Granular Rules Research Articles

Single sample-oriented attribute reduction for rule learning with formal concept analysis

As an effective tool for data mining, formal concept analysis can yield interpretable decision rules using attribute reduction. Currently, existing reduction methods within the framework of formal concept analysis result in all samples sharing the same conditional attributes, thereby overlooking the distinctions between rules extracted from different samples. To address this problem, we present a novel attribute reduction method for single samples. For this purpose, a single sample-oriented reduction framework is established by incorporating a discernibility attribute vector and an algorithm for calculating reducts based on the matrix representation of the discernibility attribute vector. Furthermore, the process of obtaining a new reduct from the original reduct is discussed by considering incremental learning. Given that attributes included in reducts are crucial conditions for decision-making, intent-minimal granular rules are generated through attribute reduction, and a reduction-based method for measuring the completeness of the rule set is discussed. Finally, to accomplish the classification task, a minimal rule-based classification model (MRCM) is proposed. The experimental results show that the MRCM has the same classification ability as the unreduced rules, and the rules with higher completeness tend to exhibit better classification performance.

Fuzzy three-way rule learning and its classification methods

Rules play a crucial role in classification tasks, driving the advancement of artificial intelligence. However, how to improve the interpretability of extracted rules while ensuring the performance of classification tasks is always a challenge, owing to the diversity of data types. Since three-way decision rules derive and explain from positive and negative aspects and provide more detailed information than general rules, this article explores fuzzy three-way rule learning from the perspective of two-way granular reduct by taking the FCA-based granular computing method as a framework. Specifically, we first present the object-induced fuzzy three-way granular rules and the object-induced two-way fuzzy three-way rules. Then, the fuzzy three-way rule-based dynamic updating method (FTRDUM) and the weight-based voting method are proposed to improve the classification performance. Finally, to illustrate the effectiveness of FTRDUM, some numerical experiments are conducted. The results show the superiority of the proposed algorithm in classification accuracy.

Extension of DBSCAN in Online Clustering: An Approach Based on Three-Layer Granular Models

In big data analysis, conventional clustering algorithms have limitations to deal with nonlinear spatial datasets, e.g., low accuracy and high computation cost. Aiming at these problems, this paper proposed a new DBSCAN extension algorithm for online clustering, which consists of three layers, considering DBSCAN, granular computing (GrC), and fuzzy rule-based modeling. Firstly, making use of DBSCAN algorithms’ advantages at extracting structural information, spatial data are clustered via DBSCAN into structural clusters, which are subsequently described by structural information granules (IG) via GrC. Secondly, based on the structural IGs, a series of granular models are constructed in the medium space, and utilized to form fuzzy rules to guide clustering on spatial data. Finally, with the help of structural IGs and granular rules, a rule-based modeling method is constructed in the output space for online clustering. Experiments on a synthetic toy dataset and a typical spatial dataset are implemented in this paper. Numerical results validate the feasibility to the proposed method in online spatial data clustering. Moreover, comparative studies with conventional methods and existing DBSCAN variants demonstrate the superiorities of the proposed method, as well as accuracy improvement and computation overhead reduction.

Fuzzy Rule-Based Classification Method for Incremental Rule Learning

Granularrules have been extensively used for classification in fuzzy datasets to promote the advancement of artificial intelligence. However, due to the diversity of data types, how to improve the readability of the extracted granular rules while ensuring efficiency is always a challenge. Since granular reduct in granular computing (GrC) can simplify real complex problem and dataset, this article carries out granular rule learning from the perspective of granular reduct by taking formal concept analysis (FCA)-based GrC method as a framework. Specifically, for achieving classification task, we first propose a method to update the granular reduct, and then explore the updating mechanism of fuzzy granular rule in a reduced dataset. Second, a novel fuzzy rule-based classification model named FRCM is presented for fuzzy granular rule learning. In order to verify the effectiveness of the proposed model, some numerical experiments for incremental learning and fuzzy rule mining are conducted to demonstrate that FRCM can achieve the state-of-the-art classification performance.

