=Paper=
{{Paper
|id=None
|storemode=property
|title=Cheating to achieve Formal Concept Analysis over a Large Formal Context
|pdfUrl=https://ceur-ws.org/Vol-959/paper24.pdf
|volume=Vol-959
|dblpUrl=https://dblp.org/rec/conf/cla/CodocedoTA11
}}
==Cheating to achieve Formal Concept Analysis over a Large Formal Context==
https://ceur-ws.org/Vol-959/paper24.pdf
Cheating to achieve Formal Concept Analysis
over a large formal context?
Victor Codocedo1,3 , Carla Taramasco2 , and Hernán Astudillo1
1
Universidad Técnica Federico Santa Marı́a, Av. España 1640. Valparaı́so, Chile.
2
École Polytechnique, 32 Boulevard Victor 75015 Paris, France.
3
LORIA, BP 70239, F-54506 Vandoeuvre-lès-Nancy, France.
Abstract. Researchers are facing one of the main problems of the In-
formation Era. As more articles are made electronically available, it gets
harder to follow trends in the different domains of research. Cheap, coher-
ent and fast to construct knowledge models of research domains will be
much required when information becomes unmanageable. While Formal
Concept Analysis (FCA) has been widely used on several areas to con-
struct knowledge artifacts for this purpose [17] (Ontology development,
Information Retrieval, Software Refactoring, Knowledge Discovery), the
large amount of documents and terminology used on research domains
makes it not a very good option (because of the high computational cost
and humanly-unprocessable output). In this article we propose a novel
heuristic to create a taxonomy from a large term-document dataset us-
ing Latent Semantic Analysis and Formal Concept Analysis. We provide
and discuss its implementation on a real dataset from the Software Ar-
chitecture community obtained from the ISI Web of Knowledge (4400
documents).
1 Introduction
Research communities are facing one of the main problems of the Information Era
and Formal Concept Analysis is not prepared to solve it. The amount of articles
available online is growing each year yielding difficult to track trends, following
ideas, looking for new terminology, etc. While some communities have under-
stood the need for an artifact representing the knowledge within the domain
(such as an ontology, a body-of-knowledge or a taxonomy) the problem remains
in its construction since it is hard (highly technical), expensive (researchers are
scarce) and complex (information is dynamic).
Automatic and semi-automatic creation of a terms taxonomy have been
widely boarded in several fields [3,4,5,13,24]. In this work we focus on the ap-
proach described by Roth et al. [19] in which a taxonomy is derived from a
corpus of documents by the use of Formal Concept Analysis (FCA). In partic-
ular, they describe an application used to “represent a meaningful structure of
?
We would like to thank Chilean project FONDEF D08I1155 ContentCompass, intra-
basal project FB/20SO/10 in the context of the Chilean basal project FB0821 and
ECOS-CONICYT project C09E08 for funding this work.
a given knowledge community in a form of a lattice-based taxonomy”. This ap-
plication is illustrated using a set of abstracts of the embryologist community
obtained from MedLine spanning 5 years where a random set of 25 authors and
18 terms were analyzed. Although the lattice-based taxonomy obtained was a
fair representation of the domain, real-size corpora of research communities are
rather much larger than this example.
Handling large datasets has been defined as one of the open problems in the
community of FCA4 for two main reasons: first, the computational costs involved
in the calculation of the concept lattice can make the use of FCA prohibitive
and second, the concept lattice structure yielded could be so complex that its
use may be impossible [10].
Iceberg lattices [21] help in improving readability by eliminating “not rep-
resentative” data, but useful information, such as “emerging behaviors [12,15],
is lost in the process. Stabilized lattices (using a stability measure [16]) also
improves readability by eliminating “noisy elements” from data, but being a
post-process tool it also raises computational costs.
We describe in this document a novel heuristic to create a lattice-based tax-
onomy from a large corpus using Formal Concept Analysis and a widely used
Information Retrieval technique called Latent Semantic Analysis (LSA). In par-
ticular, we describe a process to compress a formal context into a smaller reduced
context in order to obtain a lattice of terms that can be used to describe the
knowledge on a given research domain. We illustrate our approach using a real-
size dataset from a research community of Computer Sciences.
The remainder of this paper proceeds as follows: Section 2 explains the basis
of FCA, section 3 presents our approach and section 4, a case study over a real
dataset from a research community. Section 5 presents the results and a com-
parison of the obtained taxonomy with a human-expert handmade thesaurus.
Finally, the conclusions are described in section 6.
2 Formal Concept Analysis
Formal Concept Analysis, originally developed as a subfield of applied mathe-
matics [23], is a method for data analysis, knowledge representation and infor-
mation management. It organizes information in a lattice of formal concepts.
