A Customer Purchase Incidence Model
Applied to Recommender Services

Andreas Geyer-Schulz


, Michael Hahsler

, and Maximillian Jahn


Universita¨t Karlsruhe (TH), D-76128 Karlsruhe, Germany,
andreas.geyer-schulz@em.uni-karlsruhe.de,

Wirtschaftsuniversita¨t Wien, A-1090 Wien, Austria,
Michael.Hahsler@wu-wien.ac.at
Abstract. In this contribution we transfer a customer purchase incidence model
for consumer products which is based on Ehrenberg’s repeat-buying theory to
Web-based information products. Ehrenberg’s repeat-buying theory successfully
describes regularities on a large number of consumer product markets. We show
that these regularities exist in electronic markets for information goods, too, and
that purchase incidence models provide a well founded theoretical base for re-
commender and alert services.
The article consists of two parts. In the first part Ehrenberg’s repeat-buying theory
and its assumptions are reviewed and adapted for web-based information markets.
Second, we present the empirical validation of the model based on data collected
from the information market of the Virtual University of the Vienna University of
Economics and Business Administration from September 1999 to May 2001.
1 Introduction
In this article we concentrate on an anonymous recommender service of the correlation-
type made famous by Amazon.com applied to an information broker. It is based on con-
sumption patterns for information goods (web-sites) from market baskets (web-browser
sessions) which we treat as consumer purchase histories with unobserved consumer
identity. In Resnick and Varian’s design space [25] this recommender service is charac-
terized as:
1. The contents of a recommendation consists of links to web-sites.
2. It is an implicit service based on observed user behavior.
3. The service is anonymous.
4. The aggregation of recommendations is based on identifying outliers with the help
of a stochastic purchase incidence model.
5. A sorted list of recommended related web sites is offered to the user of a web site
(see figure 1).

In R. Kohavi, B.M. Masand, M. Spiliopoulou, and J. Srivastava, editors, WEBKDD 2001
- Mining Log Data Across All Customer Touch Points, Third International Workshop, San
Francisco, CA, USA, August 26, 2001, Revised Papers, Lecture Notes in Computer Science
LNAI 2356, pages 25-47. Springer-Verlag, July 2002.

2Fig. 1. An Anonymous Recommender Based on “Observed Purchase Behavior”.
This recommender service is part of the first educational and scientific recommender
system integrated into the Virtual University of the Vienna University of Economics
and Business Administration (http://vu.wu-wien.ac.at) since September 1999. A full
description of all recommender services of this educational and scientific recommender
system can be found in Geyer-Schulz et al. [16].
For example, figure 1 shows the recommended list of web-sites for users interested
in web-sites related to the Collaborative Filtering Workshop 1996 in
Berkeley. The web-site (in figure 1 the Collaborative Filtering Work-
shop 1996 in Berkeley) in the yellow box (gray in print) is the site for which
related web-sites have been requested. For every recommended web-site a Vote Box
allows users to evaluate the quality of this recommendation.

3In Geyer-Schulz et al. [15] we have presented the architecture of an information
market and its instrumentation for collecting data on consumer behavior. We consider
an information broker with a clearly defined system boundary and web-sites consisting
of one or more web-pages as information products. The information broker contains
only meta-data on the information products (including an external link to the home
page of each information product). Clicking on an external link (which leaves the in-
formation broker and leads to the home page of a web-site) is equated as “purchasing
an information product”. In marketing, we assume that a consumer will only repeat-
edly purchase a product or a product combination if he is sufficiently content with it.
The rationale that this analogy holds even for free information products stems from
an analysis of the transaction costs of a user of an information broker. Even free in-
formation products burden the consumer with search, selection and evaluation costs.
Therefore, in this article we derive recommendations from products which have been
used (= purchased) together in a session repeatedly across different sessions (= buying
occasions) (see Bo¨hm et al. [8]). Data collection (logging) is restricted entirely to the
information broker, only clicking on an external link is logged. This implies that on the
one hand in the logs of the information broker usage behavior for information prod-
ucts can not be observed (and is in fact completely irrelevant as far as repeat-buying
theory is concerned) and that on the other hand almost no preprocessing is necessary
to obtain sessions. Without such an instrumented architecture the purchase incidence
model may still be of use in combination with web-usage mining approaches as e.g. the
sequence miner WUM of Spiliopoulou [28]. For a recent survey of work on this area
see Srivastava et al. [29].
Such recommendations are attractive for information brokers for the following rea-
sons:
– Observed consumer purchase behavior is the most important information for pre-
dicting consumer behavior. For offline behavior this has been known for a long time
(see Ehrenberg [11]), for online behavior see Bellmann et al. [6] for a recent study.
– In traditional retail chains, basket analysis shows up to 70 percent cross-selling po-
tential (see Blischok [7]). Such recommendations facilitate “repeat-buying” which
is one of the main goals of e-commerce sites as reported in Bellmann et al. [6].
– Most important in a university environment is that such recommendations are not
subject to several incentive problems found in systems based on explicit recom-
mendations (as e.g. free-riding, bias, ...) which are analyzed by Avery and Zeck-
hauser [5]. The transaction cost of faking such recommendations is high, because
only one co-occurrence of products is counted per user-session as usual in con-
sumer panel analysis (see Ehrenberg [11]). Free-riding is impossible because by
using the information broker each user contributes usage data for the recommenda-
tions. The user’s privacy is preserved.
– And, last but not least, the transaction costs for the broker are low since high-quality
recommendations can be generated without human effort. No editor, no author, no
web-scout is needed.
Recently the problem of generating personalized recommendations from anony-
mous sessions (or market basket data) and user sessions (or consumer panels with

