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Reconstruction of Belemnite Evolution Using Formal Concept Analysis

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Reconstruction of Belemnite Evolution Using Formal Concept Analysis Reconstruction of Belemnite Evolution Using Formal Concept Analysis Radim Belohlavek,1,2∗ Martin Koˇst’´ak3†, and Petr Osicka2 1State University of New York at Binghamton, Binghamton, NY [email protected] 2Palacky University, Olomouc, Czech Republic 3Charles University, Prague, Czech Republic Abstract The paper presents results on using for- mal concept analysis in the problem of identification of taxa and reconstructing evolution from paleontological data. We present results of experiments performed with belemnites—a group of extinct marine cephalophods which seems particularly suit- Figure 1: Schematic sketch of belemnite body with able for such a purpose. We demonstrate position of the internal shell. Apical grey part repre- that the methods of formal concept analysis sents the rostrum. are capable of revealing taxa and relation- ships among them which are relevant from a paleobiological point of view. Coleoid cephalopods played an important role in the Cretaceous ecosystem in both the Northern and Southern hemispheres. Inside this diverse group, es- 1 Introduction and Problem Setting pecially belemnites were a common part of nectonic An important system-theoretic concept is that of a assemblages in marine shallower water ecosystems. taxonomy, i.e. a classification scheme arranged in They belong to coleoids with internal shell. The ex- a hierarchical structure. Biological taxonomies and ternal morphology of belemnites resembles partly Re- the methods of devising them are perhaps the best cent squids and their behavior—larger sets of spec- known and most widely studied. There exist sev- imens concentrated probably during spawning acts. eral approaches to biological classification, with phy- However, the internal shell characteristic is strongly logenetics (cladistics) and phenetics (numerical taxon- different in both, non related cephalopod groups. In omy) being perhaps the two most important methods belemnites, the internal shell is composed of three ma- [Dunn and Everitt, 2004; Jardin and Sibson, 1971; jor components—the most commonly preserved cal- Kitching et al., 1998; Sneath and Sokal, 1973]. The citic rostrum, aragonitic phragmoconus, and dorsal aim of this paper is to explore the idea of utilizing pro-ostracum. The belemnite taxonomy and system- formal concept analysis in identification of taxa and atics are based on rostrum characterstics. This hy- devising taxonomies in paleobiology. The basic idea is drodynamic and also hydrostatic organ shows gradual to identify taxa with formal concepts which are par- changes in their evolutionary history. A schematic ticular groupings of objects characterized by sharing sketch of a belemnite body is shown in Fig. 2. For [ certain properties (attributes). detailed morphologic features of rostrum see Chris- tensen, 1997; Koˇst’´aket al., 2004]. The overall goal of our research is to find out The most important morphologic features in ros- whether and how formal concept analysis may help trum morphology are size, shape and especially the in identification of taxa and reconstructing evolution part called the alveolar end and/or fracture. From from paleobiological data. In the paper, we re- quite a large systematic group (order Belemnitida), port some of our first results which we obtained from we chose a part of family Belemnitellidae—a domi- data about a particular group of extinct cephalophods nant Upper Cretaceous belemnite group. called belemnites, which are similar to modern squid. As no soft body parts are preserved for taxonomic ∗The research of R. Belohlavek was supported by grant distinction, belemnite taxonomy is mainly based on no. MSM 6198959214 and by grant no. P103/10/1056 of the following morphological characteristics of the the Czech Science Foundation. belemnite rostrum: † The research of M. Koˇst’´akwas supported by grant • Shape and size of the rostrum. no. 205/07/1365 of the Czech Science Foundation and by grant no. MSM 0021620855. • Structure of the alveolar end. • Internal structures at alveolar end. The thus introduced arrow operators form a Galois • External characteristics of the rostrum. connection between X and Y and play an important role in FCA. The set of all formal concepts of hX, Y, Ii • Structure of apex. is denoted by B(X, Y, I). That is, Belemnites are extinct cephalopod group with no ↑ ↓ descendents. Their systematic, palaeoecology, palaeo- B(X, Y, I) = {hA, Bi | A = B, B = A}. biogeography and stratigraphy is based on palaeonto- Under a partial order ≤, defined for logical research. However, the simulation of belemnite hA1,B1i, hA2,B2i ∈ B(X, Y, I) by evolution opens new options for using various math- ematical models, for which the presented cephalopod hA1,B1i ≤ hA2,B2i iff A1 ⊆ A2 (iff B2 ⊆ B1), group is particularly suitable. B(X, Y, I) happens to be a complete lattice, so- called concept lattice associated