Principle Of Justifiable Granularity Research Articles

Granular Description of Data Structures: A Two-Phase Design.

The study is concerned with a description of large numeric data with the aid of building a limited collection of representative information granules with the objective of capturing the structure of the original data. The proposed development scheme consists of two steps. First, a clustering algorithm characterized by high flexibility of coping with the diverse geometry of data structure and efficient computational overhead is invoked. At the second step, a clustering algorithm applied to the clusters already formed during the first phase, yielding a collection of numeric prototypes is involved and the numeric prototypes produced there are then generalized into their granular prototypes. The quality of granular prototypes is quantified while their build-up is supported by the mechanisms of granular computing such as the principle of justifiable granularity. In this paper, the clustering algorithms of DBSCAN and fuzzy C -means were used in successive phases of the processed approach. The experimental studies concerning synthetic data and publicly available data are covered and the performance of the developed approach is assessed along with a comparative analysis.

A Development of Granular Input Space in System Modeling.

In this paper, we elaborate on a new design approach to the development and analysis of granular input spaces and ensuing granular modeling. Given a numeric model (no matter what specific design methodology has been used to construct it and what architecture has been adopted), we form a granular input space through allocating a certain level of information granularity across the input variables. The formation of granular input space helps us gain a better insight into the ranking of input variables with respect to their precision (the variables with a lower level of information granularity need to be specified in a precise way when estimating the inputs). As a consequence, for granular inputs, the outputs of the granular model are also information granules (say, intervals, fuzzy sets, rough sets, etc.). It is shown that the process of forming granular input space can be sought as an optimization of allocation of information granularity across the input variables so that the specificity of the corresponding granular outputs of the granular model becomes the highest while coverage of data becomes maximized. The construction of granular input space dwells upon two fundamental principles of granular computing-the principle of justifiable granularity and the optimal allocation of information granularity. The quality of the granular input space is quantified in terms of the two conflicting criteria, that is, the specificity of the results produced by the granular model and the coverage of experimental data delivered by this model. In the ensuing optimization problem, one maximizes a product of specificity and coverage. Differential evolution is engaged in this optimization task. The experimental studies involve both synthetic dataset and data coming from the machine learning repository.

Top-Down Granulation Modeling Based on the Principle of Justifiable Granularity

Information granulation is an effective vehicle to explore data structure and has become a timely research topic. In this article, a top-down granulation model (T-GrM) adhering to the two fundamental requirements (coverage and specificity) in the principle of justifiable granularity is designed. Here, this principle is employed as a fundamental method for constructing information granules in each layer, and as an indicator to determine whether the obtained partitions need to be further divided in the next layer. For capturing the correlation among features and reflecting the data structure, an improved version of a T-GrM is offered by incorporating the principal component analysis. The proposed granulation models are evaluated on synthetic datasets, where the obtained experimental results offer some insights into the feasibility of the designed models and describe the effect of parameter values on the constructed information granules. The improved version of the T-GrM has the advantage of being transparent by generating information granules on several principal components, and thereby, the description of information granules is simpler and descriptive. As application examples, two real-world datasets are analyzed to exhibit the advantage of the constructed information granules.

A Two-Stage Approach for Constructing Type-2 Information Granules.

In this article, we are concerned with the formation of type-2 information granules in a two-stage approach. We present a comprehensive algorithmic framework which gives rise to information granules of a higher type (type-2, to be specific) such that the key structure of the local granular data, their topologies, and their diversities become fully reflected and quantified. In contrast to traditional collaborative clustering where local structures (information granules) are obtained by running algorithms on the local datasets and communicating findings across sites, we propose a way of characterizing granular data (formed) by forming a suite of higher type information granules to reveal an overall structure of a collection of locally available datasets. Information granules built at the lower level on a basis of local sources of data are weighted by the number of data they represent while the information granules formed at the higher level of hierarchy are more abstract and general, thus facilitating a formation of a hierarchical description of data realized at different levels of detail. The construction of information granules is completed by resorting to fuzzy clustering algorithms (more specifically, the well-known Fuzzy C-Means). In the formation of information granules, we follow the fundamental principle of granular computing, viz., the principle of justifiable granularity. Experimental studies concerning selected publicly available machine-learning datasets are reported.

