MASTER THESIS ECONOMETRICS, FREE TRACK
COUNTING YOUR CUSTOMERS: A Comparison
Between the Pareto/NBD model and the seasonal
Pareto/NBD model with different weighting
STEYN HESKES
10089233
Supervisor: Noud van Giersbergen 2nd reader: Kevin Pak
March 2017
Contents
1 Summary 2
2 Introduction 3
3 Literature Study 4
4 Data & Methods 8
4.1 Data . . . 8
4.1.1 Description . . . 8
4.1.2 Cleaning . . . 9
4.1.3 Adjustments . . . 10
4.1.4 Training- and testdata . . . 11
4.1.5 Subgroups . . . 11 4.1.6 Notes . . . 12 4.2 Techniques . . . 13 4.2.1 Pareto/NBD model . . . 13 4.2.2 BG/NBD model . . . 15 4.2.3 sPareto/NBD model . . . 16 4.3 Comparison techniques . . . 18 5 Research Method 19
6 Results & Comparison 20
7 Conclusion 26
8 Recommendations 27
1
Summary
In this thesis the main focus lies on predicting future purchasing behaviour of customers of a small retail company. To do this, different models are used and the results are compared to see which one has the best predictive performance. In more detail, it is investigated whether the Pareto/NBD model can be extended by a weighted seasonality factor and if it is possible to find an optimal weighting structure. The results show that the extension of the model, as expected, improves the performance than when it does not account for seasonality. Also, the best results are obtained when the most recent year is given a lower weight than the years before based on the seasonal weighting structure. Although there are a few undesirable fluctuations, it is concluded that the model with seasonal weighting structure is capable of giving a good prediction of future purchasing behaviour.
2
Introduction
Nowadays, society suffers from individualisation. People are self-sufficient and able to find the informa-tion and social connecinforma-tion they need by sitting on their couch at home. They do not have to leave their homes to go to a shopping mall to find a shirt they like, but search for it at Zalandoo. They do not have to leave their house to search for new furniture, they just go to Bol.com. They do not have to go outside to find the partner they will end up with, they just search for him or her online. Even specific knowledge, jobs, healthy recipes, social updates and conversations are available online. All this surfing on the internet leaves information behind about customer behaviour, which is then used by companies to better understand their customers and link actions to it.
Due to improvements in information technology, the availability of individual customer and transaction data has increased. For companies, it is fundamental to understand who their customers are, which products they buy and why they buy those specific products. But also, what kind of lifestyle they prefer, what their interests are and how these interests match the products of the company. To find the answers to these questions, companies started to collect data about their customers. They did this through the use of special customer cards that provide customers with discount and in exchange provides the company with information about the customers’ characteristics and buying behaviour. For example, Albert Heijn, a big supermarket firm, introduced the bonus card that gives customers discount on certain products they buy. To get a bonus card, the customer first has to fill in a form with basic information. Thus, the company knows who their customers are and can now use this information to, for example, improve targeting of their marketing campaigns.
Data is globally getting a more important role in obtaining insight in processes and behaviour and mak-ing decisions. Originally, datascience was mainly used in the hardcore financial sector, but more and more markets, that were not accustomed to using data science, started following this trend of applying a quantitative approach for their decision-making. A good example for this is the marketing sector. This sector was driven mostly by trial and error campaigns, but nowadays they acknowledge the power of datascience and more econometric models are applied for optimisation purposes. The reason that this did not evolve earlier lies in competitive behaviour. If companies would just publish all their specific knowledge, other companies might benefit from this information and catch up. As a result, there were not many publications on this field of expertise.
Since a few years, more scientists practice quantitative research for marketing purposes and universities are adjusting their curriculum. Students are trained to deal with marketing challenges and companies use this expertise to make a profit. Still, most of the small companies do not use this new expertise because there are not enough qualified people that are able to understand, construct and implement these quantitative marketing techniques. This for me was the reason to dive into this subject and approach a small company so I could help it use its data in a way it could benefit the company.
The company I approached was a dutch medium-sized retailer, hereinafter referred to as ”the Company”. Despite the fact that the Company is globally active and has a net turn-over of around 250 million dollars
each year, they have never implemented data sciences in their marketing strategy, that is to say they have not used statistical methods to understand or forecast their sales or to optimise their marketing campaigns to this day. Therefore this was a perfect situation for me to learn about their current market-ing strategy and compare different methods in a quantitative way to obtain the best estimation results. In this thesis, the main focus lies on estimating how many purchases customers will do in the future. There are a few models that are developed to predict the amount of future purchases based on an existing customer base. Some of those are for example the Pareto/NBD model, the BG/NBD model and the Pareto/NBD model with seasonality and HB extension. These models are explained in the Literature Study (Section 3). According to Korkmaz et al. (2013) the Pareto/NBD model and the HB model give the best results concerning the amount of future purchases. Also, according to Fader, Hardie and Lee (2007), the Pareto/NBD model and the BG/NBD model give about the same results. These methods, however, are all tested on the same CDNOW case, in which the data are quite linear and are not subjected to a clear seasonality trend. The data of the Company however, are strongly dependent on seasonality and are therefore expected to lead to different results. This is shown in Figure 1 below. Zitzlsperger et al. (2009) extended the Pareto/NBD model by correcting for seasonality and did some suggestions to improve the seasonal effects. In this thesis the focus lies mostly on the Pareto/NBD model and the suggestions of Zitzlsperger et al. (2009) to add weights to the different time cycles. This is explained in the Literature Study and the Techniques (Section 4.2). Subsequently it is tested whether the models work also in case of a real dataset and not just in case of generated data.
3
Literature Study
In today’s market it is important for a company to be able to make predictions about future purchase behaviour of its customers. With accurate knowledge about the future purchase behaviour, a manager or retailer is able to calculate the customer life value (CLV). By using this knowledge the right way, managers can make efficient strategic choices about their marketing plan, their stock capacity and the amount of employees that are needed in order to serve their clients at its best.
