunication service provider, it is known when a customer
cancels his/her contract and is therefore lost for good.3 In a noncontractual
setting, such as retail shoppers, air carrier passengers or donors for a NPO,
we cannot observe the current status of a customer-firm relationship (i.e. it
is a latent variable), but rather rely on other data, such as the transaction
history to make proper judgments. Therefore we will limit our research to
models that can handle this kind of uncertainty.
Further, because the data set only provides transaction records,4 the empha-
sis is put on models that extract the most out of the transaction history and
do not rely on incorporating other covariates, such as demographic variables,
competition activity or other exogenous variables.

1.3 Discussed Models
Table 1.1 displays an overview of the probabilistic models that are being
evaluated and applied upon the competition data within this thesis.
Firstly, the seminal work by Ehrenberg who proposed the negative binomial
2
Also known as subscription-based setting.
3
Models that explicitly model churn rates are, among others, logistic regression models
and survival models. See Rosset et al. (2003) and Mani et al. (1999) for examples of the
latter kind of models.
4
Actually it also includes detailed records of direct marketing activities, but we neglect
this data, as such data is not available for the target period. See section 2.3 for a further
reasoning.


Model Author(s) Year
NBD Ehrenberg 1959
Pareto/NBD Schmittlein, Morrison, and Colombo 1987
BG/NBD Fader, Hardie, and Lee 2005
CBG/NBD Hoppe and Wagner 2007
CBG/CNBD-k Platzer 2008
Table 1.1: Overview of Presented Models

distribution (NBD) in 1959 as a model for repeated buying is investigated in
detail in section 4.1. Further, we will evaluate the well-known Pareto/NBD
model (section 4.2) and two of its variants, the BG/NBD (section 4.3) and
the CBG/NBD (section 4.4) model, which are all extensions of the NBD
model but make additional assumptions regarding the defection process and
its heterogeneity among customer. In order to get a feeling for the forecast
accuracy of these probabilistic models, we will subsequently also benchmark
them against a simple linear regression model.
Finally, the CBG/CNBD-k model, which is a new variant of the CBG/NBD
model, will be introduced in chapter 6. This model makes diﬀering assump-
tions regarding the timing of purchases, in particular it considers a certain
extent of regularity and thereby will improve forecast quality considerably
for the competition data set. Detailed derivations for this model are provided
in appendix A.

1.4 Usage Scenarios
But before diving into the details of the present models, we try to further
increase the reader’s motivation by providing some common usage scenarios
of noncontractual relations with repeated transactions. The following list
contains usage scenarios which have already been studied in various articles
and which should give an idea of the broad ﬁeld of applications for such
models.

• Customers of the online music store CDNOW (Fader et al., 2005a).
This data set is also publicly available at http://brucehardie.com/
notes/008/, and has been used in numerous other articles (Abe, 2008;
Hoppe and Wagner, 2007; Batislam et al., 2007; Fader et al., 2005c;


Fader and Hardie, 2001; W¨bben and von Wangenheim, 2008) to bench-
u
mark the quality of various models.

• Clients of a financial service broker (Schmittlein et al., 1987).

• Members of a frequent shopper program at a department store in Japan
(Abe, 2008).

• Consumers buying at a grocery store (Batislam et al., 2007). Individual
data can be collected by providing client-cards that are being combined
with some sort of loyalty program.

• Business customers of an office supply company (Schmittlein and Pe-
terson, 1994).

• Clients of a catalog retailer (Hoppe and Wagner, 2007).

But, citing W¨bben and von Wangenheim (2008, p. 82), whenever ‘a cus-
u
tomer purchases from a catalog retailer, walks off an aircraft, checks out of a
hotel, or leaves a retail outlet, the firm has no way of knowing whether and
how often the customer will conduct business in the future’. And as such the
usage scenarios are practically unlimited.
One other example from the author’s own business experience is the challenge
to assess the number of active users of a free webservice, such as a blogging
platform. Users can be uniquely identified by a permanent cookie stored in
the browser client, when they access the site. Each posting of a new blog
entry could be seen as a transaction, and therefore these models could also
provide answers to questions like ‘How many of the registered users are still
active?’ and ‘How many blog entries will be posted within the next month
by each one of them?’.
This thesis should shed some light on how to find accurate answers to ques-
tions of this kind.

Chapter 2

DMEF Competition

2.1 Contest Details
The Direct Marketing Educational Foundation1 (DMEF) is a US based non-
profit organization with the mission ‘to attract, educate, and place top college
students by continuously improving and supporting the teaching of world-
class direct / interactive marketing’2 . The DMEF is an affiliate of the Direct
Marketing Association Inc.3 and it is also founder and publisher of the Jour-
nal of Interactive Marketing4 .
The DMEF organized a contest in 2008, with ‘the purpose [..] to compare
and improve the estimation methods and applications for [lifetime value and
customer equity modeling]’ which ‘have attracted widespread attention from
marketing researchers [..] over the past 15 years’ (May, Austin, Bartlett,
Malthouse, and Fader, 2008). The participating teams were provided with
a data set from a leading US nonprofit organization, whose name remained
undisclosed, containing detailed transaction and contact history of a cohort
of 21.166 donors over a period of 4 years and 8 months. The transaction
records included a unique donor ID, the timing, and the amount of each
single donation together with a (rather cryptic) code for the type of contact.
The contact data included records of each single contact together with the
contacted donor, the timing, the type of contact, and the implied costs of
that contact.
1
cf. http://www.directworks.org/
2
http://www.directworks.org/About/Default.aspx?id=386, retrieved on Oct. 9, 2008
3
cf. http://www.the-dma.org/
4
cf. https://www.directworks.org/Educators/Default.aspx?id=220

6

CHAPTER 2. DMEF COMPETITION 7

The first phase of the competition consisted of three separate estimation
tasks for a target period of two years:

1. Estimate the donation sum on an aggregated level.

2. Estimate the donation sum on an individual level.

3. Estimate which donors, who have made their last donation before
Sep. 1, 2004, will be donating at all during the target period.

An error measure for all 3 tasks was defined by the contest organizing com-
mittee in order to evaluate and compare the submitted calculations by the
participating teams. Closeness on an aggregated level (task 1) was simply
defined as the absolute deviation from the actual donation amount, and for
task 3 it was the percentage of correctly classified cases. The error measure
for task 2 was defined as the mean squared logarithmic error:

MSLE = (log(yi + 1) − log(î + 1))2 /21.166,
y
i

with the 1 added to avoid taking the logarithm of 0, and with 21.166 being
the size of the cohort.
The deadline for submitting calculations for phase 1 (task 1 to 3) was Sep. 15,
2008. The results for the participating teams were announced couple of
weeks afterwards and were discussed at the DMEF’s Research Summit in
Las Vegas.5

