Towards a Uni ed Taxonomy of Biclustering Methods

Abstract

Being an unsupervised machine learning and data mining technique, biclustering and its multimodal extensions are becoming popular tools for analysing object-attribute data in di erent domains. Apart from conventional clustering techniques, biclustering is searching for homogeneous groups of objects while keeping their common description, e.g., in binary setting, their shared attributes. In bioinformatics, biclustering is used to nd genes, which are active in a subset of situations, thus being candidates for biomarkers. However, the authors of those biclustering techniques that are popular in gene expression analysis, may overlook the existing methods. For instance, BiMax algorithm is aimed at nding biclusters, which are well-known for decades as formal concepts. Moreover, even if bioinformatics classify the biclustering methods according to reasonable domain-driven criteria, their classi cation taxonomies may be di erent from survey to survey and not full as well. So, in this paper we propose to use concept lattices as a tool for taxonomy building (in the biclustering domain) and attribute exploration as means for cross-domain taxonomy completion.

Keywords

• Biclustering
• taxonomy
• concept lattices
• attribute exploration

-

Biclustering is a popular family of data analysis techniques within cluster-analysis. Previously biclustering was known under the names direct clustering or subspace clustering [ 1 ]. The term biclustering was proposed by Boris Mirkin in [ 2 ], p. 296: The term biclustering refers to simultaneous clustering of both row and column sets in a data matrix. Biclustering addresses the problems of aggregate representation of the basic features of interrelation between rows and columns as expressed in the data.

The main advantage of biclustering technique lies in its ability to keep similarity of grouped objects in terms of their common attributes. So, biclustering is able to capture object similarity (homogeneity) expressed only by a subset of attributes, which allows an analyst to clearly see why certain objects were grouped together.

In the previous decade biclustering methods became extremely popular for gene expression analysis analysis in bioinformatics. Here, genes which demonstrate similar properties only in a subset of observable situations are considered to be within a bicluster along with those situations. The rst rather comprehensive survey in the eld was done in [ 3 ]. Even though the survey was limited only to biclustering in bioinformatics, the eld came to its maturity to have its own classi cation of the methods. The authors classi ed biclustering techniques according to several properties: biclustering type, biclustering structure, the way of bicluster generation, and the algorithmic strategy.

As it often happens, researchers from the bioinformatics domain overlooked or even rediscovered biclustering methods which have been known for decades. Thus, the notion of formal concept was known since the early 80-s [ 4 ], it corresponds to maximal inclusion unit submatrices in Boolean matrices [ 5,6,7 ]. The idea of closed sets from Formal Concept Analysis and from frequent itemset mining were not considered in the bioinformatics domain. However, there are numerous e cient algorithms and applications, which can be treated as special cases of biclustering-based ones. To the best of our knowledge there is no any biclustering technique mentioned in bioinformatics which exploits ordered bicluster hierarchies. Thus, in Formal Concept Analsis, biclusters (formal concepts) are hierarchically ordered by the relation \be more general concept than", which proved its helpfulness for data exploration and taxonomy building in di erent domains.

The aim of this work is two-fold: on the one hand, we are going to shed light on neighbouring domains where biclustering is actively used, and on the other hand build lattice-based taxonomies using the existing classi cations of biclustering algorithms in the literature. The main open question in this work is as follows: How to build a uni ed taxonomy of the biclustering techniques.

The rest of the paper is organised as follows. In Section 2, we shortly review previous work on biclustering, taxonomies of algorithms, and related elds. In Section 3 we give basic de nitions of FCA and biclustering (in the most general form). In Section 5 we outline several existing biclustering extensions under the the name of multimodal clustering. Section 4 is the main part of the paper which provides examples of di erent taxonomies of biclustering algorithms obtained from the literature. 2

Previous work

Construction of taxonomies of algorithms in Computer Science is not new. Thus, in [ 8,9 ] a taxonomy of string matching algorithms was built guided by domain experts according to TABASCO methodology. In those papers, it was shown that concept lattices can be a good visualisation tool paired with interactive abilities of modern computer software. Moreover, concept lattices were successfully used for epistemic taxonomy building [ 10 ] combining multiple inheritance feature with compact graphical representation.