A highly respected model to address these issues is the Pareto/NBD ”counting your customers” model that is originally proposed by Schmittlein et al. (1987). Special about this model is that it counts and identifies the customers who are currently ”active”. The meaning of ”active” is that a customer is interested in the product and there is a chance that he will purchase more products in the future. This aspect makes is possible to do three things, namely: monitoring the size and growth rate of a firm’s ongoing customer base, evaluating a new product’s success based on the pattern of trial and repeat purchases, and targeting a subgroup of customers for advertising purposes and promotions. To illustrate these issues, suppose you are in charge of the Company. You have a list of customers that have in the past bought products one or more times, as well as information about the frequency and timing of each
Figure 1: Daily purchases
customer’s transactions. With this information, Schmittlein et al. (1987) suggest that as a starting point you, as a director of the company, can now ask himself four simple question:
1. How many retail customers does the firm now have?
2. How has this customer base grown over the past year?
3. Which individuals on this list most likely represent active or inactive customers?
4. How many purchases should be expected next year by those on the list, both individually and collectively?
Of course, there are numerous examples of how executives gather the right information to deal with these questions. Department stores uses charge card records, dentists, physiotherapists and other medi-cal professionals use files, the Albert Heijn uses the AH Bonus Card and the Company that is examined in this thesis uses a special Customer card. To help find the answers to these questions, Schmittlein et al. (1987) introduced the Pareto/NBD model. This model is based on five assumptions, which are described in the Research Method section in this thesis. In order to estimate the model properly, you need to know three characteristics: the customer’s ”recency”, the customer’s ”frequency” and the time observed. The recency tells us how long ago the last purchase of a certain customer was. The larger a customer’s recency, the more likely that this customer has lost his interest, died, or in other words:
became ”inactive”. The frequency tells us how many repeat purchases a certain customer made in a specific time period. The time observed is the time between the customers first purchase and now. The moment of the first purchase is when the customer entered the database of the Company.
Although this model has proven to be a great model for predicting customer purchasing behaviour, there are a few disadvantages. First, the model is quite hard to use and understand. Second, the likelihood function used to optimise the Pareto/NBD model’s parameters is very complex since it uses numerous evaluations of the Gaussian hypergeometric function, which makes the Pareto/NBD procedure very de-manding from a computational standpoint. However, as was stated by Schmittlein et al. (1987), this is now less important since computers are getting stronger and are capable of making the computations more easily. Third, the people that use the model, such as database marketeers, CRM analysts and other marketing professionals, are not familiar with this kind of technique, which makes is hard for them to use it properly and makes it susceptible to misinterpretations. Last, one has to be especially careful with interpreting the outcome of the model. According to Lozier and Olver (1995) the precision of some of the procedures used in the model can vary significantly over the parameter space. This could cause problems for the optimisation as the model searches for the maximum of the likelihood function. According to Fader et al. (2005) the only published research reporting a successful implementation of the Pareto/NBD model using standard maximum likelihood estimation is Reinartz and Kumar (2003). These authors recognised the computational burden of the model and proposed a three-step method of moments estimation procedure. This was also proposed earlier by Schmittlein and Peterson (1994). This procedure is computational less difficult than MLE procedures, but still not very easy. Besides, it does not have the desired statistical properties like the MLE.
There have been several attempts to make the model more user friendly. One of the best known is introduced by Fader, Bruce, Hardie & Ka Lok Lee (2005). They simplified some of the Pareto/NBD assumptions, which makes it a lot more easy to implement these assumptions into the model. This model is called the Beta Geometric/ Negative Binomial Distribution model. The most important simplification is how and when a customer becomes inactive. The BG/NBD model assumes that a dropout occurs immediately after a purchase, instead of seeing this as a continuous process. Therefore, these dropouts can now be modelled using the beta-geometric (BG) model. The full assumptions are described in the Research Methods section. In their research, they use the same data as Schmittlein et al. (1987) did, namely the CDNOW data. The new BG/NBD model leads to similar results: both models have excellent predictive performance for the purpose of forecasting a customer’s future purchasing behaviour. Thus their goal of suggesting a model which is easier to use and produces similar results is met. Good news for the executives!
Nowadays, marketing departments specialise in targeting customers individually. This is made possible by the explosive amount of data that is spread among the internet and collected by all sorts of companies. For example Facebook, Twitter, Youtube, Google and LinkedIn collect data about people’s likes, tweets, video preferences, searches, interests, jobs and associated network.
Unfortunately, there are some limitations to the Pareto/NBD and BG/NBD model that prevent exec-utives from adding additional information to the ”counting your customers” models. The models only say something about the purchasing behaviour of individuals based on their past purchasing behaviour. There are no parameters for gender, age or any other variable that might explain differences in customer behaviour. To make up for this, Abe (2008) proposes a new model that extends the highly respected Pareto/NBD model. It builds on three assumptions of the Pareto/NBD model, namely: a Poisson purchase process, a memoryless dropout process and heterogeneity across customers. Also, it relaxes the assumption of independence of the dropout and purchase rates. Furthermore, it uses a simulation method called MCMC. The advantage of this technique is that it comes with some useful byproducts, such as customer specific statistics. With these customer specific statistics, it is possible to construct hierarchical models. This means that executives can distinct between different groups of customers on the basis of individual characteristics. This is important according to Schmittlein et al. (1987), the Pareto/NBD model should have three functions: monitoring the size and growth rate of a firm’s ongoing customer base, evaluating a new product’s success based on the pattern of trial and repeat purchases, and targeting a subgroup of customers for advertising and promotions.
Last, Zitzlsperger et al. (2009) introduced a model that deals with the seasonality trend that occurs in many sales datasets. This trend could be for example the winter and summer sale periods. Zitzlsperger adds a seasonality factor and multiplies this factor with the individual expected repeat purchases. This way, for each customer, the expected purchases are divided over the predicted time cycle based on their seasonality buying pattern from the past. Zitzlsperger concluded that this improved the performance of the Pareto/NBD model and suggested that adding weights to the different time cycles bases on their recency would increase these results even more. In this thesis, this suggestion is worked out.
