Stochastic Models of Noncontractual Consumer Relationships

Stochastic Models of Noncontractual Consumer Relationships | | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || | | | | | | || || | ||| | | | | | || || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || | | | | | || | | | | | | | | | | || | | | | | | | | | | | | | | | | | | | | || | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 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

Dedicated to my Mom & Dad

Abstract The primary goal of this master thesis is to evaluate several well-established probabilistic models for forecasting customer behavior in noncontractual settings 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 prediction 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 consumer 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 assessment 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 several 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 purchase 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.

CHAPTER 1. INTRODUCTION 3 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 predictions 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 relationship 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 emphasis 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.

CHAPTER 1. INTRODUCTION 4 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 assumptions 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;

CHAPTER 1. INTRODUCTION 5 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 questions of this kind.

Chapter 2 DMEF Competition 2.1 Contest Details The Direct Marketing Educational Foundation1 (DMEF) is a US based nonprofit 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 committee 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/

CHAPTER 2. DMEF COMPETITION 8 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 customers, 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 competition data set represents therefore only a small subset of the complete available 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

CHAPTER 2. DMEF COMPETITION 9 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 validation 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 implicitly 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 competition, 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 estimated 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.

CHAPTER 2. DMEF COMPETITION 10 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 characteristic timing patterns. These patterns range from single-time donors (e.g.

CHAPTER 3. EXPLORATORY DATA ANALYSIS 13 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 regular 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 further 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 customer 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

CHAPTER 3. EXPLORATORY DATA ANALYSIS 14 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 tremendously, 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 overall 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 donation 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.

CHAPTER 3. EXPLORATORY DATA ANALYSIS 15 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 correlation 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 aggregated 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

CHAPTER 3. EXPLORATORY DATA ANALYSIS 16 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

CHAPTER 3. EXPLORATORY DATA ANALYSIS 17 # 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%.

CHAPTER 3. EXPLORATORY DATA ANALYSIS 18 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 information regarding the contact activities throughout the target period is available (see section 2.3 for the previous discussion).

CHAPTER 3. EXPLORATORY DATA ANALYSIS 19 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).

CHAPTER 3. EXPLORATORY DATA ANALYSIS 20 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 according 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 customers. Table 4.1: NBD Assumptions In order to support the reader’s understanding of the postulated assumptions, visualizations of the aforementioned distributions are provided in figure 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 transactions follows an exponential distribution. In other words, the Poisson process with rate λ is the respective count process for a timing process with independently 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, because 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

CHAPTER 4. FORECAST MODELS 23 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

CHAPTER 4. FORECAST MODELS 24 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 nondurable consumer goods for an individual customer. The regularity of consumption 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 purchase rates across customers, in order to account for heterogeneity. If the different possible shapes of this two-parameter continuous probability are being 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 heterogeneity. He utilizes information of the complete customer base for modeling on an individual level. He thereby takes advantage of the well-established regression to the mean phenomenon. ‘[We] can better predict what the person 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.

CHAPTER 4. FORECAST MODELS 25 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 assessment 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).

CHAPTER 4. FORECAST MODELS 26 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 values 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 random sample from the theoretical distribution and subsequently calculating the empirical median.

CHAPTER 4. FORECAST MODELS 27 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.

CHAPTER 4. FORECAST MODELS 28 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 noncontractual 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 individual 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

CHAPTER 4. FORECAST MODELS 29 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 heterogeneity 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. A2 Heterogeneity in λ follows a Gamma distribution with shape parameter r and rate parameter α across customers. 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 customers. A5 The purchasing rate λ and the death rate µ are distributed 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 compared to the originating paper in order to ease comparison with the other models presented within this chapter.

CHAPTER 4. FORECAST MODELS 30 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 reasonable. 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 assumptions 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 lifetime. 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. 4.2.2 Empirical Results Again, we will apply the presented model to the DMEF data set and subsequently 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

CHAPTER 4. FORECAST MODELS 31 reasonable time11 . Estimating the model parameters requires another piece of information compared 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 expectancy 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 considerably 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 ].

CHAPTER 4. FORECAST MODELS 32 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.

CHAPTER 4. FORECAST MODELS 33 Despite the lack of plausibility of the estimated parameters, the question that matters most for our purpose is: How well does the Pareto/NBD predict 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-

CHAPTER 4. FORECAST MODELS 34 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 predictive strength are for most of the possible scenarios very similar to the Pareto/NBD results. A1 While active, the number of transactions follows a Pois- son process with rate λ. A2 Heterogeneity in λ follows a Gamma distribution with shape parameter r and rate parameter α across customers. A3 Directly after each purchase there is a constant probability p that the customer becomes inactive. A4 Heterogeneity in p follows a Beta distribution with parameters 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 constant 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 exponential 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/.

CHAPTER 4. FORECAST MODELS 35 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 distribution 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 mixture 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 referred to the exponentially distributed intertransaction times and not to the exponentially distributed lifetimes of the Pareto/NBD model.

CHAPTER 4. FORECAST MODELS 36 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. 4.3.2 Empirical Results 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 maximum 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 characteristic 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.

CHAPTER 4. FORECAST MODELS 37 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-

CHAPTER 4. FORECAST MODELS 38 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. A1 While active, the number of transactions follows a Pois- son process with rate λ. A2 Heterogeneity in λ follows a Gamma distribution with shape parameter r and rate parameter α across customers. A3 At time zero and directly after each purchase there is a constant probability p that the customer becomes inactive. A4 Heterogeneity in p follows a Beta distribution with parameters 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. 4.4.2 Empirical Results The BG/NBD assumptions imply that all single-time donors, which represent 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.

CHAPTER 4. FORECAST MODELS 39 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 fundamentally 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 immediately and do not bias the donation frequency anymore. The same consequence, a higher mean purchase rate together with a higher dropout probability, 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

CHAPTER 4. FORECAST MODELS 40 even more apparent. The Pareto/NBD states that 38.2% of the donors are active,17 , the CBG/NBD produces a similar estimate of 34.4%, whereas the BG/NBD18 assumes that 94.7% (!) have still not defected in the mid of 2006. After having analyzed the estimated parameters and their implications, we ﬁnd that the CBG/NBD model is better capable of explaining the characteristics of the DMEF data set than the BG/NBD model. As such the relatively new CBG/NBD model seems to be a valuable contribution to the domain of stochastic purchase models. 17 A donor is assumed to be active by us if her conditional probability of being active is higher than 0.5. 18 A mathematical expression for the probability of a customer being active for the BG/NBD model is given in Hoppe and Wagner (2008, section 4).

Chapter 5 Model Comparison This chapter provides an in-depth analysis of the performance of the previously presented models regarding the DMEF data set. First, we will assess the fit of these statistical models, and secondly determine their forecast ac- curacies.1 Our ultimate aim is to identify the model which will most likely provide us with a minimal mean squared logarithmic error for the target period of the contest. 5.1 Parameter Interpretation Table 5.1 provides a condensed overview of the calculated parameter estimates, together with their standard error.2 All of the estimated parameters are highly significant different from zero. Since these values are just specific parameters of the assumed heterogeneity distributions, namely of the Gamma and the Beta distribution, a display of the key statistical moments of these distributions is essential for interpreting the results. Table 5.2 displays the distribution of average lifetimes,3 and table 5.3 the distribution of the mean intertransaction times across the cohort for each model. As has already been stated in section 4.4, the CBG/NBD model seems to be the only model that results in plausible parameter es- 1 Generally speaking, a good fit to the data does not automatically guarantee an ability to extrapolate for new data, i.e. to make forecasts into the future. 2 The standard error is returned by the MLE implementation mle2 which is part of the R package bbmle (Bolker, 2008). 3 sd abbreviates standard deviation, d stands for days and t for number of transactions. 41

CHAPTER 5. MODEL COMPARISON 42 NBD Pareto/NBD BG/NBD CBG/NBD r (se) 0.48 (0.01) 0.66 (0.01) 0.40 (0.01) 1.11 (0.05) α (se) 499 (10) 515 (11) 332 (8) 552 (19) s (se) 0.471 (0.03) β (se) 767 (69) a (se) 0.78 (0.10) 0.38 (0.02) b (se) 6.26 (1.00) 0.67 (0.04) Table 5.1: Estimated Model Parameters timates which do not conﬂict with our ﬁndings from the exploratory data analysis phase. mean median sd NBD ∞ ∞ - Pareto/NBD 1,629 d 3,785 d 1,117 d BG/NBD 9.1 t 13.3 t 9.0 t CBG/NBD 2.7 t 3.8 t 3.0 t Table 5.2: Statistical Summary of Fitted Life Times mean median sd NBD 1,048 d 2,413 d 723 d Pareto/NBD 781 d 1,395 d 634 d BG/NBD 836 d 2,324 d 527 d CBG/NBD 496 d 688 d 523 d Table 5.3: Statistical Summary of Fitted Intertransaction Times 5.2 Data Fit The models’ abilities to explain the observed transaction patterns are subject of this section. This task has already been done partially in the preceding chapter 4, but we will now provide a complete side-by-side comparison of all four models to gain an accurate overview. Table 5.4 groups the cohort of 21,166 donors according to their number of transactions within the complete observed training period of over 4.5 years. The actual size of each of these groups is being compared to the expected sizes that have been calculated by the distinct models. The closer these

CHAPTER 5. MODEL COMPARISON 43 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 Pareto/NBD 10,642 3,933 2,173 1,358 899 615 430 1,114 BG/NBD 10,461 4,248 2,231 1,338 858 574 395 1,060 CBG/NBD 10,647 3,939 2,186 1,368 905 617 429 1,075 Table 5.4: Comparison of Actual vs. Expected Count Data for the Complete Time Span numbers are to the actual count data, the better is the model fit, at least on an aggregated level. A first look at these numbers reveals that all models with the exception of the BG/NBD model nearly perfectly fit the share of single-time donors. Other than that, a fairly big mismatch regarding the other groups can be detected for all of the models. Interestingly, all models display a bias into the same direction. The number of donors that re-donate once more (1), and also the number of frequent donors (5+) are all overestimated, whereas the remaining groups (2, 3, 4) are all underestimated. Actual vs Fitted Frequency of Repeat Transactions 10000 Observed NBD Pareto/NBD 8000 BG/NBD χ2NBD = 366.1 CBG/NBD χ Pareto/NBD 2 = 391.5 χ2BG/NBD = 487.2 6000 Frequency χ2CBG/NBD = 363.7 4000 2000 0 0 1 2 3 4 5 6 7+ # Transactions Figure 5.1: Fitted Distributions There are several possible causes for this phenomena. Probably the most

