Identifying the Mastery Concepts in Linear Algebra by Using FCM-CM Algorithm

2010 ◽  
Vol 44-47 ◽  
pp. 3897-3901
Author(s):  
Hsiang Chuan Liu ◽  
Yen Kuei Yu ◽  
Jeng Ming Yih ◽  
Chin Chun Chen
Keyword(s):  
Objective Function ◽  
Linear Algebra ◽  
Fuzzy Clustering ◽  
Distance Function ◽  
Mahalanobis Distance ◽  
Euclidean Distance ◽  
Clustering Algorithm ◽  
Clustering Algorithms ◽  
Euclidean Distance Function ◽  
Fuzzy C Means Algorithm

Euclidean distance function based fuzzy clustering algorithms can only be used to detect spherical structural clusters. Gustafson-Kessel (GK) clustering algorithm and Gath-Geva (GG) clustering algorithm were developed to detect non-spherical structural clusters by employing Mahalanobis distance in objective function, however, both of them need to add some constrains for Mahalanobis distance. In this paper, the authors’ improved Fuzzy C-Means algorithm based on common Mahalanobis distance (FCM-CM) is used to identify the mastery concepts in linear algebra, for comparing the performances with other four partition algorithms; FCM-M, GG, GK, and FCM. The result shows that FCM-CM has better performance than others.

Construct Concept Structure for Linear Algebra Based on Cognition Diagnosis and Clustering with Mahalanobis Distances

2011 ◽  
Vol 211-212 ◽  
pp. 756-760
Author(s):  
Chin Chun Chen ◽  
Yuan Horng Lin ◽  
Jeng Ming Yih ◽  
Shu Yi Juan
Keyword(s):  
Linear Algebra ◽  
Mahalanobis Distance ◽  
Euclidean Distance ◽  
Clustering Algorithms ◽  
Knowledge Structure ◽  
Interpretive Structural Modeling ◽  
Mahalanobis Distances ◽  
Fuzzy C Means ◽  
Euclidean Distance Function ◽  
Fuzzy C Means Algorithm

Euclidean distance function based fuzzy clustering algorithms can only be used to detect spherical structural clusters. The purpose of this study is improved Fuzzy C-Means algorithm based on Mahalanobis distance to identify concept structure for Linear Algebra. In addition, Concept structure analysis (CSA) could provide individualized knowledge structure. CSA algorithm is the major methodology and it is based on fuzzy logic model of perception (FLMP) and interpretive structural modeling (ISM). CSA could display individualized knowledge structure and clearly represent hierarchies and linkage among concepts for each examinee. Each cluster of data can easily describe features of knowledge structures. The results show that there are five clusters and each cluster has its own cognitive characteristics. In this study, the author provide the empirical data for concepts of linear algebra from university students. To sum up, the methodology can improve knowledge management in classroom more feasible. Finally, the result shows that Algorithm based on Mahalanobis distance has better performance than Fuzzy C-Means algorithm.

Construct Knowledge Structure of Linear Algebra

2011 ◽  
Vol 211-212 ◽  
pp. 793-797
Author(s):  
Chin Chun Chen ◽  
Yuan Horng Lin ◽  
Jeng Ming Yih ◽  
Sue Fen Huang
Keyword(s):  
Knowledge Management ◽  
Linear Algebra ◽  
Fuzzy Clustering ◽  
Mahalanobis Distance ◽  
Clustering Algorithms ◽  
Knowledge Structure ◽  
Interpretive Structural Modeling ◽  
Cognitive Characteristics ◽  
Fuzzy C Means ◽  
Fuzzy C Means Algorithm

Apply interpretive structural modeling to construct knowledge structure of linear algebra. New fuzzy clustering algorithms improved fuzzy c-means algorithm based on Mahalanobis distance has better performance than fuzzy c-means algorithm. Each cluster of data can easily describe features of knowledge structures individually. The results show that there are six clusters and each cluster has its own cognitive characteristics. The methodology can improve knowledge management in classroom more feasible.

