Piotr Artiemjew

On classification of data by means of rough mereological granules of objects and rules

Granulation of knowledge has turned an effective tool in data classification. We propose the approach to classification of data which extends our earlier methods by considering granules of either objects or decision rules obtained either from the original training set or from its granular reflection. Members of a granule vote for the decision class of that object. We present results of tests which show that this method usually gives results at least as good as the exhaustive classifier built on rough set principles.

On Granular Rough Computing with Missing Values

Granular Computing as a paradigm in Approximate Reasoning is concerned with granulation of available knowledge into granules that consists of entities similar in information content with respect to a chosen measure and with computing on such granules. Thus, operators acting on entities in a considered universe should factor through granular structures giving values similar to values of same operators in non---granular environment. Within rough set theory, proposed 25 years ago by Zdzislaw Pawlak and developed thence by many authors, granulation is also a vital area of research. The first author developed a calculus with granules as well as a granulation technique based on similarity measures called rough inclusions along with a hypothesis that granules induced in data set universe of objects should lead to new objects representing them, and such granular counterparts should preserve information content in data. In this work, this hypothesis is tested with missing values in data and results confirm the hypothesis in this context.

On Classifying Mappings Induced by Granular Structures

In this work the subject of granular computing is pursued beyond the content of the previous paper [21]. We study here voting on a decision by granules of training objects, granules of decision rules, granules of granular reflections of training data, and granules of decision rules induced from granular reflections of training data. This approach can be perceived as a direct mapping of the training data on test ones which is induced by granulation of knowledge on the training data. Some encouraging results were already presented in [21], and here the subject is pursued systematically. Granules of knowledge are defined and computed according to a previously used scheme due to Polkowski in the framework of theory of rough inclusions. On the basis of presented results, one is justified in concluding that the presented methods offer a very good quality of classification, comparable fully with best results obtained by other rough set based methods, like templates, adaptive methods, hybrid methods etc.

Granular Computing: Granular Classifiers and Missing Values

Granular computing is a paradigm destined to study how to compute with granules of knowledge that are collective objects formed from individual objects by means of a similarity measure. The idea of granulation was put forth by Lotfl Zadeh: granulation is inculcated in fuzzy set theory by the very definition of a fuzzy set and inverse values of fuzzy membership functions are elementary forms of granules. Granulation is an essential ingredient of humane thinking and it is playing a vital role in cognitive processes which are studied in cognitive informatics as emulations by computing machines of real cognitive processes in humane thinking. Rough inclusions establish a form of similarity relations that are reflexive but not necessarily symmetric; in applications presented in this work, we restrict ourselves to symmetric rough inclusions based on the set DIS(u,v) = {a epsi A : a(u) ne a(v)} of attributes discerning between given objects u, v without any additional parameters. Our rough inclusions are induced in their basic forms in a unified framework of continuous t-norms; in this work we apply the rough inclusion muL induced from the Lukasiewicz t-norm L(x, y) = max{0, x + y - 1} by means g(|DIS(u,v)|/|A|) = |IND(u,v)|/|A| where g is the function that occurs in the functional representation of L and IND(u,v) = U xU\ DIS(u, v). Granules of knowledge induced by rough inclusions are formed as neighborhoods of given radii of objects by means of the class operator of mereology. L.Polkowski in his feature talks at conferences 2005, 2006 IEEE GrC, put forth the hypothesis that similarity of objects in a granule should lead to closeness of sufficiently many attribute values on objects in the granule and thus averaging in a sense values of attributes on objects in a granule should lead to a new data set, the granular one, which should preserve information encoded in the original data set to a satisfactory degree. This hypothesis is borne out in this work with tests on real data sets. We also address the problem of missing values in data sets; this problem has been addressed within rough set theory by many authors, e.g., Grzymala-Busse, Kryszkiewicz, Rybinski. We propose a novel approach to this problem: an object with missing values is absorbed in a granule and takes part in determining a granular object; then, at classification stage, objects with missing values are matched against closest granular objects. We present details of this approach along with tests on real data sets.

On Granular Rough Computing: Factoring Classifiers Through Granulated Decision Systems

The paradigm of Granular Computing has quite recently emerged as an area of research on its own; in particular, it is pursued within rough set theory initiated by Zdzislaw Pawlak. Granules of knowledge consist of entities with a similar in a sense information content. An idea of a granular counterpart to a decision/information system has been put forth, along with its consequence in the form of the hypothesis that various operators, aimed at dealing with information, should factorize sufficiently faithfully through granular structures [7], [8]. Most important such operators are algorithms for inducing classifiers. We show results of testing few well-known algorithms for classifier induction on well---used data sets from Irvine Repository in order to verify the hypothesis. The results confirm the hypothesis in case of selected representative algorithms and open a new prospective area of research.