CHAPTER 5. MODEL COMPARISON 44 apparent one is that the actual group sizes do not decrease gradually. The drop in group sizes from 3 to 4 is only 17%, but from 4 to 5 it is a decrease of 59%. The overly large amount of people who donated 4 times can be explained by the existence of regular, yearly donors (see section 3.4) in combination with an overall time period of 4.66 years. And because none of the models accounts for any kind of regularity, they are all not capable of fitting this deviation. Figure 5.1, which resembles figure 2 of Fader et al. (2005a, p. 281), visualizes the bias of our four models. Additionally, the chart includes the calculated χ2 statistics, which can act as a measure for the fit to the actual distribution. According to the ranking of these values, the CBG/NBD model provides the best fit. Though to our surprise, the much simpler NBD model performs nearly as good as CBG/NBD and clearly outperforms the Pareto/NBD and the BG/NBD models. Another assessment of the overall data fit can be made by comparing the calculated loglikelihood (abbr. LL) values. The higher this value is the better does the model approximate the data. This method has the advantage that it operates on an individual and not on an aggregated level. Table 5.5 displays the results of this comparison. Rank Model LL I. Pareto/NBD -245,674.2 II. CBG/NBD -245,702.2 III. BG/NBD -245,833.0 IV. NBD -246,552.5 Table 5.5: Comparison of Calculated Loglikelihood Values According to this measure, the ranking of the models is different than before. The Pareto/NBD and the CBG/NBD show the best performance, whereas the BG/NBD is slightly behind and the NBD model finishes last in explaining the DMEF data. 5.3 Forecast Accuracy In order to compare the forecast accuracy of several models we need to split our data set into a calibration period and a validation period. The former is used to estimate the model parameters and the latter is necessary for assessing the difference between the predicted and the actual values. If not

CHAPTER 5. MODEL COMPARISON 45 stated otherwise, then we will choose a calibration period of 3.5 years and a validation period of 1 year in the following. In section 5.3.4 we will select different time splits in order to test the stability of our findings. Time Split Calibration Validation Period Period 2002 2003 2004 2005 2006 Figure 5.2: Default Time Split As there is no single ‘best’ method to assess the forecast accuracy, several different techniques and measures are being presented. Ultimately, the error measure defined by the DMEF contest committee will certainly be our decision criteria for the final submitted model. 5.3.1 Cumulative Repeat Transactions One of the basic managerial questions that have been stated in the introduc- tory section of this thesis is ‘How many transactions can I expect from my client`le in the future?’. Although the strength of the investigated models is e the modeling on an individual level, it is also expected that the cumulative numbers provide a good estimate for the overall transaction volume. Table 5.6 provides a comparison of these numbers for each model, and shows that on an aggregated level the CBG/NBD performs best but is still considerably off from the actual value (18.7% deviance). The next section will give some insight on the cause of this misfit of the cumulative estimate. Actual NBD Pareto/NBD BG/NBD CBG/NBD 6,047 11,088 7,351 7,219 7,179 +83.4% +21.6% +19.4% +18.7% Table 5.6: Comparison of Number of Overall Transactions Within Validation Pe- riod

CHAPTER 5. MODEL COMPARISON 46 5.3.2 Grouped by Transaction Count Along the lines of Fader et al. (2005a, p. 281) a visualization of the conditional expectations is provided in figure 5.3, together with the associated data table 5.7. The cohort is grouped by their number of transactions during the 3.5 year calibration period. Thereafter, the average predicted number of transactions during the validation period is compared to the actual average number for each group. The closer the estimates are, the better is the forecast ability of the model. 0 1 2 3 4 5 6 7+ Actual 0.04 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 0.10 0.29 0.50 0.71 0.91 1.11 1.32 2.24 BG/NBD 0.11 0.29 0.49 0.69 0.87 1.05 1.25 2.04 CBG/NBD 0.11 0.29 0.48 0.68 0.87 1.05 1.26 2.13 Group Size 10,988 3,910 2,683 1,730 731 392 239 493 Table 5.7: Comparison of Actual vs. Predicted avg. Number of Donations During the Validation Period Despite the surprisingly good data fit of the NBD model, that we observed in section 5.2, the model is not able to extrapolate into the future. Due to the lack of a defection process, the NBD model simply assumes that the past transaction frequencies can be applied to the future, and therefore the number of transactions are tremendously overestimated. All the other models provide considerably better and quite similar4 results. Surprisingly, the deviations of all models display the same direction again. This is a strong indicator of an underlying systematic mechanism that has not been taking into consideration by any of these models. First of all, the large group of donors with no repetitive donations are more than 3 times overestimated, i.e. 0.11 expected transactions versus 0.04 actual transactions. This presumably indicates that the defection process has not been modeled correctly, and that too many donors are still being considered active although they have defected long time ago. On the other hand, the number of transactions of the frequent donors (6+) are predicted by 10% to 20% too low, indicating that the underestimated defection process goes hand in hand with 4 For this reason we do not reproduce the BG/NBD model within figure 5.3 as it would clutter the chart. As can be seen from the data table, its numbers are within the close range of the Pareto/NBD and the CBG/NBD model.

CHAPTER 5. MODEL COMPARISON 47 Conditional Expectation of Future Transactions 5 Avg # Transactions in Validation Period Actual 4 NBD Pareto/NBD CBG/NBD 3 2 1 0 0 1 2 3 4 5 6 7+ # Transactions in Training Period Figure 5.3: Conditional Expectations an underestimated transaction frequency. This is the very same bias that we have already concluded in the previous section. Furthermore, the solid line representing the observed data in ﬁgure 5.3 reveals an unexpected slight bend at group 3 and 4. The average number of future transactions for the cohort group that donated 3 times seems slightly higher than expected and group 4 seems to be slightly too low. A possible explanation might again lie in the detected regularity within the donation behavior. A person who consequently donates once per year will most likely fall into group 3. This is due to the chosen length of 3.5 years in combination with the observed seasonality (see section 3.3) as the strong fourth quarter starts shortly after the calibration period ends. And such a regular donor will, unless he/she defects, make exactly one donation within the following year. On the other hand, someone who donated 4 times is probably not such a regular yearly donor but rather had a higher transaction frequency. If we additionally make the plausible assumption that an irregular donor is more likely to defect sooner than later, i.e. there is a negative correlation between regularity and dropout probability, then this slight bend in the curve is a logical consequence of the chosen time frame and the observed regularity.

CHAPTER 5. MODEL COMPARISON 48 5.3.3 Individual Level Forecasts All presented models are capable of making conditional estimates for each of the 21,166 donors based upon their individual past transaction records. But for each one of them the estimates will likely deviate from the actual value to some extent. The question is, how do we aggregate these individual errors into a single overall figure? Several measures are common in the referred papers, each one of them having their particular advantages. Probably the most basic form is the mean absolute deviation (abbr. MAE). The root mean squared error (abbr. RMSE), which builds the average over the squared individual errors, is also fairly simple and thus similarly popular. The main obstacle of the RMSE is that it puts a strong emphasis on the proper fit of all data points including any potential single outliers. Minimizing the RMSE therefore commonly results in a mediocre fit, because it is sensitive to these outliers and does not focus on the dominant patterns of the data. The median of squared errors (see for example W¨bben and von Wangenheim, 2008, p. 88) resolves this issue and u is robust regarding these outliers. Fader et al. (2005a, p. 282) interestingly suggested the correlation between estimated and actual data as a performance quantity. The correlation is a measure for the linear relation between two variables, and as such only provides information whether two variables change in unison but not whether these two values are actually close together. Hoppe and Wagner (2007, p. 85) used the geometric mean relative absolute error (GMRAE) to evaluate different models. The GMRAE is a relative measure which compares a model with some other particular benchmark model. In their article the NBD model acted as such a benchmark. Nevertheless, the contest committee decided to use the mean squared logarithmic error, which has been defined as followed5 2 MSLE = (log(yi + 1) − log(î + 1)) /21.166 y (5.1) i 2 yi + 1 = log( ) /21.166. i yi + 1 ˆ The MSLE takes the square of the logarithmic of the relative error, as opposed to the absolute error. As such it puts much more emphasis on the 5 Note, that y depicts the actual donation amount in dollars. For now we assume that each transaction has the same amount of $1, and use this error measure also to assess the forecasting accuracy regarding the number of transactions.

CHAPTER 5. MODEL COMPARISON 49 accurate estimate of the dominant group of donors with low transaction vol- umes, and is less sensitive regarding large values. In a separate simulation study, which generated artificial transaction records according to the assumptions of the BG/NBD model, we could show that the MSLE measure favors forecasts that systematically underestimate. In particular, the MSLE could be lowered by another 5% simply by subtracting 25% of the individual estimates. This is a quite surprising result, especially as we know the exact data creating mechanism in this simulation and therefore can exclude any systematic error. The same effect can also be identified for the estimates of the DMEF data set. Therefore, we certainly take advantage of this finding, and try to determine an optimal multiplication factor for our estimates in order to further minimize the MSLE. One possible explanation for this effect might lie in the following numerical example: If there is a 50% chance of y = 0 transactions and a 50% chance of y = 1 transaction occurring, then the naive guess x for the outcome would ˆ naturally be y = 0.5·0+0.5·1 = 0.5. This estimate also minimizes the expected ˆ RMSE. But, as can be shown by simple analytical derivatives, the expected √ MSLE is minimal for y = 2 − 1 = 0.414, i.e. for a 17% lower estimate! For ˆ the competition we tried to take advantage of this particular characteristic of the MSLE, and applied an ‘optimal’ factor to our estimates (see section 6.4.2 for the final model). Table 5.8 provides a condensed overview of various error measures for the four presented models. The result of the best model regarding a specific measure is printed in bold figures. MSLEopt denotes the ‘optimal’ MSLE that can be achieved by applying a multiplication factor (ratio) to the calculated estimates. The optimal ratio is found a posteriori by simply calculating the MSLE for all ratios with a precision of two digits behind the comma within the range (0, 2). MSLE RMSE MAE Corr MSLEopt (ratio) NBD 0.1587 0.849 0.415 0.597 0.0901 (0.37) Pareto/NBD 0.0977 0.653 0.359 0.628 0.0879 (0.66) BG/NBD 0.0963 0.651 0.362 0.640 0.0880 (0.68) CBG/NBD 0.0959 0.650 0.360 0.639 0.0878 (0.68) Table 5.8: Error Measures on Individual Level The table provides several insights. First of all, we have different rankings for different error measures. There is no single overall best model for the data set at hand. Regarding the MSLE and the RMSE, the CBG/NBD model