In addition to the existing term biclustering, there are several others like coclustering or simultaneous clustering. Triclustering, Triadic FCA, multimodal clustering, clustering of Boolean tensors, closed n-sets, relational clustering and several other techniqes are all examples of possible biclustering extensions. 3 3.1

Basic de nitions and models

Formal Concept Analysis

De nition 1. A formal context K = (G; M; I) consists of two sets G and M and a relation I between G and M . The elements of G are called the objects and the elements of M are called the attributes of the context. The notation gIm or (g; m) 2 I means that the object g has attribute m.

De nition 2. For A

G, let

A0 := fm 2 M j(g; m) 2 I for all g 2 Ag and, for B

M , let

B0 := fg 2 Gj(g; m) 2 I for all m 2 Bg:

These operators are called derivation operators or operators for K = (G; M; I). concept-forming Proposition 1. Let (G; M; I) be a formal context, for subsets A; A1; A2 and B M we have G

Similar properties hold for subsets of attributes.

De nition 3. A closure operator on set S is a mapping ' : 2S ! 2S with the following properties: 1. ''X = 'X (idempotency) 2. X 'X (extensity) 3. X Y ) 'X 'Y (monotonicity)

For a closure operator ' the set 'X is called closure of X.

A subset X G is called closed if 'X = X. Let (G; M; I) be a context, one can prove that operators ( )00 : 2G ! 2G; ( )00 : 2M ! 2M are closure operators.

De nition 4. A formal concept of a formal context K = (G; M; I) is a pair (A; B) with A G, B M , A0 = B and B0 = A. The sets A and B are called the extent and the intent of the formal concept (A; B), respectively. The subconceptsuperconcept relation is given by (A1; B1) (A2; B2) i A1 A2 (B1 B2).

This de nition says that every formal concept has two parts, namely, its extent and intent. This follows an old tradition of the Logic of Port Royal (1662), and is in line with the International Standard ISO 704 that formulates the following de nition: \A concept is considered to be a unit of thought constituted of two parts: its extent and its intent." De nition 5. The set of all formal concepts of a context K together with the order relation I forms a complete lattice, called the concept lattice of K and denoted by B(K).

De nition 6. Implication A ! B, where A; B M holds in context (G; M; I) if A0 B0, i.e., each object having all attributes from A also has all attributes from B. 3.2

Biclustering

In the rst survey on biclustering techniques [ 3 ], bicluster is de ned as a submatrix of an input object-attribute matrix. That is for a given matrix A 2 Rm n, a bicluster b is a pair (X; Y ), where X f1; : : : ; mg and Y f1; : : : ; ng. The bicluster should ful l a certain homogeneity property, which varies from method to method, e.g., it may be allowed to contain only 1s inside the corresponding submatrix (bicluster) in Boolean case.

For instance, for analysing large markets of context advertisement, we propose the following FCA-based de nition of a bicluster [ 11,12 ].

De nition 7. If (g; m) 2 I, then (m0; g0) is called an object-attribute or OAbicluster with density (m0; g0) = jI\(m0 g0)j .

jm0j jg0j

Here are some basic properties of oa-biclusters.

Proposition 2.

1. 0 1. 2. oa-bicluster (m0; g0) is a formal concept i 3. if (m0; g0) is a oa-bicluster, then (g00; g0) = 1.

(m0; m00).

In gure 1 you can see the example of the oa-bicluster for a particular pair (g; m) 2 I of a certain context (G; M; I). In general, only the regions (g00; g0) and (m0; m00) are full of non-empty pairs, i.e. have maximal density = 1, since they are object and attribute formal concepts respectively. Some black cells indicate non-empty pairs which one may found in such a bicluster. Therefore, the density parameter is a bicluster quality measure which shows how many non-empty pairs the bicluster contains.

De nition 8. Let (A; B) 2 2G real number, such that 0 the constraint (A; B)

min min.

2M be a oa-bicluster and min be a nonnegative

1, then (A; B) is called dense if it satis es 4

Existing taxonomies and their analysis

Since formal concept is a natural notion of bicluster for Boolean data and was rediscovered or reused in bioinformatics, one may suppose that the taxonomy of FCA algorithms is a part of the taxonomy of biclustering algorithms. In fact, paper [ 13 ] proposed such a taxonomy (see g.2).