Management of Abstract Algebra Concepts Based on Knowledge Structure

2013 ◽  
Vol 284-287 ◽  
pp. 3537-3542
Author(s):  
Chin Chun Chen ◽  
Yuan Horng Lin ◽  
Jeng Ming Yih
Keyword(s):  
Mahalanobis Distance ◽  
Clustering Algorithm ◽  
Fuzzy Theory ◽  
Clustering Algorithms ◽  
Knowledge Structures ◽  
Interpretive Structural Modeling ◽  
Fuzzy C Means ◽  
Integrated Method ◽  
Combined Algorithm ◽  
Fuzzy C Means Algorithm

Knowledge Management of Mathematics Concepts was essential in educational environment. The purpose of this study is to provide an integrated method of fuzzy theory basis for individualized concept structure analysis. This method integrates Fuzzy Logic Model of Perception (FLMP) and Interpretive Structural Modeling (ISM). The combined algorithm could analyze individualized concepts structure based on the comparisons with concept structure of expert. Fuzzy clustering algorithms are based on Euclidean distance function, which can only be used to detect spherical structural clusters. A Fuzzy C-Means algorithm based on Mahalanobis distance (FCM-M) was proposed to improve those limitations of GG and GK algorithms, but it is not stable enough when some of its covariance matrices are not equal. A new improved Fuzzy C-Means algorithm based on a Normalized Mahalanobis distance (FCM-NM) is proposed. Use the best performance of clustering Algorithm FCM-NM in data analysis and interpretation. Each cluster of data can easily describe features of knowledge structures. Manage the knowledge structures of Mathematics Concepts to construct the model of features in the pattern recognition completely. This procedure will also useful for cognition diagnosis. To sum up, this integrated algorithm could improve the assessment methodology of cognition diagnosis and manage the knowledge structures of Mathematics Concepts easily.

AN EXTENDED FUZZY CLUSTERING ALGORITHM AND ITS APPLICATION

1995 ◽  
Vol 05 (02) ◽  
pp. 239-259
Author(s):  
SU HWAN KIM ◽  
SEON WOOK KIM ◽  
TAE WON RHEE
Keyword(s):  
Fuzzy Clustering ◽  
Clustering Algorithm ◽  
Clustering Algorithms ◽  
Main Memory ◽  
Color Image Segmentation ◽  
Occurrence Rate ◽  
Secondary Memory ◽  
Worst Case ◽  
Memory Space ◽  
Fuzzy Clustering Algorithm

For data analyses, it is very important to combine data with similar attribute values into a categorically homogeneous subset, called a cluster, and this technique is called clustering. Generally crisp clustering algorithms are weak in noise, because each datum should be assigned to exactly one cluster. In order to solve the problem, a fuzzy c-means, a fuzzy maximum likelihood estimation, and an optimal fuzzy clustering algorithms in the fuzzy set theory have been proposed. They, however, require a lot of processing time because of exhaustive iteration with an amount of data and their memberships. Especially large memory space results in the degradation of performance in real-time processing applications, because it takes too much time to swap between the main memory and the secondary memory. To overcome these limitations, an extended fuzzy clustering algorithm based on an unsupervised optimal fuzzy clustering algorithm is proposed in this paper. This algorithm assigns a weight factor to each distinct datum considering its occurrence rate. Also, the proposed extended fuzzy clustering algorithm considers the degree of importances of each attribute, which determines the characteristics of the data. The worst case is that the whole data has an uniformly normal distribution, which means the importance of all attributes are the same. The proposed extended fuzzy clustering algorithm has better performance than the unsupervised optimal fuzzy clustering algorithm in terms of memory space and execution time in most cases. For simulation the proposed algorithm is applied to color image segmentation. Also automatic target detection and multipeak detection are considered as applications. These schemes can be applied to any other fuzzy clustering algorithms.

scholarly journals Research on the Method of Coke Optical Tissue Segmentation Based on Adaptive Clustering