CHAPTER 5. MODEL COMPARISON 50 performs best. But surprisingly, despite its irritating parameter values, the BG/NBD performs only marginally worse with respect to the MSLE, and even outperforms all other models in terms of correlation. By multiplying our results with 0.68 we can further reduce the MSLE by 8%. All in all, the BG/NBD and the CBG/NBD produce very similar estimates.6 But since the CBG/NBD results in far more plausible parameter estimates our top choice for the DMEF competition would currently be the CBG/NBD model (combined with a multiplication factor of 0.68). 5.3.4 Robustness For the DMEF contest we will ultimately need to calibrate our model based upon the full length of 4.66 years and forecast the following target period of 2 years. In order to gain some confidence regarding our findings from the preceding section we will now try out several different time splits. The following table 5.9 contains the results on an individual level as well as on an aggregate level (SUM) for a time split of ‘3 years to 1 year’, ‘2.5 years to 1 year’ and ‘2.5 years to 2 years’. We will only consider validation periods whose lengths are multiples of one year in order to diminish problems occurring from the strong seasonal influence that have already been noticed in section 3.3. MSLE RMSE MAE Corr MSLEopt SUM Pareto/NBD 0.1132 0.648 0.409 0.606 0.0969 (0.59) +35% BG/NBD 0.1095 0.636 0.398 0.626 0.0949 (0.61) +31% CBG/NBD 0.1096 0.636 0.398 0.625 0.0949 (0.61) +31% 3 Years Calibration, 1 Year Validation Pareto/NBD 0.1157 0.672 0.425 0.610 0.1053 (0.67) +19% BG/NBD 0.1157 0.671 0.425 0.613 0.1053 (0.67) +19% CBG/NBD 0.1160 0.672 0.426 0.610 0.1055 (0.67) +20% 2.5 Years Calibration, 1 Year Validation Pareto/NBD 0.2319 1.189 0.740 0.622 0.1879 (0.56) +28% BG/NBD 0.2323 1.187 0.741 0.625 0.1880 (0.56) +28% CBG/NBD 0.2331 1.190 0.742 0.622 0.1882 (0.56) +29% 2.5 Years Calibration, 2 Year Validation Table 5.9: Error Measures for several Time Splits Again, the results show neither a clear winner nor a loser. For all scenarios, 6 The correlation between these two estimates is actually 0.998.

CHAPTER 5. MODEL COMPARISON 51 the optimal adjustment factor is somewhere between 0.56 and 0.67, and in all cases it improves the MSLE signiﬁcantly. 5.4 Simple Forecast Benchmarks So far, we have obtained an impression of the comparative performance of the presented stochastic models. But how good are these models really? This section will benchmark the models against a very simple heuristic estimate and also against a basic linear regression model. As we will see, the results give a rather disillusioning answer to the raised question. A basic heuristic estimate is to assign each donor the same number of transactions for the following year as in the preceding year, and adjust this estimate by a factor that corresponds to the decrease in contact costs. Figure 3.8 from the exploratory data analysis depicts that the contact costs decreased by 33% within the validation period.7 Additionally, we calibrate a linear regression model, which models the number of future transactions to its past number of transactions, as well as its mixed eﬀect with recency (= T −tx ; see section 4.1.2). For this purpose we have to further split our calibration period of 3.5 years into a 2.5 year period for the input data and a 1 year period for the response variable. This yields the following model. y = 0.112 + 0.364 · x − 0.0005 · x · (T − tx ), ˆ Variable x denotes the number of transactions within the previous 2.5 years, and y is the estimated number of transactions for the following year. For ˆ those donors, who did not donate at all within the past 2.5 years we further assumed that they have defected and as such will not donate again. Table 5.10 now contains the surprising results for these very simple models. The linear model performs better than all other models regarding the MSLE, the RMSE, the correlation and regarding the optimized MSLE. And also our heuristic is able to beat the Pareto/NBD model, at least regarding the MSLE and the MAE measure. However, the good performance of the heuristic is 7 We assume that managers can assess their contact costs a year ahead, and therefore can use this information for their managerial heuristic. But even if we did not know the exact decrease, we could have guessed that the downward trend in contact costs will continue.

CHAPTER 5. MODEL COMPARISON 52 likely a result of the vastly underestimated transaction numbers and not from a good explanation of the overall data structure as can be seen from the corresponding correlation and MSLEopt values. MSLE RMSE MAE Corr MSLEopt SUM Heuristic 0.0962 0.661 0.258 0.615 0.0909 (0.70) -22% LM 0.0863 0.642 0.262 0.644 0.0861 (0.93) -31% Pareto/NBD 0.0977 0.653 0.359 0.628 0.0879 (0.66) +22% BG/NBD 0.0963 0.651 0.362 0.640 0.0880 (0.68) +19% CBG/NBD 0.0959 0.650 0.360 0.639 0.0878 (0.68) +19% Table 5.10: Error Measures for Benchmark Models W¨bben and von Wangenheim (2008) recently published an interesting ar- u ticle ‘Instant Customer Base Analysis: Managerial Heuristics Often “Get It Right” ’ with results along the same line. They demonstrate for several data sets that heuristic assessments by marketing experts can perform as good as the far more complex probabilistic models, especially when it comes to classifying the customers according to their activity status. 5.5 Error Composition The conclusion that a simple linear regression model is able to outperform the far more complex probabilistic models pushes our motivation to search for further improvements regarding our models. Some of the further approaches that we investigate are: 1. Separating the long-living but rarely donating customers from the cohort, in order to improve the validity of the estimated parameters. See the related remarks in section 4.2. 2. Removing the ﬁrst and subsequently also the second year from the transaction records, in order to put a stronger emphasis on more recent data. All the models implicitly assume stationarity in the parameters, and this assumption might be violated for long histories. Schmittlein et al. (1987, p. 18) suggest to use only two years of data, even if more data is available to cope with this issue. 3. Scaling the time units from days to months, in order to remove some of the inherent noise in the data (compare ﬁgures 3.9 and 3.10).

CHAPTER 5. MODEL COMPARISON 53 All of these attempts succeed in improving the results of the CBG/NBD model with respect to MSLE. However, none of them is able to outperform the far simpler linear regression model.8 A possible room for improvement might lie in an analysis of the error structure. We need to find out why our models perform so poorly, and especially detect those donors who cause the most problems for these models. It can be assumed that there is some underlying systematic mechanism within the error structure, which subsequently would help us in improving our estimates, if we were able to take such a systematic into account. We first start out by charting the overall distribution of the errors across the cohort and plot a Lorenz curve for this purpose. Figure 5.4 displays the cumulated share of errors with respect to the MSLE against the cumulated share of donors. This provides an impression of the (in)equal distribution of the MSLE across the 21,166 donors. Lorenz Curve for Individual Errors 1.0 Cumulative Share of Error 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Cumulative Share of Donors Figure 5.4: Lorenz Curve for Individual Errors of 1 Year BG/NBD Forecast The chart reveals that 50% of the donors account for over 90% of the cumulated errors and that 20% account for over 75% of the errors.9 Therefore, only a fraction of the cohort is responsible for the main part of the errors. The natural follow-up question is certainly, which donors are the ones for which the models perform so poorly. In order to find an answer to that question, we display the timing patterns of the 10 worst under- as well as 10 worst overestimated donors in figure 5.5. 8 Regarding the RMSE measure and the correlation, only the rescaling of time helps to surpass the regression model. 9 Which is interestingly quite close to the well-known, far more general 80-20 Pareto principle.

CHAPTER 5. MODEL COMPARISON 54 Timing Patterns for the Timing Patterns for the 10 Worst Underestimated Donors 10 Worst Overestimated Donors | | | | | | ||||| | | ||| | |||| || | | | | | |||||||||||||||||||||||||||||| || | | ||||||||||||| | |||| || |||| | || ||| |||| | | | | | || | | | |||||||||| |||||||||||||||| |||||||||||| | | | || | ||||||||||||||| || | | || | | | | ||| | | | | || | | | | | | | ||||||||||||||| || | || || | | |||||| ||| | | | | | | | ||||||||||||||| | | | || | ||| | ||||||||||| | | | | | | | |||||||||||||| |||||||||||| ||||||||||||||||||||||| | | | ||||| | | ||||||||||||||||||||||||||||||||| || | | | || ||||||| | ||| |||| | | | | | || | | | | || | | || | | | | | || | ||| | | | | Calibration Period Validation Period Calibration Period Validation Period Figure 5.5: Timing Patterns of Worst Estimates Regarding BG/NBD The left chart displays those donors for whom the BG/NBD model made too pessimistic estimates regarding their number of transactions. These donors had a rather low transaction frequency throughout the calibration period, but then started to donate frequently (and also regularly). Unfortunately, the patterns themselves do not provide any hint for this change in behavior, and therefore there is not much that can be done in order to improve the estimates for this kind of pattern change.10 On the contrary, the right chart, which contains those donors who have been overestimated, does reveal a highly interesting pattern in the transaction timings. Basically, all of the displayed donors stopped donating during the calibration period. But it seems that the stochastic model is not able to detect this defection, otherwise it would not have vastly overestimated the future number of transactions. This detected inability is even more astonish- ing since anybody who looks at this chart will conclude by simple intuition that these donors have very likely already defected at the end of the calibration period. The reason for this is the apparent regularity within the timings. One might assume that these overestimated donors had some kind of stand- ing order with which the money is transferred each month and that at some point the donors decided to cancel that order. Hence, for these donors, an inactive period of 32 days (i.e. one day more than the maximum length of a month) would already be a strong signal for a change in behavior. But why are the models not able to detect this change if it is that obvious? 10 Possibly more insight can be gained by comparing these patterns with the corresponding contact records for these donors.

CHAPTER 5. MODEL COMPARISON 55 The answer to this question lies within the critical assumption A1, which is being shared by all of the presented probabilistic models. A1 postulates that the number of transactions follows a Poisson process, which is equivalent to the statement that the intertransaction times are exponentially distributed. Thus, a timing pattern is modeled which contains absolutely no regularity at all, and is characterized by being completely random and memoryless. Therefore, these models interpret the gap of inactivity at the end of the calibration period for these regular donors as a ‘longer than average’ but still normal intertransaction period. It is this particular inability of the presented models to incorporate for any observed regularity which causes the poor estimates. Recency as well as frequency are two important pieces of information in order to assess the critical status of activity, but by additionally taking into account the regularity, results could be vastly improved. This statement does not only hold true for stochastic models but can be generalized to all kinds of RF-based models that try to estimate the state activity for a given cohort. The following chapter will present an eﬀort to incorporate regularity into the CBG/NBD model.