2.2 Data Set
The data set contains records of 53,998 donations for 21,166 distinct donors,
starting from Jan. 2, 2002, until Aug. 31, 2006. Each of these donors made
their initial donation during the first half of 2002, as this is the criteria
for donors for being included into the cohort. The record of each donation
contains a unique identifier of the donor, and the date and dollar amount of
that donation. Additionally, the type of contact that can be linked with this
transaction is given. See table 2.1 for a sample of the transaction records.
Furthermore, detailed contact records with their related costs were provided.
These 611,188 records range from Sep. 10, 1999, until Aug. 28, 2006. Each
5
cf. http://www.researchsummit.org/


id date amt source
8128357 2002-02-22 5 02WMFAWUUU
9430679 2002-01-10 50 01ZKEKAPAU
9455908 2002-04-19 25 02WMHAWUUU
9652546 2002-04-02 100 01RYAAAPBA
9652546 2003-01-06 100 02DEKAAGBA
9652546 2004-01-05 100 04CHB1AGCB
.. .. .. ..
13192422 2005-02-11 50 05HCPAAICD
13192422 2005-02-16 50 05WMFAWUUU

Table 2.1: Transaction Records

contact record contains an identiﬁer of the contacted donor, the date of
contact, the type of contact and the associated costs for the contact. See
table 2.2 for a sample of these contact records.
id date source cost
9652546 2000-07-20 00AKMIHA28 0.2800000
9430679 2000-07-07 00AXKKAPAU 0.3243999
9455908 2000-07-07 00AXKKAPAU 0.3243999
11303542 2000-07-07 00AXKKAPAU 0.3243999
11305422 2000-01-14 00CS31A489 0.2107999
11261005 2000-01-14 00CS31A489 0.2107999
.. .. .. ..
11335783 2005-09-01 06ZONAAMGE 0.4068198
11303930 2005-09-01 06ZONAAMGE 0.4068198

Table 2.2: Contact Records

According to May et al. (2008), ‘the full data set, including 1 million cus-
tomers, 17 years of transaction and contact history, and contact costs, will
be released for general research purposes’, and should become available at
https://www.directworks.org/Educators/Default.aspx?id=632. The compe-
tition data set represents therefore only a small subset of the complete avail-
able data that has been provided by the NPO after the competition.

2.3 Game Plan
Before starting out with the model building, an in-depth exploratory analysis
of the data set is performed, in order to gain a deeper understanding of its


key characteristics. Various visualizations provide a comprehensive overview
of these characteristics and help comprehend the outcomes of the modeling
process.
As mentioned above, our main emphasis is on winning task 2, i.e. on finding
the ‘best’ forecast model that will subsequently provide the lowest MSLE for
the target period. But of course no data for the target period is available
before the deadline of the competition, and therefore we have to split the
provided data into a training period and a validation period. The training
data is used for calibrating the model and its parameters, whereas the valida-
tion data enables us to compare the forecast accuracy among the models. By
choosing several different lengths of training periods, as has also been done
by Schmittlein and Peterson (1994), Batislam et al. (2007) and Hoppe and
Wagner (2007), we can further improve the robustness of our choice. After
picking a certain model for the competition, the complete provided data set
is used for the final calibration of the model.
Despite the fact that a strong causal relation between contacts and actual
donations can be assumed, we will not include the contact data into our
model building. The main reason is that such data is not available for the
target period and also cannot be reliably estimated. Therefore, we implic-
itly assume that direct marketing activities will have a similar pattern as
in the past and simply disregard this information. The same assumption is
being made regarding all other possible exogenous influences, such as com-
petition, advertisement, public opinion, and so forth, due to the absence of
such information.
All the probabilistic models under investigation try to model the purchase
opportunity as opposed to the actual purchase amount.6 The amount per
donor is estimated in a separate step and is simply multiplied with the es-
timated number of future purchases (see section 6.4.1). This approach is
feasible, if we assume independence between purchase amount and purchase
rate, respectively between purchase amount and defection rate (Schmittlein
and Peterson, 1994, p. 49).
Providing an estimate for task 3 is directly derived from task 2. This is done
by assuming that any customer with an estimated number of purchases of
0.5 or higher will actually make a purchase within the target period. Task 1
could be deduced from task 2 as well by simply building the sum over all
individual estimates.
6
Donations and purchases as well as donors and consumers or clients will be referred
to as synonymously within this thesis.


All of our following calculations and visualizations are carried out with the
statistical programming environment R (R Development Core Team, 2008),
which is freely available, well documented, widely used in academic research,
and which further provides a large repository of additional libraries. Unfor-
tunately, the presented probabilistic models are not yet part of an existent
library. Hence, the programming of these models needs to be done by our-
selves. But thanks to the published estimates regarding the CDNOW data
set7 within the originating articles we are able to verify the correctness of
our implementations.

7
http://brucehardie.com/notes/008/

Chapter 3

Exploratory Data Analysis

In this chapter an in-depth descriptive analysis of the contest data set is
undertaken. Several key characteristics are being outlined and concisely vi-
sualized. These ﬁndings will provide valuable insight into the succeeding
model ﬁtting process in chapter 4.

3.1 Key Summary

No. of donors 21,166
Cohort time length 6 months
Available time frame 4 years 8 months
Available time units days
No. of zero repeaters: absolute; relative 10,626; 50.2%
No. of rep. donations: mean; sd; max 1.55; 2.93; 55
Donation amount: mean; sd; max $39.31; $119.32; $10,000
Time between donations: mean; sd; max 296 days; 260 days; 1626 days
Time until last donation: mean; sd 460 days; 568 days

Table 3.1: Descriptive Statistics

The data set consists of a rather large, heterogeneous cohort of donors.
Heterogeneity can be observed in the donation frequency, in the donation
amount, in the time laps between succeeding donations, and in the overall
recorded lifetime.

11

CHAPTER 3. EXPLORATORY DATA ANALYSIS 12

On the one hand, the majority (50.2%) did not donate at all after their initial
donation. On the other hand, some individuals donated very frequently, up
to 55 times. The amount per transaction ranges from as little as a quarter of
a dollar up to $10,000. And the observed standard deviation of the amount
is 3 times larger than its mean. These simple statistics already make it clear
that any model that is being considered to fit the data should be able to
account for such a kind of heterogeneity.
It can also be noted that the covered time span of the records is considerably
long (like is the target period of 2 years). This implies that people who are
still active at the end of the 4 year and 8 month period are rather loyal, long-
term customers. But it also means that assuming stationarity regarding the
underlying mechanism and thereby regarding the model parameters might
not prove true.

Various Timing Patterns

11382546 | | | | |

11371770 | | | || | | | | | | | | | | |

11359536 | | |

11343894 | |

11329984 |
Donor ID

11317401 |

11303989 |

11292547 | |

11281342 | | | | | | |

11270451 |

11259736 |

10870988 ||||||||||||||||||||||||||||||||||||||||||||

2002 2003 2004 2005 2006
Time Scale

Figure 3.1: Timing Patterns for 12 Randomly Selected Donors

An important feature of the data set is that donation (as well as contact)
records are given with their exact timing, and they are neither aggregated
to longer time spans nor condensed to simple frequency numbers. Therefore
the information of the exact timing of the donations can and also should be
used for our further analysis. A first ad-hoc visualization (see figure 3.1) of
12 randomly selected donors already displays some of the differing charac-
teristic timing patterns. These patterns range from single-time donors (e.g.


ID 11259736), over sporadic donors (e.g. ID 11359536) to regular donors who
have already defected (see ID 10870988 at the bottom of the chart). Thus,
the high number of single-time donors and also the observed defection of reg-
ular donors suggests that models should be considered in particular which
can also account for such a defection process.