4individual purchase histories) has attracted considerable attention. Personalization is
achieved by taking the user’s current navigation history into account. Mobasher has
studied two variants of computing recommendations from anonymous session data,
namely PACT (profile aggregation based on clustering transactions) and ARHP (as-
sociation rule hypergraph partitioning) in [23] and [24]. Lawrence et al. [21] report on
a hybrid recommender based on clustering association rules from purchase histories
combined with the product taxanomy and the profit contribution of a product. Gaul and
Schmidt-Thieme [14] construct recommender systems from frequent substructures of
navigation histories. Lin et al. [22] develop an association rule mining algorithm with
adaptive support for a collaborative recommender system.
However, anonymous recommendations based on consumption or usage patterns
nevertheless have the following two problems which we address in this article with the
help of Ehrenberg’s repeat-buying theory [11]:
– Which co-occurrences of products qualify as non-random?
– How many products should be recommended?
Ehrenberg’s repeat-buying theory provides us with a reference model for testing
for non-random outliers, because of the strong stationarity and independence assump-
tions in the theory discussed in section 2. What makes this theory a good candidate
for describing the consumption behavior for information products is that it has been
supported by strong empirical evidence in several hundred consumer product markets
since the late 1950’s. Ehrenberg’s repeat-buying theory is a descriptive theory based
on consumer panel data. It captures how consumers behave, but not why. Several very
sophisticated and general models of the theory (e.g. the Dirichlet model by Goodhardt
et al. [18]) exist and have a long tradition in marketing research. However, for our
purposes, namely identifying non-random purchases of two information products, the
simplest model – the logarithmic series distribution (LSD) model – will prove quite
adequate. In this setting the LSD model removes all random co-purchases from the rec-
ommendation lists. It acts like a filter for noise. For a survey on stochastic consumer
behavior models, see e.g. Wagner and Taudes [32].
One of the main (conceptual) innovations of this paper is that we explain how we can
apply a theory for analyzing purchase histories from consumer panels to mere market
baskets.
2 Ehrenberg´s Repeat-Buying Theory and Bundles of Information
Products
Of the thousand and one variables which might affect buyer behavior, it is found that
nine hundred and ninety-nine usually do not matter. Many aspects of buyer behavior
can be predicted simply from the penetration and the average purchase frequency of an
item, and even these two variables are interrelated. A.S.C. Ehrenberg (1988) [11].
In purchasing a product a consumer basically makes two decisions: when does he
buy a product of a certain product class (purchase incidence) and which brand does he

5pu
rch
ase
pr
oce
ss
for
pr
od
uct
x
bo
ug
ht
by
us
er
j
products bought
by user i
products bought
by user j
ta purchase
incidence
Fig. 2. Purchase Incidences as Independent Stochastic Processes.
buy (brand choice). Ehrenberg claims that almost all aspects of repeat-buying behavior
can be adequately described by formalizing the purchase incidence process for a single
brand and by integrating these results later (see figure 2).
In a classical marketing context Ehrenberg’s repeat-buying theory is based on pur-
chase histories from consumer panels. The purchase history of a consumer is the se-
quence of the purchases in all his market baskets over an extensive periods of time (a
year or more) for a specific outlet. For information products, the purchase history of
a consumer corresponds e.g. to the sequence of sessions of a user in a personalized
environment of a specific information broker. Note, however, a purchase history could
be a sequence of sessions recorded in a cookie, in a browser cache, or in a personal
persistent proxie-server, too.
A market basket is simply the list of items (quantity and price) bought in a specific
trip to the store. In a consumer panel the identity of each user is known and an individ-
ual purchase history can be constructed from market baskets. For information products
the corresponding concept is a session which contains records of all information prod-
ucts visited (used) by a user. In anonymous systems (e.g. most public web-sites) the
identity of the user is not known. As a consequence no individual purchase history can
be constructed.
Very early in the work with consumer panel data it turned out that the most useful
unit of analysis is in terms of purchase occasions, not in terms of quantity or money
paid. A purchase occasion is coded as yes, if a consumer has purchased one or more
items of a product in a specific trip to a store. We ignore the number of items bought or
package sizes and concentrate our attention on the frequency of purchase. For informa-
tion products we define a purchase occasion as follows: a purchase occasion occurs if
a consumer visits a specific information product at least once in a specific session. We
ignore the number of pages browsed, repeat visits in a session, amount of time spent at
a specific information product, ... Note, that this definition of counting purchases or in-