Chapter 6 CBG/CNBD-k Model 6.1 Regularity The following list summarizes the key findings regarding regularity that have been identified so far: • The time between two succeeding donations cannot be considered totally random for the DMEF data set. It rather seems that the timing process follows, at least for some of the donors, a certain pattern. This result has been observed during our exploratory data analysis in chapter 3. Firstly, the plot of the observed intertransaction times (see figure 3.9) shows that there is a dead period of at least one month right after a donation, during which hardly anybody makes a following donation. Secondly, the figure indicates that there are some donors who adhere to a monthly rhythm and some who follow an annual rhythm. • In section 4.1.1, which investigated the NBD assumptions and their implications, we pointed out that modeling the negative binomial distribution is equivalent to assuming totally random transaction timings. Such an assumption seems to be violated for certain usage scenarios, in particular for purchase data for goods that are being consumed with a certain regularity. • And finally, it is demonstrated that the presented models, which are all based upon the NBD model, are indeed unable to fit certain characteristics of the data set (see section 5.2). Additionally, they provide rather mediocre results when extrapolating into the future as has been 56

CHAPTER 6. CBG/CNBD-K MODEL 57 shown by comparison to some benchmark models (see section 5.3 and 5.4). Section 5.5 identifies that it is in particular the regular donors who contribute the most to the forecast error. These results justify that special attention should be directed towards the regularity, and that an attempt to incorporate some kind of regularity into stochastic models should be undertaken. But, what is regularity and moreover, how can it be measured? The observed timings can fall anywhere between totally random patterns (i.e. Poisson processes) and ‘clockwork-like’, deterministic patterns (Wheat and Morrison, 1990, p. 87). A regularity measure should therefore provide a single figure that indicates its location between these two extremes. A common method to assess the regularity is to fit a Gamma distribution to the observed intertransaction times and subsequently inspect the estimated shape parameters. Dunn, Reader, and Wrigley (1983, p. 252) reprint H.C.S. Thom’s approximation of the MLE of the shape parameter r as followed: 1 4 r = Y −1 (1 + ˆ 1 + Y ), with (6.1) 4 3 arithmetic mean Y := log . geometric mean Additionally, Wagner and Taudes (1986, p. 243) provide a test statistic and an associated theoretical distribution which enables marketers to adequately test whether an observed process is Poisson. If the estimated shape parameters for intertransaction timings are close to 1, then the Poisson assumption does not need to be rejected for these customers. This results directly from the fact that the corresponding exponential distribution equals the Gamma distribution with shape parameter 1. But a problem arises, when it comes to applying this measure to real world data, because a rather long history of at least 5 or more transactions is required for each individual, otherwise the estimates would be biased. Unfor- tunately, such long transaction records are commonly not available for each customer. Hoppe and Wagner (2007, p. 83) for example applied this test for the Poisson assumption to purchase data from a catalog retailer and had to restrict the test to those 10% of the customers with at least 5 purchases. Their calculations showed that for only 5% of those frequent buyers (in absolute numbers: for 8 customers) the Poisson assumption had to be rejected. Therefore, the test affirmed them to hold on to the NBD assumption.

CHAPTER 6. CBG/CNBD-K MODEL 58 The same test for the Poisson assumption is now being applied to the DMEF data set. Only 8% of all donors had 5 or more donations. For these 1,728 donors the shape parameter r has been estimated according to equation 6.1. Its distribution across these donors is displayed in figure 6.1. As we can see, the median of r for these frequent donors is significantly higher than 2, which is once more a strong indicator for the already detected regularity within the data. Distribution of Estimated Gamma Shape Parameters r = 1 ⇒ Exponential IPTs r = 2 ⇒ Erlang−2 IPTs 0 2 4 6 8 10 Shape Parameter r Figure 6.1: Distribution of the Estimated Gamma Shape Parameters for the In- tertransaction Times of Donors with 5 or more Donations Wheat and Morrison (1990) introduce another regularity measure to the field of consumer behavior. This new measure relaxes the problematic con- straint of long transaction records and thereby allows statements regarding a larger share of the cohort. Wheat and Morrison also assume that the intertransaction times are distributed according to a Gamma distribution, but additionally assume that all customers share the same shape parameter r. They define the following simple measure M , which requires only two intertransaction times for each individual (T1 , T2 ). T1 M= (6.2) T1 + T2 They show that under the posed assumptions M follows a Beta(r, r) distribution. Hence, M is uniformly distributed within interval (0, 1) in the case of exponentially distributed interevent times (r = 1). The actual estimate for r is then given by: 1 − 4 · var(M ) r= ˆ (6.3) 8 · var(M )

CHAPTER 6. CBG/CNBD-K MODEL 59 with var(M ) being the estimated variance of M . This estimate of r serves again as a measure for the observed regularity, but not on an individual level but for the regularity of the complete cohort. Figure 6.2 depicts the respective smoothed histogram of the observed distribution of M for the DMEF data set. Additionally, two theoretical distributions for r = 1 and r = 2 are being displayed to ease interpretation of the curve. This chart is now able to take 33% of the cases into account, as only 3 donations are required per individual anymore. Regularity Measure M 2.5 Actual Distribution of M Distribution of M for r=2 Distribution of M for r=1 2.0 1.5 Density 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 M Figure 6.2: Distribution of Regularity Measure M for the Intertransaction Times of Donors with 3 or more Donations Because the (smoothed) histogram is not uniformly distributed but displays a high peak at 0.5, it is once more shown that the observed data does not follow the Poisson assumption. Furthermore, equation 6.3 results in an estimate for r of 2.1. Concluding our ﬁndings regarding regularity within the DMEF data set, we have to reject the assumption that the intertransaction times follow an exponential distribution. But both ﬁgures, 6.1 as well as 6.2, already suggest a possible alternative distribution. Namely the Gamma distribution with a shape parameter of 2 instead of 1, a distribution that is commonly known as the Erlang-2 distribution.

CHAPTER 6. CBG/CNBD-K MODEL 60 The family of Erlang-k distributions is a special case of the Gamma distribution with the shape parameter being restricted to positive integers. The shape parameter r is then set to the value k. An Erlang-k distributed variable can be seen as the sum of k i.i.d.1 variables that follow an exponential distribution. Another interpretation is that the interevent times are exponentially distributed, but only every k-th event is being observed or counted, therefore the term censored counting process is used for such models (Chatfield and Goodhardt, 1973, p. 829). Figure 6.3 displays the distribution of Erlang-k for several different values of k . The rate parameter has also been set to k for each row, as this results in an equal mean across all four drawn examples and thereby helps comparison. Erlang−1 | | | | 0.0 0.4 0.8 | | | | | | | | | | | || | | | ||| | | || || | | | | || | | | | | | | || | | | | 0 1 2 3 4 5 Erlang−2 | | | | | | | | | 0.0 0.4 0.8 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 0 1 2 3 4 5 Erlang−3 | | | | | | | | | | 0.0 0.4 0.8 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 0 1 2 3 4 5 Erlang−100 | | | | | | | | | 0.0 0.4 0.8 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 0 1 2 3 4 5 Figure 6.3: Erlang-k Distributions The chart gives an idea of the respective shapes for different values of k but also of the resulting timing patterns, which are drawn on the right hand side. As we can see, it is nearly impossible to distinguish the sample patterns of a Poisson process (first row) with the patterns resulting from an Erlang-2 distribution only by means of visual inspection. Such a task would become even more difficult when we are faced with different rate parameters for each individual. Even the Erlang-3 samples look totally random despite the 1 i.i.d. = independent and identically distributed

CHAPTER 6. CBG/CNBD-K MODEL 61 clear peak and the dead period for the theoretical distribution. This implies that the calculation of the aforementioned regularity measures is necessary to detect such light ‘hidden’ levels of regularity. The Erlang-100, on the other hand, resembles with its pattern the observed monthly donors that we encountered in figure 5.5 for the DMEF data set. For several reasons the family of Erlang-k distributions seems to be a good choice to incorporate regularity. Firstly, it is possible to model a specific degree of regularity by setting k according to the observed estimates (see equation 6.3). Secondly, the Erlang-k distribution is, due to its relation to the Poisson process, mathematically relative easy to handle, as opposed to the Weibull or lognormal distribution (Chatfield and Goodhardt, 1973, p. 828). And finally, it is possible to describe the following plausible behavioral story that results in Erlang-k distributed interpurchase times. Even if a user consumes a certain good in a Poisson manner, i.e. at totally random times, but only every k-th consumption results in a purchase of a new package of that good, then the observed waiting time between two purchases will be distributed according to an Erlang-k distribution. The following section will postulate a new model variant which assumes Erlang-k intertransaction times. 6.2 Assumptions Table 6.1 displays the respective assumptions of the herewith newly presented CBG/CNBD-k model. These stated assumptions differ from those of the CBG/NBD model only with respect to A1 and A6. A1 now postulates the more general Erlang-k distribution for modeling intertransaction times. This is opposed to the exponential waiting times that have been assumed for all other presented stochastic models so far. It is important to point out that the integer parameter k is not being estimated by the model itself, but has to be determined a priori. For the special case of k = 1 the CBG/CNBD-k model is equivalent to the CBG/NBD model. Assumption A6 needs to be added because the modeled timing process is not memoryless anymore and depends on the lapsed time since the last transaction. Therefore, the timing of the first event can be modeled more accurately if the timing of the previous one is known. The cohort of the DMEF data set consists in particular of those donors who made their first donation within the first half of 2002. Because this event is defined as time 0 for all individu-

CHAPTER 6. CBG/CNBD-K MODEL 62 A1 While active, transactions of customers occur with Erlang-k (rate parameter λ) distributed waiting times. A2 Heterogeneity in λ follows a Gamma distribution with shape parameter r and rate parameter α across customers. A3 At time zero and directly after each transaction there is a constant probability p that the customer becomes inactive. A4 Heterogeneity in p follows a Beta distribution with parameters a and b across customers. A5 The transaction rate λ and the dropout probability p are distributed independently of each other. A6 The observation period of each individual starts out with a transaction at time zero. Table 6.1: CBG/CNBD-k Assumptions als, we postulate A6 accordingly. If the cohort is built by some other criteria, e.g. by the date of first contact, and this date constitutes time 0, then we would need to adapt A6 consequently. But it needs to be considered, that such a change in the assumption A6 would result in different mathematical derivations than those that are presented here. The idea of modeling Erlang-k interpurchase times is not new at all to the field of consumer behavior. In 1971, Herniter also observed dead periods within his histograms of observed interpurchase times. At that time he was the first to suggest the family of Erlang distributions for fitting such histograms appropriately. By analyzing the ratio of variance to mean, also known as coefficient of variation (CV), Herniter further concluded that an Erlang-2 provides the best fit for his data sets. Two years later Chatfield and Goodhardt (1973) investigated this approach in detail. Firstly, they derived some basic results regarding the probability distribution of the counting process that corresponds to Erlang-2 interpurchase times. They coined the term condensed Poisson distribution for this resulting distribution. This naming reflects its close relationship to the Pois- son distribution. But as opposed to the Poisson, its variance is now smaller