A formal concept is constituted by its extension (the objects that compose the
concept) and its intension (the attributes that objects share). Objects and at-
tributes are placed as rows and columns (resp.) in a cross-table or formal context
where each cell indicates whether the object of that row have the attribute of that
column. In what follows, we describe the Formal Concept Analysis framework
as synthesized by Wille [22].
4
http://www.upriss.org.uk/fca/problems06.pdf
2.1 Framework
Let G be a set of objects, M a set of attributes and I a binary relation between G
and M (I ⊆ (G × M )) indicating by gIm that the object g contains the attribute
m and K = (G, M, I) be the formal context defined by G, M and I. For A ⊆ G
and B ⊆ M it is defined the derivation operator (0 ) as follows:
A0 = {m ∈ M | gIm, ∀g ∈ A}, with A ⊆ G (1)
0
B = {g ∈ G | gIm, ∀m ∈ B}, with B ⊆ M (2)
A formal concept of the formal context K is defined by (A, B) with A ⊆ G,
B ⊆ M , A0 = B and B 0 = A, where A is called the extent and B is called the
intent of the concept. The set of all formal concepts is defined as L(G, M, I).
For two formal concepts (A1 , B1 ), (A2 , B2 ) ∈ K, the hierarchy of concepts is
given by the relation subconcept-superconcept as follows:
(A1 , B1 ) ≤ (A2 , B2 ) ⇐⇒ A1 ⊆ A2 ( ⇐⇒ B1 ⊇ B2 ) (3)
Where (A1 , B1) is called the subconcept and (A2 , B2 ) is called the supercon-
cept.
B(K) = (L(G, M, I), ≤) is the complete lattice or concept lattice of context
K
2.2 Iceberg Concept Lattices
Let (A, B) be a concept of B(K), its support is defined as:
|A|
supp(A, B) = (4)
|G|
Given a threshold minsupp ∈ [0, 1], the concept (A, B) is called a “frequent
concept” if supp(A, B) ≥minsupp.
An Iceberg lattice [21] is the set of all frequent concepts for a given min-
supp.
2.3 Stability
Stability was proposed by Kuznetsov in [14,16] as a mechanism to prune “noisy
concepts”. It was extended by Roth et al. We use and provide their definition
from [19] and [15]:
Let K = (G, M, I) be a formal context and (A, B) be a formal concept of K.
The stability index, σ, of (A, B) is defined as follows:
|{C ⊆ A | C 0 = B}|
σ(A, B) = (5)
2|A|
Stability measures how much the intent of a concept depends on particular
objects of its extent, meaning that if the formal context changes and some objects
disappear, then stability indicates how likely it is for a concept to remain in the
concept lattice. Stability can also be used to construct a stabilized lattice for a
given threshold similarly to an iceberg lattice.
Analogous to definition 5, the extensional stability of a concept (A, B)
can be defined as:
|{D ⊆ B | D0 = A}|
σe (A, B) = (6)
2|B|
Extensional stability measures how likely is for a concept to remain if some
attributes are eliminated from the context. We will use both definitions in this
work differentiating them as intensional stability (on (5)) and extensional sta-
bility (on (6)).
3 Reducing a large formal context
Different from Roth’s approach [19], we are not interested in tracking groups of
people working on groups of topics, but rather in the relations among topics.
These relations occur in the articles that authors write, where topics or terms
can appear in sets and each one can appear one or more times. To elaborate:
Given a corpus of articles G, a list of terms M and the relation among them
I ⊆ (G × M ) indicating by gIm that the article g contains the term m, the
document-article formal context is defined as:
KO = (G, M, I) (7)
3.1 Rationale
Even for a small set of terms, the amount of articles for a small research com-
munity can reach thousands of articles making the processing of KO impossible
or useless. The problem gets worse over time, because it can be expected that
each year hundreds of articles will be added to the corpus.
What happens with terms over time? In taxonomy evolution, as described in
[18], symmetric patterns arise: some fields will progress or decline; some fields
will contain more or less concepts (enrichment or impoverishment); and some
fields will merge or split. In any case, it is not expected that the amount of terms
would vary greatly.
Latent Semantic Analysis (LSA) or Latent Semantic Indexing (LSI) [6] is a
technique used commonly in Information Retrieval (IR) as a tool for indexation,
clusterization and query answering. LSA is based on the idea that for a given
set of terms and documents, the relation among terms can be explained by a set
of dimensions whose size is much smaller than the amount of documents. We
exploit this feature of LSA to construct a reduced formal context of dimen-
sions and terms having as conditions that information regarding relations of
terms cannot be lost and that it has to produce a coherent taxonomy using less
computational time. In what follows, we provide a brief description of LSA to
elaborate on how we used it to produce a reduced formal context. For further
reading, please refer to [6].