6formation product usage is basic for this article and crucial for the repeat-buying theory
to hold. One of the earliest uses of purchase occasions is due to L. J. Rothman [30].
Analysis is carried out in distinct time-periods (such as 1-week, 4-week, quarterly
periods) which ties in nicely with other standard marketing reporting practices. A par-
ticular simplification from this time-period orientation is that most repeat-buying results
for any given item can be expressed in terms of penetration and purchase frequency.
The penetration

is the proportion of people who buy an item at all in a given
period. Penetration is easily measured in personalized recommender systems. In such
systems it has the classical marketing interpretation. For this article, penetration is of
less concern because in anonymous public Internet systems we simply cannot determine
the proportion of users who use a specific web-site at all.
The purchase frequency
is the average number of times these buyers buy at least
one item in the period. The mean purchase frequency
is itself the most basic measure
of repeat-buying in the Ehrenberg’s theory [11] and in this article.
In the following we consider anonymous market baskets as consumer panels with
unobserved consumer identity – and as long as we work only at the aggregate level
(with consumer groups) everything works out fine as provided that Ehrenberg’s assump-
tions on consumer purchase behavior hold.
Figure 2 shows the main idea of purchase incidence models: a consumer buys a
product according to a stationary Poisson process which is independent of the other
buying processes. Aggregation of these buying processes over the population under the
(quite general) assumption that the parameters  of the Poisson distributions (the long-
run average purchase rates) follow a truncated  -distribution results in a logarithmic
series distribution (LSD) as Chatfield et al. [9] have shown.
The logarithmic series distribution (LSD) describes the following frequency distri-
bution of purchases (see Ehrenberg [11]), namely the probability that a specific product
is bought a total of  ,  ,  , ...,  times without taking into account the number of non-
buyers.


 purchases 
 






ff (1)
Mean purchase frequency
fi




fl




(2)
The variance is:
ffi















! "
# $&%

('
"*),+


-


.


-

(3)
One important characteristic of the LSD is that ffi
0/

. For more details on the
logarithmic series distribution, we refer the reader to Johnson and Kotz [19]. The loga-
rithmic series distribution results from the following assumptions about the consumers’
purchase incidence distributions:
1. The share of never-buyers in the population is not specified. In our setting of an
Internet information broker with anonymous users this definitely holds.

72. The purchases of a consumer in successive periods follow a Poisson distribution
with a certain long-run average  . The purchases of a consumer follow a Poisson
distribution in subsequent periods if a purchase tends to be independent of previous
purchases (as is often observed) and a purchase occurs in such an irregular manner
that it can be regarded as if random (see Wagner and Taudes [32]).
3. The distribution of  in the population follows a truncated  -distribution so that the
frequency of any particular value of  is given by

' 

 
  , for   ,
where  is a very small number,  a parameter of the distribution, and a constant,
so that 

'ff 

 
    .
A  -distribution of the  in the population may have the following reason (see
Ehrenberg [11, p. 259]): If for different products 

fi
ffifl 

 ! "
the average pur-
chase rate of

is independent of the purchase rates of the other products, and
#
%"#%$%&'$%()$+*,$.-/-/-
)
is independent of a consumer’s total purchase rate of buying all
the products, then it can be shown that the distribution of  must be  . These inde-
pendence conditions are likely to hold approximately in practice (see e.g. [4], [10],
[26], [27]).
4. The market is in equilibrium (stationary). This implies that the theory does not hold
for the introduction of new information products into the broker.
Next, we present Chatfield’s proof in detail because the original proof is marred by
a typesetting error:
1. The probability 0  that a buyer makes  purchases is Poisson distributed:

'


,1
2. We integrate over all buyers in the truncated  -distribution:
0


+2




'


31



'ff4


 


31
2



'
%

$
4
)


'

 


31
2



'
%


$

ffi 
)


! 


fl

'


! 


fl

'



'

 


31

 


fl

'

2



'
%


$

ffi 
)

*
 


   

'

 


31

 


fl

'

2



'
%


$

ffi 
)

*
 


   

'



! 


 


 


fl 


31

 


fl

2



'
%


$

ffi 
)

*
 


   

'



! 


fl 
Since  is very small, for  ff and setting 5
 ! 


   this is approximately

80




31







2



'
5

'

 5



31









&



!













0

'




 



with





$

.
3. If  0    for ff  , by analyzing the recursion we get 0

'
"
# $ %

('
" )
and
0


'
" 

# $ %

'
" )
. (However, this is the LSD. q.e.d.)
Next, consider for some fixed information product
in the set  of information
products in the broker the purchase frequency of pairs of

 
 with

 \
. The
probability 0 


 that a buyer makes  purchases of products
and

together in
the same session in the observation period which follow independent Poisson processes
with means  and  is [20]: 0 


!fiff


ff
ffifl
ffi



 fl
. For our recommender services
for product
we need the conditional probability that product

has been used under the
condition that product
has been used in the same session. Because of the independence
assumption it is easy to see that the conditional probability 0 
"!

 is again Poisson
distributed by
0


#!

 
0



$


0






ff


ff
 fl





ffifl

fiff


ff
ffifl


'ff




31
 0




This is not the end of the story. In our data, sessions do not contain the identity of the
user – it is an unobserved variable. However, we can identify the purchase histories of
sets of customers (market segments) in the following way: For each information product

the purchase history for this segment contains all sessions in which
has been bought.
For each pair of information products

 
the purchase history for this segment contains
all sessions in which

 
has been bought. The stochastic process for the segment



%

– & customers which have bought product
and an other product

– is represented by
the sum of & independent random Bernoulli variables which equal  with probability
0 , and ' with probablity 

0( . The distribution of the sum of these variables tends to a
Poisson distribution. For a proof see Feller [12, p. 292]. (And to observe this aggregate
process at the segment level is the best we can do.) If we assume that the parameters 

9of the segments’ Poisson distributions follow a truncated  -distribution, we can repeat
Chatfields proof and establish that the probability of  purchases of product pairs (x,i)
follow a logarithmic series distribution (LSD).
However, we expect that non-random occurrences of such pairs occur more often
than predicted by the logarithmic series distribution and that we can identify non-
random occurrences of such pairs and use them as recommendations. For this pur-
pose we estimate the logarithmic series distribution for the whole market (over all con-
sumers) from market baskets, that is from anonymous web-sessions. We compute the
mean purchase frequency
and solve equation 2 for