CHAPTER 6. CBG/CNBD-K MODEL 63 than its mean, hence the term condensed has been preceded. Secondly, they followed Ehrenberg (1959) and also assumed a Gamma mixture of purchase frequencies across the customers. The derived distribution has been termed consistently condensed negative binomial distribution (CNBD). It needs to be taken into consideration that Chatfield and Goodhardt (1973) assumed an arbitrary starting time for the counting process, thereby the condensed Poisson distribution assumes a so called asynchronous counting distribution (abbreviated to a.c.d.). By contrast, A6 postulates a so called synchronous counting distribution (s.c.d.) which arises when the start of the counting coincides with an event (cf. Haight, 1965). Nevertheless, the naming of the present model intentionally contains the term condensed for two reasons. One the one hand, the resulting counting distribution is ‘condensed’ just as well, if we examine its coefficient of variation. On the other hand, an asynchronous counting had to be assumed for the target period in order to keep mathematical complexity within limits. But after Chatfield and Goodhardt applied the CNBD on several data sets with ‘more regular than random’ purchase patterns, they concluded that the gained improvement is hardly noticeable and further stated that the added complexity is not worth the effort for practical uses. This conclusion seems rather surprising, but is justified by the dominance of the Gamma- heterogeneity in comparison to the variance of the individual Poisson distributions (Chatfield and Goodhardt, 1973, p. 834). The latter one generally plays a minor role in explaining the variance. This dominance can be inspected numerically by decomposing the overall variance (α2 r + αr) into the variance of the Gamma (α2 r) and the average variance of the Poisson distributions (αr). For example, applied to the DMEF data set (see section 4.1) this calculation reveals that 99.8% of the variance is indeed contributed by the fitted Gamma distribution. It can be assumed that it has been Chatfield and Goodhardt’s pessimistic conclusion regarding the poor practicability that kept marketers rather away from further applying the CNBD model. But, Schmittlein and Morrison (1983) demonstrate that the CNBD is indeed able to outperform the NBD model significantly, in particular when the number of nonbuyers is large. Furthermore, Chatfield and Goodhardt’ conclusions have been based upon the observed fit on an aggregated level, whereas our focus within this work is on the disaggregated level. And finally, the importance of an accurate timing model is considerably higher for forecasting noncontractual relations than for simply finding a good fit to the data. In section 5.5 we reasoned that any information regarding the regularity improves the judgments of the

CHAPTER 6. CBG/CNBD-K MODEL 64 activity status, which are otherwise solely based upon recency and frequency. And because a defected customer will not make any further transactions at all, no matter how many times he used to purchase before, a misjudgment regarding the status results in a tremendous error of the predictions on an individual level. To the best of our knowledge, this work is the first published attempt to join the CNBD with some sort of defection process. This deficiency is surprising because even Schmittlein et al. (1987, p. 18) themselves have already pointed out a possible extension towards the CNBD. Theoretically, any of the NBD based models can be extended to Erlang-k interevent times. We choose the CBG/NBD because it provides the best results regarding the contest data. But also because the model’s derivation is very well documented and traceable in Hoppe and Wagner (2008), and thereby made the deductions of the CBG/CNBD-k actually feasible for us at all in the first place. The key mathematical expressions of the CBG/CNBD-k model are provided in full detail together with their derivations in appendix A. These derivations follow closely the notation and argumentation used in Hoppe and Wagner (2008). Unfortunately, we do not succeed in deriving exact closed formulas of the decisive expressions of the unconditional and the conditional expectations of future transactions. Nevertheless, an approximation is suggested, which is known to be biased but which still is able to provide superior estimates as the next section will show. 6.3 Comparison of Models Following the same proceeding than in chapter 5 ‘Model Comparison’, the performance of the CBG/CNBD-k is being compared with other models by applying it to the DMEF data set. The estimate 6.3 of r from the pre- ˆ vious section 6.1 suggests that a CBG/CNBD-2 should provide the best fit, i.e. Erlang-2 intertransaction times are being modeled. Additionally, a CBG/CNBD-3 is also being fitted to the data to see how the results change when a stronger degree of regularity is being assumed. 6.3.1 Parameter Interpretation Table 6.2 contains the estimated parameters when applying the model variants on the full time range of the provided DMEF data set. The new model

CHAPTER 6. CBG/CNBD-K MODEL 65 CBG/NBD CBG/CNBD-2 CBG/CNBD-3 r (se) 1.11 (0.05) 1.83 (0.06) 1.93 (0.05) α (se) 552 (19) 323 (9) 210 (5) a (se) 0.38 (0.02) 0.62 (0.02) 0.71 (0.02) b (se) 0.67 (0.04) 0.76 (0.03) 0.84 (0.03) Table 6.2: Estimated Model Parameters uses the same four parameters as the CBG/NBD and the BG/NBD model. The parameters r and α still describe the heterogeneity of the transaction frequency λ across the cohort, whereas λ is now the rate parameter of the Erlang-k distribution, with its expected mean being λ/k. Hence, it is necessary to multiply the rate parameter α with the associated integer k, if we want to make a direct comparison of the distribution of transaction frequencies. mean median sd BG/NBD 9.1 t 13.3 t 9.0 t CBG/NBD 2.7 t 3.8 t 3.0 t CBG/CNBD-2 2.2 t 2.4 t 3.1 t CBG/CNBD-3 2.2 t 2.3 t 3.2 t Table 6.3: Statistical Summary of Fitted Life Times mean median sd BG/NBD 836 d 2,324 d 527 d CBG/NBD 496 d 688 d 523 d CBG/CNBD-2 354 d 428 d 478 d CBG/CNBD-3 327 d 392 d 454 d Table 6.4: Statistical Summary of Fitted Intertransaction Times Table 6.3 and 6.4 provide the related properties for the modeled life times and intertransaction times. As can be seen, the expected life time drops even further to an average of 2.2 ‘survived’ transactions, resembling the observed average number of donations of 1.55 even closer. Interestingly, the Erlang-2 and Erlang-3 assumptions do result in very similar model parameters regarding a and b. Along with the drop in life time, a shorter expected intertransaction period is now being modeled compared to the previous models. The median waiting time is now close to a one year period. All in all, the estimated parameters seem to better represent the underlying data and its key characteristics.

CHAPTER 6. CBG/CNBD-K MODEL 66 6.3.2 Data Fit Table 6.5 and figure 6.4 give an impression of the CBG/CNBD-k models’ ability to fit the DMEF data on an aggregated level. 0 1 2 3 4 5 6 7+ Actual 10,626 3,579 2,285 1,612 1,336 548 348 832 CBG/NBD 10,647 3,939 2,186 1,368 905 617 429 1,075 CBG/CNBD-2 10,592 3,952 2,217 1,402 931 633 436 1,023 CBG/CNBD-3 10,570 3,998 2,228 1,401 927 628 431 983 Table 6.5: Comparison of Actual vs. Theoretical Count Data Actual vs Fitted Frequency of Repeat Transactions 10000 Observed CBG/NBD CBG/CNBD−2 8000 CBG/CNBD−3 χ2CBG/NBD = 363.7 χ CBG/CNBD−2 2 = 302.9 χ2CBG/CNBD−3 = 307.8 6000 Frequency 4000 2000 0 0 1 2 3 4 5 6 7+ # Transactions Figure 6.4: Fitted Distributions The calculated χ2 test statistics indicate an improvement in comparison to the CBG/CNBD model. The drop of χ2 from around 360 to nearly 300 is mainly due to the closer fit of those classes that previously showed the biggest relative offsets from the actual values. These are group 3, group 4 and the group of the heavy donors (7+). Nevertheless, the models are still not quite able to explain the large size of group 4 which likely stems from the regular yearly donors (see the discussion in section 5.2).

CHAPTER 6. CBG/CNBD-K MODEL 67 A comparison of the loglikelihood values is slightly more complex because the calculation requires the exact intertransaction times (t1 −t0 , t2 −t1 , . . . , tx −tx−1 ) as opposed to the timing tx of the last transaction (see section A.4 in the appendix for the exact formulas). Fortunately, this information is available for the DMEF data set and thus the following ranking table can be provided. Rank Model LL I. CBG/CNBD-2 -242,738.5 II. CBG/CNBD-3 -243,924.0 III. Pareto/NBD -245,674.2 IV. CBG/NBD -245,702.2 V. BG/NBD -245,833.0 VI. NBD -246,552.5 Table 6.6: Comparison of Calculated Loglikelihood Values Hence, the maximized loglikelihood values of the estimated CBG/CNBD-k models clearly surpass the related values of the classic models. And among the CBG/CNBD-k models the CBG/CNBD-2 provides the best fit with respect to this measure, which is also the expected result corresponding to our assessment of r in the preceding section. In summary, these results prove ˆ that the consideration of regularity does indeed provide a better fit to the present data set, and that the extra effort is thereby justified. 6.3.3 Forecast Accuracy As has been demonstrated for the NBD model, a decent fit to the observed data does not necessarily imply that the model is capable of providing sound estimates for the future. Therefore, the crucial evaluation criterion in our context is again the capability of making such predictions. Table 6.7 and figure 6.5 compare the model’s estimates with the actual data throughout the 1 year calibration period. The data table displays that the CBG/CNBD-2 makes nearly a perfect assessment for the large group of donors, who have not made any repetitive donations within the training period. The new model also outperforms all other so far presented models for group 1 and 2. Unfortunately, the model is neither capable to repair the notable underestimation of the frequent donors (6+) nor capable to explain the bend curve for group 4. In particular, the latter defect should have been fixed to some extent by incorporating regular-

CHAPTER 6. CBG/CNBD-K MODEL 68 0 1 2 3 4 5 6 7+ Actual 0.04 0.20 0.43 0.69 0.75 1.06 1.54 2.44 CBG/NBD 0.11 0.29 0.48 0.68 0.87 1.05 1.25 2.10 CBG/CNBD-2 0.04 0.17 0.38 0.60 0.78 0.95 1.19 2.05 CBG/CNBD-3 0.02 0.13 0.33 0.55 0.71 0.86 1.09 1.87 Group Size 10,988 3,910 2,683 1,730 731 392 239 493 Table 6.7: Comparison of Actual vs. Theoretical Average Number of Donations per Donor during the Validation Period Conditional Expectation of Future Transactions 3.0 Avg # Transactions in Validation Period 2.5 Actual CBG/NBD CBG/CNBD−2 2.0 CBG/CNBD−3 1.5 1.0 0.5 0.0 0 1 2 3 4 5 6 7+ # Transactions in Training Period Figure 6.5: Conditional Expectations ity because the bend curve can be attributed to the regular yearly donors. The reason is that the provided mathematical expressions for the future estimates are not exact. As can be seen in section A.8.5 of the appendix, some simplifying approximations need to be made in order to make the derivations feasible. Among others, the exact duration since the last recorded transaction of each donor is neglected and thus the model is unable to simulate the yearly rhythm appropriately. Finally, table 6.8 contains the most important figures with respect to the DMEF contest. Both CBG/CNBD-k models considerably outperform all other models with respect to the DMEF data set. Incorporating regularity therefore results in significantly improved estimations on a disaggregated level for the present case. This statement is affirmed by several different error measures, like the MSLE, the RMSE and the correlation. Additionally, separate calculations