4
Data & Methods
First, in Section 4.1 the data and data manipulation processes used to obtain the results are described. Second, in Section 4.2 the techniques to create and improve the prediction models are described. Last, in Section 4.3 measures to compare the performance of the different models are described.
4.1
Data
The Company followed the trend of collecting as many data as possible about their customers. To do this, they introduced a customer card to collect more detailed information about their customers. The data collected this way is stored and analysed by other companies than the Company itself. Thus, the company does not have full control over their own data. This was actually the first time they asked such a company for these data.
First, the data directly received from the company or the raw data is described in detail. Second, the cleaning process is described and illustrated. Since there has never been a check on these data by the Company itself, it is expected that the data might not be complete and clean. Therefore, reliability checks and secure cleaning should be done first, otherwise decent analysis is not possible. Third, adjustments to make the data ready for modelling are described. Last, some notes of unexpected findings in the data are given.
4.1.1 Description
The company data can be divided into four different sections, which are stored in different databases: The Customer data, the Transaction data, the Retailer data and the Merchandise data. These data are connected by the customer ID or customer card number. In this thesis, only the Customer data, consisting of 1.577.413 customers, and the Transaction data from 2009 till 2014 are used. Therefore, only these data are described. The transaction data is divided into two groups; 2009 till 2013 as training data and 2014 as test data. The training data is used to estimate the model’s parameters. Based on these parameter estimations, a prediction on the number of weekly purchases is made and compared with the test data. The better the estimation describes the test data, the higher the prediction performance of the model.
The Customer data gives information about each customer that has been registered in a store of one of the brands the company distributes. Customers are triggered to get a customer card because it gives them discount. Through the use of this card, information about characteristics and purchasing behaviour is shared with the Company. The most important data given by the Customer data and Transaction data is described as follows:
• Customer ID: every customer has its own unique Customer ID. This Customer ID is not only used in the Customer data, but also in the Transaction data. When a customer purchases at one of the
stores of the Company, while using his or her Customer’s ID, it is possible to connect information about these transactions to information about the customer.
• Registration time: this is when the customer subscribed at one of the stores.
• Country: Because the company does not only distribute their brands in the Netherlands, differences between countries can be interesting. For example, the Company distributes in the Netherlands, Belgium, Spain, Germany and France. In this thesis only Dutch data is used.
• Gender: male or female.
• Date of Birth: to what age (group) do the customers of the company belong?
The Transaction data gives information about each purchase made by registered customers at stores of brands of the company. The most important information given in the Transaction data is:
• Customer ID: as mentioned before, the Customer ID is important because it makes it possible to link a certain purchase to the customer who bought the product. With this information, we know which customer bought which products for what value and with what frequency.
• Transaction time: this gives information about when the transaction occurred. This is important to know, because this information tells something about when a customer did his or her first purchase and with what frequency and recency.
• Store name: because the company has more than one brand, by knowing the name of the store, we can distinguish between different brands. This is necessary because the different brands will have different target groups and probably different sales patterns.
4.1.2 Cleaning
As the data was never used before, it was important to check if it was ”clean” before using it. Also, because the data consists of more than one and a half million customers, there is a high risk of mistakes. Two cleaning operations have been performed. First, all people older than 99 years old were removed, since it is not clear whether these people are really that old and still active, or maybe there was something wrong with their birthdates. Second, a check of the data has been performed to see if there were any obvious fallacies. For example, all different entities have been checked for undesirable or strange values. Also, a check was done for outliers shown by Figure 2. In this plot, the daily purchases are plotted from 2009 up until 2014. There is one big outlier present with a value of more than one hundred times the average sales. The Company did not recognise this result because it did not initiate any special marketing actions around that time. The company that stored the data also did not recognise the large amount of transactions that day and where these transactions came from. Especially strange was that all of these transactions were done at exact the same time. Did the system fail whereby some of the transactions where not stored properly and then just imported later on the same time on the same day
to make the data complete? Or was it another flaw? They could not explain it. Therefore, the day of the outlier is removed from the dataset.
Figure 2: Original daily purchases
4.1.3 Adjustments
After the cleaning phase, the data are adjusted to the requirements that the data have to meet in order to be able to use them in the models. The requirements that have to be met are:
• A customer has done at least one transaction from 2009 until 2013;
• The Customer ID must be available, otherwise it is impossible to link the customer database to the transaction database;
• The Date of Birth from a customer must be available;
• The Gender of a customer must be available;
• The Country of a customer must be available, as the Company asked to only look at Dutch customers;
• The customer data are not Transferred or Deleted at any point in time. When data is transferred, the future transactions are connected to another CustomerID. Unfortunately, the Company could’t find out to which account. Deleted accounts are accounts of customers that are not active in the
system anymore. Therefore, Transferred and/or Deleted accounts might be incomplete and are therefore not taken into account;
• From all transactions, the store must be available;
• The daily sales numbers are aggregated to weekly sales numbers.
On the basis of these requirements, only 113.621 customers with a total of 454.426 transactions re-main. All customers and corresponding transactions that did not meet the requirements are left out as mentioned before.
4.1.4 Training- and testdata
From the remaining data, two types of training data are constructed. This is done in the same manner as Fader et al. (2005) constructed their data. The first training set, the one without restrictions, is based on customers that made purchases between 2009 and 2013. Unfortunately, we do not know whether customers made purchases before 2009, because these data are simply not available. Therefore, we assume that the first purchase from a customer within the available data is the customer’s first purchase. The second training set, the one with restrictions, consists of customers that did their first purchase between 2009 and 2011. Customers that did their first purchases later than 2011, are not taken into account. Therefore, this training set is smaller than the first one. According to Fader et al. (2005), this should give better results, because when a customer is new to the company in the last month of the training set and does two purchases, he will have a very high purchasing frequency. If there would have have been more time to observe this customer, maybe the first repeat purchase would occur after two or more years. Therefore, customers that are not observed for quite a long time might give wrong information.
