Rough Inclusion Research Articles

On topological covering-based rough spaces

Rough set theory, a mathematical tool to deal with vague concepts, has originally described the indiscernibility of elements by equivalence relations. Covering-based rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to coverings. Recently, some topological concepts such as subbase, neighborhood and separation axioms have been applied to study covering-based rough sets. However, the topological space on covering-based rough sets and the corresponding topological properties on the topological covering-based rough space are not studied. This paper studies some of these problems. We defined open sets, closed sets, rough inclusion, rough equality on covering-based rough sets and some of their properties are studied. Then, we give the definition of topology on covering-based rough sets. Finally, we study the properties of rough homeomorphisms. This research not only can form the theoretical basis for further applications of topology on covering-based rough sets but also lead to the development of the rough set theory and artificial intelligence. Key words: Topology, rough Sets, covering, artificial Intelligence.

Granular computing in the frame of rough mereology. A case study: Classification of data into decision categories by means of granular reflections of data

This work extends the authors' contribution to the International Conference on Rough Sets and Current Trends in Computing (RSCTC 2008) held at the University of Akron in October 2008. It is dedicated to the topic of granular computing, formalized within the theory of rough mereology, as proposed by Polkowski; as an application of the idea of a granular reflection of data and of classifiers induced from it (Polkowski, 2005), we give an account of recent results in this area. A scheme for classifier construction based on factoring a classifier through a granular reflection of data; voting by granules of training objects; voting by granules of decision rules induced from the training set; voting by granules induced from a granular reflection of data, is presented here. In voting cases, voting is based on weights computed by means of rough inclusions induced from residual implications of continuous t-norms. The results show a high effectiveness of this approach as witnessed by the reported tests with some well--known data sets from University of California, at Irvine (UCI) repository whose results are compared against the standard rough set exhaustive classifier whose accuracy is indicated under the radius of “nil.” © 2011 Wiley Periodicals, Inc.

Lethal short-limbed platyspondylic dwarfism, torrance type

A male fetus was delivered at 20 weeks gestation following an ultrasound diagnosis of a lethal skeletal dysplasia. This was the first baby for a non-consanguineous northern European Australian couple with no relevant family history. At post-mortem there was micromelia with bowing of the long bones, brachydactyly, a narrow, bell-shaped thorax but normal trunk length. There was mild macrocephaly, a low nasal bridge but no clover leaf skull. Internally the lungs were hypoplastic and the temporal lobes of the brain were larger than expected with a posterior link. Radiology showed small platyspondylic vertebral bodies with poor ossification, shortening of long limb bones, curved radii and irregular and flared metaphyses. Histology of bones revealed disorganisation of growth plates with some chondrocytes showing large rough endoplasmic reticulum inclusion bodies. There was no mutation of the FGFR3 gene and the karotype was normal. These features are of lethal short-limbed platyspondylic dwarfism, Torrance type (OMIM# 151210), an auto-somal dominant condition due to a defect in the COL2A1 gene. This results in a defect in a protein that forms type II collagen that is essential for normal bone development. The risk of recurrence is the same as the general population although there is a theoretical risk of germline mosaicism.

Probing model tumor interfacial properties using piezoelectric cantilevers

Invasive malignant breast cancers are typically branchy and benign breast tumors are typically smooth. It is of interest to characterize tumor branchiness (roughness) to differentiate invasive malignant breast cancer from noninvasive ones. In this study, we examined the shear modulus (G) to elastic modulus (E) ratio, G/E, as a quantity to describe model tumor interfacial roughness using a piezoelectric cantilever capable of measuring both tissue elastic modulus and tissue shear modulus. The piezoelectric cantilever used had two lead zirconate titanate layers to facilitate all-electrical elastic (shear) modulus measurements using one single device. We constructed model tissues with tumors by embedding one-dimensional (1D) corrugated inclusions and three-dimensional (3D) spiky-ball inclusions made of modeling clay in gelatin. We showed that for smooth inclusions, G/E was 0.3 regardless of the shear direction. In contrast, for a 1D corrugated rough inclusion G/E was 0.3 only when the shear was parallel to corrugation and G/E increased with an increasing angle between the shear direction and the corrugation. When the shear was perpendicular to corrugation, G/E became >0.7. For 3D isotropic spiky-ball inclusions we showed that the G/E depended on the degree of the roughness. Using the ratio s/r of the spike length (s) to the overall inclusion radius (r) as a roughness parameter, we showed that for inclusions with s/r larger than or equal to 0.28, the G/E ratio over the inclusions was larger than 0.7 whereas for inclusions with s/r less than 0.28, the G/E decreased with decreasing s/r to around 0.3 at s/r=0. In addition, we showed that the depth limit of the G/E measurement is twice the width of the probe area of the piezoelectric cantilever.