A Hierarchical Approach to Interpretability of TS Rule-Based Models

Interpretability of fuzzy rule-based models has always been of significant interest to the research community and the research in this area led to a number of far-reaching results. In this study, we briefly revisit the methodology and concepts of interpretability of Takagi–Sugeno (T-S) rule-based models and develop a conceptual framework involving several levels at which rules are interpreted. The layers at which interpretability is positioned are structured hierarchically by starting with the initial fuzzy set level (originating from the design of the rules), moving to information granules of finite support (where interval calculus is engaged) and finally ending up with symbols built at the higher level. As T-S rule-based models are endowed with local functions forming the conclusion parts of the rules, with the use of the principle of justifiable granularity, we develop a way of forming an interpretable conclusion in the form of information granule. To facilitate interpretability of conditions of the rules, multidimensional fuzzy sets (coming as a result of clustering) are decomposed into a Cartesian product of 1-D fuzzy sets and the quality of the resulting decomposition is evaluated. The quality of granular rules is assessed by analyzing the relationship between specificity of condition and conclusion information granules. The rules emerging at the level of symbols are further interpreted by engaging linguistic approximation, which helps approximate a collection of linguistic terms of subconditions producing a linguistic summarization in the form <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</i> (inputs are <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> ) consisting of a certain linguistic quantifier <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</i> . The performance of summarization is provided in the form of ranking of the relevance of the rules. Experimental studies using publicly available data are completed and analyzed.

VM consolidation improvement approach using heuristics granular rules in cloud computing environment

Since cloud computing has been developing as a successful business and growing interest of organizations to make use of the environment, the volume of provided services is highly demanding. Cloud computing environment has high maintenance costs, therefore, the resources are limited; due to rapid progress and the highly demanded volume of utilization of resources, ascertaining an appropriate approach for controlling and managing resources has become essential in this area. Therefore, an algorithm must be designed to perform better in all QoS. A significant factor in current exploratory methods is that each method owns distinctive features and perhaps does not estimate all QoSes. For example, one specific method has low energy consumption but does not consider SLA violation. Concerning the aims of the present study, the granular model is designed which uses various criteria involved in computations; in practice, by entering the values given in the past and discovering the relationships among them, the most desirable classification is produced. The membership function and the inference rule are derived from the data provided by the actual workload. Experimental analysis of the results showed that this method works out better than all other related methods and avails consistent performance in all QoS criteria.

Granule-Based-Classifier (GbC): A Lattice Computing Scheme Applied on Tree Data Structures

Social robots keep proliferating. A critical challenge remains their sensible interaction with humans, especially in real world applications. Hence, computing with real world semantics is instrumental. Recently, the Lattice Computing (LC) paradigm has been proposed with a capacity to compute with semantics represented by partial order in a mathematical lattice data domain. In the aforementioned context, this work proposes a parametric LC classifier, namely a Granule-based-Classifier (GbC), applicable in a mathematical lattice (T,⊑) of tree data structures, each of which represents a human face. A tree data structure here emerges from 68 facial landmarks (points) computed in a data preprocessing step by the OpenFace software. The proposed (tree) representation retains human anonymity during data processing. Extensive computational experiments regarding three different pattern recognition problems, namely (1) head orientation, (2) facial expressions, and (3) human face recognition, demonstrate GbC capacities, including good classification results, and a common human face representation in different pattern recognition problems, as well as data induced granular rules in (T,⊑) that allow for (a) explainable decision-making, (b) tunable generalization enabled also by formal logic/reasoning techniques, and (c) an inherent capacity for modular data fusion extensions. The potential of the proposed techniques is discussed.

A dynamic rule-based classification model via granular computing

As an effective tool for data representation and processing, granular computing has been incorporated into formal decision contexts for finding granular reducts to achieve the task of mining granular rules. However, the classification performance of granular rules has not been evaluated, and this type of method is not suitable for dynamic data. To solve this problem, the current study updates granular reducts and evaluates the obtained granular rules in terms of classification performance. Concretely, we first give a theoretical analysis of updating granular reducts and granular rules and then present a novel dynamic rule-based classification model (DRCM) based on the updating mechanism. Finally, we discuss the feasibility of the proposed model and compare it with several popular classification algorithms. The conducted experiments demonstrate that the granular reducts can improve the classification ability to a certain extent and that DRCM can achieve better classification performance on some consistent datasets.