Information Granules-Based BP Neural Network for Long-Term Prediction of Time Series

Long-term time series prediction is a challenging and essential task both in theory and practice. Recently, information granulation is shown to be an appropriate tool for the long-term forecast. Though some models for the long-term prediction problem have been proposed using information granulation recently, there is still a growing need to develop new prediction approaches for time series data based on information granule, which can capture the dynamic trend change with high accuracy. In this article, a long-term prediction approach, based on back-propagation neural network and information granule, is proposed. First, the individual numerical intervals for the time series are obtained by using the principle of justifiable granularity in information granule. Then, an automatic linear trend extraction method is developed to extract the trend change, which is inherited in granules. Finally, a hierarchy of neural network is constructed to carry out prediction by using information granule as input. Experiments using publicly available time series datasets demonstrate that the proposed approach can achieve better performance than the existing models for long-term prediction.

An Interval Type-2 Fuzzy Risk Analysis Model (IT2FRAM) for Determining Construction Project Contingency Reserve

Determining contingency reserve is critical to project risk management. Classic methods of determining contingency reserve significantly rely on historical data and fail to effectively incorporate certain types of uncertainties such as vagueness, ambiguity, and subjectivity. In this paper, an interval type-2 fuzzy risk analysis model (IT2FRAM) is introduced in order to determine the contingency reserve. In IT2FRAM, the membership functions for the linguistic terms used to describe the probability, impact of risk and the opportunity events are developed, optimized, and aggregated using interval type-2 fuzzy sets and the principle of justifiable granularity. IT2FRAM is an extension of a fuzzy arithmetic-based risk analysis method which considers such uncertainties and addresses the limitations of probabilistic and deterministic techniques of contingency determination methods. The contribution of IT2FRAM is that it considers the opinions of several subject matter experts to develop the membership functions of linguistic terms. Moreover, the effect of outlier opinions in developing the membership functions of linguistic terms are reduced. IT2FRAM also enables the aggregation of non-linear membership functions into trapezoidal membership functions. A hypothetical case study is presented in order to illustrate the application of IT2FRAM in Fuzzy Risk Analyzer© (FRA©), a risk analysis software.

Granular fuzzy pay-off method for real option valuation

In the last decade, the fuzzy pay-off method has emerged as a widely used alternative approach for real option valuation thanks to its simplicity that makes it easily approachable by practitioners from various application domains. In this study, a new direction for real option valuation is pursued by proposing a granular fuzzy pay-off method. We motivate the proposal by discussing how the extension of the original approach with granular representation can further improve the fuzzy pay-off method. The design of the granular fuzzy pay-off method is founded on the principle of justifiable granularity, which in turn relies on numeric data to build information granules that are semantically sound and experimentally justified. To illustrate the method, a case study in R&D investment in manufacturing is worked out. The extended granular fuzzy pay-off method improves performance and usability in cases with uncertainty.

Uncertain Data Modeling Based on Evolving Ellipsoidal Fuzzy Information Granules

Dealing with uncertain data requires effective methods to properly describe their real meaning in terms of a tradeoff between interpretability and generality on the process of knowledge formation based on data abstraction. This article proposes an online granulation process based on evolving ellipsoidal fuzzy information granules (EEFIG) and the principle of justifiable granularity (PJG) for data streams parameterization. The granulation process consists in the information granule development taking into consideration the data stream with a simplified optimal granularity allocation. In the sequel, an evolving Takagi–Sugeno fuzzy model based on the ellipsoidal granules is proposed for data reconstruction and one-step ahead prediction from past data numerical evidence. Experimental studies concerning clustering, data granulation, and time-series forecasting are performed to illustrate the effectiveness of the proposed method.