4.1.5 Subgroups
To see if the findings from Fader et al. (2005) also apply to the data of the Company, the data are divided into several groups. The customers are split based on gender and age characteristics and the time of a customer’s first purchase. In the first place, the data are divided by gender. Second, the data are divided by age into three different agegroups: 0 - 34, 35 - 64 and 65 - 99. The groups are chosen this way according to the Company’s experience that these groups are quite homogenous. Last, the data are divided by the time of the customer’s first purchase. First, all available customers that did a purchase in the training set interval are taken into account. This way, the time of a customer’s first purchase is not a condition. Second, only the customers that did their first purchase in the first half of the training period are taken into account. By doing this, the model is based on a fixed customer base at the half of the training period and has another half of the training period to ”calibrate” for this selection of customers. By constructing these different groups of customers, the different datasets contain different amounts of customers, different characteristics of customers and transactions so that you can see if the training data
Table 1: Characteristics of the data
Trainingset Gender Age Customers Purchases Purchases/Customers Unrestricted Males 0-34 8085 23176 2.87 Restricted Males 0-34 2958 11178 3.78 Unrestricted Males 35-64 69511 279908 4.03 Restricted Males 35-64 34998 188291 5.38 Unrestricted Males 65-99 31811 137653 4.33 Restricted Males 65-99 18282 101117 5.53 Unrestricted Females 0-34 521 1432 2.75 Restricted Females 0-34 153 540 3.53 Unrestricted Females 35-64 3055 9802 3.21 Restricted Females 35-64 1116 4894 4.39 Unrestricted Females 65-99 638 2455 3.85 Restricted Females 65-99 280 1563 5.58
with restriction performs better or worse than the training data without restrictions. The groups are shown in Table 1.
When you look at the results, the first thing that stands out is that the vast amount of the customers are male and older than 35 years old. Only 4% of all customers in the database is female, most of which are of a younger age class. Thus it appears that the brands of the Company are more interesting for males than for females. Also, Males that are older than 34 years have a higher amount of purchases per customer. This underlines that the Company’s target audience consists of male adults.
When you compare the training dataset with restriction to the training dataset without restriction for a customer’s first purchase, we find that there are relatively a lot more young people new to the brand than older people. From the age class 0 to 34 years old the number of transactions and customers in the data is doubled during the second time period. From the other two age classes, the number of transactions is not doubled. This counts especially for males, who represent the largest part of the customer base of the Company. The number of customers grew harder than the number of new transactions for males, in absolute numbers. This probably means that customers that were already known to the company in the first half of the training set buy relatively less products at the Company.
4.1.6 Notes
There are some remarkable findings in the dataset. First, there are many customers in the database that did not do any transaction from 2009 to 2014. These customers could have died, are not active anymore or have made purchases without using their CustomerID. So the active customer base of the Company is far smaller than I expected at first sight. Second, many customer’s have been transferred to another account. Unfortunately, it is not registered to which account. Therefore, these data are useless. Third, 2009 until 2012 are quite stable years. Total sales is almost the same every year. But comparing the sales of these years with 2013 and 2014, you find a big difference. Total sales is bigger in 2013 and the first half of 2014 compared to the years before. The sales level of the last half of 2014 is about the same
level as 2009 till 2012. The increase in sales is likely to influence the estimation results. The last remark is based on the number of transactions each day. We find that in the weekend, a lot more transactions are done than during the workweek. Therefore, as mentioned before, the daily sales are aggregated to weekly sales.
4.2
Techniques
In this section, the models that are used to estimate the number of future transactions based on the company’s customer base are described. First, in Section 4.2.1 the Pareto/NBD model and accessory assumptions are described. Second, in Section 4.2.2. the BG/NBD model is described. Last, in Section 4.2.3 the sPareto/NBD model with average and exponential weighting is described.
4.2.1 Pareto/NBD model
The Pareto/NBD model, or Pareto / Negative Binomial Distribution model, describes repeat-buying behaviour in a non-contractual setting. The model’s foundation was constructed by Schmittlein et al. in 1987 and is later expanded by others like Fader et al. The model uses three types of history purchase data as primary input. These are the customer’s ”recency”, ”frequency” and the length of time over which we have observed his purchasing behaviour. The recency is the time between the last purchase of the customer and now and is denoted by ti. The frequency is the total number of repeat purchases
divided by the time between his first purchase and his last purchase. This value is denoted by λi. The
time a customer is observed is denoted by Ti. Furthermore, the model is based on draws from two
events. The first draw determines for how long a customer remains active. A customer is active as long as he or she is expected to have a chance higher than zero to buy products of the Company. The second draw determines the frequency of how many purchases a customer does while he or she remains active. Accordingly, the Pareto/NBD model is based on the following five assumptions:
1. Whilst a customer is active, the number of purchases he or she makes is determined by a Poisson process with purchase rate λ. Thus the time between purchases is distributed exponentially with purchase rate λ. P [xi|λi, τi> Ti] = e−λiTi(λ iTi)xi xi! where τi> Ti
2. Each customer remains active during a time being exponentially distributed with death rate µi as
follows:
f (τi|µi) = µie−µiτi
3. The purchasing rate λi is distributed according to a Gamma distribution across the population of
customers as follows: g(λi|r, α) = αr Γ(r)λ r−1 i e −αλi
where r, α > 0 and E[λi|r, α] = αr
4. Like the purchasing rate, the death rate µiis distributed according to a Gamma distribution across
the population of customers as follows:
g(µi|s, β) = βs Γ(s)µ s−1 i e −βµi
where s, β > 0 and E[µi|s, β] = βs
5. The purchasing rate λi and the death rate µiare considered to be distributed independently from
each other.