3.2 Distribution of Individual Donation Be-
havior

Distribution of Numbers of Donations
12000

50.2%
8000
# Donors

4000

16.9%
10.8%
7.6% 6.3%
2.6% 3.9%
1.6%
0

1 2 3 4 5 6 7 8+

# Donations

Figure 3.2: Histogram of Number of Donations per Donor

Figure 3.2 displays once more the aforementioned 50.2% of single-time donors,
i.e. donors who have never made any additional transaction after their initial
donation in the ﬁrst half of 2002. Aside from these single-time donors, a fur-
ther large share of donors must be considered as ‘light’ users. In particular
42% donate less than 6 times which corresponds to an average frequency of
about or even less than once a year. And only as little as 8% of the cus-
tomer base (in total 1733 people) can be considered frequent donors, with 6
or more donations. However, these 8% actually account for over half of the
transactions (51,5%) in the last year of the observation period, and therefore
are of great importance for our estimates into the future.
It it is important to point out that a low number of recorded donations can
result from two diﬀerent causes. Either this low number really stems from a
(very) low donation frequency, i.e. people just rarely donate. Or this stems
from the fact that people defected, i.e. turned away from the NPO and will


not donate at all anymore. An upcoming challenge will be to distinguish
these two mechanism within the data.

Distribution of Donation Amounts
0.30

25
0.25

10
0.20
Relative Frequency

0.15

50
20
0.10

15

5 100
0.05
0.00

0.25 1 2 3.5 6 10 18 32 57 110 235 500 1200 3000 10000

Donation Amount − logarithmic scale

Figure 3.3: Histogram of Dollar Amount per Donation

Figure 3.3 plots the observed donation amounts. These amounts vary tremen-
dously, and range from as low as a quarter of a dollar up to a single generous
donation of $10,000. A visual inspection of the figure indicates that the over-
all distribution follows, at least to some extent, a log-normal distribution,1
but with its values being restricted to certain integers. Particularly 89% of
the 53,998 donations are accounted by some very specific dollar amounts,
namely $5, $10, $15, $20, $25, $50 and $100. The other donation amounts
seem to play a minor role. Though, special attention should be directed to
those few large donations, because the 3% of donations that exceed $100
actually sum up to 30% of the overall donation sum.
In figure 3.4 a possible relation between the average amount of a single do-
nation and the number of donations per individual is inspected.2 As we can
see, single time donors as well as very active donors (7+) tend to spend a
1
The dashed gray line in the chart represents a kernel density estimation with a broad
bandwidth.
2
Note: The widths of the drawn boxes in the chart are proportional to the square roots
of the number of observations in the corresponding groups.


Conditional Distribution of Donation Amounts
100
80
Average Donation Amount

60
40
20
0

1 2 3 4 5 6 7 8+

# Donations

Figure 3.4: Distribution of Average Donation Amounts grouped by Number of
Donations per Donor

little less money per donation. A result that seems plausible, as single time
donors rather ‘cautiously try out the product’ and heavy donors spread their
overall donation over several transactions. Nevertheless, the observed corre-
lation between these two variables is minimal and will be neglected in the
following.

3.3 Trends on Aggregated Level
This section analyzes possible existing trends within the data on an aggre-
gated level by examining time series. Most of the charts that are presented
in the following share the same layout. The connected line represents the
evolution of the particular figures for the quarters of a year, and the horizon-
tal lines are the averages over 4 of these quarters at a time. The time series
are aggregated to quarters instead of tracking the daily movements in order
to reduce the noise within these figures and to help identify the long-term
trends. The displayed percentage changes indicate the change from one year
to the next, whereas these averages cover the second half of one year and the
first half of the next year. This shifted year average has been chosen, since


the covered time range of the competition data ends slightly after the second
quarter in 2006.

Donation Sum

4e+05
2e+05
0e+00

+8% −24% −3%

2002 2003 2004 2005 2006 2007

Time

Figure 3.5: Trend in Overall Donation Sum

Inspecting the evolution of overall donation sums (figure 3.5) directly reveals
various interesting properties. First of all, it is apparent that donations show
a sharp decline immediately after the second quarter in 2002. This observed
drop is plausible, if we recall that our cohort has actually been built by
definition of new donors from the first half of 2002 and that on average only
a few following donations are being made. Further, it can be stated that the
data shows a strong seasonal fluctuation with the third quarter being the
weakest, and the fourth and first quarter being the strongest periods. About
twice as many donations occur during each of these strong quarters than
during the third quarter. It also seems that there is a downward trend in
donation sums. But the speed of this trend remains ambiguous, if a look at
the corresponding percentage changes is taken. At the beginning an increase
of 8% is recorded, then a sharp drop of 24%, which is followed by a moderate
decrease of 3% over the last year. Task 1 of the competition is the estimation
of the future trend of these aggregated donation sums for the next two years.
Considering the erratic movements this is quite a challenge.
The overall donation sum is the result of the multiplication of the number
of donations with the average donation amount. Figure 3.6, which separates
these two variables, provides some further insight into the decomposition of
the overall trend. The time series for the number of donations also displays
a strong seasonality, which has a peak around the Christmas holidays. The
continuous downward trend (-13%, -15%, -14%) in the transaction numbers
is considerably stable and hence predictable. A simple heuristic could, for
example, assume a constant decreasing rate of 14% for the next two years.
As has been noted in the preceding section, this downward trend can either
be the result from a decreasing donation frequency for each donor or might


# Donations Avg Donation Amount

50
8000

40
30
4000

20
10
−13% −15% −14% +24% −10% +12%
0

0
2002 2004 2006 2002 2004 2006

Time Time

Figure 3.6: Trend in Number of Donations and Average Donation Amount

stem from an ongoing defection process. Figure 3.7 indicates that rather
the latter of these two effects is dominant. The number of active donors
is steadily decreasing,3 whereas the average number of donations per active
donor is slightly increasing.

Percentage of Donors Average # Donations
who Have Donated Within that Year per Active Donor
0.5

2.0
0.4

1.51 1.55
1.46
1.5

1.42
27.8% 29.5%
0.3

23.5%
1.0

18.8%
0.2

0.5
0.1
0.0

0.0

2002 2003 2004 2005 2002 2003 2004 2005

Time Time

Figure 3.7: Trend in Activity

Due to the stable decline of donation numbers it can be concluded that the
erratic movement of the overall sum stems from the up and downs in the
average donation amounts. The chart on the right hand side of figure 3.6
surprisingly also shows seasonal fluctuation, and has no clear overall trend
at all, which makes it hard to make predictions into the future.
3
Note that we disregard the initial donation for this chart as otherwise the share for
2002 would simply be 100%.