The classi cation properties of the concept lattice building algorithms encoded as follows: { m1 means incremental approach; { m2 means that an algorithm uses canonicity based on the lexical order; { m3 means that an algorithm divides the set of concepts into several parts; { m4 designates that an algorithm uses hashing; { m5 means that an algorithm maintains an auxiliary tree structure; { m6 means usage of attribute cache; { m7 encodes that an algorithm computes intents by subsequently computing intersections of object intents (i.e., fgg0 \ fhg0); { m8 means that an algorithm computes intersections of already generated intents; { m9 encodes that an algorithm computes intersections of non-object intents and object intents; { m10 means that an algorithm uses supports of attribute sets.

We formed a context based on Table II \Overall comparison of the biclustering algorithms" [ 3 ] and build its concept-based taxonomy in Fig.3.

Originally, the authors used several criteria to classify the existing (reviewed) biclustering algorithm: type of bicluster, structure of biclusters, type of bicluster discovery, and algorithmic strategy.

Thus, with respect to the de nition of bicluster (its type) the authors di erentiate between 1) biclusters with constant values, 2) biclusters with constant values on rows or columns, 3) biclusters with coherent values, and 4) biclusters with coherent evolutions.

The biclusters were classi ed into one of 9 classes according to their structure. a) Single Bicluster b) Exclusive row and column biclusters (rectangular diagonal blocks after row and column reorder). c) Non-Overlapping biclusters with checkerboard structure. d) Exclusive-rows biclusters. e) Exclusive-columns biclusters. f) Non-Overlapping biclusters with tree structure. g) Non-Overlapping non-exclusive biclusters. h) Overlapping biclusters with hierarchical structure. i) Arbitrarily positioned overlapping biclusters.

Di erent biclustering methods pursue di erent goals in terms of the number of discovered biclusters. Thus, they may identify one bicluster at a time or be targeted to discovering one set of biclusters at a time, or they can follow simultaneous bicluster identi cation, which means that the biclusters are discovered all at the same time. All the three types are possible values of Discovery type attribute in the proposed taxonomy.

Since in many cases the biclustering enumeration is a hard task (the corresponding counting problem may belong to #P complexity class), di erent algorithmic enumeration strategies were proposed. Thus, Madeira and Oliviera sort out several categories: 1) Iterative Row and Column Clustering Combination, 2) Divide and Conquer, 3) Greedy Iterative Search, 4) Exhaustive Bicluster Enumeration, and 5) Distribution Parameter Identi cation. e h t r o f

One of the reasonable questions here is: Why should we build diagrams instead of looking at tables? The answer is we need two complementary views, object-attribute descriptions in tables and ordered clusters of objects such that the objects inside a particular cluster (formal concept) share the same attributes. It is not easy to nd such clusters with respect to permutations of rows and columns manually even for small contexts. Moreover, by examining the concept lattice of a certain taxonomy we can nd useful attribute dependencies, which can help to discover the underlying taxonomy's domain.

The previous classi cation done by Madeira and Oliviera was extended and completed almost 11 years later in [ 14 ]. We build the corresponding line diagram in Fig.4.

In fact, the number of classi ed methods were extended to 47 from 16.

The authors slightly redesigned the proposed classi cation criteria. Thus, they split the analysed methods into two categories: metric-based and non-metric based. We counted this split as two corresponding attributes in the related formal context. However, we also decided to include all the mentioned evaluation metrics into our analysis like \Measure:MSR" meaning Mean Squared Residue.

The remaining criteria have been changed or extended by the authors. For instance, instead of bicluster types, now eight patterns has been proposed: 1. Constant; 2. Constant columns; 3. Coherent values; 4. Additive coherent values; 5. Multiplicative coherent values; 6. Simultaneous coherent values; 7. Coherent evolutions; 8. Negative correlations.

The sub-taxonomy based on bicluster structure now contains only six criteria: row exhaustive, column exhaustive, non-exhaustive, row exclusive, column exclusive, and non-exclusive.

By means of terms \exhaustive" and \exclusive" it is possible to describe the desired structure. Thus, exhaustive means where all genes (conditions) should belong to some bicluster, i.e. to be covered by it. Exclusive means whether a gene (condition) has to belong no more than one bicluster; e.g., in non-exclusive case overlapping is allowed.