As mentioned before, the notation used to represent the frequency, recency and the length of time over which we have observed the purchasing behaviour of a customer, is (X = x, tx, T ), respectively.
Hereby, x is the number of transactions that are observed during the time period that the customer is followed, thus (0, T ], tx(0 < tx≤ T ) is the time of the last transaction the customer made.
Now the model is optimised by estimating the associated model parameters. To do this the Maxi-mum Likelihood Estimation (MLE) is applied. The likelihood for an individual i with purchase history (xi, ti, Ti), Li= L(r, α, s, β|xi, ti, Ti), is shown to be equal to Li= Γ(r + xi)αrβs Γ(r) 1 (α + Ti)r+xi(β + Ti)s + ( s r + s + xi )A0 where, for α ≥ β A0= F (r + s + xi, s + 1; r + s + xi+ 1;α+tα−β x) (α + tx)r+s+xi −F (r + s + xi, s + 1; r + s + xi+ 1; α−β α+T) (α + T )r+s+xi and for α ≤ β A0= F (r + s + xi, r + xi; r + s + xi+ 1;β+tβ−α x) (β + tx)r+s+xi −F (r + s + xi, r + xi; r + s + xi+ 1; β−α β+T) (β + T )r+s+xi
where F is the Gaussian hypergeometric function.
As the model is constructed, the model parameters (r, α, s, β) can be estimated by maximum likeli-hood. Suppose we have N customers. From these customers, we know customer i had Xi = xipurchases
in period (0, Ti], with the last purchase occurring at txi. Taken this into account, the log-likelihood
function is given by LL(r, α, s, β) = N X i=1 ln[L(r, α, s, β|Xi= xi, txi, Ti)]
In this thesis, the log-likelihood function is maximised by minimising −LL by the fmincon routine from MATLAB. This gives the values of the parameters (r, α, s, β) that maximise the likelihood.
Based on the parameter estimations, the expected number of repeat purchases in a period of length t for a randomly chosen customer can be calculated as follows:
E[X(t)|r, α, s, β] = rβ α(s − 1) " 1 − β β + t s−1#
Since we are interested in the expected total number of repeat purchases rather than repeat purchases from each single customer, the total number of transactions is calculated by multiplying the estimation from a single customer by the total number of customers that did their first purchase on that specific date for each first-purchase date.
Because we are interested in the conditional expectations of the model, the chance of a customer being active must be determined. Schmittlein et al. (1987) provided an expression for this possibility based on information about their past behaviour. This expression is defined as follows:
P (active|r, α, s, β, X = x, tx, T ) = 1 + s r + s + x (α + T )r+x(β + T )sA0 −1
Now that all necessary information is provided, the conditional expectation can be computed. Hereby, E(Y (t)|X = x, tx, T ) denotes the expected number of purchases in period (T, T + 1] for an customer from
which we know the three characteristics: (X = x, tx, T ) or ”frequency”, ”recency” and time observed.
The conditional expectation is then given by
E(Y (t)|X = x, tx, T, r, α, β, a, b) = (r + x)(β + T ) (α + T )(s − 1) " 1 − β + T β + T + t s−1# ×P (active|r, α, s, β, X = x, tx, T )
where P (active|r, α, s, β, X = x, tx, T ) is given above.
4.2.2 BG/NBD model
The beta-geometric/NBD model has been developed by Peter S. Fader, Bruce G. S. Hardie and Ka Lok Lee in 2005 and is an attractive alternative to the Pareto/NBD model in most applications. According to the authors, the BG/NBD model is easier to implement and its parameters are easier obtained using simple tools such as Microsoft Excel.
The BG/NBD model is quite similar to the Pareto/NBD model. The only difference consists of the conception of when customers becomes inactive. The Pareto/NBD model assumes that customers can become inactive at any point in time, independent of the timing of actual purchases. The BG/NBD model decides directly after a customer’s purchase if the customers become inactive or not using the beta-geometric model.
assumptions are slightly different. Some assumptions are the same, some are slightly different as shown below.
1. Same as the Pareto/NBD model
2. Same as the Pareto/NBD model
3. After a customer has done a purchase, he or she has a chance p of becoming inactive. This chance is determined according to a geometric distribution with the following pdf:
P(inactive immediately after jth transaction)
= p(1 − p)j−1
where j = 1, 2, 3, ...
4. Heterogeneity in p follows a beta distribution with the following pdf:
f (p|a, b) = p
α−1(1 − p)b−1
B(a, b)
where 0 ≤ p ≤ 1 and B(a, b) is the beta function, which can also be described as B(a, b) = Γ(a)Γ(b)/Γ(a, b).
5. Same as the Pareto/NBD model
In this thesis, the BG/NBD is not used further, since the results are about the same as the Pareto/NBD model. Also, because the computational power of computers has improved a lot since the invention of the Pareto/NBD model, there are now less problems with estimating the parameters of the model.
4.2.3 sPareto/NBD model
Schmittlein et al. (1987), Fader et al. (YEAR) and others already concluded that the Pareto/NBD model has strong predictive performance of purchases in a non-contractual setting from a fixed customers base. Unfortunately, the model did not yet account for seasonality effects. Zitzlsperger et al. (2009) introduced an extension to the Pareto/NBD model to deal with seasonality. This model is called the sPareto/NBD model.
The sPareto/NBD model uses the resulting sets of parameters estimated by the Pareto/NBD model. These parameters are used to calculate the Conditional Expectations (CE) of future purchases for all customers in the customer base following the methodology of Schmittlein et al. (1987) as described in Section 4.2.2. The model is extended with seasonality effects that are described by the Seasonality Factor. This is done as follows:
1. The seasonality cycle time is determined based on the whole customer base. The seasonality cycle in the data of the Company is one year, because every year a new clothing line is introduced in the same month and the sales period occurs in the same holiday seasons. The seasonality cycle time is denoted by tc.
