Hierarchical structure invariance and optimal approximation for proximity data

Based on granular space, the hierarchical structure invariance and optimal approximation for proximity data are presented, and four results are obtained as follows. Firstly, an improved algorithm for computing the hierarchical structure and the related min-transitive closure of a fuzzy proximity relation are given, and the properties of key point sequence are studied by introducing the key points and key values of a fuzzy proximity relation. Secondly, two basic concepts, the hierarchical structure invariance and isomorphism of fuzzy proximity relations are introduced, and some examples are given to illustrate that the hierarchical structure of a fuzzy proximity relation is variant under three typical triangular norms. Thirdly, the hierarchical structure invariance theorem of fuzzy proximity relation under a mapping or transformation is given. Finally, for a given fuzzy proximity relation, a mathematical model for obtaining the optimal approximation in order to keep its hierarchical structure is established through the use of the linear combination of its minimum fuzzy proximity relation and min-transitive closure. These results help us understand the hierarchical structures, and provide theories and methodologies for the structural analysis and applications of proximity data.

[1]  Sargur N. Srihari,et al.  Fast k-nearest neighbor classification using cluster-based trees , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[2]  Geert Wets,et al.  A generalized multiple layer data envelopment analysis model for hierarchical structure assessment: A case study in road safety performance evaluation , 2009, Expert Syst. Appl..

[3]  B. Deviren,et al.  Hierarchical structures of correlations networks among Turkey’s exports and imports by currencies , 2012 .

[4]  Bernard De Baets,et al.  Algorithms for computing the min-transitive closure and associated partition tree of a symmetric fuzzy relation , 2004, Eur. J. Oper. Res..

[5]  Tang Xu.  Study on Fuzzy Granular Space Based on Normalized Equicrural Metric , 2008 .

[6]  Witold Pedrycz,et al.  Collaborative clustering with the use of Fuzzy C-Means and its quantification , 2008, Fuzzy Sets Syst..

[7]  Jessica K. Hodgins,et al.  Hierarchical Aligned Cluster Analysis for Temporal Clustering of Human Motion , 2013, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[8]  Bernard De Baets,et al.  General results on the decomposition of transitive fuzzy relations , 2010, Fuzzy Optim. Decis. Mak..

[9]  Narendra Ahuja,et al.  Location- and Density-Based Hierarchical Clustering Using Similarity Analysis , 1998, IEEE Trans. Pattern Anal. Mach. Intell..

[10]  Bo Zhang,et al.  The structure analysis of fuzzy sets , 2005, Int. J. Approx. Reason..

[11]  Luis Garmendia,et al.  A method to make some fuzzy relations T-transitive , 1999, Int. J. Intell. Syst..

[12]  Christian Döring,et al.  Data analysis with fuzzy clustering methods , 2006, Comput. Stat. Data Anal..

[13]  Bo Zhang,et al.  Fuzzy reasoning model under quotient space structure , 2005, Inf. Sci..

[14]  Bo Zhang,et al.  Fuzzy tolerance quotient spaces and fuzzy subsets , 2010, Science China Information Sciences.

[15]  Javier Montero,et al.  Computing a T-transitive lower approximation or opening of a proximity relation , 2008, Fuzzy Sets Syst..

[16]  Zhu Ping,et al.  Cluster Analysis Based on Fuzzy Quotient Space , 2008 .

[17]  Delbert Dueck,et al.  Clustering by Passing Messages Between Data Points , 2007, Science.

[18]  B. De Baets,et al.  Transitive approximation of fuzzy relations by alternating closures and openings , 2003 .

[19]  Jordi Recasens,et al.  How to Make $T$-Transitive a Proximity Relation , 2009, IEEE Transactions on Fuzzy Systems.

[20]  Ching-Hsue Cheng,et al.  Fuzzy hierarchical TOPSIS for supplier selection , 2009, Appl. Soft Comput..

[21]  Witold Pedrycz,et al.  Analytic Hierarchy Process (AHP) in Group Decision Making and its Optimization With an Allocation of Information Granularity , 2011, IEEE Transactions on Fuzzy Systems.

[22]  Joachim M. Buhmann,et al.  Optimal Cluster Preserving Embedding of Nonmetric Proximity Data , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[23]  Ping Zhu,et al.  The structural clustering and analysis of metric based on granular space , 2010, Pattern Recognit..

[24]  Bernard De Baets,et al.  On the existence and construction of T-transitive closures , 2003, Inf. Sci..

[25]  Ping Zhu,et al.  Cluster Analysis Based on Fuzzy Quotient Space: Cluster Analysis Based on Fuzzy Quotient Space , 2010 .

[26]  Ping Zhu,et al.  Hierarchical Clustering Problems and Analysis of Fuzzy Proximity Relation on Granular Space , 2013, IEEE Transactions on Fuzzy Systems.