CHAPTER 6. CBG/CNBD-K MODEL 69 MSLE RMSE MAE Corr MSLEopt SUM Heuristic 0.0962 0.661 0.258 0.615 0.0909 (0.70) -22% LM 0.0863 0.642 0.262 0.644 0.0861 (0.93) -31% Pareto/NBD 0.0977 0.653 0.359 0.628 0.0879 (0.66) +22% BG/NBD 0.0963 0.651 0.362 0.640 0.0880 (0.68) +19% CBG/NBD 0.0959 0.650 0.360 0.639 0.0878 (0.68) +19% CBG/CNBD-2 0.0831 0.632 0.293 0.660 0.0818 (0.84) -11% CBG/CNBD-3 0.0816 0.637 0.275 0.663 0.0814 (0.94) -24% Table 6.8: Error Measures have shown that a modification of chosen training and calibration period lengths does not change the overall ranking either. The deviance of the cumulative number of estimated transactions suggest that the CBG/CNBD-k models are likely to be biased and tend to underestimate the actual number. This can be reasoned by the simplifications that are made for the derivations. Nevertheless, the calculated optimized MSLE and the correlation numbers indicate that, regardless of this systematic underestimation, the CBG/CNBD-2 and CBG/CNBD-3 models are still more capable of modeling the number of future transactions for each donor. 6.4 Final Model The details and calculations of the final model, which have been used for our contest submission, are presented within this section. 6.4.1 Estimation of Monetary Component All the presented probabilistic models make predictions for the future number of transactions. A missing piece for the computation of customer lifetime values is therefore the estimation of the donation amounts. In chapter 3, several characteristics of the observed donation amounts are being identified. Firstly, donation amounts vary tremendously across donors. Secondly, donation amounts normally take certain even integer values. And thirdly, average donation amounts change over time but it is impossible to detect a clear trend. Several different approaches of estimating donation amounts are tried out

CHAPTER 6. CBG/CNBD-K MODEL 70 and evaluated with respect to the resulting MSLE for the calibration period. Schmittlein and Peterson (1994, p. 56) propose a model which combines individual past amounts with the average over the complete cohort to make individual estimates. A much simpler method is to take the last, the average or the median of the observed donation amounts for each donor as an estimate for future transaction values. The calculations for the validation period indicate that the mean over the past donation amounts provides the best estimate with the lowest corresponding MSLE measure for the DMEF data set. Therefore, this simple assessment is combined with the CBG/CNBD-k model. 6.4.2 Submission to DMEF Competition For our ﬁnal model we choose the CBG/CNBD-2 model for the number of future transactions,2 and take the past average dollar amounts as an estimate for each future donation. Additionally, an optimal multiplication factor is determined in order to minimize the MSLE (see the related discussion in section 5.3.3 and also the bracketed optimal ratios within table 6.8). With respect to the calibration period, the optimal ratio is set to 0.25. The parameters r, α, a and b of the CBG/CNBD-2 model have been of course calibrated by using the complete provided DMEF data set of 4 years and 8 months. Subsequently, the number of transactions within the target period of 2 years have been estimated for each donor based upon their past transaction records (x, tx , T ). Then the number of transactions are being multiplied with the corresponding average past donation amounts in order to derive a lifetime value (for the target period) for each donor. This value is further multiplied by the determined optimal ratio (0.25) in order to produce an estimate that will hopefully minimize the MSLE. This results in our submitted estimates for task 2 of the contest. In addition, we simply assumed that any donor with an estimated number of transactions of more than 0.5 will be actually donating within the target period and that all others will not. This provides our estimates for task 3. 2 The idea for the CBG/CNBD-k model emerged only two days prior to the submission deadline of the contest. In the remaining limited time span we therefore focused on the special case of Erlang-2 distributed intertransaction times. It was only after the contest that we succeeded in providing the necessary analytical results for the more general Erlang- k case. Nevertheless, later calculations showed that the CBG/CNBD-2 did indeed provide the best estimate among the family of CBG/CNBD-k models.

CHAPTER 6. CBG/CNBD-K MODEL 71 Finally we simply guess the solution to task 1, which is an estimate of the cumulated donation amounts of all donors, by assessing the further trend in donation sum in ﬁgure 3.5. The CBG/CNBD-k model is not being used for this purpose, because of the known overall bias which can lead to poor estimates on an aggregated level.

Chapter 7 Conclusion Within this thesis we provided a thorough analysis of several popular probabilistic purchase models for noncontractual consumer relationships. Their corresponding assumptions regarding the underlying behavior were presented and underwent a critical review. All of the presented existent models share the same problematic implications of the NBD model with respect to the randomness of the implied transaction timings. This lack of face validity has been disputed ever since Ehrenberg’s ﬁrst introduction of the NBD model, nevertheless numerous papers concluded that this model is indeed able to explain observed count patterns in real world data very well. However, as has been argued in this thesis, the importance of an accurate timing model is much higher if forecasts are being made on a disaggregated level in noncontractual setting. This is due to the fact, that the current status of activity functions as a knock-out criterion for future transactions, and its estimate is therefore crucial for making accurate forecasts. As a consequence, we suggest to incorporate the regularity within the observed timing patterns into the model building. By that, the assessment of the signiﬁcance of any observed frequency and especially recency information could be improved. In the following, a new probabilistic model variant, the CBG/CNBD-k model, was outlined, which allows to account for an arbitrary extent of regularity. We also succeeded in providing exact derivations for several key mathematical expressions, such as the likelihood, the probability distribution of purchase frequencies, and the crucial probability of being active. Though, a closed-form expression of the expected number of transactions could not be deduced, but instead a heuristic approximation was suggested which made the calculations feasible. 72

CHAPTER 7. CONCLUSION 73 This newly introduced model was subsequently applied to donation records of a US nonprofit organization. This data set was provided by the Direct Marketing Educational Foundation as part of a lifetime value contest. A detailed exploratory data analysis revealed, among other findings, the inherent regularity in the timing patterns. In particular, the presence of donors who make monthly, and donors who make yearly donations became apparent. After fitting all presented models to the provided data set, it could be concluded that the CBG/CNBD-k model is capable of considerably outper- forming existent models with respect to parameter plausibility, data fitting, and forecasting accuracy. This finding was further attested by the final outcome of the DMEF contest. Out of 25 participating teams, ranging from software and consulting companies to university institutions, the herewith introduced model finished at the exceptional second place regarding the forecast accuracy on a disaggregated level, only marginally behind the winning model. In particular, the CBG/CNBD-k was able to clearly exceed all other participating stochastic models. The presented idea of extending the NBD to the CNBD model can theoretically be carried out to all other NBD-based models, as such a Pareto/CNBD- k, as well as a BG/CNBD-k are thinkable. Although the analytical complexity is significantly raised by this extension, it has been shown that also a simplified, biased model is able to improve the forecast quality. A further promising extension could be the modeling of a varying degree of regularity across the cohort, as has also been noticeable for the DMEF data set. However, more generally speaking, we hope that we were able to make the case for incorporating regularity into the consideration when modeling consumer behavior, not just for the stochastic kind. Even further, the inherent dynamics and patterns of the actual transaction timings potentially contain valuable information. Therefore it seems negligent to disregard such information by reducing given transaction data to simple recency and frequency statistics in the first place.

Appendix A Derivation of CBG/CNBD-k A.1 Assumptions A1 While active, transactions of customers occur with Erlang-k (rate parameter λ) distributed waiting times. A2 Heterogeneity in λ follows a Gamma distribution with shape parameter r and rate parameter α across customers. A3 At time zero and directly after each transaction there is a constant probability p that the customer becomes inactive. A4 Heterogeneity in p follows a Beta distribution with parameters a and b across customers. A5 The transaction rate λ and the dropout probability p are distributed independently of each other. A6 The observation period of each individual starts out with a transaction at time zero. These assumptions diﬀer from the CBG/NBD model only regarding the mod- iﬁed assumption A1 and the newly introduced assumption A6. 74

APPENDIX A. DERIVATION OF CBG/CNBD-K 75 A.2 Erlang-k The Erlang-k distribution with parameters k and λ is defined by the probability density 1 fΓ (t|k, λ) = λk tk−1 e−λt ∀t > 0; k ∈ N+ , λ > 0. (A.1) (k − 1)! The Erlang-k is a specialization of the more general Gamma distribution, with the restriction of k being an integer. If k = 1, then we are dealing with the exponential distribution again. The Erlang-k distribution can also be seen as the sum of k i.i.d. exponentially distributed random variables with parameter λ. Therefore, the corresponding counting process of events with Erlang-k distributed waiting times can be deduced from the Poisson process straightforward. Under the assumption that an event actually occurred at time zero the probability of encountering x events until time t is k−1 Pk (X(t) = x) = PP (X(t) = kx + j). (A.2) j=0 This result is straightforward if we take a look at figure A.1, which renders the relation between a Poisson process (t0 , t1 , t2 , ..) and the timing of Erlang-k (t0 , t1 , t2 , ..). We consider the occurrence of an event as the k-th realization of an corresponding exponentially distributed process (tx = tkx ). Therefore, the probability of encountering x events until time t, is the sum of the probabilities of encountering kx, kx+1, .., kx+k−1 Poisson events. The remark that we start counting with an event at time zero is important, since we are not dealing with a memoryless process anymore, as has been the case for exponentially distributed timings. Being memoryless implies that the chances of the event to occur within the near future remains constant and is independent of the time that has past sine the last event. On the other hand, the Erlang-k distribution clearly has a peak unequal to 0 (for k > 1). The absence of a memoryless process is thus the reason, why we had to postulate assumption A6 for our model. Haight (1965) distinguished between counting processes that start out with an event at time zero and those who do not. He termed them synchronous and asynchronous counting processes. Chatfield and Goodhardt (1973) studied the asynchronous counting of Erlang-k events and termed the resulting process condensed Poisson process.