Donation Sum Contact Costs
4e+05

25000
2e+05

10000
0e+00

+8% −24% −3% +25% −16% −33%

0
2002 2003 2004 2005 2006 2007 2002 2003 2004 2005 2006 2007

Time Time

# Contacts Avg Contact Cost

0.6
50000

0.4
20000

0.2
0.0

−3% −30% −7% +22% +19% −24%
0

2002 2003 2004 2005 2006 2007 2002 2003 2004 2005 2006 2007

Time Time

Figure 3.8: Trend in Contacts

A possible explanation for the observed trends and movements might be
contained in the contact records which have been provided by the organizing
committee. Each donation is linked to a particular contact, but certainly
not each contact resulted in a donation. Therefore, it seems logical that
the amount of contacts and the associated expenses have a strong inﬂuence
on the donation sums. The displayed time series from ﬁgure 3.8 strongly
support this assumption. And again, the same seasonal variations in the
number of contacts as well as in their average costs can be detected as before.
Furthermore, the increase in donation sums in 2003/2004 can now be linked
to the tremendous increase of 25% in contact spending during that period.
On the other hand, the NPO has been able to cut costs in 2005/2006 by
33% (mostly due to a 24% drop in average contact costs) without hurting
the generated contributions.
Unfortunately, it is not possible to take any advantage out of this detected
relation between donations and contacts for the contest, because no informa-
tion regarding the contact activities throughout the target period is available
(see section 2.3 for the previous discussion).


3.4 Distribution of Intertransaction Times

Overall Distribution of Intertransaction Times
4000

1

12
3000
Count

2000
1000

24
0

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51

# Months in between Donations

Figure 3.9: Histogram of Intertransaction Times in Months

The disaggregated availability of transaction data on a day-to-day base allows
an inspection of the observed intertransaction times, i.e. the lapsed time
between two succeeding donations for an individual.4 Figure 3.9 depicts the
overall distribution of this variable. The distribution contains two peaks, the
ﬁrst and also highest peak represents waiting times of one month and the
second peak represents one year intervals. Further, we see that only very few
times (1.4%) donations occur within a single month. It seems that there is
a dead period of one month, which marks the time until a donor is willing
to make another transaction. It is also interesting to note that in 5% of the
cases we have a waiting period of more than 24 months and that there are
even values higher than 4 years. This is an indicator that some customers
can remain inactive for a very long period and nevertheless can still possibly
be persuaded to make another donation. This particular characteristic of
the data set will make it hard to model the defection process correctly in the
following, as some long-living customers just never actually defect but are
rather ‘hibernating’ and can be reactivated at anytime5 .
Figure 3.10 shows that light and frequent donors have a diﬀering distribution
of intertransaction times, with the former one donating approximately every
4
Also commonly termed as interpurchase times or interevent times.
5
Compare further the lost-for-good versus always-a-share discussion in Rust, Lemon,
and Zeithaml (2004, p. 112).


year, and the latter one donating regularly each month. As we will see,
this particular observed regularity will play a major role in the upcoming
modeling phase.

Intertransaction Times for Light Donors (2, 3 or 4 Donations)
300

Yearly Donations (~8%)
Count

8814 Donors , 18352 Donations
150
0

0 76 178 292 406 520 634 748 862 976 1103 1243 1383 1524

# Days in between Donations

Interpurchase Times for Frequent Donors (5 or more Donations)

Monthly Donations (~10%)
Count

400

1733 Donors , 14480 Donations
0

0 76 178 292 406 520 634 749 870 994 1126 1385

# Days in between Donations

Figure 3.10: Intertransaction Times Split by Frequency

Chapter 4

Forecast Models

4.1 NBD Model

4.1.1 Assumptions

As early as 1959, Andrew Ehrenberg1 published his seminal article ‘The
Pattern of Consumer Purchase’ (Ehrenberg, 1959), in which he suggested the
negative binomial distribution (abbr. NBD) as a fit to aggregated count data
of sales of non-durable consumer goods.2 Since then Ehrenberg’s paper has
been cited numerous times in the marketing literature and various models
have been derived based upon his work, proving that his assumptions are
reasonable and widely applicable.
Besides the sheer benefit that a well fitting probability distribution is found,
Ehrenberg further provides a logical justification for choosing that particular
distribution. He argues that each consumer purchases according to a Poisson
process and that the associated purchase rates vary across consumers accord-
ing to a Gamma distribution.3 Now, the negative binomial distribution is
exactly the theoretical distribution that arises from such a Gamma-Poisson
mixture. Table 4.1 summarizes the postulated assumptions of Ehrenberg’s
model.
1
See http://www.marketingscience.info/people/Andrew.html for a brief summary of
his major achievements in the field of marketing science.
2
In other words, a discrete distribution is proposed that is supposed to fit the data
displayed in figure 3.2 on page 13.
3
Actually, he assumed a χ2 -distribution in Ehrenberg (1959) but this is simply a special
case of the more general Gamma distribution.

21

CHAPTER 4. FORECAST MODELS 22

A1 The number of transactions follows a Poisson process
with rate λ.

A2 Heterogeneity in λ follows a Gamma distribution with
shape parameter r and rate parameter α across cus-
tomers.
Table 4.1: NBD Assumptions

In order to support the reader’s understanding of the postulated assump-
tions, visualizations of the aforementioned distributions are provided in fig-
ure 4.1, 4.2 and 4.3 for various parameter constellations.
The Poisson distribution is characterized by the relation that its associated
mean and also its variance are equal to the rate parameter λ. Further, it
can be shown that assuming a Poisson distributed number of transactions is
equivalent to assuming that the lapsed time between two succeeding transac-
tions follows an exponential distribution. In other words, the Poisson process
with rate λ is the respective count process for a timing process with indepen-
dently exponential distributed waiting times with mean 1/λ (Chatfield and
Goodhardt, 1973).
The exponential distribution itself is a special case of the Gamma distribution
with its shape parameter being set to 1 (see the middle chart in figure 4.3).
An important property of exponentially distributed random variables is that
it is memoryless. This means that any provided information about the time
since the last event does not change the probability of an event occurring
within the immediate future.
P (T > s + t | T > s) = P (T > t) for all s, t ≥ 0.
For the mathematical calculations such a property might be appealing, be-
cause it simplifies some derivations. But applied on sales data, this implies
that the timing of a purchase does not depend on how far in the past the
last purchase took place. A conclusion that is quite contrary to common
intuition which would rather suggest that nondurable consumer goods are
purchased with certain regularity. If a consumer buys for example a certain
good, such as a package of detergent, he/she will wait with the next purchase
until that package is nearly consumed. But the memoryless property even
further implies that the most likely time for another purchase is immediately
after a purchase has just occurred (Morrison and Schmittlein, 1988, p. 148).4
4
This can also be depicted from the middle chart of figure 4.3, as the density function


0.4 Negative Binomial Distribution

0.4

0.4
r=1 r=1 r=3
0.3

0.3

0.3
p = 0.4 p = 0.2 p = 0.5
0.2

0.2

0.2
0.1

0.1

0.1
0.0

0.0

0.0
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

Figure 4.1: Probability Mass Function of the Negative Binomial Distribution for
Different Parameter Values

Poisson Distribution
0.4

0.4

0.4
0.3

0.3

0.3

lambda = 0.9 lambda = 2.5 lambda = 5
0.2

0.2

0.2
0.1

0.1

0.1
0.0

0.0

0.0

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

Figure 4.2: Probability Mass Function of the Poisson Distribution for Different
Parameter Values

Gamma Distribution
0.5

0.5

0.5

shape = 0.5 shape = 1 shape = 2
0.4

0.4

0.4

rate = 0.5 rate = 0.5 rate = 0.5
0.3

0.3

0.3
0.2

0.2

0.2
0.1

0.1

0.1
0.0

0.0

0.0

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

Figure 4.3: Probability Density Function of the Gamma distribution for Different
Parameter Values