Three-way decision based on improved aggregation method of interval loss function

Given a hybrid incomplete information table consisting both of the incomplete information table and loss function, the loss function of each object is denoted as an interval. The loss function of similarity class is defined as the aggregation of interval loss functions of all objects in its similarity class. However, existing optimistic aggregation method utilizes the union of interval loss functions of all objects in similarity class, which make the length of the aggregated interval loss function too broad. Pessimistic aggregation method utilizes the intersection of interval loss functions of all objects in similarity class, which make the length of the aggregated interval loss function too narrow. In order to get reasonable aggregated interval loss function, we propose a new aggregation method based on the principle of justifiable granularity, which make the length of the aggregated interval loss function neither too broad nor too narrow. That is, the aggregation result includes the interval loss functions of all objects in similarity class as much as possible and the aggregation result is as specific as possible. Based on the proposed aggregation method, we propose a new three-way decision model. Examples and experimental results demonstrate the effectiveness of the proposed method.

Stabilizing the information granules formed by the principle of justifiable granularity

The principle of justifiable granularity (PoJG) balances coverage and specificity to optimize information granularity. Although numerous studies have successfully formed information granules (IGs) using this principle, the stability of these IGs rarely receives attention. This paper analyzes the stability of such an IG’s modal, performance and upper or lower bound. First, we define some concepts to quantify the stability. Then, by the use of the binomial distribution, the central limit theorem and the union bound, we prove some theorems, which rely on several reasonable hypotheses and reveal the relations between the data size and the stability of the IG’s modal, performance and upper or lower bound. Furthermore, we put forward an algorithm built on the theorems to generate stable IGs by building data that have the proper size. Finally, we analyze its time complexity, applications and limitations. Experiments indicate the reliability of this algorithm when it is applied to several probability distributions and real datasets with a large scale of evidence.

Hierarchical granular hotspots detection

We present a hierarchical model based on the extended fuzzy C-means (EFCM) clustering algorithm to develop a granular view of hotspots on a geographic map. The objective is to establish an overview of the spatial distribution of a phenomenon when the relevant data are partitioned into different datasets. The EFCM algorithm is applied to each dataset to detect local hotspots, represented as circles, on the map. The local hotspots constitute information granules at lower level of abstraction in the model. A weighted EFCM algorithm is then applied to a dataset formed by the centers of all the local hotspots to extract circular prototypes, defined as global hotspots, which constitute information granules at the higher level, and hence, they deliver a global overview of the spatial distribution of the phenomenon on the map. Two indices related to the essential criteria of the principle of justifiable granularity are used. The results demonstrate that the most justifiable overview is obtained by using the radius of the local hotspot as weight. Comparisons with a hierarchical model based on FCM algorithm show that our algorithm gives a better granular view of the phenomenon with respect to the latter.

Fuzzy granular classification based on the principle of justifiable granularity

Fuzzy rule-based classifiers have been used in many applications showing better performance due to simple modular architectures and significant interpretability. Generally, researchers design this type of classifiers along with two incompatible directions: some of them devote to mine more rules from data for enhancing the accuracy of classifiers and others focus on reducing the number of rules for simplifying the classifier with high accuracy. However, from the perspective of practicability, the tradeoff between the accuracy of model and the number of rules including in the model should be carefully considered when designing classifiers. Further, these classifiers also encounter two evident limitations in the process of design: one is that the contribution of each sample versus individual attributions is equally important and another is that the concept of information granularity is not considered. These two limitations result in the reduction of accuracy of the ensuing classifier. To alleviate these two limitations and make compromise between the accuracy and the number of rules of formed classifier, in this study, a novel method is proposed to construct a fuzzy classifier by means of the principle of justifiable granularity with weighted data. The proposed method involves two main stages: the first stage concerns with the formation of an initial classification model. The initial model is constructed by engaging a synergy of Fuzzy C-Means clustering (FCM) and the principle of justifiable granularity with weighted data. The second stage focuses on the refinement of the initial classification model. The number of rules of the completely formed fuzzy classification model is equal to that of classes of experimental data. A series of experiments concerning synthetic datasets and ten UCI datasets are also implemented to exhibit the feasibility and effectiveness of the proposed classification method as well as reveal the impact of information granularity on resulting classification model.

