Fuzzy Partition Research Articles

Fuzzy clustering algorithms with distance metric learning and entropy regularization

Clustering has been used in various fields, such as image processing, data mining, pattern recognition, and statistical analysis. Generally, clustering algorithms consider all variables equally relevant or not correlated. Nevertheless, the pattern of data samples in the multidimensional space can be geometrically complicated, e.g., clusters may exist in different subsets of features. In this regard, new soft subspace clustering algorithms have been proposed, in which the correlation and relevance of variables are considered to improve their performance. Since regularization-based methods are robust for initializations, the approaches proposed introduce an entropy regularization term for controlling the membership degree of the objects. Such regularizations are popular due to high performance in large-scale data clustering and low computational complexity. These three-step iterative algorithms provide a fuzzy partition, a representative for each cluster, and the relevance weight of the variables or their correlation by minimizing a suitable objective function. Several experiments on synthetic and real datasets, including their application to the segmentation of noisy image textures, demonstrate the usefulness of the proposed clustering methods.

An evolving concept in the identification of an interval fuzzy model of Wiener-Hammerstein nonlinear dynamic systems

This paper presents a new approach for identifying interval fuzzy models, which enables estimating fuzzy model structures, parameters, and upper and lower bounds simultaneously and online. It is based on a filtered recursive least squares method combined with an incrementally evolving Gaussian clustering. The proposed method generates interval fuzzy models online, on the fly, and in an evolving manner. This means that the algorithm starts with no a priori information, evolves the structure of the model, adjusts the model parameters simultaneously, and computes the upper and lower intervals simultaneously. The fuzzy partitioning of the input–output data space is based on the eGauss + method; the parameters of the local linear models, which together form the fuzzy model, are determined using a recursive least squares method. The interval fuzzy model, used in a predictive manner as a single or multi-level predictor, can be successfully applied in on-line monitoring, fault detection, and the control of dynamic systems. The proposed identification procedure was used to identify the fuzzy interval model of two different processes: a simplified Hammerstein-type nonlinear dynamic process and a realistic industrial continuously stirred tank reactor.The main contribution and advantage of the proposed new method is the identification of an interval fuzzy model in an online manner, which means that the structural and parametric identification of nonlinear systems is done simultaneously and from the data stream. The structural and parametric uncertainties are modeled and integrated into the upper and lower fuzzy models, which form the fuzzy interval in which the measured data samples of the process output are located with a certain probability. The approach is limited to processes that have Wiener-Hammerstein structures.

An Enhanced Fuzzy Segmentation Framework for extracting white matter from T1-weighted MR images

BackgroundWhite matter atrophy computed from Magnetic Resonance (MR) images is a clinical indication of a broad spectrum of neurological disorders. Accurate segmentation of white matter from MR images is necessary to estimate the white matter volume. Most of the techniques in literature used for the segmentation of white matter are computationally slow. The repeatability of segmentation results and consistency of performance on different input images are often poor. ObjectivesA computationally simple fuzzy clustering technique termed Enhanced Fuzzy Segmentation Framework (EFSF) for segmenting the white matter from the T1-Weighted MR images is proposed in this paper. MethodologyIN EFSF, the fuzzy membership function and prototype value are derived from the generic objective function of FCM using the method of Lagrange’s multiplier. The membership and prototype values are updated iteratively. The clustered image is obtained by replacing each grey level in the input image with the prototype value of the cluster with the largest membership value in the corresponding row of the fuzzy partition matrix. The pixels in the clustered image whose values are equal to the largest prototype value belong to the white matter region. ResultsOn 100 test images, the Dice Similarity Index (DSI) and the computational time (in sec) shown by EFSF are 0.8051 ± 0.0577 and 0.6522 ± 0.0502, respectively. ConclusionEFSF offers high segmentation accuracy and is computationally fast. Segmentation results offered by EFSF have good repeatability on the same MR slice and consistency on MR slices from various regions of the brain.

Multimodel System Identification Based on New Fuzzy Partitioning Similarity Measure

The problem of identifying unstructured nonlinear systems is generally addressed on the basis of multi-model representations involving several linear local models. In the present work, local models are combined to get a global representation using incremental fuzzy clustering. The main contribution is a novel vector similarity measure defined in the System Working Space (SWS) that combines the angular deviation and the usual Euclidean distance. Such a combination makes the new metric highly discriminating leading to a better partitioning of the operating space providing, thereby, a higher accuracy of the model. The developed partitioning method is first evaluated by performing linear local model (LLM) based identification of a academic benchmark multivariable nonlinear system. Then, the performances of the identification method are evaluated using experimental tropospheric ozone data. These evaluations illustrate the supremacy of the new method over the standard Euclidian-distance based partitioning approach.