3.2 Latent Semantic Analysis
Given a list of m terms and a corpus of n documents, let A be a term-document
matrix of rank-min(m,n) as defined in 8, where aij is the weight5 of the term i in
the document j. The Single-Value Decomposition of matrix A (in equation (9))
produces its factorization in three matrices where Σ contains the single-values
of matrix A at the diagonal in descending order and the columns of matrices U
and V are called left and right singular vectors of A.
Am×n = [aij ] ; i = [1..m], j = [1..n] (8)
Am×n = Um×m · Σm×n · Vn×n T
(9)
A0m×n = Um×k · Σk×k · Vk×n
T
(10)
Since singular values drops quickly, we can create a new approximation of
matrix A using k min(m, n) as shown in (10). Matrix A0 ≈ A is the closest
k-rank matrix approximation to A by the Frobenius measure [11]. Two new
matrices can be calculated:
Bm×k = Um×k · Σk×k (11)
Cn×k = Vn×k · Σk×k (12)
where B holds the vector-space representation in k dimensions of terms; and
C the one of documents. Both of these matrices are used for clusterization since,
on them, similar elements are closer on each dimension. In particular, each di-
mension on B (each column) has a Gaussian-like distribution where terms group
around the mean value (see figure 1), except for dimension 0 (the different be-
havior in figure 1(b)) where terms have almost the same value6 . We exploit this
feature to define a conversion-function that allows us to construct the reduced
context.
3.3 A probabilistic-based conversion-function
Which terms are related within a given dimension? Since each dimension holds
continuous values, it is hard to define a region for them. Nevertheless, we know
5
Several weighting functions can be used, being the most used frequency of term and
term frequency-inverse document frequency (tf.idf)
6
We do not use the information in this dimension for our analysis and exclude it from
our results.
90 300
80
250
70
60 200
50
Hits
Hits
150
40
30 100
20
50
10
0 0
0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.2 0.1 0.0 0.1 0.2 0.3 0.4
Coordinate Value Coordinate Value
(a) Distribution of values in dimension 1 (b) Distribution of values in all dimensions
Fig. 1. Distribution of Dimensions’ values in matrix B
that such a region has to be centered at the mean value of the dimension. Hence,
we define a “belonging region” centered at the mean with a modifiable width.
Terms in this region are related because they belong in the dimension and hence,
the pair dimension-term will appear in the reduced context. The width of the
“belonging region” is a parameter that allows us to manage the density of the
context. The conversion function is defined as:
(
1 Gk (x) ∈ [α, 1 − α]
bl (x, k) = (13)
0 otherwise
where function Gk is the probability density function (PDF) for dimension k
and α ∈ ]0,0.5[ defines the limits of the “belonging region”.
3.4 Creating the reduced context
For a document-article formal context KO as defined in (7) (original context)
and a term-document matrix A as defined in (8) analogous to KO :
Given the factorization of matrix A as defined in (10), the vector-space rep-
resentation of its terms in k dimensions B as defined in (11) and a conversion-
function bl (x, k) as defined in (13):
Let D be the set of k dimensions in B
IR ⊆ (D × M ) = {(j, i) : ∀j ∈ D ∧ ∀i ∈ M ⇐⇒ bl (Bij , j) = 1} (14)
we define the reduced context of KO as KR = (D, M, IR ).
Notice the inversion of pair (j, i) and Bij performed to respect LSA con-
ventions that require term-document matrices and FCA that uses document as
objects and terms as attributes. In the reduced context we say “dimension j con-
tains term i if the evaluation of the conversion-function bl over the value of the
coordinate j for the term i is 1”.
Summarizing, in order to get a reduced context, the values for α and k must
be found.
3.5 Related approaches
Similar techniques have been proposed before. Gajdos et al[9] used LSA to re-
duce complexity in the structure of the lattice by eliminating noise in the formal
context. While this approach is useful, it does not reduce the amount of data,
but it “tunes it” to get a clearer result. Snasel et al. [20,9] proposed a matrix-
reduction algorithms based on NMF. 7 and SVD8 . While they state that these
methods are successful to reduce the amount of concepts obtained using FCA,
they do not describe a real life use of their technique (their experiment was
performed over a 17x16 matrix) neither do they discuss about the performance
of their approach. Kumar and Srinivas [1] approach consists of using fuzzy K-
Means clustering 9 to reduce the attributes in a formal term-document context.
