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Summary DCT Given a finite set of points in  an Euclidean space the \emph{spanning tree} is a tree of minimal length having the given points as vertices. The length of the tree is the sum of the distances of all connected  point pairs of the tree. The clustering tree with a given  length of a given finite set of points is the spanning tree of an appropriately chosen other set of points approximating the given set of points with minimal sum of square distances among all spanning trees with the given length. DCM A matrix of real numbers is said to be column monotone orderable if there exists an ordering of columns of the matrix such that all rows of the matrix become monotone after ordering. The {\emph{monotone sum of squares of a matrix}} is the minimum of sum of squares of differences of the elements of the matrix and a column monotone orderable matrix where the minimum is taken on the set of all column monotone orderable matrices. Decomposition clusters of monotone orderings of a matrix is a clustering ofthe rows of the matrix into given number of clusters such that thesum of monotone sum of squares of the matrices formed by the rowsof the same cluster is minimal.DCP A matrix of real numbers is said to be column partitionable if there exists a partition of the columns such that the elements belonging to the same subset of the partition are equal in each row. Given a partition of the columns of a matrix the partition sum of squares of the matrix is the minimum of the sum of square of differences of the elements of the matrix and a column partitionable matrix where the minimum is taken on the set of all column partitionable matrices. Decomposition  of the rows of a matrix into clusters of partitions is the minimization of the corresponding partition sum of squares given the number of clusters and the sizes of the subsets of the partitions.

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Acta Mathematica Hungarica
Authors: Villő Csiszár, Péter Hussami, János Komlós, Tamás F. Móri, Lídia Rejtő, and Gábor Tusnády

Abstract

There is a uniquely defined random graph model with independent adjacencies in which the degree sequence is a sufficient statistic. The model was recently discovered independently by several authors. Here we join to the statistical investigation of the model, proving that if the degree sequence is in the interior of the polytope defined by the Erdős–Gallai conditions, then a unique maximum likelihood estimate exists.

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