Cophenetic-based fuzzy clustering of time series by linear dependency

In this work, a new approach to cluster large sets of time series is presented. The proposed methodology takes into account the dependency among the time series to obtain a fuzzy partition of the set of observations. A two-step procedure to accomplish this is presented. First, the cophenetic distances, based on a time series linear cross-dependency measure, are obtained. Second, these distances are used as an input of a non-Euclidean fuzzy relational clustering algorithm. As a result, we obtain a robust fuzzy procedure capable of detecting groups of time series with different types of cross-dependency. We illustrate the usefulness of the stated methodology through some Monte Carlo experiments and a real data example. Our results show that the methodology proposed in this work substantially improves the hard partitioning clustering alternative.

A Summary of F-Transform Techniques in Data Analysis

Fuzzy transform is a technique applied to approximate a function of one or more variables applied by researchers in various image and data analysis. In this work we present a summary of a fuzzy transform method proposed in recent years in different data mining disciplines, such as the detection of relationships between features and the extraction of association rules, time series analysis, data classification. After having given the definition of the concept of Fuzzy Transform in one or more dimensions in which the constraint of sufficient data density with respect to fuzzy partitions is also explored, the data analysis approaches recently proposed in the literature based on the use of the Fuzzy Transform are analyzed. In particular, the strategies adopted in these approaches for managing the constraint of sufficient data density and the performance results obtained, compared with those measured by adopting other methods in the literature, are explored. The last section is dedicated to final considerations and future scenarios for using the Fuzzy Transform for the analysis of massive and high-dimensional data.

Three-Mode Fuzzy Co-Clustering Based on Probabilistic Concept and Comparison with FCM-Type Algorithms

Three-mode fuzzy co-clustering is a promising technique for analyzing relational co-occurrence information among three mode elements. The conventional FCM-type algorithms achieved simultaneous fuzzy partition of three mode elements based on the fuzzy c-means (FCM) concept, and then, they often suffer from careful tuning of three independent fuzzification parameters. In this paper, a novel three-mode fuzzy co-clustering algorithm is proposed by modifying the conventional aggregation criterion of three elements based on a probabilistic concept. The fuzziness degree of three-mode partition can be easily tuned only with a single parameter under the guideline of the probabilistic standard. The characteristic features of the proposed method are compared with the conventional algorithms through numerical experiments using an artificial dataset and are demonstrated in application to a real world dataset of MovieLens movie evaluation data.

A classification algorithm based on multi-dimensional fuzzy transforms

We present a new classification algorithm for machine learning numerical data based on direct and inverse fuzzy transforms. In our previous work fuzzy transforms were used for numerical attribute dependency in data analysis: the multi-dimensional inverse fuzzy transform was used to approximate the regression function. Also here the classification method presented is based on this operator. Strictly speaking, we apply the K-fold cross-validation algorithm for controlling the presence of over-fitting and for estimating the accuracy of the classification model: for each training (resp., testing) subset an iteration process evaluates the best fuzzy partitions of the inputs. Finally, a weighted mean of the multi-dimensional inverse fuzzy transforms calculated for each training subset (resp., testing) is used for data classification. We compare this algorithm on well-known datasets with other five classification methods.

Soft subspace clustering of interval-valued data with regularizations

Data analysis plays an indispensable role in understanding different phenomena. One of the vital means of handling these data is to group them into a set of clusters given a measure of similarity. Usually, clustering methods deal with objects described by single-valued variables. Nevertheless, this representation is too restrictive for representing complex data, such as lists, histograms, or even intervals. Furthermore, in some problems, many dimensions are irrelevant and can mask existing clusters. In this regard, new interval-valued data clustering methods with regularizations and adaptive distances are proposed. These approaches consider that the boundaries of the interval-valued variables have the same and different importance for the clustering process. The algorithms optimize an objective function alternating three steps for obtaining the representatives of each group, a fuzzy partition, and the relevance weights of the variables. Experiments on synthetic and real data sets corroborate the robustness and usefulness of the proposed methods.

Spatial robust fuzzy clustering of COVID 19 time series based on B-splines

The aim of the work is to identify a clustering structure for the 20 Italian regions according to the main variables related to COVID-19 pandemic. Data are observed over time, spanning from the last week of February 2020 to the first week of February 2021. Dealing with geographical units observed at several time occasions, the proposed fuzzy clustering model embedded both space and time information. Properly, an Exponential distance-based Fuzzy Partitioning Around Medoids algorithm with spatial penalty term has been proposed to classify the spline representation of the time trajectories. The results show that the heterogeneity among regions along with the spatial contiguity is essential to understand the spread of the pandemic and to design effective policies to mitigate the effects.