Approximation of Fuzzy Sets by Interval Type-2 Trapezoidal Fuzzy Sets.

In this paper, we propose a gradient-based method to approximate a fuzzy set through a trapezoidal fuzzy set (TFS). By adding some constraints in the formulated optimization problem, the major characteristics of the fuzzy set such as the core, the major part of the support, and the shape of the membership function could be preserved; also the form of the optimized result as a TFS is guaranteed. We regard the optimized TFS as the "skeleton" (blueprint) of the original fuzzy set. Based on this skeleton, we further extend the TFS to a higher type, that is, an interval type-2 TFS (IT2 TFS), so that more information about the original fuzzy set could be captured but the number of the parameters used to describe the original fuzzy set is still maintained low (nine parameters are required for an IT2 TFS). The principle of justifiable granularity is used to ensure that the formed type-2 information granule exhibits a sound interpretation. Both synthetic fuzzy sets and those constructed by the fuzzy C -means algorithm applied to the publicly available data have been used to demonstrate the usefulness of the proposed approximation methods.

Facial semantic descriptors based on information granules

In this paper, we investigate a granular data description for facial components in which a characterization of facial components is presented by a collection of information granules. Firstly, the facial landmark detector is utilized to extract facial components automatically. Secondly, semantic concepts are formed by involving various mechanisms of fuzzy clustering based on these detected landmarks. A collection of numeric prototypes can be sought as a blueprint of the descriptors. Consequently, the information granules are being formed around the prototypes that are engaged by the fundamental ideas of Granular Computing, especially the principle of justifiable granularity. Multiple experiments on Multi-PIE facial database illustrate the proposed facial semantic descriptors based on information granules not only can characterize the key semantics of facial components of data, but also can improve the semantic classification performance in comparison with human perception.

Hidden Markov Models Based Approaches to Long-Term Prediction for Granular Time Series

In time series forecasting, a challenging and important task is to realize long-term forecasting that is both accurate and transparent. In this study, we propose a long-term prediction approach by transforming the original numerical data into some meaningful and interpretable entities following the principle of justifiable granularity. The obtained sequences exhibiting sound semantics may have different lengths, which bring some difficulties when carrying out predictions. To equalize these temporal sequences, we propose to adjust their lengths by involving the dynamic time warping (DTW) distance. Two theorems are included to ensure the correctness of the proposed equalization approach. Finally, we exploit hidden Markov models (HMM) to derive the relations existing in the granular time series. A series of experiments using publicly available data are conducted to assess the performance of the proposed prediction method. The comparative analysis demonstrates the performance of the prediction delivered by the proposed model.

Sequential three-way classifier with justifiable granularity

Sequential three-way decisions approach has been demonstrated as an effective methodology of human problem solving with the aid of multiple levels of granularity. Searching an appropriate information granularity for decision or classification is a crucial issue. In this paper, inspired by the principle of justifiable granularity, we investigate a novel classification algorithm, called sequential three-way classifier with justifiable subspace. The major contribution of this study is threefold. First, in training model, with an investigation of the essence of information granularity in rough sets theory, the justifiable attribute subspace is located in an interval with local and global notions. Second, in light of the advantages of attribute reduction technology, the local and global attribute subspaces are determined by core and reduct, respectively. Third, a novel dynamic tri-partition-based predicting strategy is presented with the aid of neighborhood monotonic property. Finally, several experiments are undertaken to verify the effectiveness of the proposed method. Compared with several state-of-the-art classifiers, the proposed algorithm generally exhibits a better classification performance involving fewer attributes.

Clustering Homogeneous Granular Data: Formation and Evaluation.