, the parameter of the LSD. By
comparing the observed repeat-buying frequencies with the theoretically expected fre-
quencies we identify outliers and use them as recommendations.
The advantage of this approach is that the estimation of the LSD is computationally
efficient and robust. The limitation is that we cannot analyze the behavior of different
types of consumers (e.g. light and heavy buyers) which would be possible with a full
negative binomial distribution model (see Ehrenberg [11]).
What kind of behavior is captured by the LSD-model? Because of the independence
assumptions the LSD-model estimates the probability that a product pair has been used
at least once together in a session by chance  -times in the relevant time period. This
can be justified by the following example: Consider that a user reads – as his time al-
lows – some Internet newspaper and that he uses an Internet-based train schedule for his
travel plans. Clearly, the use of both information products follows independent stochas-
tic processes. And because of this, we would hesitate to recommend to other users who
read the same Internet newspaper the train schedule. The frequency of observing this
pair of information products in one session is as expected from the prediction of the
LSD-model. Ehrenberg claims that this describes a large part of consumer behavior in
daily life and he surveys the empirical evidence for this claim in [11].
Next, consider complementarities between information products: Internet users usu-
ally tend to need several information products for a task. E.g. to write a paper in a for-
eign language the author might repeatedly need an on-line dictionary as well as some
help with LATEX, his favorite type-setting software. In this case, however, we would not
hesitate to recommend a LATEX-online documentation to the user of the on-line dic-
tionary. And the frequency of observing these two information products in the same
session is (far) higher than predicted by the LSD-model.
A recommendation for an information product
simply is an outlier of the LSD-
model – that is an information product that has been used more often at least once
together in the same session with product
in the observation period as could have been
expected from independent random choice acts. A recommendation reveals a comple-
mentarity between information products.
The main purpose of the LSD-model in this setting is to separate non-random co-
occurrences of information products (outliers) from random co-occurrences (as ex-
pected from the LSD-model). We use the LSD-model as a benchmark for discovering
regularities.
Table 1 shows the algorithm we use for computing recommendations. In step 1 of
the algorithm repeated usage of two information products in a single session is counted
once as required in repeat-buying theory. In step 2 of the algorithm we discard all fre-

10
Table 1. Algorithm for computing recommendations.
1. Compute for all information products x in the market baskets the frequency distributions for
repeat-purchases of the co-occurrences of with other information products in a session,
that is of the pair


 with  
\ . Several co-occurrences of a pair


 in a single
session are counted only once.
2. Discard all frequency distributions with less than
observations.
3. For each frequency distribution:
(a) Compute the robust mean purchase frequency  by trimming the sample by removing
percent (e.g. 2.5%) of the high repeat-buy pairs.
(b) Estimate the parameter  for the LSD-model from
 



 fffi

fl
with either a bisection or Newton method.
(c) Apply a ffi

-goodness-of-fit test with a suitable  (e.g. "! $# or "! &% ) between the ob-
served and the expected LSD distribution with a suitable partitioning.
(d) Determine the outliers in the tail. (We suggest to be quite conservative here: Outliers at
r are above
('
)+*
)
.)
(e) Finally, we prepare the list of recommendations for information product , if we have a
significant LSD-model with outliers.
quency distributions with a small number of observations, because no valid model can
be estimated. This implies that in this case no recommendations are given. For each
remaining frequency distribution, in step 3, the mean purchase frequency, the LSD pa-
rameter and the outliers are computed.
Note that high repeat-buy outliers may have a considerable impact on the mean
purchase frequency and thus on the parameter of the distribution. By ignoring these
high repeat-buy outliers by trimming the sample (step 3a) and thus computing a ro-
bust mean we considerably improve the chances of finding a significant LSD-model.
This approach is justified by the data shown in column V of table 4 as discussed in
section 4.1.
In step 3d outliers are identified by the property that they occur more often as pre-
dicted by the cumulated theoretically expected frequency of the LSD-model. Several
less conservative options for determining the outliers in the tail of the distribution are
discussed in the next section. These options lead to variants of the recommender service
which exhibit different first and second type errors.
3 A Small Example: Java Code Engineering & Reverse
Engineering
In figure 3 we show the first -, candidates for recommendations of the list of 117
web-sites generated for the site Java Code Engineering & Reverse Engi-
neering by the algorithm described in table 1 in the last section. Table 2 shows
the statistics for this recommendation list. In the table as well as in the following we
denote with &/.
103254( the observed frequency distribution for  repeat-buys, by .
60724,
the observed relative frequency distribution, by .


fi8  the density function of the

11
Java Code Engineering & Reverse Engineering
Persons using the above web-site used the following web-sites too:
1. Free Programming Source Code
2. Softwareentwicklung: Java
3. Developer.com
4. Java-Einfuehrung
5. The Java Tutorial
6. JAR Files
7. The Java Boutique
8. Code Conventions for the Java(TM) Programming Language
9. Working with XML: The Java(TM)/XML Tutorial
10. Java Home Page
11. Java Commerce
=== Cut ==================================================
12. Collection of Java Applets
13. Experts Exchange
=== Cut ==================================================
14. The GNU-Win32 Project
15. Microsoft Education: Tutorials
16. HotScripts.com
...
Fig. 3. List of web-sites with cuts.
estimated LSD-model, by
03254  the cumulative relative frequency distribution and
by


 8  the distribution function of the estimated LSD-model. For a sample of &
observations, the expected number of observations with  repeat buys is &/.