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Huaiguang Liu ◽  
Liheng Zhang ◽  
Shiyang Zhou ◽  
Li Fang
Keyword(s):  
Mahalanobis Distance ◽  
Euclidean Distance ◽  
Clustering Algorithm ◽  
Adaptive Threshold ◽  
Segmentation Method ◽  
Tissue Segmentation ◽  
Adaptive Clustering ◽  
Morphological Filtering ◽  
Key Factor ◽  
The Relationship

The microstructure is the key factor for quality discriminate of coke. In view of the characteristics of coke optical tissue (COT), a segmentation method of coke microstructures based on adaptive clustering was proposed. According to the strategy of multiresolution, adaptive threshold binarization and morphological filtering were carried out on COT images with lower resolution. The contour of the COT body was detected through the relationship checking between contours in the binary image, and hence, COT pixels were picked out to cluster for tissue segmentation. In order to get the optimum segmentation for each tissue, an advanced K -means method with adaptive clustering centers was provided according to the Calinski-Harabasz score. Meanwhile, Euclidean distance was substituted with Mahalanobis distance between each pixel in HSV space to improve the accuracy. The experimental results show that compared with the traditional K -means algorithm, FCM algorithm, and Meanshift algorithm, the adaptive clustering algorithm proposed in this paper is more accurate in the segmentation of various tissue components in COT images, and the accuracy of tissue segmentation reaches 94.3500%.

Kernel-Based Fuzzy Local Information Clustering Algorithm Self-integrating Non-Local Information

2021 ◽  
Author(s):  
Qiuyu Song ◽  
Chengmao Wu ◽  
Xiaoping Tian ◽  
Yue Song ◽  
Xiaokang Guo
Keyword(s):  
Fuzzy Clustering ◽  
Clustering Algorithm ◽  
Integration Method ◽  
Clustering Algorithms ◽  
Noise Removal ◽  
Local Information ◽  
Gaussian Kernel ◽  
Distance Metric ◽  
High Noise ◽  
Non Local

Abstract The application of fuzzy clustering algorithms in image segmentation is a hot research topic nowadays. Existing fuzzy clustering algorithms have the following three problems: (1)The parameters of spatial information constraints can$'$t be selected adaptively; (2)The image corrupted by high noise can$'$t be segmented effectively; (3)It is difficult to achieve a balance between noise removal and detail preservation. In the fuzzy clustering based on the optimization model, the choice of distance metric is very important. Since the use of Euclidean distance will lead to sensitivity to outliers and noise, it is difficult to obtain satisfactory segmentation results, which will affect the clustering performance. This paper proposes an optimization algorithm based on the kernel-based fuzzy local information clustering integrating non-local information (KFLNLI). The algorithm adopts a self-integration method to introduce local and non-local information of images, which solves the common problems of current clustering algorithm. Firstly, the self-integration method solves the problem of selecting spatial constraint parameters. The algorithm uses continuous self-learning iteration to calculate the weight coefficients; Secondly, the distance metric uses Gaussian kernel function to induce the distance to further enhance the robustness against noise and the adaptivity of processing different images; Finally, both local and non-local information are introduced to achieve a segmentation effect that can eliminate most of the noise and retain the original details of the image. Experimental results show that the algorithm is superior to existing state-of-the-art fuzzy clustering-related algorithm in the presence of high noise.

scholarly journals An Adaptive Multiobjective Genetic Algorithm with Fuzzy c-Means for Automatic Data Clustering

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Ze Dong ◽  
Hao Jia ◽  
Miao Liu
Keyword(s):  
Genetic Algorithm ◽  
Fuzzy Clustering ◽  
Clustering Algorithm ◽  
Majority Vote ◽  
Clustering Algorithms ◽  
Nsga Ii ◽  
Number Of Clusters ◽  
Automatic Data ◽  
Multiobjective Genetic Algorithm ◽  
Fuzzy Clustering Method

This paper presents a fuzzy clustering method based on multiobjective genetic algorithm. The ADNSGA2-FCM algorithm was developed to solve the clustering problem by combining the fuzzy clustering algorithm (FCM) with the multiobjective genetic algorithm (NSGA-II) and introducing an adaptive mechanism. The algorithm does not need to give the number of clusters in advance. After the number of initial clusters and the center coordinates are given randomly, the optimal solution set is found by the multiobjective evolutionary algorithm. After determining the optimal number of clusters by majority vote method, the Jm value is continuously optimized through the combination of Canonical Genetic Algorithm and FCM, and finally the best clustering result is obtained. By using standard UCI dataset verification and comparing with existing single-objective and multiobjective clustering algorithms, the effectiveness of this method is proved.