Nevertheless, the Poisson distribution has proven to be an accurate model
for a wide range of applications, like the decay of radioactive particles, the
occurrence of accidents or the arrival of customers in a queue. But in all
these cases the memoryless property withstand basic face validity checks. It
seems plausible for example that the particular arrival time of one customer
in a queue is absolutely independent of the arrival of the next customer, as
they both do not interact with each other. The fact that a customer has just
arrived does not influence the arrival time of the next one. Therefore, it can
be argued that queuing arrivals are indeed a memoryless process.
But, as has been argued above, this is not the case for purchases of non-
durable consumer goods for an individual customer. The regularity of con-
sumption of a good does lead to a certain extent of regularity regarding its
purchases. Ehrenberg has been aware of this defect (Ehrenberg, 1959, p. 30)
but simply required that the observed periods should not be ‘too short, so
that the purchases made in one period do not directly affect those made in
the next’ (ibid., p. 34).
Assumption A2 postulates a Gamma distribution for the distribution of pur-
chase rates across customers, in order to account for heterogeneity. If the
different possible shapes of this two-parameter continuous probability are be-
ing considered, then it is safe to state that such an assumption adds some
substantial flexibility to the model. But besides the added flexibility and its
positive skewness no behavioral story is being provided in Ehrenberg (1959)
in order to justify the choice of the Gamma distribution.
Nevertheless, Ehrenberg applies a powerful trick by explicitly modeling het-
erogeneity. He utilizes information of the complete customer base for model-
ing on an individual level. He thereby takes advantage of the well-established
regression to the mean phenomenon. ‘[We] can better predict what the per-
son will do next if we know not only what that person did before, but what
other people did’ (Greene, 1982, p. 130 reprinted from Hoppe and Wagner,
2007, p. 80). Schmittlein et al. (1987, p. 5) similarly stated that ‘while
there is not enough information to reliably estimate [the purchase rate] for
each person, there will generally be enough to estimate the distribution of
[it] over customers. [..] This approach, estimating a prior distribution from
the available data, is usually called an empirical Bayes method’.
So, despite a possibly violated assumption A15 and a somewhat arbitrary
assumption A2, the negative binomial distribution proves to fit empirical
reaches its maximum for value zero.
5
See section 6.1 and also Herniter (1971) for some further empirical evidence.


market data very well (Dunn et al., 1983; Wagner and Taudes, 1987; Chatfield
and Goodhardt, 1973).

4.1.2 Empirical Results

In the following the NBD model is applied on the data set from the DMEF
competition. First, we will estimate the parameters, then analyze how well
the model fits the data on an aggregated level, and finally we will calculate
individual estimates.6
Ehrenberg suggests an estimation method for the parameters α and r that
only requires the mean number of purchases m and the proportion share
of non-buyers p0 (Ehrenberg, 1959). However, with modern computational
power the calculation of a maximum likelihood estimation (abbr. MLE) does
not pose a problem anymore. The MLE method tries to find those parameter
values, for which the likelihood of the observed data is maximized. It can be
shown that this method has the favorable property of being an asymptotically
unbiased, asymptotically efficient and asymptotically normal estimator.
The calculation of the likelihood for the NBD model requires two pieces of
information per donor: The length of observed time T , and the number of
transactions x within time interval (0, T ]. This time span differs from donor
to donor, because the particular date of the first transaction varies across the
cohort. It needs to be noted that x does not include the initial transaction,
because that transaction occurred for each person of our cohort by definition.
As we will see later on, the upcoming models will also require another piece of
information for each donor, namely the recency, i.e. the timing tx of the last
recorded transaction.7 The set of information consisting of recency, frequency
and a monetary value is often referred to as RFM variables and is commonly
(not only for probabilistic models) the condensed data base of many customer
base analyses. The layout of the transformed data can be depicted from
table 4.2. The displayed information is read as followed: The donor with
the ID 10458867 made no additional transactions throughout the observed
period of 1605 days after his initial donation of 25.42 dollars. Further, donor
9791641 made five donations (one initial and four repetitive ones) which sum
up to 275 dollars during an observed time span of 1687 days, whereas the
last donation occurred 1488 days after the initial one. That is, the donor did
6
Again note that we only model the number of donations for now, and make an assess-
ment for the amount per donation in a separate step in section 6.4.1.
7
With this notation we closely follow the variable conventions used in Schmittlein et al.
(1987) and Fader et al. (2005a).


not donate during the last 199 days (= T −tx = 1687−1488) of the observation
anymore.

id x tx T amt
10458867 0 0 1605 25.42
10544021 1 728 1602 175.00
10581619 7 1339 1592 80.00
.. .. .. .. ..
9455908 0 0 1595 25
9652546 4 1365 1612 450
9791641 4 1488 1687 275

Table 4.2: DMEF Data Converted to RFM

Applying the MLE method on the transformed data results in the following
parameter estimates
r = 0.475 = shape parameter, and
α = 498.5 = rate parameter,

for the DMEF data set, with both parameters being highly significant. The
general shape of the resulting Gamma distribution can be depicted from the
left chart of figure 4.3, i.e. it is reversed J-shaped. This implies that the
majority of donors have a very low donation frequency, with the mode being
at zero, the median being 0.00042 and the mean being 0.00095 (= r/α). In
terms of average intertransaction times, which are simply the reciprocal val-
ues of the frequencies, this result implies an average time period of 1,048 days
(=2.9 years) between two succeeding donations, and that half of the donors
are donating less often than every 2,406 days (=6.6 years).8 If we consider
that the majority of donors has not redonated at all during the observation
period, these long intertransaction times are obviously a consequence of the
overall low observed donation frequencies.
The next step is an analysis of the model’s capability to represent the data.
For this purpose the actual observed number of donations are being compared
with their theoretical counterparts that are calculated by the NBD model.
Table 4.3 contains the result.
As can be seen, a nearly perfect fit for the large share of non-repeaters is
achieved. However, the deviations of the estimated group sizes increase for
8
The median of the Gamma distribution is approximated by generating a large ran-
dom sample from the theoretical distribution and subsequently calculating the empirical
median.


0 1 2 3 4 5 6 7+
Actual 10,626 3,579 2,285 1,612 1,336 548 348 832
NBD 10,617 3,865 2,183 1,379 918 629 439 1,135
Table 4.3: Comparison of Actual vs. Theoretical Count Data

the more frequent donors, which indicates that the model is not fully able to
explain the observed data.
Attention is now turned to the predictive accuracy of the NBD model on
an individual level. For this purpose the overall observation period of 4
years and 8 months needs to be split into a calibration period of 3.5 years
and a validation period of 1 year. Due to the shorter time range for the
calibration, the estimate parameters (r = 0.53, α = 501) are now slightly
different compared to our results from above. Subsequently, a conditional
estimate is being calculated for each individual for a one year period. These
estimates take their respective observed frequencies x and time spans T into
account. Table 4.4 displays a small subset of such estimates with x365 being
the actual number and x365Nbd being the estimated number of transactions.
For example, the donor with ID 10581619 donated 6 times within the first
3.5 years but only made a single donation in the following year, whereas the
NBD model predicted approximately 2.5 donations during that period.9

id x tx T x365 x365Nbd
10458867 0 0 1179.5 0 0.0011
10544021 1 728 1176.5 0 0.4226
10581619 6 1079 1166.5 1 2.5303
.. .. .. .. .. ..
9455908 0 0 1169.5 0 0.0011
9652546 3 1001 1186.5 1 1.2657
9791641 3 777 1261.5 1 1.2657

Table 4.4: Individual NBD Forecasts for a Data Split of 3.5 Years to 1 Year

Table 4.5 contains these numbers in an aggregated form. It compares the
actual with the average expected number of donations during the validation
period split by the associated number of donations during the calibration
period. For example, those people that did not donate at all within the first
3.5 years donated in average 0.038 times in the following year, whereas the
NBD model only predicted an average of 0.001 donations. On the other hand,
as can also be depicted from the table, the future donations of the frequent
9
Note that the model estimates are not restricted to integer numbers.


donors are being vastly overestimated. Overall, the NBD model estimates
11,088 donations for the 21,166 donors, which is nearly twice as much as the
observed 6,047 donations during the validation period.