 8

  ,
the expected number of observations with at least  repeat buys is &



fi8


  
&




.




fi8


 .
The observed mean purchase frequency in table 2 is 


, . After trimming the
highest 

 percentile (ignoring two observations with 7 and 8 repeat-buys, respec-
tively), the robust mean purchase frequency is 

 ,  and the estimated parameter

of the LSD-model is '


  . Visual inspection of figures 4 and 5 shows that the esti-
mated LSD-model properly describes the empirical data. This impression is supported
by comparing the columns &/.
07254  and &/.


fi8  in table 2 as well as looking at the


-values in the second part of table 2. The

goodness-of-fit test for the trimmed data
is highly significant with a

-value of 

'

which is considerably below 


  , the
critical value at
0 '

'

.
Table 2 also shows, that fitting a LSD-model to the original, untrimmed data results
in a higher parameter

( '


, ) and leads to a model with a higher  -value ( 

 ,
) than
the model obtained from the trimmed data. This indicates that ignoring high repeat-
buy outliers improves the fit of the LSD-model. However, experimentation with several
trimming percentiles in the range from 1.0 to 10 % indicated that ignoring 2.5 % of the
observations contributed to an improved model fit, whereas ignoring additional obser-

12
vations did not lead to further improvement. In addition, in the evaluation the quality
of recommendations proved rather insensitive to trimming. Our current experience sug-
gests that 2.5 % of trimming is a robust choice for this parameter. However, additional
experience with different data sets would be welcome.
All outliers whose observed repeat-purchase frequency is above the theoretically
expected frequency are candidates to be selected as recommendations. In figure 5 (with
a logarithmic y-axis) we explore three options of determining the cut-off point for such
outliers:
Option 1. Without doubt, as long as the observed repeat-purchase frequency is above





 8


 & , we have detected outliers. Look for &/.
03254

 
/
&








fi8

 *
in table 2. In our example, this holds for all co-purchases with more than  repeat-
buys which correspond to the top   sites shown as recommendations in figure 3.
For 0  &/.


07254

    is less than &








 8

& 
  

'

' in table
2, so we can not expect outliers in this class. This is the most conservative choice.
Inspecting these recommendations shows that all of them are more or less directly
related to Java programming, which is probably the task in which students use the
example site.
Option 2. Discounting any model errors, as long as the observed repeat-purchase fre-
quency &/.


10724

  is above the theoretically expected frequency &/.


fi8

& is a
less conservative option. For the example, we select all co-purchases with more
than  occurrences as recommendations. Here this coincides with the option de-
scribed above. See the top   sites in figure 3.
Option 3. If we take the cut, where both cumulative purchase frequency distributions
cross, we get   recommendations regarding all co-purchases occurring more than
twice as nonrandom – see the top   sites in figure 3. However, it seems, that
web-sites 12 and 13, namelyCollection of Java Applets and Experts
Exchange seem to be not quite so related to Java programming.
The last three web-sites shown in figure 3 are not used as recommendations by any
of the three explored options. And in fact they seem to be of little or no relevance for
Java programming.
We have implemented the most conservative approach, namely option 1, in the re-
commender service based on a check of the face validity of the recommendations for
a small sample of information products (25 products). We think that, at least in cases
with a considerable number of candidates for the recommendation list, this is a suitable
approach.
Instead of the 3 simple cutoff algorithms described above we consider the following
error threshold procedure. We can determine for each class of products with  repeat-
buys individually the probability of recommending a random web-site by dividing the
theoretically expected number of occurrences by the observed number of occurrences
( .


fi8



.



07254
 in table 2). If this quotient exceeds 1 the probability is 1. Consider,
for example, the number of product combinations which have been bought 8 times to-
gether in table 2. Theoretically, we would expect that this is a chance event roughly in
one out of ten cases ( '

'

 ). Now, we have observed 5 product combinations with 4
repeat-buys. Unfortunately, theoretically 




product combinations can be expected

13
Table 2. Statistics for web-site Java Code Engineering & Reverse Engineering.
# Web-site: wu01_74 (Mon May 7 16:48:37 2001)
# Heuristic: Distr=NBD - Case 4: NBD heuristic var>mean
# Total number of observations: 117
# Sample mean=1.56410256410256 and var=1.64456233421751
# Estimate for q=0.566835385131836
#
# Robust estimation: Trimmed begin: 0 / end: 0.025 (2 observations)
# Robust estimation: Number of observations: 115
# Robust mean=1.46086956521739
# Robust estimate for q=0.511090921020508
# Plot:
# r repeat-buys nf(x_obs) nf(x_exp) n(1-F(x_obs,r)) n(1-F(x_exp,r))
1 87 83.565 117 117
2 17 21.355 30 33.435
3 2 7.276 13 12.080
4 5 2.789 11 4.804
5 3 1.140 6 2.015
6 1 0.486 3 0.874
7 1 0.213 2 0.389
8 1 0.095 1 0.176
# Identifying non-random co-occurrences:
# Option 1: mixed intersection (f(x_obs) with 1-F(x_exp) at:
# 3 (leaves 11 nonrandom outliers)
# Option 2: f(x) intersection at: 3 (leaves 11 nonrandom outliers)
# Option 3: 1-F(x) intersection at: 2 (leaves 13 nonrandom outliers)
# Chi-square test (q=0.566835385131836; 117 observations):
# Class obs exp chi-square
# 1 87 79.269 0.754
# 2 17 22.466 1.330
# 3 13 15.071 0.285
# -------
# Chi-square value: 2.369
# Chi-square test (robust) (q=0.511090921020508; 115 observations):
# Class obs exp chi-square
# 1 87 82.137 0.288
# 2 17 20.990 0.758
# 3 11 11.794 0.053
# -------
# Chi-square value: 1.099
# Chi-square test threshold with alpha=0.01 w/1 d.f.: 10.828
# Chi-square test threshold with alpha=0.05 w/1 d.f.: 3.841
# -> LSD with alpha=0.05