In their approach, documents are categorized in k clusters using the cosine sim-
ilarity measure. Cheung et al. [2] introduced term-document lattices complexity
reduction by defining a set of equivalence relations that allows to reduce the set
of objects. Finally, Dias et al. introduced JBOS [7] (junction based on objects
similarity) which proposed a similar method where objects where group into pro-
totype objects by calculating its similarity according to certain weights assigned
manually to attributes.
4 Case Study: Software Architecture Community
The Software Architecture Corpus (SAC) was composed by extracting metadata
from papers retrieved by the ISI Web of Knowledge search engine 10 using the
query ”software architecture”. It is assumed that the keyword ”software archi-
tecture” is present in each paper on their titles and/or abstracts.
While the search engine retrieved 4701 articles, not all of them have an
abstract to work with. Those are excluded from our analysis leaving 4565 articles
spanning from 1990 to 2009 (retrieved documents span from 1973 to 2009).
4.1 Term list
A term list was assembled by using Natural Language Processing over the arti-
cles’ titles and abstracts. In order to avoid common words, a stopword list and a
lexical tagger were used as a filter. A list of candidate terms was then manually
filtered to obtain a final list of 120 terms, which included words and multi-words
(such as “Unified Model Language”). Table 1 shows a sample of selected terms.
Each term was looked up on each document and its frequency of use was
calculated. Then, a weighting measure was applied (tf.idf11 ) to each value. The
7
Non-negative matrix factorization
8
Single-Value Decomposition
9
K-Means Clustering is a classic clustering technique for vector-space models
10
http://isiwebofknowledge.com
11
Term Frequency-Inverse Document Frequency is a weighting measure commonly
used on IR based on the notion that term infrequency on a global scale makes it
important.
Table 1. Top 10 more frequent terms
Term Frequency
design 1710
development 1450
component 1253
process 1083
implementation 1006
datum 874
requirement 869
analysis 851
framework 817
control 801
term-document matrix Aw = aij was constructed using the final list of terms
(M) and the corpus of documents (G) where aij represents the weight of term i
in document j. We defined the relation I ⊆ (G × M ) = {(j, i) : ∀j ∈ G ∧ ∀i ∈
M ⇐⇒ aij > 0} to build up the original context KO = (G, M, I) describing
that a document contains a term only if its weight on it is over 0. The formal
context KO was used later to compare our reductions.
4.2 Reducing the SAC
As we stated at the end of section 3.3, two parameters had to be set in order to
create the reduced context. Sadly, in LSA there is not a known method to find
the best value for k, and not knowing that, it is not possible to find a good value
for α. We defined a set of goals to observe which were the values of k and α that
best accomplished them. The goals defined were:
– Low Execution time
– High Stability
– Few Concepts in the final lattice
Using three fixed values for k we reduced several contexts and processed them
through FCA in order to find the best value for α. As shown in figure 2, it was
found that higher values of α (close to 0.5) yields the best results. Repeating
the experience with 3 fixed values for α (0.45, 0.47 and 0.49) to find the best
value for k we found a trade-off between stability and execution time as it can
be observed in figure 3. Higher values of k yield higher stability but also a high
execution time, and vice-versa. Since stability drops fast on k=60 and in the
same value the execution time grows greatly, we selected it to obtain our results.
α was set on 0.45 and 0.47.
2000 0.30
K=10 K=10
K=15 K=15
K=20 K=20
0.25
1500
0.20
Mean Stability
1000
time
0.15
0.10
500
0.05
0
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.00
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
alpha alpha
(a) alpha vs Execution Time (b) alpha vs Stability
Fig. 2. Fixed K, Variable α
1.0 1.0
alpha=0.45 alpha=0.45
alpha=0.47 alpha=0.47
alpha=0.49 0.9 alpha=0.49
0.8
0.8
0.7
0.6
100 mean stab
0.6
time
0.4
0.5
0.4
0.2
0.3
0.00 20 40 60 80 100 120 0.20 20 40 60 80 100 120
K K
(a) K vs Execution Time Normalized (b) K vs Top 100 Mean Stability
Fig. 3. Variable K, Fixed α
5 Results and Discussion
Table 2 shows a comparative of the characteristics of the lattices yielded from
two reduced contexts (KR ) and the original context (KO ). The lattices were
processed using the FCA suite Coron System12 .