0 1 2 3 4 5 6 7+
Actual 0.038 0.20 0.43 0.69 0.75 1.06 1.54 2.44
NBD 0.001 0.42 0.84 1.27 1.69 2.11 2.53 4.68
Table 4.5: Comparison of Actual vs. Theoretical Average Number of Donations
per Donor during the Validation Period

A possible explanation for the poor performance of the NBD model is the
long overall time period, in combination with the assumption that all donors
remain active. The upcoming section will present a model that explicitly
takes a possible defection process into account.

4.2 Pareto/NBD Model

4.2.1 Assumptions

In 1987, Schmittlein, Morrison, and Colombo introduced the Pareto/NBD
model to the marketing science community (Schmittlein et al., 1987). It is
nowadays a well known, and well studied stochastic purchase model for non-
contractual settings and has even further ‘received growing attention among
researchers and managers within recent years’ (Fader et al., 2005a, p. 275).
Schmittlein et al. explicitly try to tackle the problem of a nonobservable
defection process. For various reasons existing customers may decide to quit
a business relation, e.g. stop purchasing a product or buying at a shop. The
reasons can range from a change in personal taste or attitudes, over changes
in personal circumstances, such as marriages, newborns, illnesses, or moving
to other places, to the very deﬁnitive form of defection, namely death. But
regardless of the actual cause, the fundamental problem in a noncontractual
customer relationship is that the organization will generally not be notiﬁed
of that defection. Hence the organization relies on other indicators to assess
the current activity status.
Building a stochastic model for a nonobservable dropout process on an in-
dividual level is a challenging task. Especially if we consider that a drop
out can only occur a single time per customer. And even then, it is still


not possible to verify whether this event has really occurred. Looking at the
various timing patterns (see ﬁgure 3.1 on page 12) gives an impression on the
inherent diﬃculty of estimating which of these donors are still active after
August 2006, let alone of building a stochastic parametric model.
But the Pareto/NBD succeeds in solving this dilemma. It uses the same
smart technique like the NBD model already does for modeling individual
purchase frequencies (see end of section 4.1.1), and applies this trick to the
defection process. In particular it assumes some sort of individual stochastic
dropout process, and makes assumptions regarding the form of heterogene-
ity across all customers at the same time. Thereby, the information of the
complete customer base can be used for modeling the individual customer.
The assumptions of the Pareto/NBD regarding consumer behavior are sum-
marized in table 4.6.10

A1 While active, the number of transactions follows a Pois-
son process with rate λ.

tomers.

A3 Customer lifetime is exponentially distributed with death
rate µ.

A4 Heterogeneity in µ follows a Gamma distribution with
shape parameters s and rate parameter β across cus-
tomers.

A5 The purchasing rate λ and the death rate µ are dis-
tributed independently of each other.
Table 4.6: Pareto/NBD Assumptions

A1 and A2 are identical with the already presented NBD model and hence the
same concerns regarding these assumptions apply again (see section 4.1.1).
Assumption A3 now postulates an exponentially distributed lifetime with a
10
For consistency reasons the ordering and wording of the assumptions is changed com-
pared to the originating paper in order to ease comparison with the other models presented
within this chapter.


certain ‘death’ rate µ for each customer. This assumption is justified by
Schmittlein et al. because ‘the events that could trigger death (a move, a
financial setback, a lifestyle change, etc.) may arrive in a Poisson manner’
(Schmittlein et al., 1987, p. 3). On the one hand, this seems entirely rea-
sonable. On the other hand, it is also hard to verify because the event of
defection is not observable. And even if the event was observable, defection
just occurs a single time for a customer and therefore reveals hardly any
information on the underlying death rate µ. But by making specific assump-
tions regarding the distribution of µ across customers (A4) an estimation
of the model for the complete customer base becomes feasible. Heterogene-
ity is again assumed to follow the flexible Gamma distribution, but with
two different parameters than for the purchase frequency. And because a
Gamma-Exponential mixture results in the Pareto distribution, the overall
model is termed Pareto/NBD model.
Finally, assumption A5 requires independence between frequency and life-
time. It is for example assumed that a heavy purchaser has neither a longer
nor a shorter lifetime expectancy than less frequent buyers. This assumption
is necessary in order to simplify the fairly complex mathematical derivations
of the model. Schmittlein et al. provide some reasoning for this assumption
and Abe (2008, p. 19) present some statistical evidence that λ and µ are
indeed uncorrelated.


Again, we will apply the presented model to the DMEF data set and subse-
quently evaluate its forecasting accuracy.
Several different methods for estimating the four parameters r, α, s and β of
our model are available. A two-step estimation method which tries to fit the
observed moments is suggested in Schmittlein et al. (1987) and described in
detail in Schmittlein and Peterson (1994, appendix A2). Nevertheless, the
MLE method seems to be more reliable for a wide range of data constellations.
But despite the ongoing increase in computational power, the computational
burden for calculating the maximum likelihood estimates are still challenging
(Fader et al., 2005a, p. 275). The bottleneck is the evaluation of the Gaussian
Hypergeometric function, which is part of the likelihood function, and as such
needs to be evaluated numerous times for each customer and for each step of
the numerical optimization procedure. An efficient and fast implementation
of that function is essential to make the estimation procedure complete in


reasonable time11 .
Estimating the model parameters requires another piece of information com-
pared to the NBD model, which is the actual timing of the last transaction
tx .12 Schmittlein et al. (1987) prove that tx is a sufficient information for the
model and that the actual timing of the preceding transactions (t1 ,..,tx−1 ) is
not required for calculating the likelihood. This is due to the memoryless
property of the assumed Poisson process.
The MLE method applied on the DMEF data set results in the following
parameter estimates

r = 0.659, α = 514.651, and
s = 0.471, β = 766.603,

with all four parameters being highly significant. The shape parameters
for both Gamma distributions (r and s) are well below 1 and therefore the
resulting distributions of the purchase rate λ and the death rate µ can again
be depicted from the outer left chart of figure 4.3. The resulting average time
√
between two transactions (α/r) is 781 days with a standard deviation (α/ r)
of 634 days and a median of 1,395 days. The corresponding theoretical
average lifetime (β/s) across the cohort is 1,629 days (=4.5 years) with a
√
standard deviation (β/ s) of 1,117 days and a median of 3,785 days (=over
10 years).
Comparing these numbers with the NBD results shows that due to the added
defection possibility the intertransaction time has dropped from 1,024 days
to 787 days. In other words, most of the active donor wait over two years
until they make another donation. Further, the average donor has a life ex-
pectancy of over 4 years, which is nearly as long as the provided time span.
These estimates still seem too high in comparison with our findings from the
exploratory data analysis. Assessing the theoretical standard deviations, it
can further be concluded that the overall extent of heterogeneity is consid-
erably high within the data set. In short, the estimated parameters suggest
that we are dealing with a heterogeneous, long living, rarely donating cohort
of donors.
11
Many thanks go to Dr. Hoppe, who provided us with a R wrapper package for the
impressively fast Fortran-77 implementation of the Gaussian Hypergeometric function
developed by Zhang et al. (1996). See http://jin.ece.uiuc.edu/routines/routines.html for
their source code. It was this contribution that made the herewith presented calculations
feasible for us.
12
By convention tx is set to 0, if no (re-)purchase has occurred within time span (0, T ].