Fuzzy Study on the Winning Rate of Football Game Betting

This study aims to find variables that affect the winning rate of the football team before a match. Qualitative variables such as venue, match importance, performance, and atmosphere of both teams are suggested to predict the outcome. Regression analysis is used to select proper variables. In this study, the performance of the football team is based on the opinions of experts, and the team atmosphere can be calculated with the results of the previous five games. ELO rating represents the state of the opponent. Also, the selected qualitative variables are expressed in fuzzy numbers using fuzzy partitions. A fuzzy regression model for the winning rate of the football team can be estimated by using the least squares method and the least absolute method. It is concluded that the stadium environment, ELO rating, team performance, and importance of the match have effects on the winning rate of Korean National Football (KNF) team from the data on 118 matches.

Research on Condition Evaluation of Gearbox Based on PCA-FCM

In order to evaluate the state of the gearbox, this paper proposes to use PCA-FCM algorithm to divide the state of the gearbox. By decomposing the vibration signal of the gearbox and taking the principal component, the state of the gearbox is divided. At the same time, this paper considers the fuzzy partition and geometric structure partition of the data set to determine the classification, which is verified by the wind turbine gearbox simulation experiment platform. The experimental results show that the gearbox state assessment based on PCA-FCM algorithm can effectively divide and identify the operation state of the gearbox, which makes the operation state division of the gearbox more scientific, and provides a theoretical basis for subsequent fault diagnosis and maintenance strategy.

Computational modeling and water analysis of parameter estimation by uncertainty quantification in water resources

Falling losses from aquatic disease is the main public condition goal in developed countries. The present study terms the application of Physical Sciences and Environmental Sciences techniques to solve the real-life problem associated with the society. Water samples are taken from a river, pond, bore well, and municipality at Kabisthalam in Kumbakonam Taluk, Tanjore District, Tamil Nadu, South India. The current work defines the expansion of mathematical typical to expect the interaction parameters which were used to determine the quantitative characteristics of the water. Analysis of Variance (ANOVA) was employed to identify the level of importance in interaction parameters on their characteristics. The effect of these parameters on interaction has been investigated using experimental designs. The new results have been correlated using the Fuzzy Evidence Theory. The salient aspect of the work is that a very simplified analytic output of the fuzzy model is achieved when all fuzzy sets in the fuzzy partition of the output space have the same power (the area under the membership function), and the determination of basic probability assignments associated with fuzzy Dempster-Shafer belief structure using fuzzy focal elements.

Quantitative-integration-based TSK fuzzy classification through improving the consistency of multi-hierarchical structure

In traditional hierarchical fuzzy classifiers, there exist some main problems. These issues include the output of the previous training layer influencing with the input of the next layers, the inconsistency between the intra-layer and inter-layer, and the improvement of consequent parameters. Based on the above troubles, a novel quantitative-integration-based hierarchical Takagi–Sugeno–Kang (TSK) is proposed to handle the challenges in existing hierarchical fuzzy classifiers. As a novel hierarchical structure, the proposed classifier is built in a stacked manner. Each base building unit consists of an optimized zero-order TSK fuzzy classifier. For good interpretability of each base building unit, the antecedent parameters are solved by random selections of input features, random combinations of fuzzy rules, divisions of fuzzy partitions and generations of cluster centers. In addition, an improved method based on classical ridge regression is proposed to solve the problem of the consistency between intra-layer and inter-layer. In order to enhance the classification performance of the fuzzy classifier, the input of each layer is optimized to effectively resolve the problem that the output of the previous layer affects the input of the next base building unit. Moreover, a method that randomly selects a part of the sample from the original training set is adopted to open the manifold structure of input set. Furthermore, the experimental results indicate that this method can indeed make the QI-TSK-FC fuzzy classifier obtain good or comparative classification performance. To improve the generalization ability of the fuzzy classifier, a method that quantitatively solves the integrated output is considered so that the QI-TSK-FC model can quickly obtain the satisfactory optimal solution. Finally, the classification performance and interpretability of the proposed fuzzy classifier are verified through the MIT-BIH sleep datasets.