APPENDIX A. DERIVATION OF CBG/CNBD-K 76 t t0 = 0 t1 t2 3·2 × × × × × × × × × - t0 t1 t2 t3 t4 t5 t6 t7 t8 3·2+1 × × × × × × × × × - t7 t8 3·2+2 × × × × × × × × × - t7 t8 Figure A.1: Illustration for Erlang-3 distributed interevent times. P3 (X(t) = 2) is the probability of encountering 6, 7 or 8 Poisson events. A.3 Individual Likelihood The likelihood of parameters λ and p for a particular purchase pattern (t1 , . . . , tx , T ) can be deduced analogous to the referred papers. It is the likelihood of the observed interevent periods (t1 − t0 , t2 − t1 , . . . , tx − tx−1 ), times the probability of having ‘survived’ time 0 and the ﬁrst x−1 purchases, times the probability of seeing no transaction within (tx , T ]. Whereas the latter can result from a customer that defected immediately after the last purchase, or from a customer whose next transaction simply happens to be after time T . L(λ, p|t1 , . . . , tx , T ) = (1 − p)fΓ (t1 |k, λ) · · · (1 − p)fΓ (tx − tx−1 |k, λ) · p + (1 − p)P (X(T − tx ) = 0|k, λ) Inserting the Erlang-k pdf A.1 and our previous result A.2, it follows that

APPENDIX A. DERIVATION OF CBG/CNBD-K 77 L(λ, p|t1 , . . . , tx , T ) = λk tk−1 e−λt1 1 λk (tx − tx−1 )k−1 e−λ(tx −tx−1 ) = (1 − p)x · ··· (k − 1)! (k − 1)! k−1 · p + (1 − p) PP (X(T − tx ) = j|λ) j=0 ˜ t := = (1 − p)x λkx e−λtx (1/(k−1)!)x (tx −tx−1 )k−1 · · · (t1 −0)k−1 k−1 −λ(T −tx ) λj (T − tx )j · p + (1 − p)e j=0 j! k−1 ˜ ˜ λj (T − tx )j = t · p(1 − p)x λkx e−λtx + t · (1 − p)x+1 λkx e−λT (A.3) j=0 j! An important difference of this result from the likelihood methods of models with exponential timing is that we still have the actual timing of the trans- ˜ actions t1 , ..., tx (which we subsumed into variable t) in our final formula. (x, tx , T ) is therefore not a sufficient statistic anymore for the likelihood. But, as we will see shortly, we do not need these timings for the estimation of the parameters, and therefore actually do not impose any extra requirements regarding the input data. A.4 Aggregate Likelihood In order to take assumptions A2 and A4 regarding the distribution of λ and p into account, we need to mix in the gamma- and beta-distribution by means of integration. L(r, α, a, b|t1 , ..., tx , T ) = 1 ∞ ˜ = t· p(1−p)x λkx e−λtx fΓ (λ|r, α)fB (p|a, b) dλ dp 0 0 1 ∞ k−1 ˜ (T −tx )j λj +t· (1−p)x+1 λkx e−λT fΓ (λ|r, α)fB (p|a, b) dλ dp 0 0 j=0 j! (A.4) Due to assumption A5 we can solve these integrals separately and will for this purpose use the following definitions and results from Hoppe and Wagner

APPENDIX A. DERIVATION OF CBG/CNBD-K 78 (2008, section 2): ∞ αr · (r)i IΓ (i, j, r, α) := λi e−λj fΓ (λ|r, α) dλ = (A.5) 0 (j + α)r+i 1 B(a + i, b + j) IB (i, j, a, b) := pi (1−p)j fB (p|a, b) dp = (A.6) 0 B(a, b) B(a, b) denotes the Beta-Function, and (r)x the Pochhammer’s symbol: Γ(a)Γ(b) B(a, b) = (A.7) Γ(a + b) Γ(r + x) (r)x = (A.8) Γ(r) Furthermore, we can easily see by considering Γ(a+1) = aΓ(a) that a B(a + 1, b + x) = · B(a, b) (A.9) b+x (r)x+y =(r + x)y · (r)x (A.10) holds. Therefore: L(r, α,a, b|t1 , ..., tx , T ) = ˜ = t · IB (1, x, a, b) · IΓ (kx, tx , r, α) k−1 ˜ (T − tx )j + t · IB (0, x + 1, a, b) · IΓ (kx + j, T, r, α) (A.11) j=0 j! ˜ (b)x+1 = t· · αr (r)kx (a + b)x+1 r+kx k−1 a 1 (T − tx )j (r + kx)j · + (A.12) b+x α + tx j=0 j! (α + T )r+kx+j For the Erlang-2 case this is L(r, α,a, b|t1 , ..., tx , T ) = ˜ (b)x+1 = t· · αr (r)2x (a + b)x+1 r+2x r+2x r+2x+1 a 1 1 1 · + + (T −tx )(r+2x) b+x α+tx α+T α+T (A.13) ˜ with t being t1 · (t2 − t1 ) · · · (tx − tx−1 ).

APPENDIX A. DERIVATION OF CBG/CNBD-K 79 A.5 Parameter Estimation A well-known parameter estimation method, which is under considerably general conditions asymptotically optimal (i.e. unbiased and efficient), is the maximum likelihood estimation (MLE). This method tries to find a parameter set (r, α, a, b) at which the likelihood reaches its global maximum for some given data (ti,1 , ..., ti,x , Ti )i=1..N . (ˆ, α, a, ˆ = argmax L(r, α, a, b|(ti,1 , ..., ti,x , Ti )i=1..N ) r ˆ ˆ b) r,α,a,b N = argmax L(r, α, a, b|ti,1 , ..., ti,x , Ti )) r,α,a,b i=1 ˜ As can be seen, w can now simply disregard the cumulative term ti for the exact timing patterns, since this multiplicative factor has no effect on the location of the maximum, i.e. on the estimated parameters. Therefore, we can remain to (x, tx , T ) as input data for our further calculations. To circumvent problems with numerical precision, it is common to actually optimize the logarithmic of the likelihood, which transforms the multiplication (of very small numbers) into a sum. N (ˆ, α, a, ˆ = argmax r ˆ ˆ b) log(L(r, α, a, b|ti,1 , ..., ti,x , Ti )) (A.14) r,α,a,b i=1 A.6 Probability Distribution of Purchase Fre- quencies We now try to deduce an expression for P (X(t) = x|r, α, a, b), i.e. the probability distribution of the purchase frequencies conditional on the (estimated) parameters, and will again closely follow the mathematical derivation from Hoppe and Wagner (2008, section 3.3). For a single customer (with given λ and p) the probability of encountering x transactions until time t can be split into two cases. Either the customer simply just had x transactions and is still active at time t, or he/she would have had more than x transactions but defected immediately after the x-th purchase. P (X(t) = x|λ, p) = (1 − p)x+1 P (X(t) = x) + p(1 − p)x P (X(t) ≥ x) (A.15)

APPENDIX A. DERIVATION OF CBG/CNBD-K 80 Using P (X(t) ≥ x) = 1 − P (X(t) < x) and result A.2, we derive kx+k−1 x+1 P (X(t) = x|λ, p) = (1 − p) PP (X(t) = j) j=kx kx−1 + p(1 − p)x 1 − δx>0 PP (X(t) = j) . (A.16) j=0 Note that we added the Kronecker-Delta, which is 1 for x > 0 and 0 otherwise, to correctly consider the case x = 0 for which the second summation term simply becomes the dropout probability p at time zero. Again we mix in our heterogeneity assumptions: P (X(t) = x|r, α, a, b) = 1 ∞ = P (X(t) = x|λ, p)fΓ (λ|r, α)fB (p|a, b) dλ dp 0 0 1 ∞ kx+k−1 x+1 (λt)j −λt = (1 − p) fB dp e fΓ dλ 0 0 j=kx (j)! 1 ∞ kx−1 (λt)j −λt + p(1 − p)x fB dp 1 − δx>0 e fΓ dλ (A.17) 0 0 j=0 (j)! and apply the results A.6 and A.5: P (X(t) = x|r, α, a, b) = kx+k−1 j t = IB (0, x + 1, a, b) · IΓ (j, t, r, α) j=kx j! kx−1 tj + IB (1, x, a, b) · 1 − δx>0 IΓ (j, t, r, α) j=0 j! kx+k−1 j B(a, b + x + 1) t αr (r)j = · B(a, b) j=kx j! (α + t)r+j kx−1 B(a + 1, b + x) tj αr (r)j + · 1 − δx>0 (A.18) B(a, b) j=0 j! (α + t)r+j Considering the probability distribution for the negative binomial distribution tj αr (r)j PNBD (X(t) = j) = , (A.19) j! (α + t)r+j

APPENDIX A. DERIVATION OF CBG/CNBD-K 81 we can also write P (X(t) = x|r, α, a, b) = kx+k−1 B(a, b + x + 1) = · PNBD (X(t) = j) B(a, b) j=kx kx−1 B(a + 1, b + x) + · 1 − δx>0 PNBD (X(t) = j) . (A.20) B(a, b) j=0 Thus, for the Erlang-2 case this expression is P (X(t) = x|r, α, a, b) = B(a, b + x + 1) = · (PNBD (X(t) = 2x) + PNBD (X(t) = 2x + 1)) B(a, b) 2x−1 B(a + 1, b + x) + · 1 − δx>0 PNBD (X(t) = j) . (A.21) B(a, b) j=0 A.7 Probability of Being Active As Schmittlein et al. (1987) pointed out, one of the key expressions of models of this kind is the probability of a single customer still being active at the end of the observation period, based on his past transaction history. That is, we ask for P (τ > T | t1 , .., tx , T, r, α, a, b) with τ being the unobserved customer lifetime. P (τ > T | t1 , .., tx , T, λ, p) = 1 − P (τ ≤ T | t1 , .., tx , T, λ, p) p =1− P (X(T − tx ) = 0) p =1− k−1 p + (1 − p) j=0 PP (X(T − tx ) = j) k−1 (1 − p) j=0 PP (X(T − tx ) = j) = k−1 p + (1 − p) j=0 PP (X(T − tx ) = j) ˜ By expanding this term with t(1 − p)x λkx e−λtx , and comparing the denominator with equation A.3 it follows that k−1 λj (T −tx )j ˜ t(1 − p)x+1 λkx e−λT j=0 j! P (τ > T | t1 , .., tx , T, λ, p) = (A.22) L(λ, p | t1 , .., tx , T )