These conclusions indicate that the fitted model does not fully take advantage
of the dropout possibility. According to the estimated model, 38.2% of the
donors are still active in the mid of 2006, which is a high number compared
to the 18.8% that actually made a donation in 2005 (see figure 3.7). On
the other hand, figure 3.9 indicates that there are indeed some donors with
intertransaction times of four years and more. In separate calculations, that
are not being presented here, it could be verified that this rather small group
of long-living, ‘hibernating’, ‘always-a-share’ donors has a significant effect
on the estimated parameter values. This occurs because the overall model
tries to fit the complete cohort including these outliers altogether.13
But, at what point does a customer finally defect? Maybe the postulated
concept of activity, which is that a customer can be either active or is lost for
good, is too shortsighted, too simple for the data set? Alternative approaches
that allow customers to switch between several states of activity back and
forth, such as Markov Chain models (cf. Jain and Singh, 2002, p. 39 for an
overview), might be more appropriate, especially when we consider the long
time span of the observation period.
Figure 4.4 depicts the estimated distributions for the donation frequency λ as
well as for the estimated death rate µ. The axes on top of the charts display
the related average intertransaction times respectively the average lifetime,
both being measured in number of days. The short vertical line segment at
that top axis represents the corresponding mean value.

Distribution of Purchase Frequency Distribution of Death Rate
Inf 250 125 83.3 62.5 50 Inf 250 125 83.3 62.5 50
100

100

shape = 0.66 shape = 0.47
80

80

rate = 515 rate = 767
60

60
40

40
20

20
0

0

0.000 0.005 0.010 0.015 0.020 0.000 0.005 0.010 0.015 0.020

Figure 4.4: Estimated Distribution of λ and µ across Donors
13
Nevertheless, for our final chosen model, the CBG/CNBD-k, these outliers did not
pose a relevant problem anymore and therefore we did not split up the data set in the
following.


Despite the lack of plausibility of the estimated parameters, the question
that matters most for our purpose is: How well does the Pareto/NBD pre-
dict future transactions for the DMEF data set? Did the forecast improve
compared to the NBD model or did we possibly overfit the training data?
For now, we will only reproduce the comparison on an aggregated level in
table 4.7. These numbers reveal that for the large share of no-repeaters
the Pareto/NBD surprisingly provides inferior results by making overly op-
timistic forecasts. But for all other groups the model succeeds in providing
a much closer fit to the actual transaction counts.

0 1 2 3 4 5 6 7+
Actual 0.038 0.20 0.43 0.69 0.75 1.06 1.54 2.44
NBD 0.001 0.42 0.84 1.27 1.69 2.11 2.53 4.68
Pareto/NBD 0.102 0.23 0.50 0.71 0.91 1.11 1.32 2.24
Table 4.7: Comparison of Actual vs. Theoretical Average Number of Donations
per Donor during the Validation Period

All further assessments of this model’s accuracy are deferred to chapter 5,
which provides a detailed, extensive comparative analyses of all presented
models.

4.3 BG/NBD Model

4.3.1 Assumptions

18 years after the introduction of the Pareto/NBD model, Fader, Hardie,
and Lee (2005a) call attention to the discrepancy between the raised scientific
interest in that model, measured in terms of citations, and the small numbers
of actual implementations. They argue that it is the inherent mathematical
complexity and the computational burden of the Pareto/NBD that keeps
practitioners from applying it to real world data.
As a solution Fader et al. introduce an alternative model which makes a
slightly different assumption regarding the dropout and termed it the Beta-
geometric/NBD (abbr. BG/NBD) model. They succeed in simplifying the
mathematical key expressions of the model and further demonstrate that an
implementation is nowadays even possible with standard spreadsheet appli-


cations, such as MS Excel.14 Further, they show that despite this change
in the assumptions, the accuracy of the resulting fit and the individual pre-
dictive strength are for most of the possible scenarios very similar to the
Pareto/NBD results.


tomers.

A3 Directly after each purchase there is a constant probabil-
ity p that the customer becomes inactive.

A4 Heterogeneity in p follows a Beta distribution with pa-
rameters a and b across customers.

A5 The transaction rate λ and the dropout probability p are
distributed independently of each other.
Table 4.8: BG/NBD Assumptions

The assumed behavioral ‘story’ regarding the dropout process is modified by
Fader et al. in that respect that an existent customer cannot defect at an
arbitrary point in time but only right after a purchase is being made. This
modification seems to be plausible to some extent, because the customer is
most likely to have either a positive or a negative experience regarding the
product or service right after the purchase. And this extent of satisfaction
will have a strong influence on the future purchase decisions.
Assumption A3 claims that the probability p of such a dropout remains con-
stant throughout an individual customer lifetime. As such, lifetime measured
in number of ‘survived’ transactions results in a geometric distribution. This
distribution can be seen as the discrete analogue to the continuous expo-
nential distribution since it is also characterized by being memoryless. This
means that the number of already ‘survived’ transactions does not effect the
drop out probability p for the upcoming transaction. This assumption also
seems reasonable since it is possible to find arguments in favor of high early
14
The Microsoft Excel implementation of the BG/NBD model can be downloaded from
http://www.brucehardie.com/notes/004/.


drop out probabilities (e.g. customer is still trying out the product) as well as
high drop out probabilities later on (e.g. customer becomes tired of a certain
product and is more likely to switch for something new).15
A4 is an assumption regarding the heterogeneous distribution of the dropout
rate. But as opposed to the death rate µ, the constant drop out probability
p is bound between 0 and 1, and therefore the Beta distribution which shares
the same property is considered. As can be depicted from figure 4.5, this dis-
tribution is, like the Gamma distribution, also fairly flexible and is defined
by two shape parameters. Aside from its provided flexibility no particular
justification for the Beta distribution is being provided. The resulting mix-
ture distribution is generally referred to as the Betageometric distribution
(BG).