Chi-MFlexDT:Chi-square-based multi flexible fuzzy decision tree for data stream classification

In recent years, a number of classification algorithms using fuzzy decision trees have been proposed. A key issue encountered in all of these algorithms is how to select the division feature. In traditional algorithms, it is common to use the information gain criterion. There is a claim that the use of information-based criteria leads to an unfair selection of features that have large number of values while not being suitable for division. This problem causes to decrease accuracy of classification in stream data. However, other algorithms for classifying stream data and extracting fuzzy rules have been proposed but the problem of unfairly selecting features with large amount of values still persists. This paper, present a new algorithm to solve problems of selecting splitting feature in fuzzy decision tree for classification of stream data. In proposed algorithm, we extend multi flexible fuzzy decision tree (MFlexDT) with chi-square based fuzzy partitioning of values. In this paper, we also extend traditional chi-square statistical to fuzzy chi-square statistical data distribution. In evaluating the proposed algorithm, tree depth and accuracy are two factors that affect performance. Experimental results show that the new algorithm has been able to improve the performance of other algorithms.

An image segmentation method based on a modified local-information weighted intuitionistic Fuzzy C-means clustering and Gold-panning Algorithm

The image segmentation method based on clustering analysis has the advantages of small sample space constraints and strong universality. As an unsupervised clustering algorithm, the fuzzy C-means clustering algorithm is widely used in practical engineering. However, it is still some shortcomings: the fuzzy C-means clustering algorithm is difficult to interpret the noise effectively, which makes it more sensitive to the noise, and the selection of key parameters has to be made by trial and error experiments, reducing the adaptability of the algorithm. Besides, its iteration process is heavily influenced by the initial clustering centers and easy to fall into local optimum. Therefore, an intuitionistic Fuzzy C-means clustering method, based on local-information weight, is proposed in this paper. By introducing the local-information weight, the proposed algorithm adjusts the local-information influence weight adaptively in fuzzy partition, which enhances its robustness to noisy images. Furthermore, a novel swarm intelligence algorithm, called the Gold-Panning Algorithm, is proposed to optimize the initial clustering centers and key parameters in the clustering algorithm. By utilizing the Gold-Panning Algorithm, the adaptability of the proposed clustering algorithm is further improved. In this paper, the proposed methods are explained in detail and compared with the existing methods to demonstrate its superior performance.

A wrapper methodology to learn interval-valued fuzzy rule-based classification systems

Learning an interval-valued fuzzy rule-based classification system is a challenge as its success directly depends on the interval-valued fuzzy partition used. In fact, the learning of an interval-valued fuzzy system usually starts by creating a partition composed of numerical fuzzy sets, which are used to build an initial fuzzy classifier. Then, it is augmented with interval-valued fuzzy sets whose shape is subsequently optimized to improve the system’s performance. However, as in this methodology the fuzzy rules are learned using numerical fuzzy sets, the benefits of the interval-valued fuzzy sets may not be fully exploited. In this paper we define a new learning methodology that avoids building the initial fuzzy classifier but directly learns interval-valued fuzzy rules. To do so, we define a wrapper methodology to learn the interval-valued fuzzy partitions such that they lead to an interval-valued fuzzy rule-based classification system as accurate as possible. Moreover, our new method allows one to represent each membership function using the most proper type of fuzzy set for the sake of modeling the uncertainty in the best possible manner. Consequently, the antecedents of the rules can be formed of only numerical fuzzy sets, only interval-valued fuzzy sets or a mixture of both. The quality of the proposal is compared versus four state-of-the-art fuzzy classifiers like FARC-HD, IVTURS, FURIA and FARC-HD using an inference based on a generalization of the Choquet integral. We also compare our new approach besides its numerical fuzzy counterpart to clearly show the benefits of the usage of interval-valued fuzzy sets. Specifically, the average accuracy rate of our new method is 81.17%, which is at least 0.66% better than the remainder state-of-the-art fuzzy classifiers.

Fuzzy image clustering incorporating local and region-level information with median memberships