The attribute algorithmic strategy has been altered in its original form from [ 3 ]. The attribute Discovery from [ 3 ] has been renamed to Strategy, but the values remain the same: one bicluster at a time, set of biclusters at a time, and simultaneous bicluster identi cation. e h t r o f e h T

In the beginning of 2000s it was unusual that data analysts and biologists can miss existing biclustering methods (like FCA), which were not applied in the bioinformatics domain yet. However, later FCA was successfully applied in the domain of gene expression analysis [ 15,16,17,7,18 ], formal concepts were rediscovered by [ 6,19 ] in bioinformatics, approximate greedy [ 20 ] and fast [ 21,12 ] methods for dense bicluster discovery in Boolean setting appeared.

However, even the recent taxonomy from [ 14 ] does not include any of them.

To overcome incompleteness caused by the bioinformatics domain view restriction, an attempt to extend the taxonomy of Madeira and Oliveira was done in [ 11 ].

In addition to the existing criteria, the attribute bicluster values type was added taking two values: binary and numeric. The former shows whether the method is able to nd patterns in Boolean object-attribute tables and the latter indicates whether the input entries from R can be processed by an algorithm. Another important criteria is whether an algorithm based on the notion of closure (operator) from FCA and Closed Frequent Itemset Mining, implicitly or explicitly. The corresponding formal context is given below.

FCA-related biclustering

BiMax

Box biclustering FCA Freq. Closed Itemsets Association rules Fault-tolerant concepts OA-biclusters ecexh l.reov ray t it irc :tsceoynp i:ttsceoynpw .i:tttrrrcbuA i:lteeaynbpu iil:lrsceeoxpu lii:lrsceopum :.lteeayunpm

T T S V C C V { fType:Additive coherent val.g ! fStruct:Non-Exhaustiveg, sup = 20 { fMeasure:MSRg ! fMetric-based, Struct:Non-Exhaustiveg, sup = 18 { fType:Additive coherent val., Struct:Non-Exhaustive, Struct:Non-exclusiveg ! fMetric-basedg, sup = 18 { fStrategy:Oneg ! fStruct:Non-Exhaustive, Struct:Non-exclusiveg, sup = 17 { fType:Coherent values, Struct:Non-Exhaustiveg ! fStruct:Non-exclusiveg, sup = 15 { fStrategy:One setg ! fStruct:Non-Exhaustiveg, sup = 13 { fMeasure:Varg ! fMetric-based, Struct:Non-Exhaustive, Struct:Non-exclusiveg, sup = 8 { fType:Negative correlationsg ! fStruct:Non-Exhaustive, Struct:Non-exclusiveg, sup = 7 { fMetric-based, Struct:Non-Exhaustive, Strategy:Simultg ! fType:Additive coherent val., Struct:Non-exclusiveg, sup = 7

Since we deal with implications, their con dence measure is equal to 1. The size of the whole set of implications in Duquenne-Gigues base is 105.

If we start attribute exploration for the same context, then the rst question in a row is the following:

Is it true, that when biclustering technique has attribute \Strategy:One set", that it also has attribute \Struct:Non-exhaustive"?

An expert can either agree with the implication fStrategy:One set ! Struct:Non-exhaustiveg or disagree. In the latter case, (s)he needs to provide a counterexample: a biclustering technique which follows discovery strategy \one set of biclusters at a time" but does not result in biclusters of the structure type \exhaustive". There is also an option to stop Attribute Exploration process at every step. 5

Multimodal clustering and closed n-sets

Since the eld of biclustering is a subdomain of multimodal or relational clustering, the taxonomy can be extended by applying similar criteria to n-ary relation and tensor clustering algorithms.

Thus, the notion of formal concept was generalised for triadic [ 27 ] and polyadic case [ 28 ]. There are e cient algorithms to nd triconcepts [ 29 ] and poliadic concepts (closed n-sets) [ 30 ]. There exist relaxations of triconcept and poliadic concept notions, triclusters and n-clusters, which allow for certain entries inside such n-dic concept to be zeros [ 20,31,32 ]; the theoretical and experimental comparison is done in [ 33 ]. There are also methods for mining closed patterns in n-ary relations [ 34 ]. Two biclustering approaches can be used for mining two formal contexts simultaneously, which shares either set of attributes or objects; this results in pseudotriclusters [ 35 ]. As for purely biological applications of triclustering we may suggest reading, for example, [ 36 ] and [ 37 ].

