Jaroslaw Stepaniuk

Tolerance Approximation Spaces

We generalize the notion of an approximation space introduced in [8]. In tolerance approximation spaces we define the lower and upper set approximations. We investigate some attribute reduction problems for tolerance approximation spaces determined by tolerance information systems. The tolerance relation defined by the so called uncertainty function or the positive region of a given partition of objects have been chosen as invariants in the attribute reduction process. We obtain the solutions of the reduction problems by applying boolean reasoning [1]. The solutions are represented by tolerance reducts and relative tolerance reducts.

Information granules: Towards foundations of granular computing,

We introduce basic notions related to granular computing, namely the information granule syntax and semantics as well as the inclusion and closeness (similarity) relations of granules. Different information sources (units, agents) are equipped with two kinds of operations on information granules: operations transforming tuples of information granules definable by a given agent into information granules definable by this agent and approximation operations for computing by agents approximations of information granules delivered by other agents. More complex granules are constructed by means of these operations and approximation operations from some input information granules. The construction of information granules is described by expressions called terms. We discuss a problem of synthesis of robust terms, i.e., descriptions of information granules, satisfying a given specification. This is an important problem for granular computing and its applications for spatial reasoning or knowledge discovery and data mining. © 2001 John Wiley & Sons, Inc.

Modeling rough granular computing based on approximation spaces

The results reported in this paper create a step toward the rough set-based foundations of data mining and machine learning. The approach is based on calculi of approximation spaces. In this paper, we present the summarization and extension of our results obtained since 2003 when we started investigations on foundations of approximation of partially defined concepts (see, e.g., [2,3,7,37,20,21,5,42,39,38,40]). We discuss some important issues for modeling granular computations aimed at inducing compound granules relevant for solving problems such as approximation of complex concepts or selecting relevant actions (plans) for reaching target goals. The problems discussed in this article are crucial for building computer systems that assist researchers in scientific discoveries in many areas such as biology. In this paper, we present foundations for modeling of granular computations inside of system that is based on granules called approximation spaces. Our approach is based on the rough set approach introduced by Pawlak [24,25]. Approximation spaces are fundamental granules used in searching for relevant complex granules called as data models, e.g., approximations of complex concepts, functions or relations. In particular, we discuss some issues that are related to generalizations of the approximation space introduced in [33,34]. We present examples of rough set-based strategies for the extension of approximation spaces from samples of objects onto a whole universe of objects. This makes it possible to present foundations for inducing data models such as approximations of concepts or classifications analogous to the approaches for inducing different types of classifiers known in machine learning and data mining. Searching for relevant approximation spaces and data models are formulated as complex optimization problems. The proposed interactive, granular computing systems should be equipped with efficient heuristics that support searching for (semi-)optimal granules.

Knowledge discovery by application of rough set models

The amount of electronic data available is growing very fast and this explosive growth in databases has generated a need for new techniques and tools that can intelligently and automatically extract implicit, previously unknown, hidden and potentially useful information and knowledge from these data. These tools and techniques are the subject of the field of Knowledge Discovery in Databases. In this Chapter we discuss selected rough set based solutions to two main knowledge discovery problems, namely the description problem and the classification (prediction) problem.

Approximation Spaces, Reducts and Representatives

The main objective of this chapter is to discuss different approaches to searching for optimal approximation spaces. Basic notions concerning rough set concept based on generalized approximation spaced are presented. Different constructions of approximation spaces are described. The problems of attribute and object selection are discussed.

Granular Computing: a Rough Set Approach

We discuss information granule calculi as a basis of granular computing. They are defined by constructs like information granules, basic relations of inclusion and closeness between information granules as well as operations on them. The exact interpretation between granule languages of different information sources (agents) often does not exist. Hence (rough) inclusion and closeness of granules are considered instead of their equality. Examples of all the basic constructs of information granule calculi are presented. The construction of more complex information granules is described by expressions called terms. We discuss the synthesis problem of robust terms, i.e., descriptions of information granules, satisfying a given specification in a satisfactory degree. We also present a method for synthesis of information granules represented by robust terms (approximate schemes of reasoning) by means of decomposition of specifications for such granules. The discussed problems of granular computing are of special importance for many applications, in particular related to spatial reasoning as well as to knowledge discovery and data mining.

Information Granules and Rough-Neural Computing

In this chapter we discuss the foundations of rough-neural computing (RNC). We introduce information granule systems and information granules in such systems. Information granule networks, called approximate reasoning schemes (AR schemes), are used to represent information granule constructions. We discuss the foundations of RNC using an analogy of information granule networks with neural networks. RNC is a basic paradigm of granular computing (GC). This paradigm makes it possible to tune AR schemes to construct relevant information granules, e.g., satisfying a given specification to a satisfactory degree. One of the goals of our project is to develop methods based on rough-neural computing for computing with words (CW).

Information Granules in Distributed Environment

We propose to use complex information granules to extract patterns from data in distributed environment. These patterns can be treated as a generalization of association rules.

Adaptive multilevel rough entropy evolutionary thresholding

Abstract In this study, comprehensive research into rough set entropy-based thresholding image segmentation techniques has been performed producing new and robust algorithmic schemes. Segmentation is the low-level image transformation routine that partitions an input image into distinct disjoint and homogenous regions using thresholding algorithms most often applied in practical situations, especially when there is pressing need for algorithm implementation simplicity, high segmentation quality, and robustness. Combining entropy-based thresholding with rough set results in the rough entropy thresholding algorithm. The authors propose a new algorithm based on granular multilevel rough entropy evolutionary thresholding that operates on a multilevel domain. The MRET algorithm performance has been compared to the iterative RET algorithm and standard k-means clustering methods on the basis of β -index as a representative validation measure. Performance in experimental assessment suggests that granular multilevel rough entropy threshold based segmentations – MRET – present high quality, comparable with and often better than k-means clustering based segmentations. In this context, the rough entropy evolutionary thresholding MRET algorithm is suitable for specific segmentation tasks, when seeking solutions that incorporate spatial data features with particular characteristics.

Approximation Spaces in Rough-Granular Computing

We discuss some generalizations of the approximation space definition introduced in 1994 [24, 25]. These generalizations are motivated by real-life applications. Rough set based strategies for extension of such generalized approximation spaces from samples of objects onto their extensions are discussed. This enables us to present the uniform foundations for inducing approximations of different kinds of granules such as concepts, classifications, or functions. In particular, we emphasize the fundamental role of approximation spaces for inducing diverse kinds of classifiers used in machine learning or data mining.