14
0
10
20
30
40
50
60
70
80
90
1 2 3 4 5 6 7 8
N
um
be
r o
f p
ai
rs
b
ou
gh
t r
ti
m
es
r number of repeat buys in t
n f(x_obs)
n f(x_exp)
Fig. 4. Plot of frequency distribution.
0.01
0.1
1
10
100
1000
1 2 3 4 5 6 7 8
N
um
be
r o
f p
ai
rs
b
ou
gh
t r
ti
m
es
r number of repeat buys in t
n f(x_obs)
n f(x_exp)
n (1-F(x_obs))
n (1-F(x_exp))
Fig. 5. Plot of log frequency distribution.

15
Table 3. Finding classes with less than 0.40 random observations.
repeat-buys




7





&




&



  Class shown
1 87 83.565 0.961 0
2 17 21.355 1.256 0
3 2 7.276 3.638 0
4 5 2.789 0.558 0
5 3 1.140 0.380 1
6 1 0.486 0.486 0
7 1 0.213 0.213 1
8 1 0.095 0.095 1
Java Code Engineering & Reverse Engineering
Persons using the above web-site used the following web-sites too:
1. Free Programming Source Code
2. Softwareentwicklung: Java
4. Java-Einfuehrung
5. The Java Tutorial
6. JAR Files
Fig. 6. List of web-sites selected by class.
from pure chance. In this class we observe now a mixture of random product combina-
tions and non-random product combinations, but we are not able to distinguish them.
However, we can specify a threshold for the chance of falsely presenting a random co-
occurrence, e.g. below 0.40. Table 3 summarizes this procedure. That is we pick only
those classes with  observed repeat-buys where the probability of falsely presenting an
outlier is below  '

 ' .
In the example, we would then present the web-sites in classes 5, 7, and 8, but not
the web-site in class 6. That is, we would present web-sites 1, 2, 4, 5, and 6 as shown
in figure 6. Web-site 3 (Developer.com, a site definitely not exclusively devoted to
Java programming) and others from figure 3 are not shown because of the high proba-
bility of presenting random web-sites.
Considering the theoretically expected number of occurrences .




fi8
 of a class as
the error to give a recommendation for a random co-occurrence (the type II error) leads
to a thresholding strategy based on the specification of acceptable expected type II error

: We add to the recommender products from the tail of the distribution as long as the
expected type II error of the recommender is smaller than

. If there is more than one
product in a class  , the type II error of this class is divided by the number of products
in the class and products in a class may be picked in arbitrary sequence. This strategy
has been implemented and used for the evaluation of the recommender in section 4.3.

16
Table 4. Detailed results for observation period 1999-09-01 – 2001-05-07.
I II III IV V VI
 no ffi

Sign. Sign. Sign. Not

undef. ( 
classes) + "! -%   "! $# (trim) sign.
A 1128 66 0 0 0 0 1194
Obs. # (0) (63) (0) (0) (0) (0) (63)
B 1374 0 0 0 0 0 1374
 # (0) (0) (0) (0) (0) (0) (0)
C 2375 0 0 0 0 0 2375


; 
(0) (0) (0) (0) (0) (0) (0)
D 201 617 105 46 15 372 1356


; 
(0) (605) (105) (46) (15) (145) (916)
E 3 86 222 194 93 253 851


 (0) (86) (222) (194) (93) (253) (848)

5081 769 327 240 108 625 7150
(0) (754) (327) (240) (108) (398) (1827)


 indicates lists with at least # outlier.
In the analysis of outliers there is still room for improvement as e.g. by developing
statistical tests for identifying outliers. For example, the choice of the threshold value
should be analyzed in terms of the tradeoff between errors of type I and II for a sample
of co-occurrence lists all of whose entries have been completely evaluated by experts
with regard to their usefulness as recommendations.
4 First Empirical Results
To establish that a recommender service based on Ehrenberg’s repeat buying-theory is
supported by empirical evidence, we proceed as follows:
1. In section 4.1 we investigate how well the LSD-model explains the actual data for
fl

' information products.
2. However, that the LSD-model fits the data well does not yet mean that the out-
liers we have identified are suitable recommendations for a user. In section 4.2 we
present the results of a first small face evaluation experiment whose result suggests
that these outliers are indeed valuable recommendations.
The data set used for the example given in section 3 and in this section is from the
anonymous recommender services of the Virtual University of the Vienna University
of Economics and Business Administration (http://vu.wu-wien.ac.at) for the observa-
tion period from 1999-09-01 to 2001-03-05. Co-occurrences have been observed for