Conclusion and future work

Even though the taxonomy building of a particular sub eld of Data Analysis or Computer Science is not as laborious as devising Carl Linnaeus' pre-phylogenetic taxonomy, this is not an easy task to merge several such existing taxonomies and build a uni ed one. Similarly to new species discovery, new algorithms can be proposed and since they can contain new speci c features, new classi cation attributes may be needed.

One of the possible schemes of taxonomy maintaining here could be done in terms of Attribute Exploration [ 5 ].

At a certain moment the group of expert xes the set of existing biclustering methods and proposes suitable criteria for their classi cation. A person or a team which has proposed a new biclustering method should classify the method according to the chosen scheme, then it should be validated by experts. Such a team can propose an extra criterion for method classi cation. If a person or a team is going to propose a new method for an unexplored combination of classi cation attributes, it is possible to run attribute exploration to see which prospective types of methods are missing to date. By means of Object Exploration, it may become clear that some attributes are missing, e.g. it is evident that formal concepts or Boolean biclusters is only a particular case of bicluster type with constant values and we need at least one new attribute, Boolean entry values.

Since a taxonomy may be used not only for classi cation itself, but as a search index for potential users, we may suggest using several ways of interactive visualisation: tree-based (TABASCO-like), concept lattice based (line diagrams), object-attribute tables, and nested line diagrams. The latter can help when someone is interested in a special main set of attributes, which should be shown in the outer taxonomy on the line diagram; the inner taxonomy can be shown if the method-seeker needs a ner level granularity or more detailed description inside of the selected node from the outer taxonomy.

It is important to note that taxonomies can be considered as a special case of ontologies, and here FCA was successfully used both for ontology merging and completion [ 38,39 ].

There are two main tasks for our future studies: 1) unifying the existing bicluster taxonomies, and 2) creation a taxonomy of multimodal clustering techniques. Even though there are several good tools for building and managing concept lattices like Concept Explorer, we need to rely on more exible tools with extensible components. In particular we hope that FCART can become our tool of choice in the near future [ 40 ].

Acknowledgements. We would like to thank Sergei Obiedkov and Derrick Kourie for a piece of advice and their earlier work on the topic. This work was supported by the Basic Research Program at the National Research University Higher School of Economics in 2015-2016 and performed in the Laboratory of Intelligent Systems and Structural Analysis. The rst author was also supported by Russian Foundation for Basic Research (grant #13-07-00504).

References

1. Hartigan , J.A. : Direct Clustering of a Data Matrix . Journal of the American Statistical Association 67 ( 337 ) ( 1972 ) 123 { 129

2. Mirkin , B. : Mathematical Classi cation and Clustering . Kluwer, Dordrecht ( 1996 )

3. Madeira , S.C. , Oliveira , A.L. : Biclustering algorithms for biological data analysis: A survey . IEEE/ACM Trans. Comput. Biology Bioinform . 1 ( 1 ) ( 2004 ) 24 { 45

4. Wille , R.: Restructuring lattice theory: An approach based on hierarchies of concepts . In Rival , I., ed.: Ordered Sets. Volume 83 of NATO Advanced Study Institutes Series . Springer Netherlands ( 1982 ) 445 { 470

5. Ganter , B. , Wille , R.: Formal Concept Analysis: Mathematical Foundations . Springer, Berlin/Heidelberg ( 1999 )

6. Barkow , S. , Bleuler , S. , Prelic , A. , Zimmermann , P. , Zitzler , E.: Bicat: a biclustering analysis toolbox . Bioinformatics 22 ( 10 ) ( 2006 ) 1282 { 1283

7. Kaytoue , M. , Duplessis , S. , Kuznetsov , S. , Napoli , A. : Two fca-based methods for mining gene expression data . In Ferr, S., Rudolph , S., eds.: Formal Concept Analysis. Volume 5548 of Lecture Notes in Computer Science . Springer Berlin Heidelberg ( 2009 ) 251 { 266

8. Cleophas , L.G. , Watson , B.W. : Taxonomy-based software construction of SPARE time: a case study . IEE Proceedings - Software 152 ( 1 ) ( 2005 ) 29 { 37