APPENDIX A. DERIVATION OF CBG/CNBD-K 82 Building the double integral P (τ > T | t1 , .., tx , T, r, α, a, b) = 1 ∞ P (τ > T | t1 , .., tx , T, λ, p)fΓ (λ | r, α)fB (p | a, b) dλ dp (A.23) 0 0 and using the following result from Hoppe and Wagner (2008, section 3.2.3) L(λ, p | t1 , .., tx , T )fΓ (λ | r, α)fB (p | a, b) f (λ, p | t1 , .., tx , T ) = , (A.24) L(r, α, a, b | t1 , .., tx , T ) yields P (τ > T | t1 , .., tx , T, r, α, a, b) = ˜ t 1 = · (1 − p)x+1 fB (p|a, b) dp L(r, α, a, b | t1 , .., tx , T ) 0 ∞ k−1 (T − tx )j j · λkx e−λT λ fΓ (λ|r, α) dλ 0 j=0 j! k−1 ˜ (T − tx )j = t · IB (0, x + 1, a, b) · IΓ (kx + j, T, r, α) j=0 j! /L(r, α, a, b|t1 , .., tx , T ). (A.25) Comparing this with equation A.11, we can see that the numerator is actually one of the summation terms of the aggregated likelihood function in the A A denominator. And considering A+B = (1 + B )−1 the fraction can be reduced to P (τ > T |t1 , .., tx , T, r, α, a, b) = −1 ˜ t · IB (1, x, a, b) · IΓ (kx, tx , r, α) = 1+ k−1 (T −tx )j ˜ t · IB (0, x + 1, a, b) · j=0 IΓ (kx + j, T, r, α) j! (A.26) ˜ Fortunately, the term t cancels out and therefore, we still do not require the information on the exact timing of the transactions for carrying out our calculations. We resolve the integral functions, extract common terms and

APPENDIX A. DERIVATION OF CBG/CNBD-K 83 use the relation (r)kx+j = (r)kx · (r + kx)j and yield P (τ > T |x, tx , T, r, α, a, b) = B(a + 1, b + x) αr (r)kx (α + T )r+kx = 1+ · · B(a, b + x + 1) (α + tx )r+kx αr (r)kx k−1 −1 (T − tx )j / (r + kx)j (α + T )j j=0 j! k−1 −1 r+kx a α+T (T − tx )j (r + kx)j = 1+ / . (A.27) b+x α + tx j=0 j! (α + T )j Thus, for Erlang-2: P (τ > T |x, tx , T, r, α, a, b) = r+2x −1 a α+T T − tx = 1+ / 1 + (r + 2x) (A.28) b+x α + tx α+T A.8 Expected Number of Transactions In order to arrive at a closed form solution for the predicted number of transactions for a single customer with given purchase history E(Y (T, T + t)|x, tx , T, r, α, a, b), we try to follow the same steps as in Hoppe and Wagner (2008, section 3.5). Unfortunately, we do not succeed. Nevertheless, we come up with an heuristic approximation, and provide some reasoning for our simpliﬁcations. As the calculations for the DMEF competition have shown, such an approach can still outperform existing models which assume a Poisson process. A.8.1 Unconditional Expectation for Condensed Pois- son The expected number of transactions for an active customer with exponentially distributed interevent times is known to be E(X(t)|λ) = λt. The asynchronous counting process for Erlang-2 waiting times has an expectation of E(X(t)|λ) = λt/2 (Chatﬁeld and Goodhardt, 1973, p. 829). Simi- larly, we will now prove that the generalization for Erlang-k E(X(t)|λ) = λt/k

APPENDIX A. DERIVATION OF CBG/CNBD-K 84 also holds true. Let us recall that asynchronous counting for Erlang-k can also be seen as a censored counting of a Poisson process, where every k-th event is being counted. As we start the counting independent of a particular event, the recording of r censored events can either arise from recording rk, rk+1, rk−1,..., rk+k−1 or rk−k+1 uncensored events. Or, if we take a look at it the other way around, then rk +j (0 ≤ j ≤ k) uncensored events result in either r (with probability k−j ) or r+1 (with probability k ) censored events k j to be counted. Therefore ∞ E(X(t)|λ) = rPC (r) r=1 ∞ k−1 k − |j| = r PP (kr + j) r=1 j=−k+1 k ∞ 1 = krPP (kr) k r=1 k−1 ∞ ∞ j k−j + krPP (kr − k + j) + krPP (kr + j) j=1 k2 r=1 k2 r=1 =:Tj Tj can be reduced to j Tj = (kr−k+j)PP (kr−k+j) + (k−j)PP (kr−k+j) k2 k−j + (kr+j)PP (kr+j) − jPP (kr+j) k2 j = (kr−k+j)PP (kr−k+j) + (k−j)PP (kr−k+j) k2 k−j + (kr−k+j)PP (kr−k+j) − jPP (j) − jPP (kr−k+j) + jPP (j) k2 1 = (kr − k + j)PP (kr − k + j), k and we receive our previously stated result for the unconditional expected number for asynchronous counting: ∞ k−1 ∞ 1 1 E(X(t)|λ) = krPP (kr) + (kr − k + j)PP (kr − k + j) k r=1 j=1 k r=1 ∞ 1 λt = rPP (r) = (A.29) k r=1 k

APPENDIX A. DERIVATION OF CBG/CNBD-K 85 A.8.2 Unconditional Expectation for Grouped Poisson For a synchronous counting process with Erlang-k waiting times the derivation of the expectation is more difficult. Using result A.2, we can deduce ∞ ∞ k−1 E(X(t)|λ) = rPG (r) = r PP (rk + j) r=1 r=1 j=0 k−1 ∞ 1 = rkPP (rk + j) k j=0 r=1 k−1 ∞ ∞ 1 = (rk + j)PP (rk + j) − j PP (rk + j) k j=0 r=1 r=1 ∞ r=0 PP (rk+j)−PP (j) ∞ k−1 k−1 ∞ 1 = rPP (r) − rPP (r) − j PP (rk + j) − PP (j) k r=0 r=0 j=0 r=0 k−1 ∞ 1 = λt − j PP (rk + j) . k j=1 r=0 For k = 2 it is possible to find a simple closed form for the unconditional expected number for synchronous counting. ∞ 1 E(X(t)|λ) = λt − PP (2r + 1) 2 r=0 ∞ 1 (λt)2r+1 = λt − e−λt 2 r=0 (2r + 1)! λt 1 −λt = − e sinh(λt) (A.30) 2 2 The result for the synchronous counting process (A.30) differs from the asynchronous result (A.29) only by an additional subtraction term that converges for Erlang-2 to 1/4 for t → ∞. Hence, for a long time horizon we can assess the error that we make, if we use the simpler formula A.29. A.8.3 Expectations for Condensed NBD Schmittlein and Morrison (1983) published some findings regarding the condensed negative binomial distribution, but only considered the Erlang-2 case.

APPENDIX A. DERIVATION OF CBG/CNBD-K 86 They state a formula for the higher moments of the unconditional expectation, in particular r E(X|r, α) = , and (A.31) 2α r r 1 α r Var(X|r, α) = + 1− + , (A.32) 4α 8 α+2 4α2 but also derived a formula for the conditional expectation. Due to its complexity, we will not reproduce this result here, but rather want to point out two important characteristic differences to the NBD that Schmittlein and Morrison noted. First, the expected number of future transactions is not linear regarding the observed number of transactions anymore, and second, the result now does depend on any elapsed time between the observation and the prediction period. Both of these statements already indicate that deriving a formula for the conditional expectation of CBG/CNBD-k model will be anything but trivial. A.8.4 Unconditional Expectation for CBG/CNBD-k Unfortunately, we did not succeed in deriving a closed form for the expression E(X(t) | r, α, a, b). We could derive a (rather complex) expression for E(X(t) | λ, p) for k = 2, but subsequently incorporating heterogeneity would have required solving double integrals of the form 1 ∞ √ pv4 (1 − p)v5 λv1 e−λ(v3 +v2 1−p)t dλ dp. (A.33) 0 0 Nevertheless, we proceed with our calculations by using some simple heuristic modifications to the results of Hoppe and Wagner (2007). They define v1 v4 t G(v1 , v2 , v3 , v4 | α, t) := 1 − 2 F1 (v1 , v2 + 1; v3 + a; ) v4 + t v4 + t (A.34) with 2 F1 being the Gaussian hypergeometric function, and stated b E(X(t)|r, α, a, b) = · G(r, b, b, α | α, t) (A.35) a−1 for the unconditional expected number of transactions until time t for their CBG/NBD model.

APPENDIX A. DERIVATION OF CBG/CNBD-K 87 Recalling our findings that the expectation for asynchronous counting is simple 1/k of the corresponding Poisson process (see equation A.29), and that the synchronous counting only differs by some term that becomes a constant for long time horizon, we simply approximate the expected number of transactions for the CBG/CNBD-k model with ˆ 1 b E(X(t)|r, α, a, b) = · · G(r, b, b, α | α, t). (A.36) k a−1 A.8.5 Conditional Expectation for CBG/CNBD-k But even if we come up with a proper solution for the unconditional expectation, the next hurdle is to calculate the expected number of future transactions, based on a given purchase history. Due to the fact that as opposed to the exponential distribution the Erlang-k distribution is not memoryless, we can not use the relation E(Y (T, T + t)|x, tx , T, r, α, a, b) = E(X(t)|τ > T, λ, p) · P (τ > T |x, tx , T, λ, p), (A.37) as it is the case for the CBG/NBD model. Recency (T − tx ) actually does influence the expected number of future transactions (i.e. the first multiplication term), and not just the probability of still being active. Assuming that the customer has survived the last transaction, a longer time period since the last transaction actually makes it more likely that the next transaction will take place soon. Therefore, we will systematically underestimate future transactions, if we still use this relation for CBG/CNBD-k. Nevertheless, we proceed with our heuristic simplifications, and again adapt the findings of Hoppe and Wagner. They derived a+b+x E(Y (T, T + t)|x, tx , T, r, α, a, b) = · G(r+x, b+x, b+x, α+T | α, t) a−1 · P (τ > T | x, tx , T, r, α, a, b) (A.38) for the CBG/NBD model. In their erratum (Wagner and Hoppe, 2008) to Batislam et al. (2007) they note that it is possible to derive the result for the forecast by updating the parameters (r, α, a, b) to (r + x, α + T, a, b + x). We use our exact derivation (A.27) for P (τ > T |x, tx , T, r, α, a, b), and combine this with our approximation for the expectation from the previous section. Additionally, we will update the parameters from (r, α, a, b) to (r + kx, α +

APPENDIX A. DERIVATION OF CBG/CNBD-K 88 T, a, b + x), since we encountered kx uncensored events within (0, T ]). Hence, we conclude: ˆ 1 a+b+x E(Y (T, T + t)|x, tx , T, r, α, a, b) = · k a−1 · G(r + kx, b + x, b + x, α + T | α, t) · P (τ > T |x, tx , T, r, α, a, b) (A.39) A.9 Concluding Remarks Despite the fact that we are just able to derive a biased approximation, we demonstrate in the main part of this thesis that this formula is still able to outperform classic models based on the Poisson assumption regarding individual forecasts. It is assumed that the crucial part for a correct prediction is a proper assessment of whether a customer is still active or not (in particular when faced with rather long prediction periods). It seems that the error that we get by approximating the expected number of transactions is less then the gained precision for the assessment of whether a customer is still active or not.

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