17


, information products. After elimination of web-sites which ceased to exist in the
observation period co-occurrences for    ' information products remain available for
analysis.
4.1 Fit of Data to LSD-Models
Table 4 summarizes the results of applying the algorithm for computing recommenda-
tions presented in table 1. If the sample variance is larger than the sample mean this
may indicate that a negative binomial distribution (NBD)-model (and thus its LSD-
approximation) is appropriate (see Johnson and Kotz [19, p.138]). This heuristic sug-
gests   candidates for an LSD-model (see table 4, row E).
The rows of the table represent the following cases:
A The number of observations is less than 10. In this row we find co-occurrence lists
either for very young or for very rarely used web-sites. These are not included
into the further analysis. Cell (A/II) in table 4 contains lists which have repeated
co-occurrences despite the low number of observations. In this cell good recom-
mendation lists may be present (4 out of 5).
B No repeat-buys, just one co-occurrence. These are discarded from further analyses.
C Less than 4 repeat-buys and trimmed sample mean larger than variance. Trimming
outliers may lead to the case that only the observations of class 1 (no repeat-buys)
remain in the sample. These are discarded from further analyses.
D More than 3 repeat-buys and trimmed sample mean larger than variance. In cell (D/I)
after trimming only class 1 web-sites remain in the trimmed sample (no repeat-
buys). As a future improvement, the analysis should be repeated without trimming.
In cell (D/II) the  -test is not applicable, because less than 3 classes remain.
E (Trimmed) sample variance larger than sample mean. For cell (E/I) we recommend
the same as for cell (D/I). For cell (E/II) we observe the same as for cell (D/II).
The columns I – VI of table 4 have the following meaning:
I The parameter

of the LSD model could not be estimated. For example, only a single
co-occurrence has been observed for some product pairs.
II The

goodness-of-fit test could not be computed, because of lack of observations.
III, IV The

goodness-of-fit test is significant at
0 '

'

or
0 '

'  , respectively.
V The

goodness-of-fit test is significant at
 '

' using the trimmed data. All
high repeat-buy pairs in the 

 percentile have been excluded from the model esti-
mation.
VI The

-test is not significant.
As summarized in table 5 we tested the fitted LSD-model for the frequency distribu-
tions of co-occurrences for  '' information products. For ,  information products,
that is more than 50 percent, the estimated LSD-models pass a

goodness-of-fit test
at
0 '

' .

18
Table 5. Summary of results for observation period 1999-09-01 – 2001-05-07.
n %
Information products 9498 100.00
Products bought together
with other products 7150 75.28
Parameter  defined 2069 21.78
Enough classes for ffi

-test 1300 13.69
LSD with   "! $# (robust) 675 7.11
LSD not significant 625 6.58
LSD fitted, no ffi

-test 703 7.40

# and no ffi

-test 66 0.69
4.2 Face Validation of Recommendations
In order to establish the plausibility of the recommendations identified by the recom-
mender service previously described we performed a small scale face validation ex-
periment. The numbers in parenthesis in table 4 indicate the number of lists for which
outliers were detected. From these lists  '' lists of recommendations were randomly
selected. Each of the   
recommendations in these lists was inspected for plausibil-
ity. Plausible recommendations were counted as good recommendations by pressing the
affirmative symbol (a hook) in the Vote Box shown in figure 1 in the introduction of this
paper.
This small scale face validation experiment of inspecting recommendations for
plausibility led to a quite satisfactory result:
– For the  lists for which a significant LSD-model could be fitted,  

fl % of the
recommendations were judged as good recommendations.
– 
 lists for which an LSD model was not significant contained 



 % good rec-
ommendations.
– Only  

  % good recommendations were found in the 44 lists for those LSD-
models where no

test could be computed which is a significantly lower percent-
age.
Surprisingly, the class of models where the LSD model was not significant contains
a slightly higher number of recommendations evaluated as good. However, a number of
(different) reasons may explain this:
– First, we might argue that even if the LSD-model is insignificant, it still serves its
purpose, namely to identify non-random outliers as recommendations.
– A close inspection of frequency distributions for these lists revealed the quite un-
expected fact that several of these frequency distributions were for information
products which belong to the oldest in the data set and which account for many

19
observations. The reasons for this may be explained e.g. by a shift in user behavior
(non-stationarity) or too regular behavior as e.g. for cigarettes in consumer markets
(see Ehrenberg [11]). If too regular behavior is the reason that the LSD-model is
insignificant, again, we still identified the non-random outliers.
– Another factor which might contribute to this problem is that several web-sites in
this group belong to lists integrated in the web-sites of other organizational units.
These lists, at least some of them, contain web-sites which have been carefully
selected by the web-masters of these organizational units for their students. For
example, the list of web-sites for student jobs is integrated within the main web-
site of the university. The recommendations for such lists seem to reflect mainly the
search behavior of the users. A similar effect is known in classic consumer panel
analysis, if the points of sale of different purchases are not cleanly separated. This
implies that e.g. purchases in a supermarket are not distinguished from purchases
from a salesman. In our analysis, the purchase occasions are in different web-sites,
namely the broker system and the organizational web-site with the embedded list.
Ehrenberg’s recommendation is to analyse the data separately for each purchase
occasion.
Also, the fact that the data set contains information products with different age may
explain some of these difficulties. However, to settle this issue further investigations are
required.
4.3 A First Comparison with a Simple Association Rule Based Recommender
System
Compared to several other recently published recommender systems (e.g. [21], [24],
[23]) which combine several data mining techniques, most noteably association-rule
mining algorithms and various clustering techniques, the recommender system dis-
cussed here is very simple. Computing the frequency distribution corresponds exactly to
the identification of frequent itemsets in association-rule mining algorithms whose most
famous representative is Agrawal’s a-priori algorithm ([2], [3]) with support and confi-
dence of ' . Recent improvements of these algorithms include TITANIC of Stumme et
al. [31] for highly correlated data sets, the association rule mining algorithm with adap-
tive support of Lin et al. [22], a graph-based approach for association rule mining by
Yen and Chen [33], and a new approach for the online generation of association rules
by Aggrawal and Yu [1]. TITANIC efficiently exploits the concept lattice and is based
on concept analysis (see Ganter and Wille [13]).
The difference between our system and the association rule framework is in the
way, “significant” item pairs are identified. In our model “significant” item pairs are
outliers with regard to the LSD model, in association rule approaches “significant pairs”
have more than a specified amount of support and more than a specified amount of
confidence.
We have evaluated these two simple recommender systems on the VU data set for
the observation period of 2001-01-01 to 2001-06-30. For the purpose of comparing
our recommender system with a simple association rule based recommender system
we have randomly drawn association lists for 300 information products with a total of