12
http://coron.loria.fr/
Table 2. Comparison of characteristics
α = 0, 45 α = 0, 47 Original
Objects 60 60 4565
Attributes 120 120 120
Density [%] 17,24 10,59 6,59
Concepts 6309 1207 170606
Coincidental Intents 3029 815 -
Mean attributes per concept 20,52 12,6 7,91
Intensional Stability 0,2170 0,3041 0.3995
Extensional Stability 0.2277 0.3211 0.1103
100 Top Int. Stab. 0,9061 0,7576 1
100 Top Ext. Stab. 0.9515 0.8287 0.9837
Levels 10 7 10
Time [s] 6,869 1,145 2865,723
Time to reduce [s] 39,333 39,325 -
Results shows that using LSA before FCA performs a clear reduction in
the formal context from a size of 4565 × 120 (original context) to 60 × 120
(reduced context), specifically a reduction of 76 times the amount of data to
be processed. It also lowers the amount of concepts yielded in the final
lattice (27 and 141 times for α equal to 0.45 and 0.47 resp.), and because of that
the time required to calculate the full concept lattice is considerably
reduced, even considering the time required to create the reduced contexts.
Stability gives more clues about the good quality of the reduction. Figure 4
shows intensional and extensional stability distribution. As it can be observed,
the original context’s lattice has a better intensional stability than the reduced
contexts but a worst extensional stability. Mean values for these two measures
are shown in table 2.
Since we have eliminated redundant data, each dimension is almost equally
important meaning that in reduced contexts we cannot afford to eliminate a
subset of them without affecting greatly the structure of the lattice obtained. In
this case, we have eliminated a big part of the noise (k=60 was in fact a very good
choice). On the other hand, the growth in extensional stability reflects that the
structure of the reduced lattices is not tied to some specific terms. Some terms
can be removed and the structure of the lattice would not vary greatly, which is
what happens each year (see section 3.1).
5.1 A Software Architecture Taxonomy
Figure 5 shows the reduced notation of the lattice for the reduced context (k=60
and α = 0.45). This lattice was drawn with Coron-drawer13 a set of scripts
specially written for large lattices. For the sake of space and simplicity we provide
13
http://code.google.com/p/coron-drawer/
7 6
a=0.45 a=0.45
a=0.47 a=0.47
original original
6
5
5
4
4
hits
hits
3
3
2
2
1
1
0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Stability Stability
(a) Intensional Stability (b) Extensional Stability
Fig. 4. Stability distribution (k=60)
Fig. 5. Filtered Lattice (K=60, α = 0.45, minsupp=0)
a small comparison of the terms in the reduced lattice-based taxonomy with a
human-expert handmade thesaurus of Software Architecture [8].
Software Architecture Thesaurus Comparison The thesaurus contains
494 elements (we call them elements to differentiate them from lattice’s con-
cepts and taxonomy’s terms) organized in a hierarchical fashion. They have at
most one parent and the hierarchy has multiple roots. The thesaurus is exhaus-
tive and comprises mainly definitions of Software Architecture’s concepts and
entities (such as framework’s names or important authors in the domain). The
comparison shows:
– From our 120 term list, 50 terms (42%) match exactly with a term on the
thesaurus. 25 terms (21%) have a semi-match, meaning that they are part
of a term on the thesaurus (database in our hierarchy and shared database
in the thesaurus) and 45 (37%) terms do not have a simile in the thesaurus.
– The three main concepts design, analysis and framework (with support over
50%) found in our taxonomy, also remain being main elements in the the-
saurus.
– Even when some elements in the thesaurus are not found in our taxonomy,
they actually exists as relations among terms. For instance, the relation
among the terms design and pattern describe the thesaurus’ element design
pattern. This is also true for design decision, information view, knowledge
reuse, quality requirements, business methodology and several more elements.
6 Conclusions
In this work we have presented a method and a technique to apply Formal
Concept Analysis (FCA) to large contexts of data in order to obtain a lattice-
based taxonomy. We have outlined that large-size datasets are not suitable to
be processed by FCA and that, this fact is an important problem in the domain.
The solution presented here, is based on an Information Retrieval technique
called Latent Semantic Analysis which is used to reduce a term-document matrix
to a much smaller matrix where terms are related to a set of dimensions instead
of documents. Using a probabilistic approach, this matrix is converted into a
binary formal context where FCA can be applied.
The approach was illustrated with a case study using a research domain from
computational sciences called Software Architecture. The corpus created for this
domain consists of more than 4500 documents and 120 terms. We have com-
pared the characteristics of the lattice obtained through FCA from the original
formal context of terms and documents and the reduced contexts generated by
our approach. We have found that not only our approach is considerably more
economic in execution time as well as in the amount of concepts obtained in
the final lattice but intensional and extensional stabilities give us elements to be
certain of the quality of our approach.
A small comparison with a human expert handmade thesaurus of the com-
munity of Software Architecture is provided in order to illustrate that a real and
coherent taxonomy can be obtained using our approach.
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