Rough approximations based on intersection of indiscernibility, similarity and outranking relations

Rough sets theory has proved to be a useful mathematical tool for dealing with the vagueness and granularity in information tables. Classical definitions of lower and upper approximations were originally introduced with reference to an indiscernibility relation. However, indiscernibility relation is still restrictive for many applications. Many real-world problems deal with assignment of some objects to some preference-ordered decision classes. And, the objects are described by a finite set of qualitative attributes and quantitative attributes. In this paper, we construct the indiscernibility relation for the subset of nominal attributes, the outranking relation for the subset of ordinal attributes, and the similarity relation for the subset of quantitative attributes. Then the global binary relation is generated by the intersection of indiscernibility relation, outranking relation and similarity relation. New definitions of lower and upper approximations of the upward and downward unions of decision classes are proposed based on the global relation. We also prove that the lower and upper approximation operations satisfy the properties of rough inclusion, complementarity, identity of boundaries, and monotonicity.

Peak surface motion due to scattering of a plane harmonic SH wave by a randomly corrugated scatterer

Scattering of a plane harmonic elastic SH wave by a rough elastic inclusion embedded within an elastic half space is investigated for an anti-plane strain model by using a direct boundary integral equation method. The scatterer’s interface is assumed to be of an elliptical shape perturbed in a random fashion. The perturbations are assumed to have either uniform or normal probability distributions. The peak perturbation amplitude is assumed to be arbitrary. Surface displacement spectral amplitude is evaluated for different scatterers, impedance contrasts, angles of incidence, and range of frequencies. Corresponding contour plots are used to identify the frequencies of the peak surface spectral amplitudes for each model. The results clearly demonstrate that the presence of small roughness in the inclusion’s interface may change significantly the peak surface motion frequencies. These changes depend upon the inclusion geometry, the nature of the interface roughness, and the impedance contrast of the materials. For certain frequencies, the presence of the inclusion roughness is important for the surface response and modeling may require detailed description of the corrugated inclusion. Lack of such analysis may result in considerable error in the surface motion amplification estimates.

Scattering of plane harmonic P, SV and Rayleigh waves by a completely embedded corrugated elastic inclusion

Scattering of a plane harmonic P, SV or Rayleigh wave by a corrugated elastic inclusion completely embedded in a two-dimensional isotropic half or full space is investigated by using a direct boundary integral equation method. The corrugated scatterer is generated by superimposing a sinusoidal perturbation of arbitrary amplitude and period to a smooth elliptical shape. Extensive testing of the numerical results is presented. The displacement fields are evaluated along the half space and rough inclusion surfaces for a wide range of frequencies and corrugation shapes. The results clearly demonstrate the importance of the inclusion roughness on the ensuing motion. The surface response is found to strongly depend upon the frequency and polarization of the incident wave and the shape of the inclusion. For both models and low frequency incident waves the difference between the rough and smooth inclusion response is small. Moderate differences between the two responses are obtained at a higher frequency for vertical P and SV incidences or an oblique incident P wave. However, at the higher frequency for an oblique SV or a Rayleigh wave incidence, the half-space surface response is very sensitive upon the inclusion roughness especially for shorter corrugation periods. Similarly, for the full-space problem the inclusion roughness may produce significant effects in the resulting motion for both P and SV incident waves even at moderate frequencies.