2. The seasonality cycle time is divided in u blocks. In this case 52 blocks (52 weeks).
3. The blocks have a certain time length, dependent on tc and u as follows: tu=tuc.
4. Since the seasonality pattern might be different depending on the kind of customer, an average individual seasonal purchase pattern is determined as in Zitzlsperger et al. (2009). For each time block tu, the number of total purchases is counted and divided by the total number of purchases
in the accessory cycle time tc.
The seasonality factor is calculated in two different ways. As an average of all years, where each year has the same weight, and as a weighted moving average involving all periods as follows:
1.
SFtcuk=
xtcuk
P xtctotal
where SFtcukis the Seasonality Factor (SF) for each time block k, xtcukis the number of purchases
in each time block k and xtctotal is the total number of purchases in one cycle.
2. SFt∗ (cj)= ω · SFt(cj)+ √ ω · SFt(cj−1)+ ... + x √ ω · SFt(cj−x) ω +√ω + ... +√xω where SFt∗
(cj) is the weighted Seasonality Factor (WSF) for each time block k, SFt(cj)is the time
block for each time cycle, x the number of time cycles, ω the weighting factor and j denotes the time cycle. A value of ω above one gives recent periods more weight, a value of ω smaller than one overweights older periods.
As in Zitzlsperger et al. (2009), the Seasonality (S) is defined as follows:
S[SF∗|tc, u, X = x, tx, T ]
where X is the information about the recency and frequency of all customers. The Seasonality Factor (SF) is now multiplied with the Conditional Expectations (CEs) for individual purchases calculated with the Pareto/NBD model. The expected number of purchases in the prediction period is multiplied by the seasonality as follows:
CEseasonali = E[X∗|r, α, β, X = x, tx, T, T∗] × S[SF∗|tc, u, X = x, tx, T ]
CEseasonaltotal = i X seasonal CEi
4.3
Comparison techniques
To compare the predictive performance of the models, two measures that examine the forecasts are used. Furthermore, the plots of the Pareto/NBD model and the seasonal Pareto/NBD model with different weightings will be plotted against the actual purchases. The two measures are the Mean Absolute Percentage Error (MAPE) and the Mean Squared Percentage Error (MSPE). The measures are defined as follows: 1. MAPE: M AP E = 1 n n X i=1 ANi− F Ni ANi
where F Ni is the number of forecasted future purchases, ANi is the number of actual future
purchases and n is the number of predicted ”blocks”.
2. MSPE: M SP E = 1 n n X i=1 ANi− F Ni ANi 2
5
Research Method
The Pareto/NBD model has been tested and extended mostly based on the CDNOW customer database. Looking at the results of the studies that use these data, but different techniques, the weekly purchases graph based on the CDNOW data looks quite linear compared to the weekly sales graphs from the Company. As mentioned before, the weekly purchases of the Company are obviously susceptible on seasonal effects (see Figure 1). This property of the data is hard to explain using the Pareto/NBD model, since it does not account for seasonality. Also, as in the researchpaper of Fader et al. (2005), the training set of the data is limited to customers that did their first purchase the second half of the training set period. Does this give better results? To check this, samples based on age and gender are taken from the data and checked for two types of training data: first with customers that made their first purchase during the whole training set interval, second with only customers that made their first purchases during the first half of the training set interval. The performance of the two types of training data sets is measured using the MAPE. As an extra advantage, optical differences between the seasonality behaviour of different groups of customers can be evaluated. Furthermore, Zitzlsperger et al. (2009) found great results using the Pareto/NBD model with seasonality extension. They suggested to add weights to the time cycles based on their recency. In this thesis, it is investigated if adding these weights improve prediction results and what the optimal weighting scheme is based on the underlying data.
The Pareto/NBD model is compared with the seasonal Pareto/NBD model. Furthermore, the seasonal Pareto/NBD is weighted by different values of ω. These values determine the weight of the more recent years compared to the less recent years. Does the different weighting improve the results and what value of ω gives optimal results? The data is divided into two groups; the training data and the testing data. The training data is used to estimate the model parameters and the test data is used to compare the prediction results to the real data. Like Fader et al. (2005), the training data is constructed in two different ways. One consists of the whole customer base, and one consists only of customers that made their first purchase in the first half of the training period. According to Fader et al. (2005), the second alternative gives better results. If this is correct, only the training data with customers that made their first purchase in the first half of the training set interval is used to construct the sPareto/NBD model. The predictions of all models on the different samples is compared by two measures: the mean absolute percentage error and the mean squared percentage error. These two measures are used to compare both the training data as well as the test data to the real data.
6
Results & Comparison
In this section, the results are discussed. First, the parameters of the Pareto/NBD models with different subgroups and training sets are shown in Table 2. Second, the graphs of the Pareto/NBD versus the real data are shown and discussed. Furthermore, the results of the (weighted) seasonality factor are discussed and shown by a graph and table.
The results of the Pareto/NBD are shown in Table 2. First, the type of training set is defined. The unrestricted training set is the set without restriction. The restricted training set is the training set that is limited to customers that made their first purchase in the first half of the training set interval. Second, groups are divided by gender and age. The variables (r, s, α, β) are the estimated values of the Gamma distributions used in the Pareto/NBD model. ll shows the optimal value of the MLE. MAPE shows the value of the Pareto/NBD model compared to the real data. From the theory, we know that the expected value of the purchasing rate is E[λi|r, α] = αr and the expected value of the death rate
E[µi|s, β] = sβ. These values are also given in Table 1. Looking at this Table 2, you find that the
purchasing rate is much higher than the death rate for all groups, which seems logical. Also, there are big differences between older customers and younger customers. Older customers have a much higher value of µ, which means they are more likely te stay active for a long time, whereas younger people have a higher chance of becoming inactive. Also, younger customers have a higher purchasing rate (because the time between purchases is shorter, as shown in Table 1). Older customers make purchases with a lower frequency, but, as mentioned before, are more loyal than younger customers and thus expected to remains active for a longer period of time. The most important finding is that the restricted trainingset shows better results according to the values of MAPE. Only the males between 0 and 34 years old show an odd result. Therefore, the sPareto/NBD model is estimated based on the restricted trainingset. To make these results easier to see, two plots are made: one based on the unrestricted model (Figure 3) and one on the restricted model (Figure 4). In these figures, it is shown that the unrestricted model follows the trend of 2013 quite good, but does not follow the weekly sales amounts over the other years quite well.