Image segmentation with Fuzzy C-Means (FCM) clustering algorithm is a widely researched topic. In the literature, region-level information considers more redundancy information of image and becomes a more powerful FCM technique. However, noise pixels can be over-preserved via region-level information causing FCM produce fake segmentation areas. Fuzzification of partition is also vital in FCM and Kullback–Leibler (KL) information with membership mean template is successful to reduce fuzzification. However, existence of outliers influences final segmentation result due to the sum operation during calculation of mean template. In this paper, we propose an improved FCM with adaptive local and region-level information as well as KL information with locally median membership degrees. Firstly, followed by region-level information, a new distance measure incorporating both local and region-level information is proposed where local information of the image of region-level information is acquired to smooth the over-preserved noise pixels. Secondly, we renounce the sum operation in membership mean template and propose that locally median template is more reasonable to be considered as prior probability in KL information. Thirdly, adaptive constraints for conventional FCM and local as well as region-level information are introduced. In the beginning, differences between local variances of original image and image of region-level information in exponential form are computed. Then, reciprocals of the differences and the differences themselves are considered as constraints of conventional FCM and local as well as region-level information individually. Experiments of grayscale and color image segmentation show that the proposed method FALRCM (Fuzzy Adaptive Local and Region-level information C-Means) achieves better performance in terms of fuzzy partition coefficient (VPC), fuzzy partition entropy (VPE), Segmentation Accuracy (SA), mean Intersection-over-Union (mIoU), Peak Signal-to-Noise Ratio (PSNR) and visual effects compared with several state-of-the-art FCM variants. For example, in grayscale noise image segmentation, average VPC and mIoU are up to 99.92% and 97.59% with standard deviation of 0.05% and 2.46% respectively while the method with the closest performance provides VPC and mIoU of 95.27% and 91.76% with standard deviations of 15.59% and 3.53% respectively. The proposed method FALRCM achieves better robustness of noise and lower partition fuzzification. In the end, limitations are shown in discussion and application fields as well as future studies are proposed in conclusion part.

E-fuzzy feature fusion and thresholding for morphology segmentation of brain MRI modalities

Brain tumor segmentation is a significant procedure in medical image processing. Effective and efficient segmentation is always a key concern for the radiologists due to the presence of low illumination in imaging modalities of Magnetic Resonance (MR) imaging. Some of the challenging issues addressed in this paper are the sensitivity of noise, variations, and non-standardization in inter-slice intensity, and intensity inhomogeneity. A reliable segmentation method for the brain tumor segmentation is necessary for efficient measurement of the tumor. The foremost objective of the research is to fuse the different modalities of MRI images by considering the most useful features to obtain the best segmentation. Sufficient diagnostic information in clinical applications is not provided by single modality images. Therefore, combining the features of different modalities of images is vital. Recently, many researchers applied many techniques for fusing medical images, still many issues are to be addressed. Hence, to combine different modality of images, a novel fusion method is proposed. Our proposed work contains four stages (i) pre-processing, (ii) feature extraction (iii) feature fusion, and (iv) segmentation. In the pre-processing step, the image quality is increased and the unwanted noise is removed using average filtering. In the feature extraction stage, the Extended Grey Level Co-occurrence Matric (E-GLCM) is introduced to extract the suitable intensity and texture features of brain tumor MRI images. To fuse the correct features for efficient segmentation, the Enhanced Fuzzy Radial Basis function Neural Network (E-FRBNN) incorporates five layer inputs, fuzzy partition, front combination, inference, and output. Finally, the segmentation is carried by thresholding based segmentation approach with morphology operators. The simulation results of the proposed segmentation procedure acquire competitive performance when compared with the existing techniques for the BRATS 2015 dataset.

Design of Fuzzy Ensemble Architecture Realized With the Aid of FCM-Based Fuzzy Partition and NN With Weighted LSE Estimation

Neural networks (NNs) with least square error (LSE) estimation form a certain type of single hidden layer feed-forward NNs. In this class of networks, the input connections (weights) and the biases of hidden neurons are generated randomly and fixed after being generated. The output connections are estimated by the LSE method rather than the back-propagation method. The random generation of the input connection weights and the hidden biases results in the larger number of hidden neurons to assure the quality of classification performance. To reduce the number of neurons in the hidden layer while maintaining the classification performance, we apply a “divide and conquer” strategy in this article. In other words, we divide an overall input space into several subspaces by using information granulation technique (Fuzzy C-Means clustering algorithm) and determine the local decision boundaries among related subspaces. A decision boundary defined in the input space can be considered as being composed of several decision boundaries defined in subspaces that form the entire input space. For the decision boundaries defined in the subspaces, their nonlinearity becomes lower in comparison with the one being encountered when considering the entire input space. Through the weighted LSE estimation instead of using the LSE estimation method, the connections of several NNs can be estimated without interfering with each other. After estimating the weights, the decision boundaries defined in the related subspaces are merged to a single decision boundary by using fuzzy ensemble technique. Several machine learning datasets and one real world application dataset are used to evaluate and validate the proposed fuzzy ensemble classifier. Based on the experimental results, the proposed classifier shows better classification performance when compared with the performance of some selected classifiers.