Before proceeding, we briefly recall the FCA terminology . Given a (formal) context , where is called a set of objects, is called a set of attributes, and the binary relation specifies which objects have which attribute, the derivation operators are defined for and as follows:
Put differently, is the set of attributes common to all objects of and is the set of objects sharing all attributes of .
If this does not result in ambiguity, is used instead of . The double application of is a closure operator, i.e., is extensive, idempotent, and monotonous. Therefore, sets and are said to be closed.
A (formal) concept of the context is a pair , where , , , and . In this case, we also have and . The set is called the extent and is called the intent of the concept . In categorical terms, is equivalently defined by its objects or its attributes .
A concept is a subconcept of if (equivalently, ). In this case, is called a superconcept of . We write and define the relations , , and as usual. If and there is no such that , then is a lower neighbor of and is an upper neighbor of ; notation: and .
The set of all concepts ordered by forms a lattice, which is denoted by and called the concept lattice of the context The relation defines edges in the covering graph of .
Problem Statement and Domain Models
Owners of a web-site are often interested in analyzing groups of users of their site. Information on these groups can help to optimize the structure and contents of the site. For example, interaction with members of each group may be organized in a special manner. In this paper we use an approach based on formal concepts  for constructing taxonomies of user groups.
For our experiments we have chosen four target websites: the site of the State University Higher School of Economics (www.hse.ru), an e-shop of household equipment, the site of a large bank, and the site of a car e-shop (the names of the last three sites cannot be disclosed due to legal agreements).
Users of these sites are described by attributes that correspond to other sites, either external (from three groups of sites: finance, media, education) or internal (web-pages of the site). More precisely, initial “external” data consists of user records each containing the user id, the time when the user first entered this site, the time of his/her last visit, and the total number of sessions during the period under consideration. An “internal” user record, on the other hand, is simply a list of pages within the target website visited by a particular user.
By “external” and “internal” taxonomies we mean (parts of) concept lattices for contexts with either “external” or “internal” attributes. For example, the external context has the form
where is the set of all users of the target site, is the set of all sites from a sample (not including the target one), the incidence relation is given by all pairs , such that user visited site . Analogously, the internal context is of the form , where is the set of all own pages of the target site.
A concept of this context is a pair such that is a group of users that visited together all other sites from .
Data and Their Preprocessing
We received “external” data with the following information for each user-site pair: (user id, time of the first visit, time of the last visit, total number of sessions during the period). “Internal” data have almost the same format with an additional field url page, which corresponds to a particular visited page of the target site.
Information was gathered from about 10000 sites of Russian internet (domain .ru). In describing users in terms of sites they visites we had to tackle the problem of dimensionality, since concept lattices can be very large (exponential in the worst case) in terms of attributes. To reduce the size of input data we used the following techniques.
For each user we selected only those sites that were visited by more than a certain number of times during the observation period. This gave us information about permament interests of particular users. Each target site was considered in terms of sites of three groups: newspaper sites, financial sites, and educational sites. Some pages can be merged (as attributes) according to (implicit) domain ontology. For example, if users of a bank site have personal pages, it is reasonable to fuse all these pages by calling the resulting attribute “a personal web-page”. A certain observation period can be chosen; usually we took a one-month period.
However, even for large reduction of input size, concept lattices can be very large. For example, a context of size 4125225 gave rise to a lattice with 57 329 concepts.
Using Stability for Selecting Interesting Subsets of Concepts
To choose interesting groups of users we employed stability index of a concept defined in [2, 3] and considered in  (in slightly different form) as a tool for constructing taxonomies. On one hand, stability index shows the independence of an intent on particular objects of extent (which may appear or not appear in the context depending on random factors). On the other hand, stability index of a concept shows how much extent of a concept is different from similar smaller extents (if this difference is very small, then its doubtful that extent refers to a “stable category”). For detailed motivation of staibility indices see [2, 3, 4].
Definition. Let be a formal context and be a formal concept of . The stability index of is defined as follows:
. The stability index of a concept indicates how much the concept intent depends on particular objects of the extent. A stable intent (with stability index close to 1) is probably “real” even if the description of some objects is “noisy”. In application to our data, the stability index shows how likely we are to still observe a common group of interests if we ignore several users. Apart from being noise-resistance, a stable group does not collapse (e.g., merge with a different group, split into several independent subgroups) when a few members of the group stop attending the target sites.
In our experiments we used ConceptExplorer  for computing and visualizing lattices and their parts. We compared results of taking most stable concepts (with stability index exceeding a threshold) with taking an “iceberg” of a concept lattice (order filter of a lattice containing all concepts with extents larger than a threshold). The results look correlated, but nevertheless, substantially different. The set of stable extents contained very important, but not large groups of users.
In Figs. 1, 2 we present parts of a concept lattice for the site www.hse.ru described by “external” attributes which were taken to be Russian internet newspapers visited by users of www.hse.ru during one month more than 20 times. Fig. 1 presents an iceberg with 25 concepts having largest extent. Many of the concepts correspond to newspapers that are in the middle of political spectrum, read “by everybody” and thus, not very interesting in characteizing social groups.
Fig. 1. Iceberg with 25 concepts
Fig. 2 presents an ordered set of 25 concepts having largest stability index. As compared to the iceberg, this part of the concept latice contains several sociologically important groups such as readers of AIF (“yellow press”), Cosmopolitain, Expert (high professional analytical surveys) etc.
Fig. 2. Ordered set of 25 concepts with largest stability
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