Table 2: Results of the Pareto/NBD model for different subgroups
Trainingset Gender Age r s α β ll MAPE E[λi|r, α] E[µi|s, β]
Unrestricted Males 0-34 0.947 35.90 0.117 0.617 49862 1.798 8.11 58.15 Restricted Males 0-34 0.995 48.10 0.126 1.541 32460 1.867 7.90 31.21 Unrestricted Males 35-64 1.291 48.07 0.065 0.503 717910 1.684 19.89 95.57 Restricted Males 35-64 1.446 61.26 0.064 0.665 573060 1.351 22.67 92.20 Unrestricted Males 65-99 1.303 50.00 0.059 0.685 373820 2.004 22.23 72.94 Restricted Males 65-99 1.561 66.64 0.060 0.782 311850 1.533 26.23 85.21 Unrestricted Females 0-34 0.733 13.01 0.181 0.606 2953.5 0.850 4.06 21.48 Restricted Females 0-34 1.198 37.93 0.180 0.519 1550.2 0.592 6.67 73.07 Unrestricted Females 35-64 1.111 32.06 0.106 0.483 22443 2.280 10.51 66.39 Restricted Females 35-64 1.139 50.86 0.091 0.699 14404 1.957 12.50 72.75 Unrestricted Females 65-99 1.233 41.80 0.061 0.492 6070.2 1.870 20.08 84.91 Restricted Females 65-99 0.971 40.55 0.048 0.416 4604.6 1.508 20.11 97.47
In the graphs shown by Figure 5 and Figure 6, the real data, Pareto/NBD model and the sPareto/NBD model with ω = ω∗ are shown (The value of ω∗ is defined below). The Pareto/NBD and sPareto/NBD
model are based upon the restricted training set. Figure 5 shows the weekly repeat purchases and Figure 6 shows the cumulative weekly repeat purchases. The weekly purchases graph shows that the real weekly purchases (grey) have a seasonal pattern. This pattern is shown over the whole graph. Also, 2013 is a year with extraordinary sales. In 2014, the weekly sales fall back to ’normal’ levels.
On average, the Pareto/NBD model (green) follows the weekly sales level quite good, but totally misses the seasonal trend. This is logical because this model does not account for seasonality as mentioned before. When we look at the first period (2010-2011), the models follow the real data on average quite good. But when we look at 2012, it overestimates the real sales level and in 2013, it underestimates the real sales level. Furthermore, in 2014 the model overestimates the real sales level again. This is probably caused by the exceptional high weekly sales levels in 2013.
Looking at the sPareto/NBD model (blue), the model captures the seasonality in the data quite well. The seasonal trend of the real data is incorporated by the sPareto/NBD model, but despite the fact that the model includes the seasonality effect, the total number of weekly repeat purchases is still over- or underestimated in the same way as the Pareto/NBD model does: in 2012 and 2014, the sPareto/NBD sales (blue) are too high, and in 2013 the sales are too low. This is caused by the underlying Pareto/NBD model that is probably being influenced by the extraordinary sales levels in 2013, as mentioned before.
In Figure 6, the cumulative weekly repeat purchases are shown from the real data, the Pareto/NBD model and the sPareto/NBD model with ω = ω∗. The Pareto/NBD model (green) follows the aggregate sales level quite good, but does not capture the increased sales in 2013 quite well, because it is a really smooth model. The sPareto/NBD model is based on the Pareto/NBD model and shows the same results, but has a clear seasonal trend, that is also shown in the real data. Therefore, it is expected that the results from the sPareto/NBD model outperform the Pareto/NBD model.
Now that the optical results of the sPareto/NBD are shown, the value of ω is varied to find an optimal point. Values of ω that are shown are (0.01, 0.02, 0.05, 0.1, 0.5, 1, 2 and 5. The value of ω determines the
Figure 5: Weekly purchases
Table 3: My caption
Data Measure ω 0,01 0,02 0,05 0,1 0,5 1 2 5 Pareto/NBD Training MAPE 0,931 0,930 0,932 0,938 0,980 1,014 1,059 1,136 1,394 Training MSPE 1,501 1,493 1,492 1,502 1,611 1,721 1,880 2,153 2,660 Test MAPE 0,813 0,808 0,805 0,808 0,855 0,897 0,963 1,073 1,167 Test MSPE 0,992 0,985 0,985 0,995 1,106 1,217 1,377 1,649 2,038
weighting of the recent cycles compared to the preceding cycles. For each value of ω, the sPareto/NBD model is estimated and the MAPE and MSPE are calculated. Besides, the results of the underlying Pareto/NBD model are shown. The results are shown in Table 3. First, the performance within and outside the training data is indicated by the training and the test set. Second, the type of test (MAPE or MSPE) is shown. Thereafter, the results of the test for the sPareto/NBD models and Parete/NBD model are shown. For example: the mean absolute percentage error from the sPareto/NBD model with ω = 0.05 within the training data is 0.932. What directly strikes, is the performance of the each of the sPareto/NBD models, thus with seasonality, performs better than the normal Pareto/NBD model for each value of ω. Both the MAPE and the MSPE give the same results. This is shown both in the training set as in the test set. Furthermore, the lower the value of ω, the better the prediction results of the sPareto/NBD model. This means that the lower the weight of recent seasonality patterns, the better the results are for both the training set as the test set.