A Study in Granular Computing: On Classifiers Induced from Granular Reflections of Data

Granular Computing as a paradigm in the area of ApproximateReasoning/Soft Computing, goes back to the idea of L. A. Zadeh(1979) of computing with collections of similar entities. Bothfuzzy and rough set theories are immanently occupied with granulesas atomic units of knowledge are inverse images of fuzzy membershipfunctions in the first and indiscernibility classes in the otherset theory. Research on granulation in the framework of rough set theory hasstarted soon after Zadehs program manifest (T.Y. Lin, L.Polkowski,Qing Liu, A.Skowron, J.Stepaniuk, Y.Y.Yao) with various tools fromgeneral theory of binary relations (T.Y.Lin, Y.Y.Yao), roughmereology (L.Polkowski, A.Skowron), approximation spaces (A.Skowron and J. Stepaniuk), logics for approximate reasoning(L.Polkowski, M. Semeniuk-Polkowska, Qing Liu). The program of granular computing requires that granules formedfrom entities described by data should enter computing process aselementary units of computation; this program has been pursued insome aspects of reasoning under uncertainty like fusion ofknowledge, rough---neural computing, many agent systems. In this work, granules of knowledge are exploited in tasks ofclassification of data. This research is a follow---up on theprogram initiated by the first author in plenary talks at IEEEInternational Conferences on Granular Computing in Beijing, 2005,and Atlanta, 2006. The idea of this program consists in granulatingdata and creating a granular data set (called the granularreflection of the original data set); due to expected in theprocess of granulation smoothing of data, eliminating of outliers,and averaging of attribute values, classification on the basis ofgranular data is expected to be of satisfactory quality, i.e.,granulation should preserve information encoded in data to asatisfactory degre. It should be stressed, however, that theproposed process of building a granular structure involves a fewrandom procedures (factoring attributes through a granule,selection of a granular covering of the universe of objects) whichmakes it difficult for a rigorous analysis. It is the aim of this work to verify the program of granularclassification on the basis of experiments with real data. Granules of knowledge are in this work defined and computed onlines proposed by Polkowski in teh framework of rough mereology: itdoes involve usage of similarity measures called rough inclusionsalong with techniques of mereological theory of concepts. Inconsequence, definitions of granules are invariant with respect tothe choice of the underlying similarity measure. Granules of knowledge enter the realm of classification problemsin this work from a three---fold perspective: first, granulateddata sets give rise to new data sets on which classifiers aretested and the results are compared to results obtained with thesame classifiers on the original data sets; next, granules oftraining objects as well as granules of rules obtained from thetraining set vote for value of decision at a test object; this isrepeated with granules of granular reflections of granules and withgranules of rules obtained from granulated data sets. Finally, thevoting is augmented with weights resulting from the distribution ofattribute values between the test object and training objects. In the first case, the rough inclusion based on Hamming’smetric is applied (or, equivalently, it is the rough inclusionproduced from the archimedean t–norm of Łukasiewicz); inthe last two cases, rough inclusions are produced on the basis ofresidual implications induced from continuous t–norms ofŁukasiewicz, the product t–norm, and the minimumt–norm, respectively. In all cases results of experiments on chosen real data sets,most often used as a test data for rough set methods, are verysatisfactory, and, in some cases, offer results better than manyother rough set based classification methods.

Granular Computing in Decision Approximation

This book presents a study in knowledge discovery in data with knowledge understood as a set of relations among objects and their properties. Relations in this case are implicative decision rules and the paradigm in which they are induced is that of computing with granules defined by rough inclusions, the latter introduced and studiedwithin rough mereology, the fuzzified version of mereology. In this book basic classes of rough inclusions are defined and based on them methods for inducing granular structures from data are highlighted. The resulting granular structures are subjected to classifying algorithms, notably knearest neighbors and bayesian classifiers.Experimental results are given in detail both in tabular and visualized form for fourteen data sets from UCI data repository. A striking feature of granular classifiers obtained by this approach is that preserving the accuracy of them on original data, they reducesubstantially the size of the granulated data set as well as the set of granular decision rules. This feature makes the presented approach attractive in cases where a small number of rules providing a high classification accuracy is desirable. As basic algorithms used throughout the text are explained and illustrated with hand examples, the book may also serve as a textbook.

Rough mereological classifiers obtained from weak variants of rough inclusions

Granular reflections of data sets have turned out to be very effective in data classification. In this work we present results of classification of real data sets by means of an approach in which granules of objects or decision rules are built on the basis of weak variants of rough inclusions.

A New Classifier Based on the Dual Indiscernibility Matrix

A new approach to classifier synthesis was proposed by Polkowski and in this work we propose an implementation of this idea. The idea is based on usage of a dual indiscernibility matrix which allows to determine for each test object in the data, pairs of training objects which cover in a sense the given test object. A family of pairs best covering the given object pass their decisions for majority voting on decision for the test object. We present results obtained by our classifier on standard data from UCI Repository and compare them with results obtained by means of k-NN and Bayes classifiers. The results are validated by multiple cross-validation. We find our classifier on par with k-NN and Bayes classifiers.

Natural versus Granular Computing: Classifiers from Granular Structures

In data sets/decision systems, written down as pairs(U,A∪ {d}) with objects U,attributes A, and a decision d, objects aredescribed in terms of attribute---value formulas. Thisrepresentation gives rise to a calculus in terms of descriptorswhich we call a natural computing. In some recent papers,the idea of L. Polkowski of computing with granules induced fromsimilarity measures called rough inclusions have been tested. Inthis work, we pursue this topic and we study granular structuresresulting from rough inclusions with classification problem infocus. Our results show that classifiers obtained from granularstructures give better quality of classification than naturalexhaustive classifiers.