to hX, Y, Ii [Gan- 2 Formal Concept Analysis ter and Wille, 1999]. The partial order ≤ rep- Formal concept analysis (FCA) is a method for resents the subconcept-superconcept (generalization- data analysis with applications in various domains specialization) relationship between formal concepts. (see [Ganter and Wille, 1999] for foundations and That is, hA1,B1i ≤ hA2,B2i means that formal con- [Carpineto and Romano, 2004] for applications). The cept hA2,B2i (representing e.g. mammal) is more gen- input to FCA consists of a data table describing a eral than hA1,B1i (e.g. dog). Efficient algorithms for set X = {1, . , n} objects and a set Y = {1, . , m} computing B(X, Y, I) exist. of attributes. The table specifies attribute values of The concept lattice is visalized by a so-called la- the objects. The main aim in FCA is to reveal from belled line diagram (or Hasse diagram) [Ganter and the data a hierarchically organized set of particular Wille, 1999]. The diagram consists of nodes, some of clusters, called formal concepts, and a small set of which are connected by lines. The formal concepts are particular attribute dependencies, called attribute im- represented by nodes, the subconcept-superconcept plications. FCA aims at formalizing and utilizing a relationship is represented by the lines, and a particu- traditional theory of concepts, in which a concept is lar way of labeling the nodes by object and attribute understood as an entity consisting of its extent (col- names is used. In particular every node (formal con- lection of all objects to which the concept applies) cept) is connected to all of its direct predecessors (for- and its intent (collection of all attributes to which are mal concepts which are more specific) and direct suc- characteristic for the concept). For example, the ex- cessors (formal concepts which are more general). The tent of the concept dog consists of all dogs while its diagram is easily understood by users and provides a intent consists of attributes such as barks, has tail, has hierarchical view on the data. four limbs, etc. The information extracted from data In several situations, the expert finds certain for- in FCA is well-comprehensible by users because FCA mal concepts in the concept lattice interesting and works with notions which humans are used to reason relevant while others not. One reason for this is with in the ordinary life. This is an important feature that in addition to the input data (object, attributes, of FCA which makes it appealing for users. their relationship), the expert may have further knowl- In the basic setting of FCA, attributes are assumed edge regarding the objects and attributes. Accord- to be binary, i.e. a given object either has or does not ing to this knowledge, the expert may regard some have a given attribute. A data table with binary at- formal concepts interesting (relevant) and the others tributes is represented by a triplet hX, Y, Ii, called a not. In [Belohlavek and Sklenar, 2005; Belohlavek formal context, which consists of the above-mentioned and Vychodil, 2009], we developed an approach to sets X and Y of objects and attributes, and a binary handling one particular type of a background knowl- relation I between X and Y (incidence relation, to- edge which we utilize in this paper. The idea is have relation). Thus, I ⊆ X × Y , hx, yi ∈ I indicates that the attributes may not be equally important to that object x has attribute y, hx, yi 6∈ I indicates that form concepts. To give a simple example, consider x does not have y. Objects x ∈ X correspond to table the following attributes of books: hardbound, paper- rows, attributes y ∈ Y correspond to table columns, back, engineering, science, philosophy. In a reason- and I is represented by 0s and 1s in the table en- able taxonomy of books, we naturally consider the tries (i.e., 1 indicates that the corresponding object attributes engineering, science, and philosophy more does have the corresponding attribute and 0 indicates important than hardbound or paperback. As a conse- the opposite). A formal concept of hX, Y, Ii is any quence, we would consider a formal concept character- pair hA, Bi of sets A ⊆ X (extent, set of objects to ized by (i.e. with its intent consisting of) hardbound which the concept applies) and B ⊆ Y (intent, set of (all books which are hardbound) not natural (relevant, attributes characterizing the concept) such that B is interesting). One the other hand, the formal con- just the set of attributes shared by all objects from A, cept characterized by engineering (books on engineer- and A is the set of all objects sharing all attributes ing), the one characterized by science, and perhaps from B. In symbols, this can be written as A↑ = B also the one characterized by engineering and paper- and B↓ = A, where back (paperback books on engineering) would be con- ↑ sidered natural. Such a background knowledge may A = {y ∈ Y | for each x ∈ A: hx, yi ∈ I}, be represented by so-called attribute-dependency for- B↓ = {x ∈ X | for each y ∈ B : hx, yi ∈ I}.
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