A comparative study of decision implication, concept rule and granular rule

Decision implication is a basic form of knowledge representation of formal concept analysis in the setting of decision-making. Concept rules are decision implications that reveal the dependencies between condition concepts and decision concepts. Granular rules are concept rules that reveal the dependencies between condition object concepts and decision object concepts. This paper conducts a comparative study of decision implication, concept rule and granular rule. First, we conclude that both concept rules and granular rules are not complete w.r.t. decision implications, and that granular rules are not complete w.r.t. concept rules, implying that there exists information loss when studying decision implications by using only concept rules or granular rules, or when studying concept rules by using only granular rules. Next, with the help of decision implication logic, we identify accurately the information loss in concept rules and granular rules, and explore the underlying reason behind the information loss in concept rules and granular rules. Finally, by using the obtained results, we revisit some work on concept rule and granular rule, make some insightful remarks on the non-redundancy of concept rules and clarify some seemingly misleading statements on the representation of concept rules by granular rules.

Neighborhood Granule Classifiers

Classifiers are divided into linear and nonlinear classifiers. The linear classifiers are built on a basis of some hyper planes. The nonlinear classifiers are mainly neural networks. In this paper, we propose a novel neighborhood granule classifier based on a concept of granular structure and neighborhood granules of datasets. By introducing a neighborhood rough set model, the condition features and decision features of classification systems are respectively granulated to form some condition neighborhood granules and decision neighborhood granules. These neighborhood granules are sets; thus, their calculations are intersection and union operations of sets. A condition neighborhood granule and a decision neighborhood granule form a granular rule, and the collection of granular rules constitutes a granular rule library. Furthermore, we propose two kinds of distance and similarity metrics to measure granules, which are used for the searching and matching of granules. Thus, we design a granule classifier by the similarity metric. Finally, we use the granule classifier proposed in this paper for a classification test with UCI datasets. The theoretical analysis and experiments show that the proposed granule classifier achieves a better classification performance under an appropriate neighborhood granulation parameter.

A fuzzy rough set-based feature selection method using representative instances

The fuzzy rough set theory has been widely used to deal with uncertainty in real-valued or even complex data, in which one of the most concerned issues is feature selection. Since a real-world data set generally contains redundant data objects (or instances) and errors which lead to the fact that not all the instances are of equal importance, focusing on the representative instances for feature selection can not only acquire more convincing analysis results but also alleviate computational complexity in mining knowledge. At present, however, little attention has been paid on using representative instances to select features. In this paper, the issue of selecting features by using representative instances is investigated based on fuzzy rough sets and a representative instance-based feature selection approach is proposed. First, the fuzzy granular rule is employed to describe the discriminating information of an instance. Then, the representative instances are selected according to the coverage ability of the fuzzy granular rules induced by all of the instances. Furthermore, an implication relationship-preserved reduction is presented to maintain the discriminating information of the selected instances, and then a heuristic algorithm is presented to search for such a feature subset. Finally, a filter-wrapper approach is suggested to select the best subset of the features. Some numerical experiments are further conducted to show the performance of the proposed feature selection method and the results are satisfactory in terms of both efficiency and effectiveness.

An intensive study on rule acquisition in formal decision contexts based on minimal closed label concept lattices

AbstractRule acquisition is one of the main purposes in the analysis of formal decision contexts. Up to now, there have been several types of rules in formal decision contexts. However, these existing rules were investigated independently and are, to some extent, unsatisfactory for making better decision analysis since some of them (e.g. decision implications) are computationally expensive and others (e.g. decision rules and granular rules) are short of compactness. Motivated by these problems, minimal closed label concept lattice is defined to present limitary decision implications, which not only are easier to be extracted than decision implications, but also have more concise premises than decision rules and granular rules. Moreover, we discuss the pairwise inclusion and inference relationships among limitary decision implications, decision implications, decision rules and granular rules. Finally, some numerical experiments are conducted to demonstrate that the proposed limitary decision implications a...