Rough set approach to multiple criteria classification with imprecise evaluations and assignments

Dominance-based Rough Set Approach (DRSA) has been introduced to deal with multiple criteria classification (also called multiple criteria sorting, or ordinal classification with monotonicity constraints), where assignments of objects may be inconsistent with respect to dominance principle. In this paper, we consider an extension of DRSA to the context of imprecise evaluations of objects on condition criteria and imprecise assignments of objects to decision classes. The imprecisions are given in the form of intervals of possible values. In order to solve the problem, we reformulate the dominance principle and introduce second-order rough approximations. The presented methodology preserves well-known properties of rough approximations, such as rough inclusion, complementarity, identity of boundaries and precisiation. Moreover, the meaning of the precisiation property is extended to the considered case. The paper presents also a way to reduce decision tables and to induce decision rules from rough approximations.

On Knowledge Granulation and Applications to Classifier Induction in the Framework of Rough Mereology

Knowledge granulation as proposed by Zadeh consists in making objects under discussion into classes called granules; objects within a granule are similar one to another to a satisfactory degree relative to a chosen similarity measure. Rough mereology as developed by Polkowski in a series of works is especially suited to tasks of granulation as it does propose a systematic way to construct similarity measures in data sets and offers as well theoretic tools for granule formation in the form of an adaptation of the idea of mereological classes defined by Lesniewski in his mereology theory. In this article, which extends our contributions to the Special Session on Rough Mereology organized by Polkowski and Artiemjew as a part of the Conference on Rough Sets and Knowledge Technology RSKT 2008, we give a fairly detailed account of basic ideas of rough mereology, a description of basic similarity measures called rough inclusions along with the idea of granulated data sets (granular reflections of data sets); the...

Scattering of a plane harmonic SH wave by a completely embedded corrugated scatterer

AbstractAnti‐plane‐strain model for steady‐state scattering of elastic waves by a rough inclusion or a cavity embedded in a half space is considered by using a direct boundary integral equation method. The roughness of the scatterer is assumed to be periodic with arbitrary amplitude and period.Detailed testing of the numerical results is presented. The motion along the half‐space surface is evaluated for different corrugations, frequencies and impedance contrast of the materials. The importance of the scatterer roughness upon the displacement field is clearly demonstrated. It was shown that larger corrugation amplitudes, shorter corrugation periods and higher frequencies may produce significant change in the displacement field when compared with the corresponding smooth scatterer result. This effect strongly depends upon the impedance contrast of the materials. Copyright © 2008 John Wiley & Sons, Ltd.

Rough function model and rough membership function

Two pairs of approximation operators, which are the scale lower and upper approximations as well as the real line lower and upper approximations, are defined. Their properties and antithesis characteristics are analyzed. The rough function model is generalized based on rough set theory, and the scheme of rough function theory is made more distinct and complete. Therefore, the transformation of the real function analysis from real line to scale is achieved. A series of basic concepts in rough function model including rough numbers, rough intervals, and rough membership functions are defined in the new scheme of the rough function model. Operating properties of rough intervals similar to rough sets are obtained. The relationship of rough inclusion and rough equality of rough intervals is defined by two kinds of tools, known as the lower (upper) approximation operator in real numbers domain and rough membership functions. Their relative properties are analyzed and proved strictly, which provides necessary theoretical foundation and technical support for the further discussion of properties and practical application of the rough function model.

Interval rough mereology and description logic: An approach to formal treatment of imprecision in the Semantic Web ontologies

Learning and maintaining ontologies play critically important roles in the Semantic Web. Ontologies provide formal and explicit semantics to available knowledge and make possible shared understanding of a domain. A great deal of knowledge in the Semantic Web involves different sorts of imprecision. Common examples of imprecise knowledge include various statistical hypotheses, vagueness and ambiguities of natural language and many others. Consequently, ontologies must be expressive enough to handle imprecision. This paper presents an approach to formal handling of granularity as a special kind of imprecision in ontologies. It is based on Soft Computing techniques, in particular, Rough Set and Fuzzy Set theories. The approach also makes heavy use of Rough Mereology - a recently proposed paradigm for approximate reasoning - to compute degrees of imprecision of ontological concepts, relationships and assertions. It is demonstrated how the proposed mereological approach can be used together with the framework for representing uncertainty in Description Logic - family of popular logics for specifying formal ontologies. Finally, the paper introduces interval computations to the area of representing and reasoning with ontologies and shows how intervals can formally capture the imprecision incurred by computing rough inclusion (mereological) functions.