9. Cleophas , L.G. , Watson , B.W. , Kourie , D.G. , Boake , A. , Obiedkov , S.A. : TABASCO: using concept-based taxonomies in domain engineering . South African Computer Journal 37 ( 2006 ) 30 { 40

10. ROTH , C. , OBIEDKOV , S. , KOURIE , D.G.: On succinct representation of knowledge community taxonomies with formal concept analysis . International Journal of Foundations of Computer Science 19 ( 02 ) ( 2008 ) 383 { 404

11. Ignatov , D. : Models, Algorithms, and Software Tools of Biclustering Based on Closed Sets . PhD thesis , State University Higher School of Economics, Moscow ( 2010 )

12. Ignatov , D.I. , Kuznetsov , S.O. , Poelmans , J.: Concept-Based Biclustering for Internet Advertisement . In: Data Mining Workshops (ICDMW) , 2012 IEEE 12th International Conference on. (Dec 2012 ) 123 { 130

13. Kuznetsov , S.O. , Obiedkov , S.A.: Comparing performance of algorithms for generating concept lattices . J. Exp. Theor. Artif. Intell . 14 ( 2-3 ) ( 2002 ) 189 { 216

14. Pontes , B. , Giraldez , R. , Aguilar-Ruiz , J.S.: Biclustering on expression data: A review . Journal of Biomedical Informatics 57 ( 2015 /12/28) 163{ 180

15. Besson , J. , Robardet , C. , Boulicaut , J. , Rome, S.: Constraint-based concept mining and its application to microarray data analysis . Intell. Data Anal . 9 ( 1 ) ( 2005 ) 59 { 82

16. Kaytoue-Uberall , M. , Duplessis , S. , Napoli , A. : Using formal concept analysis for the extraction of groups of co-expressed genes . In Le Thi, H. , Bouvry , P. , Pham Dinh , T., eds.: Modelling, Computation and Optimization in Information Systems and Management Sciences. Volume 14 of Communications in Computer and Information Science . Springer Berlin Heidelberg ( 2008 ) 439 { 449

17. Motameny , S. , Versmold , B. , Schmutzler , R.: Formal concept analysis for the identi cation of combinatorial biomarkers in breast cancer . In Medina, R., Obiedkov , S., eds.: Formal Concept Analysis. Volume 4933 of Lecture Notes in Computer Science . Springer Berlin Heidelberg ( 2008 ) 229 { 240

18. Kaytoue , M. , Kuznetsov , S.O. , Napoli , A. , Duplessis , S. : Mining gene expression data with pattern structures in formal concept analysis . Information Sciences 181 ( 10 ) ( 2011 ) 1989 { 2001 Special Issue on Information Engineering Applications Based on Lattices.

19. Preli , A. , Bleuler , S. , Zimmermann , P. , Wille , A. , Bhlmann , P. , Gruissem , W. , Hennig , L. , Thiele , L. , Zitzler , E.: A systematic comparison and evaluation of biclustering methods for gene expression data . Bioinformatics 22 ( 9 ) ( 2006 ) 1122 { 1129

20. Mirkin , B. , Kramarenko , A. : Approximate bicluster and tricluster boxes in the analysis of binary data . In Kuznetsov, S., lzak, D., Hepting , D. , Mirkin , B., eds.: Rough Sets, Fuzzy Sets, Data Mining and Granular Computing. Volume 6743 of Lecture Notes in Computer Science . Springer Berlin Heidelberg ( 2011 ) 248 { 256

21. Ignatov , D.I. , Kuznetsov , S.O. : Concept-based recommendations for internet advertisement . In Belohlavek, R., Kuznetsov , S.O., eds. : Proc. of The Sixth International Conference Concept Lattices and Their Applications (CLA'08) , Palacky University, Olomouc ( 2008 ) 157 { 166

22. Pensa , R.G. , Boulicaut , J.: Towards fault-tolerant formal concept analysis . In: AI*IA . ( 2005 ) 212 { 223

23. Flores , J.L. , Inza , I. , Larraaga , P. , Calvo , B. : A new measure for gene expression biclustering based on non-parametric correlation . Computer Methods and Programs in Biomedicine 112(3) ( 2013 ) 367 { 397

24. Henriques , R. , Antunes , C. , Madeira , S.C. : A structured view on pattern miningbased biclustering . Pattern Recognition 48 ( 12 ) ( 12 2015 ) 3941 { 3958

25. Kaytoue , M. , Kuznetsov , S.O. , Macko , J. , Napoli , A. : Biclustering meets triadic concept analysis . Ann. Math. Artif. Intell . 70 ( 1-2 ) ( 2014 ) 55 { 79

26. Guigues , J.L. , Duquenne , V. : Familles minimales d'implications informatives rsultant d'un tableau de donnes binaires . Mathmatiques et Sciences Humaines 95 ( 1 ) ( 1986 ) 5{18 In French .