A comparative study of multigranulation rough sets and concept lattices via rule acquisition

Recently, by combining rough set theory with granular computing, pessimistic and optimistic multigranulation rough sets have been proposed to derive “AND” and “OR” decision rules from decision systems. At the same time, by integrating granular computing and formal concept analysis, Wille’s concept lattice and object-oriented concept lattice were used to obtain granular rules and disjunctive rules from formal decision contexts. So, the problem of rule acquisition can bring rough set theory, granular computing and formal concept analysis together. In this study, to shed some light on the comparison and combination of rough set theory, granular computing and formal concept analysis, we investigate the relationship between multigranulation rough sets and concept lattices via rule acquisition. Some interesting results are obtained in this paper: (1) “AND” decision rules in pessimistic multigranulation rough sets are proved to be granular rules in concept lattices, but the inverse may not be true; (2) the combination of the truth parts of an “OR” decision rule in optimistic multigranulation rough sets is an item of the decomposition of a disjunctive rule in concept lattices; (3) a non-redundant disjunctive rule in concept lattices is shown to be the multi-combination of the truth parts of “OR” decision rules in optimistic multigranulation rough sets; and (4) the same rule is defined with a same certainty factor but a different support factor in multigranulation rough sets and concept lattices. Moreover, algorithm complexity analysis is made for the acquisition of “AND” decision rules, “OR” decision rules, granular rules and disjunctive rules.

Multi-confidence rule acquisition and confidence-preserved attribute reduction in interval-valued decision systems

Rule acquisition is one of the most important objectives in the analysis of decision systems. Because of the interference of errors, a real-world decision system is generally inconsistent, which can lead to the consequence that some rules extracted from the system are not certain but possible rules. In practice, however, the possible rules with high confidence are also useful in making decision. With this consideration, we study how to extract from an interval-valued decision system the compact decision rules whose confidences are not less than a pre-specified threshold. Specifically, by properly defining a binary relation on an interval-valued information system, the concept of interval-valued granular rules is presented for the interval-valued decision system. Then, an index is introduced to measure the confidence of an interval-valued granular rule and an implication relationship is defined between the interval-valued granular rules whose confidences are not less than the threshold. Based on the implication relationship, a confidence-preserved attribute reduction approach is proposed to extract compact decision rules and a combinatorial optimization-based algorithm is developed to compute all the reducts of an interval-valued decision system. Finally, some numerical experiments are conducted to evaluate the performance of the reduction approach and the gain of using the possible rules in making decision.

On rule acquisition in decision formal contexts

Rule acquisition is one of the main purposes in the analysis of decision formal contexts. Up to now, there have existed several types of rules (e.g., the decision rules and the granular rules) in decision formal contexts. This study firstly proposes a new algorithm with less time complexity for deriving the non-redundant decision rules from a decision formal context. Then, we invesigate decision rules and the granular rules in the consistent decision formal contexts and make a contrast between the decision rule oriented knowledge reduction and the granular rule oriented knowledge reduction. Finally, some experiments are conducted to assess the efficiency of the proposed rule acquisition algorithm.

Parametric Rough Sets with Application to Granular Association Rule Mining

Granular association rules reveal patterns hidden in many-to-many relationships which are common in relational databases. In recommender systems, these rules are appropriate for cold-start recommendation, where a customer or a product has just entered the system. An example of such rules might be “40% men like at least 30% kinds of alcohol; 45% customers are men and 6% products are alcohol.” Mining such rules is a challenging problem due to pattern explosion. In this paper, we build a new type of parametric rough sets on two universes and propose an efficient rule mining algorithm based on the new model. Specifically, the model is deliberately defined such that the parameter corresponds to one threshold of rules. The algorithm benefits from the lower approximation operator in the new model. Experiments on two real-world data sets show that the new algorithm is significantly faster than an existing algorithm, and the performance of recommender systems is stable.

Granular Neural Networks With Evolutionary Interval Learning

To deal with different membership functions of the same linguistic term, a new interval reasoning method using new granular sets is proposed based on Yin Yang methodology. To make interval-valued granular reasoning efficiently and optimize interval membership functions based on training data effectively, a granular neural network (GNN) with a new high-speed evolutionary interval learning is designed. Simulation results in nonlinear function approximation and bioinformatics have shown that the GNN with the evolutionary interval learning is able to extract interval-valued granular rules effectively and efficiently from training data by using the new evolutionary interval learning algorithm.

Constructive granular systems with universal approximation and fast knowledge discovery

Conventional gradient descent learning algorithms for soft computing systems have the learning speed bottleneck problem and the local minima problem. To effectively solve the two problems, the n-variable constructive granular system with high-speed granular constructive learning is proposed based on granular computing and soft computing, and proved to be a universal approximator. The fast granular constructive learning algorithm can highly speed up granular knowledge discovery by directly calculating all parameters of the n-variable constructive granular system using training data, and then construct the n-variable constructive granular system with any required accuracy using a small number of granular rules. Predictive granular knowledge discovery simulation results indicate that the direct-calculation-based granular constructive algorithm is better than the conventional gradient descent learning algorithm in terms of learning speed, learning error, and prediction error.