Identification of Key Factors of Fire Risk of Oil Depot Based on Fuzzy Clustering Algorithm

2019 ◽  
Author(s):  
Shuyi Xie ◽  
Shaohua Dong ◽  
Guangyu Zhang
Keyword(s):  
Risk Factors ◽  
Fuzzy Clustering ◽  
Index System ◽  
Clustering Algorithm ◽  
Risk Index ◽  
Fire Risk ◽  
Key Factors ◽  
Fuzzy Clustering Algorithm ◽  
Fuzzy C Means Algorithm ◽  
Fire Risk Index

Abstract With the rapid development of the national economy, the demand for oil is increasing. In order to meet the increasing energy demand, China has established a number of oil depot in recent years, whose largest capacity reaching up to tens of millions of cubic meters. Due to the flammable and explosive nature of the stored medium, the risk of fire in the oil depot area has increased dramatically as the tank capacity of the storage tank area has increased. The intensification of the oil depot and the development of large-scale oil storage tanks have brought convenience to the national oil depot, but also brought many catastrophic consequences. In recent years, there have been many fires and explosions in the oil depot, causing major casualties and property losses, which seriously endangered the ecological environment and public safety. Based on the constructed oil depot fire risk index system, the fuzzy C-means algorithm (FCM) and fuzzy maximum support tree clustering algorithm is introduced. Through the two fuzzy clustering mathematical models, key factors in the established index system are identified. Firstly, the expert scoring method is used to evaluate the indicators in the oil depot fire risk index system, and the importance degree evaluation matrix of oil depot fire risk factors is constructed through the fuzzy analysis of expert comments. Then, the fuzzy C-means algorithm (FCM) and the fuzzy clustering tree algorithm are used to cluster the various risk indicators, and the key factors of the oil depot fire risk are identified. Through the comparative analysis and cross-validation of the results of the two fuzzy clustering methods, the accuracy of the recognition results is ensured. Finally, using an oil depot as a case study, it is found that passive fire prevention capability and emergency rescue capability are key factors that need to be paid attention to in the oil depot fire risk index. The fuzzy clustering algorithm used in this paper can digitize the subjective comments of experts, thus reducing the influence of human subjective factors. In addition, by using two fuzzy clustering algorithms to analyze and verify the key factors of the oil depot fire risk, the reliability of the clustering results is guaranteed. The identification of key factors can enable managers to predict high-risk factors in advance in the fire risk prevention and control process of the oil depot, so as to adopt corresponding preventive measures to minimize the fire risk in the oil depot, and ensure the safety of the operation of the oil depot.

scholarly journals Geometric applications of critical point theory to submanifolds of complex projective space

1974 ◽  
Vol 55 ◽  
pp. 5-31 ◽  
Author(s):  
Thomas E. Cecil
Keyword(s):  
Projective Space ◽  
Critical Point ◽  
Distance Function ◽  
Complex Projective Space ◽  
Euclidean Distance ◽  
Point Theory ◽  
Euclidean Distance Function ◽  
Complex Projective ◽  
Class Of Functions

In a recent paper, [6], Nomizu and Rodriguez found a geometric characterization of umbilical submanifolds Mn ⊂ Rn+p in terms of the critical point behavior of a certain class of functions Lp, p ⊂ Rn+p, on Mn. In that case, if p ⊂ Rn+p, x ⊂ Mn, then Lp(x) = (d(x,p))2, where d is the Euclidean distance function.

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