Beta Distribution
2.5

2.5

2.5
a = 0.5 a=1 a=2
2.0

2.0

2.0
b = 0.7 b=3 b=5
1.5

1.5

1.5
1.0

1.0

1.0
0.5

0.5

0.5
0.0

0.0

0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
2.5

2.5

2.5

a=1 a=1 a = 1.5
2.0

2.0

2.0

b=1 b = 1.5 b=2
1.5

1.5

1.5
1.0

1.0

1.0
0.5

0.5

0.5
0.0

0.0

0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Figure 4.5: Probability Density Function of the Beta distribution for Different
Parameter Values

Assumption A5 requires independence between the dropout probability and
the purchase frequency. But attention should be paid to the result that the
actual lifetime measured in days and not in number of survived purchases
is, compared to the Pareto/NBD, not independent of the purchase frequency
anymore. The more frequent a customer purchases, the more opportunities
to defect he/she will have, and because of the independence of p are λ the
15
Note that the previously made critical remarks regarding the memoryless property re-
ferred to the exponentially distributed intertransaction times and not to the exponentially
distributed lifetimes of the Pareto/NBD model.


sooner that customer will defect (Fader et al., 2005a, p. 278). Interestingly,
this fundamentally different consequence of A5 does not seem to play an
important role in the overall model accuracy.


The implementation of the BG/NBD model on top of R has been indeed fairly
straightforward, in particular because of the provided MATLAB source code
in Fader et al. (2005b) which simply had to be ‘translated’ from one statistical
programming environment to another. Also the computation of the maxi-
mum likelihood estimation itself finishes far faster than for the Pareto/NBD
because the Gaussian Hypergeometric function is not part of the optimized
likelihood function anymore.16
The MLE method produced the following parameter estimates:

r = 0.397, α = 331.8, and
a = 0.777, b = 6.262.

In accordance with the statements of Fader et al. (2005a), the overall char-
acteristic of the distribution of transaction frequency λ across donors is not
much different from the Pareto/NBD model. The corresponding mean is
slightly higher (858 days) and the standard deviation slightly lower (546
days) for our estimated BG/NBD model.
The dropout probability p varies around its mean a/(a + b) of 11%. The 11%
correspond to an average life time of 9.1 ‘survived’ donations. Considering
that the average number of donations has been 1.55 times, the underlying
data seems again to be represented rather poorly. Further, figure 4.6 depicts
the estimated distributions of λ and p and reveals that hardly any of the
donors has a lifetime of less than 5 donations. Again this result is quite
contrary to our findings from the exploratory analysis in chapter 3. It is likely
that the same concerns regarding those problematic long living customers,
that have already been raised in section 4.2, apply here too.
Additionally, the simulation results of Fader et al. (2005a, p. 279) show that
the BG/NBD model has problems mimicking the Pareto/NBD model if the
transaction rate is very low, like it is the case for the DMEF data set. The
16
It took about 15 seconds on the author’s personal laptop, which is powered by a
Intel Centrino 1.6GHz chip, to complete the calculations for the DMEF data set of 21,166
donors.


Distribution of Purchase Frequency Distribution of Drop Out Probability
Inf 250 125 83.3 62.5 50 Inf 5 2.5 1.7 1.2 1
100

10
shape = 0.4 a = 0.77
80

8
rate = 332 b = 6.26
60

6
40

4
20

2
0

0
0.000 0.005 0.010 0.015 0.020 0.0 0.2 0.4 0.6 0.8 1.0

Figure 4.6: Estimated Distribution of λ and p across Donors

upcoming model will present a variant of the BG/NBD which fortunately
can solve this issue.

4.4 CBG/NBD Model

4.4.1 Assumptions

The CBG/NBD is a modified variant of the BG/NBD model and has been
developed by Daniel Hoppe and Udo Wagner (Hoppe and Wagner, 2007).
This variant makes similar assumptions as before but inserts an additional
dropout opportunity at time zero. By doing so it resolves the rather unre-
alistic implication of the BG/NBD model that all customers that have not
(re-)purchased at all after time zero are still active. Hoppe and Wagner also
show that their modification results in a slightly better fit to the publicly
free available CDNOW data set that has been already used by Fader et al.
(2005a) as a benchmark.
Aside from providing this new variant of the BG/NBD Hoppe and Wagner
additionally contribute valuable insight by deriving their mathematic key
expressions by focusing on counting processes instead of timing processes and
thereby can reduce the inherent complexity in the derivations significantly.
For this reason the article Hoppe and Wagner (2007) is a highly recommended
reading also in terms of gaining a deeper understanding of the BG/NBD
model.
Around the same time as Hoppe and Wagner worked on their model, Batis-


lam, Denizel, and Filiztekin developed the same modification of the BG/NBD
and termed it MBG/CBG (Batislam et al., 2007), whereas the letter M stands
for modified. Within this thesis we choose to use the abbreviation CBG/NBD
instead of MBD/NBD when we refer to this kind of variant, because the term
CBG adheres a deeper meaning as it abbreviates central variant of the Be-
tageometric distribution.


tomers.

A3 At time zero and directly after each purchase there is a
constant probability p that the customer becomes inac-
tive.

A4 Heterogeneity in p follows a Beta distribution with pa-
rameters a and b across customers.

A5 The transaction rate λ and the dropout probability p are
distributed independently of each other.
Table 4.9: CBG/NBD Assumptions

As can be seen in table 4.9, assumptions A1, A2, A4, and A5 are identical to
the corresponding assumptions of the BG/NBD model. Only assumption A3
is slightly modified. It now allows for the aforementioned immediate defect
of a customer at time zero. The same constant probability p is used for this
additional dropout opportunity.


The BG/NBD assumptions imply that all single-time donors, which repre-
sent the majority of the data set, are still ‘active’ despite an inactivity period
of over 4.5 years. Taking this implausible implication into account, it can
be expected that the added dropout opportunity of the CBG/NBD model is
necessary to fit our data structure appropriately.


Our implementation on top of R results in the following parameter estimates:

r = 1.113, α = 552.5, and
a = 0.385, b = 0.668.

The related estimated distributions of λ and p can be depicted from figure 4.7.

Distribution of Purchase Frequency Distribution of Drop Out Probability
Inf 250 125 83.3 62.5 50 Inf 5 2.5 1.7 1.2 1
100

10
shape = 1.11 a = 0.38
80

rate = 552 8 b = 0.67
60

6
40

4
20

2
0

0

0.000 0.005 0.010 0.015 0.020 0.0 0.2 0.4 0.6 0.8 1.0

Figure 4.7: Estimated Distribution of λ and p across Donors

Comparing this with figure 4.6 from the previous section, we notice the fun-
damentally different shape for the distribution of the dropout probability.
It has one peak at 1, representing the single-time donors, and one peak at
0, representing those loyal, long-living donors which hardly defect at all.
The mean number of repetitive donations is now 2.7 times, and seems much
more realistic in comparison with the estimate of 9.1 donations made by the
BG/NBD model. On the other hand, the detected level of heterogeneity
within life time, measured in terms of the standard deviation of p, increased
from 0.11 to 0.34 for the CBG/NBD model at the same time.
Further, the average intertransaction time has dropped from 836 to 496 days
with the standard deviation remaining at the high level of 524 days. This is
a logical effect, since the single-timer donors are now allowed to defect im-
mediately and do not bias the donation frequency anymore. The same con-
sequence, a higher mean purchase rate together with a higher dropout prob-
ability, has been diagnosed by Hoppe and Wagner (2007) for the CDNOW
data set.
If we observe the estimates for the number of active donors at the end of
the observation period, then the difference between these models become

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