Probabilistic rough set approximations

Probabilistic approaches have been applied to the theory of rough set in several forms, including decision-theoretic analysis, variable precision analysis, and information-theoretic analysis. Based on rough membership functions and rough inclusion functions, we revisit probabilistic rough set approximation operators and present a critical review of existing studies. Intuitively, they are defined based on a pair of thresholds representing the desired levels of precision. Formally, the Bayesian decision-theoretic analysis is adopted to provide a systematic method for determining the precision parameters by using more familiar notions of costs and risks. Results from existing studies are reviewed, synthesized and critically analyzed, and new results on the decision-theoretic rough set model are reported.

Rough-fuzzy-neurocomputing based on rough mereological calculus of granules1

In this paper, we propose a scheme for merging rough, fuzzy, and cognitive (neural) computation methods. The underlying paradigm is that of rough mereology. Rough inclusions that are logical connectives of rough mereology allow for constructing granu

Special issue on Hybrid Intelligence using rough sets

The problem of imperfect knowledge under uncertain environments has been tackled for a long time by philosophers, logicians and mathematicians. Rough set theory proposed by Zdzislaw Pawlak [1] has attracted attention of many researchers and practitioners all over the world, and has a fast growing group of researchers interested in this methodology. Fuzzy set theory proposed by Lotfi Zadeh [2] helps to understand and manipulate imperfect knowledge. Fuzzy sets are defined by partial membership, in contrast to crisp membership used in classical definition of a set. Rough set theory, expresses vagueness, not by means of membership, but employing a boundary region of a set. The back bone of rough set theory is the approximation space and lower and upper approximations of a set. The approximation space is a classification of the domain of interest into disjoint categories. The lower approximation is a description of the domain objects which are known with certainty to belong to the subset of interest, whereas the upper approximation is a description of the objects which possibly belong to the subset. Any subset defined through its lower and upper approximations is called a rough set. The main advantage of rough set theory is that it does not need any preliminary or additional information about data – like probability in statistics, grade of membership in fuzzy set and so on. Readers may consult the International Rough Set Society Web page [3] for more online resources, publications etc. The Third International Conference on Hybrid Intelligent Systems (HIS’03) gathered individual researchers who see the need for synergy between various intelligent techniques. This special issue comprising of four papers is focused on hybrid intelligence using rough set theory and its applications. Papers were selected on the basis of fundamental ideas/concepts rather than the thoroughness of techniques deployed. The papers are organized as follows. In the first paper, Polkowski proposes a framework for hybridizing rough, fuzzy and neural computational models. The concept is based on rough mereology and by having rough inclusions that are logical connectives of rough mereology it is possible to construct granules of knowledge that constitute elementary objects for calculi merging rough, fuzzy and neurocomputing schemes. The idea of rough inclusions provide a bridge between rough and fuzzy theories linking them across the gap resulting from distinct approaches to model uncertainty of knowledge.

Thermal Conductivity of a Three-Dimensional Body with a Rough Inclusion

Thermal conductivity of a three‐dimensional body with a rough inclusion is studied. The problem is reduced to the problem of conjugation of harmonic functions and its exact solution is constructed. Distributions of temperature and heat‐flux density near distinctions of the interface between media are found. Results are illustrated by a numerical example.