Designing Enterprise Client/Server Systems

B. Gold-Bernstein and D. Marca Prentice Hall, Upper Saddle River, NJ, 1998, 360 pp. ISBN 0-13-890195-3, $45.00 Client/servers systems represent the vast majority of enterprise software. Yet over 80% of these projects fail. This book presents a concrete design method that provides for reliable design and implementation of multi-tier client/server systems. This book is organized into five basic parts: Context for Understanding Distributed Client/Server Systems Business Components—The Essential Model Software Components—The Implementation Model Organizational Enablers Case Study In Part 1, Chapter 1 answers the question: Why build client/server systems? by giving a general context roadmap, discussing why businesses are under pressure to change, how legacy systems inhibit change, why client/server systems enable change. What is new and not new in client/server systems, and the importance of good design and modeling are also presented. Chapter 2 discusses all aspects of models and modeling in general, including a definition of the basic structure of a model. Chapter 3 presents the Illumisys Design Method, a specific event- and object-based method for developing distributed client/server systems. In Part 2, the first stage of the Illumisys Design Method, the business requirements, are covered. Chapter 4 provides an overview on a rigorous and formal model for business requirements, termed the Essential Model. Chapter 5 discusses the Environmental Aspect—how a system interacts with the outside world and how boundaries are defined. The organization of the rules which drive the business are encompassed in the Information Aspect and are discussed in Chapter 6. This chapter is careful to note the differences between the Information Aspect and a Database Model. Chapter 7 describes the Bridge Aspect, which is a consistency component that insures that the Environmental Aspect and the Information Aspect are describing a single system. The Behavioral Aspect, as described in Chapter 8, brings together further all previously defined aspects and provides detailed descriptions of responses to specific events. Chapter 9 discusses the Event Distribution Aspect, which describes the geographical distribution of the business events—where business events actually occur. However, there are additional factors that can affect system partitioning, such as performance goals and processing volumes. Chapter 10 describes the Component Measurement Aspect, which encapsulates the natural rhythms and cycles of the business. In Part 3, the design of the distributed architecture implementation is described. Chapter 11 presents an overview of the Implementation Model, which maps the business events and responses defined in the Essential Model to the distributed architecture. A Statement of Design Metrics, which determines design tradeoffs and is used to partition the essential requirements across the distributed processor architecture, is discussed in Chapter 12. The Multi-Tier Partitioning Aspect is described in Chapter 13. Multi-tier partitioning enables: data encapsulation, granular business rules, component application architecture, and scalability. Selecting the appropriate technology, such as processors and tools, is a critical aspect of distributed architecture design. Chapter 14 defines five steps of the Technical Architecture Design that is appropriate for enterprise architecture planning. The Processor Distribution Aspect, as described in Chapter 15, provides the specifications of the software granules that will operate on each processor, and these specifications are used to build the software components. Having defined the processing and data distribution design, Chapter 16 describes the Geographical Distribution Aspect, which maps processors to geographical location. Chapter 17 defines Traffic Analysis, which measures the message traffic between processors as well as the contention on individual processors. In Part 4, an overall plan to enable change to the system is described. Organizational enablers are used and include: organizational architecture, process management, and organizational transformation. Chapter 18 describes an Organizational Architecture that easily adapts to changes in the business environment. Designing client/server systems requires a strategy for identifying and managing concurrent activities. Process Management, as described in Chapter 19, synchronizes all concurrent activities and ensures that the system delivers the required functionality. The last aspect of the Organizational enablers deals with the transformation required by business. Chapter 20 deals with Organizational Transformation—a process required to enable business change. In Part 5, all of the models are brought together to show, from beginning to end, how the Illumisys Design Method is applied to a specific business situation. This is accomplished via a case study (Fleury Market). Chapter 21 provides necessary background for the case study. Chapters 22 and 23 contain, respectively, all components for the Essential Model and the Implementation Model for the case study. Overall, the book is well written and easy to understand. This book would be an important addition for anyone interested in designing and implementing a client/server system in a business environment. This book would also be appropriate for an advanced database or special topics course. James P. Buckley Department of Computer Science University of Dayton