27. Lehmann , F. , Wille , R.: A triadic approach to formal concept analysis . In: Conceptual Structures: Applications , Implementation and Theory, Third International Conference on Conceptual Structures, ICCS '95, Santa Cruz , California, USA, August 14- 18 , 1995 , Proceedings. ( 1995 ) 32 { 43

28. Voutsadakis , G.: Polyadic concept analysis . Order 19 ( 3 ) ( 2002 ) 295 { 304

29. Jaschke, R., Hotho , A. , Schmitz , C. , Ganter , B. , Stumme , G.: TRIAS - an algorithm for mining iceberg tri-lattices . In: Proceedings of the 6th IEEE International Conference on Data Mining (ICDM 2006 ), 18 - 22 December 2006, Hong Kong , China. ( 2006 ) 907 { 911

30. Cerf , L. , Besson , J. , Robardet , C. , Boulicaut , J.: Closed patterns meet n-ary relations . TKDD 3 ( 1 ) ( 2009 )

31. Ignatov , D.I. , Kuznetsov , S.O. , Poelmans , J. , Zhukov , L.E. : Can triconcepts become triclusters? Int. J. General Systems 42 ( 6 ) ( 2013 ) 572 { 593

32. Cerf , L. , Besson , J. , Nguyen , K. , Boulicaut , J.: Closed and noise-tolerant patterns in n-ary relations . Data Min. Knowl. Discov . 26 ( 3 ) ( 2013 ) 574 { 619

33. Ignatov , D. , Gnatyshak , D. , Kuznetsov , S. , Mirkin , B. : Triadic formal concept analysis and triclustering: searching for optimal patterns . Machine Learning ( 2015 ) 1 { 32

34. Spyropoulou , E. , Bie , T.D., Boley , M. : Interesting pattern mining in multirelational data . Data Min. Knowl. Discov . 28 ( 3 ) ( 2014 ) 808 { 849

35. Gnatyshak , D. , Ignatov , D.I. , Semenov , A. , Poelmans , J.: Gaining insight in social networks with biclustering and triclustering . In: Perspectives in Business Informatics Research - 11th International Conference, BIR 2012, Nizhny Novgorod , Russia, September 24-26 , 2012 . Proceedings. ( 2012 ) 162 { 171

36. Zhao , L. , Zaki , M.J.: Tricluster: An e ective algorithm for mining coherent clusters in 3d microarray data . In: Proceedings of the ACM SIGMOD International Conference on Management of Data , Baltimore, Maryland, USA, June 14-16, 2005 . ( 2005 ) 694 { 705

37. Li , A. , Tuck , D.: An e ective tri-clustering algorithm combining expression data with gene regulation information . Gene regulation and systems biology 3 ( 2009 ) 49 { 64

38. Ganter , B. , Stumme , G.: Creation and merging of ontology top-levels . In: Conceptual Structures for Knowledge Creation and Communication, 11th International Conference on Conceptual Structures, ICCS 2003 Dresden, Germany, July 21-25 , 2003 Proceedings. ( 2003 ) 131 { 145

39. Sertkaya , B. : A survey on how description logic ontologies bene t from FCA . In: Proceedings of the 7th International Conference on Concept Lattices and Their Applications , Sevilla, Spain, October 19-21 , 2010 . ( 2010 ) 2 { 21

40. Neznanov , A. , Parinov , A. : FCA analyst session and data access tools in FCART . In Agre, G., Hitzler , P. , Krisnadhi , A.A. , Kuznetsov , S.O., eds.: Arti cial Intelligence: Methodology, Systems , and Applications - 16th International Conference, AIMSA 2014, Varna, Bulgaria, September 11-13 , 2014 . Proceedings. Volume 8722 of Lecture Notes in Computer Science., Springer ( 2014 ) 214 { 221