Figure 6: Cumulative weekly purchases
The optimal value of ω is found by minimising the value of the mean absolute percentage error and the mean squared percentage error. The results is ω = 0.01951 for the minimisation of the MAPE within the training set which gives a minimal value of MAPE of 0.93018 in the training data and 0.80770 in the test data. The result is ω = 0.03606 for the minimisation of the MSPE within the training set which gives a minimal value of MSPE of 1.49061 in the training data and 0.98334 in the test data. In the two plots below, the value of ω is plotted against the values of MAPE and MSPE, where the optimal points are the lowest points of the graphs.
If you plot the graphs for 2010 until 2013 separately, you can take a closer look at the individual years. The plots are shown in Figure 9, Figure 10, Figure 11 and Figure 12. In the first three years, the seasonal Pareto/NBD model (green) follows the real data (grey) quite good. The sales level starts low and shows two peaks at a quarter and three quarter of the year. The normal Pareto/NBD model (blue) is not capable of following this trend. Looking more closely, the peaks in the model are captured sometimes, but can also vary a little. This could be caused by the fact that in this thesis the actual week numbers are used and not the exact weeks when events like vacation occur. This could cause a peak to show up one week earlier of later.
The year 2013 shows different results. Where the sPareto/NBD model (green) goes devaluates in the middle and follows the trend of 2010 until 2012, the real data (grey) goes up to the level of the first peak and breaks the seasonality pattern that occurs in the years before by not devaluating in the middle of the year. This explains the counterintuitive results given in Table 3. Since the periodical trend in 2013
Figure 7: Mean Absolute Percentage Error Figure 8: Mean Squared Percentage Error
is different than the others, it is logical that the more weight is given to that year, the worse the model represents the data.
Figure 9: Weekly repeat purchases, 2010 Figure 10: Weekly repeat purchases, 2011
Finally, the Pareto/NBD model, three sPareto/NBD models based on different weighting patterns, among which the sPareto/NBD model with the optimal weighting pattern according to MAPE, and the real weekly sales data of the year 2014 are shown in Figure 13. Clearly, the real weekly sales start higher at the beginning of the year compared to the weekly sales of 2010, 2011, 2012 and 2013. The seasonality factor is not able to capture this since it relies on cycles where this pattern is not present. The real weekly purchases return to ’normal’ levels in the middle of the graph, where the sPareto/NBD model captures the seasonal trend quite good again. From the results of Table 3, the sPareto/NBD model performs even better in the test period, which is the year 2014, than in the training set, that consists of the years before 2014. This could be caused by the fact that the weekly purchases vary more during the training period than they do in the test period. Besides, the weekly purchase level and seasonality pattern in 2014 looks a lot like those in 2010, 2011 and 2012.
Figure 11: Weekly repeat purchases, 2012 Figure 12: Weekly repeat purchases, 2013
7
Conclusion
In this section, a detailed conclusion to the research question is formulated. First, the restricted training set that consists only of customers that made their first purchase in the first half of the training set interval has better results than the training set without this restriction. This is shown by Table 2, where for almost all subgroups, the performance based on the mean absolute percentage error shows better results for the restricted training set than the unrestricted training set. This result was also found by Fader et al. (2005).
Also, like Zitzlsperger et al. (2009), the sPareto/NBD model gives better results than the Pareto/NBD model for all values of ω. This seems logical, since the data is obviously exposed to seasonality effects, as shown in Figure 1 and Figure 5. From Figure 5 it is also clear that the Pareto/NBD model does not follow the seasonality trend at all, whereas the seasonal Pareto/NBD model follows the trend quite good. According to Table 3, all models are better at explaining the real weekly purchases outside the data (test set) than inside the data (training set). This is a remarkable result, because this means that the model is better at predicting (unknown) trends than it is in capturing known data points. This result can be explained by the deviating year 2013. The model has a hard time capturing the extraordinary increase in sales as from the second quarter of 2013 until the first quarter of 2014. This deviation is shown by Figures 9 until 12, where Figure 12 shows the seasonal pattern of 2013. The Pareto/NBD model does not include any dummy variables or other variables that have an influence on the amount of sales that year. It only relies on the customer base that grew up to 2011. Also, because the seasonal Pareto/NBD model is based on the ordinary Pareto/NBD model, it is also not capable to capture these effects. Besides, the seasonal Pareto/NBD’s weighting scheme gives counter-intuitive results. One would think that the seasonality pattern of recent years would be more representative than the pattern of former years. But in contrary, the results based on the data of the Company show that the less weight is given to recent years, and thus more weight to former years, the better the performance of the seasonal Pareto/NBD model. This is found both by the mean absolute percentage error (MAPE) as by the mean squared percentage error (MSPE). The reason for this counter-intuitive result is probably also caused by the abnormal weekly sales pattern of the year 2013. Last, an optimal value of ω can be found based on the value of the MAPE and MSPE. The optimal values are ωM AP E = 0.01951 with MAPE=0.93018 in the within the data
and 0.80770 outside the data and ωM SP E = 0.03606 with MSPE=1.49061 within the data and 0.98334
outside the data. The values are graphically shown in Figure 7 and Figure 8. Conclusion: as stated by Zitzlsperger et al. (2009) the seasonal Pareto/NBD model gives better results than the Pareto/NBD model. Furthermore, an optimal weighting pattern can be found based on the value of ω which leads to the best results within the data (training set). This solution also gives one of the best results outside the data (test set). Thus adding these weights improves the prediction performance of the model.
8
Recommendations
Since this thesis is based on data from just one company, the outcome might not be entirely representative. The remarkable outlier shown in Figure 2, the weekly purchase levels and the unruly seasonal pattern of 2013 shows that the data are not only being influenced by the number of customers and seasonality, but also by other factors. Therefore, it is recommended to combine the model of Zitzlsperger et al. (2009) with the model of Abe (2008) and include dummy data based on different years. Also, because the data are from a company that went bankrupt in 2016, there could have been more external factors present that led to the abnormal behaviour of the data in 2013 and 2014 and the counter-intuitive results. For these reasons, more research based on data of other companies should be done.
9
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