20
Table 6. Comparison of LSD-model with no trim and an acceptable expected type II error rate of
$!# and with association rules with a support of "! - "# % and a confidence of "! $# .
No. of LSD-model Association Rules
produced recommendations 145 154
correct recommendations 96 100
unknown recommendations 35 38
false recommendations 14 16
(type II error)
missed recommendations 465 461
(type I error)
1966 pairs of information products. For each pair

%
 a member of our research group
answered with yes or no to the following question “Is

a good recommendation for
?”.
For

, pairs the answer was yes, for   '' pairs the answer was no. Unfortunately,
however, for '  pairs no expert evaluation could be done, because the web-site for

ceased to exist.
This evaluation approach differs from the usual machine learning methodology of
splitting the data set in a training and a testing data set for testing the capability of
the algorithm to extract patterns. In addition we test whether frequent co-purchase (or
co-usage) is a suitable indicator for recommendations.
Table 6 shows a first result for the two suitably parametrized models. A rough com-
parison indicates that the LSD-model is at least as good as the association rule approach.
However, a complete analysis of the trade-off of the type I and II error over the param-
eter range of the two models and a sensitivity analysis with regard to misspecification
of parameters is still on the todo-list. With regard to parametrization, the type II er-
ror threshold is the only parameter of the LSD-model based algorithm. This parameter
is independent of the size of the data set. The support and confidence parameters of
association rule algorithms seem to depend on the data set size.
5 Further Research
The main contribution of this paper is that Ehrenberg’s classical repeat-buying models
can be applied to market baskets and describe – despite their strong independence and
stationarity assumptions – the consumption patterns of information products – at least
for the data set analyzed – surprisingly well. For e-commerce sites this implies that a
large part of the theory developed for consumer panels may be applied to data from
web-sites, too, as long as the analysis remains on the aggregate level.
For anonymous recommender services they seem to do a remarkable job of iden-
tifying non-random repeated-choice acts of consumers of information products as we
have demonstrated in section 3. The use of the LSD-model for identifying non-random
co-occurrences of information products constitutes a major improvement which is not
yet present in other correlation-type recommender services.
However, establishing an empirical base for the validity of repeat-buying models in
information markets as suggested in this article still requires a lot of additional evidence

21
and a careful investigation of additional data sets. We expect that such an empirical re-
search program would have a good chance to succeed because to establish Ehrenberg’s
repeat-buying theory a similar research program has been conducted by Aske Research
Ldt., London, in several consumer product markets (e.g. dentifrice, ready-to-eat cereals,
detergents, refrigerated dough, cigarettes, petrol, tooth-pastes, biscuits, color cosmetics,
...) from 1969 to 1981 (see Ehrenberg [11]).
The current version of the anonymous recommender services (and the analysis in
this article) still suffers from several deficiencies. The first is that new information prod-
ucts are daily added to the information broker’s database so that the stationarity as-
sumptions for the market are violated and the information products in the data set are of
non-homogenous age. The second drawback is that testing the behavioral assumptions
of the model, e.g. by testing the behavioral assumptions with data from the personalized
part of the VU, as well as validation either by studying user acceptance or by controlled
experiments, still has to be done. Third, for performance reasons the co-occurrence lists
for each information product do not contain time-stamps. Therefore, the development
of time-dependent e.g. alert systems has not been tried, although Ehrenberg’s theory is
in principle suitable for this task.
We expect Ehrenberg’s repeat-buying models to be of considerable help to create
anonymous recommender services for recognizing emerging shifts in consumer behav-
ior patterns (fashion, emerging trends, moods, new subcultures, ...). Embedded in a
personalized environment Ehrenberg’s repeat-buying models may serve as the base of
continuous marketing research services for managerial decision support which provide
forecasts and classical consumer panel analysis in a cost efficient way.
6 Acknowledgment
We acknowledge the financial support of the Jubila¨umsfonds of the Austrian National
Bank under Grant No. 7925 without which this project would not have been possible.
For the evaluation of the system we acknowledge support of the DFG for the project
“Scientific Libraries in Information Markets” of the DFG program SPP 1041 “V3D2:
Verteilte Verarbeitung und Vermittlung digitaler Dokumente”. Thanks to Anke Thede
and Andreas Neumann for correcting the final versions of this contribution and to one
of the anonymous reviewers whose constructive critic helped us to improve this contri-
bution considerably.
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