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  • 1
    Electronic Resource
    Electronic Resource
    A geometric approach to cluster validity for normal mixtures (1997)
    Springer
    Soft computing 1 (1997), S. 166-179 
    Details
    ISSN: 1433-7479
    Keywords: Keywords cluster validity ; EM algorithm ; generalized Dunn’s index ; mixture decomposition ; normal mixtures ; Xie-Beni index
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract  We study indices for choosing the correct number of components in a mixture of normal distributions. Previous studies have been confined to indices based wholly on probabilistic models. Viewing mixture decomposition as probabilistic clustering (where the emphasis is on partitioning for geometric substructure) as opposed to parametric estimation enables us to introduce both fuzzy and crisp measures of cluster validity for this problem. We presume the underlying samples to be unlabeled, and use the expectation-maximization (EM) algorithm to find clusters in the data. We test 16 probabilistic, 3 fuzzy and 4 crisp indices on 12 data sets that are samples from bivariate normal mixtures having either 3 or 6 components. Over three run averages based on different initializations of EM, 10 of the 23 indices tested for choosing the right number of mixture components were correct in at least 9 of the 12 trials. Among these were the fuzzy index of Xie-Beni, the crisp Davies-Bouldin index, and two crisp indices that are recent generalizations of Dunn’s index.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s005000050019
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Numerical taxonomy with fuzzy sets (1974)
    Springer
    Journal of mathematical biology 1 (1974), S. 57-71 
    Details
    ISSN: 1432-1416
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary A recently developed fuzzy clustering technique is utilized to analyze the substructure of a well known set of 4-dimensional botanical data. A solution obtained without prior knowledge of labelled pattern structure is offered in support of our contention that the technique proposed affords a comparatively reliable criterion for a posteriori evaluation of cluster validity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02339490
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Grouped coordinate minimization using Newton's method for inexact minimization in one vector coordinate (1991)
    Springer
    Journal of optimization theory and applications 71 (1991), S. 503-516 
    Details
    ISSN: 1573-2878
    Keywords: Coordinate minimization ; Newton's method ; local convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Letf(x,y) be a function of the vector variablesx ∈R n andy ∈R m. The grouped (variable) coordinate minimization (GCM) method for minimizingf consists of alternating exact minimizations in either of the two vector variables, while holding the other fixed at the most recent value. This scheme is known to be locally,q-linearly convergent, and is most useful in certain types of statistical and pattern recognition problems where the necessary coordinate minimizers are available explicitly. In some important cases, the exact minimizer in one of the vector variables is not explicitly available, so that an iterative technique such as Newton's method must be employed. The main result proved here shows that a single iteration of Newton's method solves the coordinate minimization problem sufficiently well to preserve the overall rate of convergence of the GCM sequence.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00941400
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Local convergence analysis of a grouped variable version of coordinate descent (1987)
    Springer
    Journal of optimization theory and applications 54 (1987), S. 471-477 
    Details
    ISSN: 1573-2878
    Keywords: Coordinate descent ; local linear convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract LetF(x,y) be a function of the vector variablesx∈R n andy∈R m . One possible scheme for minimizingF(x,y) is to successively alternate minimizations in one vector variable while holding the other fixed. Local convergence analysis is done for this vector (grouped variable) version of coordinate descent, and assuming certain regularity conditions, it is shown that such an approach is locally convergent to a minimizer and that the rate of convergence in each vector variable is linear. Examples where the algorithm is useful in clustering and mixture density decomposition are given, and global convergence properties are briefly discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00940196
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    Paper (German National Licenses)
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