In this paper, we develop a comprehensive conceptual and algorithmic framework to cope with a problem of clustering homogeneous information granules. While there have been several approaches to coping with granular (viz. non-numeric) data, the origin of granular data themselves considered there is somewhat unclear and, as a consequence, the results of clustering start lacking some full-fledged interpretation. In this paper, we offer a holistic view at clustering information granules and an evaluation of the results of clustering. We start with a process of forming information granules with the use of the principle of justifiable granularity (PJG). With this regard, we discuss a number of parameters used in this development of information granules as well as quantify the quality of the granules produced in this manner. In the sequel, Fuzzy C -Means is applied to cluster the derived information granules, which are represented in a parametric manner and associated with weights resulting from the usage of the PJG. The quality of clustering results is evaluated through the use of the reconstruction criterion (quantifying the concept of information granulation and degranulation). A suite of experiments using synthetic and publicly available datasets is reported to quantify the performance of the proposed approach and highlight its key features.

Local-Density-Based Optimal Granulation and Manifold Information Granule Description.

Constructing information granules (IGs) has been of significant interest to the discipline of granular computing. The principle of justifiable granularity has been proposed to guide the design of IGs, opening an avenue of pursuits of building IGs carried out on a basis of well-defined and intuitively appealing principles. However, how to improve the efficiency and accuracy of the resulting constructs is an open issue. In this paper, we present a local-density-based optimal granulation model (LoDOG), exhibiting evident advantages: 1) it can detect arbitrarily-shaped IGs and 2) it finds the optimal granulation solutions with O(N) complexity, once the leading tree structure has been constructed. We describe IGs of arbitrary shapes using a small collection of landmark points positioned on the skeleton of the underlying manifold, which contribute to approximate reconstruction capabilities of the original dataset. A dissimilarity metric is developed to evaluate the quality of the obtained reconstruction. The interpretability of LoDOG IGs is discussed. Theoretical analysis and empirical evaluations are covered to demonstrate the effectiveness of LoDOG and the manifold description.

Hybrid Learning for General Type-2 TSK Fuzzy Logic Systems

This work is focused on creating fuzzy granular classification models based on general type-2 fuzzy logic systems when consequents are represented by interval type-2 TSK linear functions. Due to the complexity of general type-2 TSK fuzzy logic systems, a hybrid learning approach is proposed, where the principle of justifiable granularity is heuristically used to define an amount of uncertainty in the system, which in turn is used to define the parameters in the interval type-2 TSK linear functions via a dual LSE algorithm. Multiple classification benchmark datasets were tested in order to assess the quality of the formed granular models; its performance is also compared against other common classification algorithms. Shown results conclude that classification performance in general is better than results obtained by other techniques, and in general, all achieved results, when averaged, have a better performance rate than compared techniques, demonstrating the stability of the proposed hybrid learning technique.

Granular data imputation: A framework of Granular Computing

Data imputation is a common practice encountered when dealing with incomplete data. Irrespectively of the existing spectrum of techniques, the results of imputation are commonly numeric meaning that once the data have been imputed they are not distinguishable from the original data being initially available prior to imputation. In this study, the crux of the proposed approach is to develop a way of representing imputed (missing) entries as information granules and in this manner quantify the quality of the imputation process and the quality of the ensuing data. We establish a two-stage imputation mechanism in which we start with any method of numeric imputation and then form a granular representative of missing value. In this sense, the approach could be regarded as an enhancement of the existing imputation techniques.Proceeding with the detailed imputation schemes, we discuss two ways of imputation. In the first one, imputation is realized for individual variables of data sets and afterwards enhanced by the buildup of information granules. In the second approach, we are concerned with the use of fuzzy clustering, Fuzzy C-Means (FCM), which helps establish a structure in the data and then use this information in the imputation process.The design of information granules invokes the fundamentals of Granular Computing, namely a principle of justifiable granularity and an allocation of information granularity. Numeric experiments concerned with a suite of publicly available data sets offer detailed insights into the main facets of the overall design process and deliver a parametric analysis of the methods.