Preliminary analysis of polyhydroxyalkanoate inclusions using atomic force microscopy

Atomic force microscopy analysis of polyhydroxyalkanoate (PHA) inclusions isolated from sonicated Ralstonia eutropha cells revealed that they exhibit two types of surface structure and shape; rough and ovoid, or smooth and spherical. Smooth inclusions possessed linear surface structures that were in parallel arrays with 7-nm spacing. Occasionally, cracks or fissures could be seen on the surface of the rough inclusions, which allowed a measurement of approximately 4 nm for the thickness of the boundary layer. When the rough inclusions were imaged at higher resolution, globular structures, 35 nm in diameter, having a central pore could be seen. These globular structures were connected by a network of 4-nm-wide linear structures. When the inclusions were treated with sodium lauryl sulfate, the boundary layer of the inclusion deteriorated in a manner that would be consistent with a lipid envelope. When the boundary layer was largely gone, 35-nm globular disks could be imaged laying on the surface of the filter beside the inclusions. These data have facilitated the development of a preliminary model for PHA inclusion structure that is more advanced than previous models.

Rough Mereology in Information Systems with Applications to Qualitative Spatial Reasoning

Rough Mereology has been proposed as a paradigm for approximate reasoning in complex information systems. Its primitive notion is that of a predicate of rough inclusion which gives for any two entities of discourse the degree in which one of them is a part of the other. Rough Mereology may be regarded as an extension of Rough Set Theory as it proposes to argue in terms of similarity relations induced from a rough inclusion instead of reasoning in terms of more strict indiscernibility relations. Rough Mereology is also a generalization of Mereology i.e. a theory of reasoning based on the notion of a part. Classical languages of mathematics are of two-fold kind: the language of set theory (naive or formal) expressing classes of objects as sets consisting of ”elements”, ”points” etc. suitable for objects perceived as built of ”atoms” and applied to structures perceived as discrete and the language of part relations suitable for e.g. continuous objects like solids, regions, etc. where two objects are related to each other by saying that one of them is a part of the other. Mereological theories for reasoning about complex structures are at the heart of Qualitative Spatial Reasoning. In this paper, we study basic aspects of Rough Mereology in Information Systems. Mereology makes the distinction between entities perceived as individuals (singletons), to which the part predicate may be applied, and entities perceived as distributive classes (sets, lists, general names etc.) of entities. This distinction is made formal and precise within Ontology i.e. Theory of Being based on the primitive notion of the copula is which is also a basic ingredient of theories for Spatial Reasoning. The practical aim of Ontology is to elaborate a system of concepts (notions, names, sets of entities) about which the reasoning is carried out. Therefore, we begin our study with an analysis of a simple rough set-based Ontology (the template ontology) in Information Systems and in this setting we present our approach to Mereology in Information Systems. In this framework we introduce Rough Mereology and we present some ways for defining rough inclusions. We demonstrate applications of Rough Mereology to approximate reasoning taking as the case subject Qualitative Spatial Reasoning. We address here some of its mereo-topological as well as mereo-geometrical aspects.

Etude du champ de déplacements d'un milieu granulaire au voisinage d'une inclusion

ABSTRACT Scale effects occur in modeling of models of cylindrical inclusions in sand. An Apparatus of Interface Visualisation has been achieved to study this phenomena. This test allows to visualise the shear mecanisms involved during an inclusion pull-out in granular medium. The results have been obtained with two kinds of sand, Hostun and Fontainebleau, and a 12 mm diameter rough inclusion. The visualisation of the sample allows to determine the grains displacements and volume variations in the shear zone.

Optimizations of Rough Set Model

Rough set methodology is based on concept (set) approximations constructed from available background knowledge represented in information systems [14]. In many applications only partial knowledge about approximated concepts is given. Hence quite often first a parametrized family of concept approximations is built and next, by parameters tuning the best, in a sense, approximation is chosen (see e.g. the variable precision rough set model [40]). In this paper we follow this approach in generalized approximation spaces. We discuss rough set model based on approximation spaces with uncertainty functions and rough inclusions. Elements of approximation space are parametrized, moreover for the proper application of such model to a particular data set it is necessary to make optimization of the parameters. We discuss not only basic properties of the mentioned model, but strategies of parameters optimization as well. We also present different notions of rough relations. Optimization of different parameters can be based on the degree of inclusion of relations defined by condition and decision attributes. Some illustration of presented methods on real life medical data set is also included.