60 research outputs found
Perturbation method for determining the group of invariance of hierarchical models
We propose a perturbation method for determining the (largest) group of
invariance of a toric ideal defined in Aoki and Takemura [2008a]. In the
perturbation method, we investigate how a generic element in the row space of
the configuration defining a toric ideal is mapped by a permutation of the
indeterminates. Compared to the proof in Aoki and Takemura [2008a] which was
based on stabilizers of a subset of indeterminates, the perturbation method
gives a much simpler proof of the group of invariance. In particular, we
determine the group of invariance for a general hierarchical model of
contingency tables in statistics, under the assumption that the numbers of the
levels of the factors are generic. We prove that it is a wreath product indexed
by a poset related to the intersection poset of the maximal interaction effects
of the model.Comment: 17pages, no figure
Markov bases and subbases for bounded contingency tables
In this paper we study the computation of Markov bases for contingency tables
whose cell entries have an upper bound. In general a Markov basis for unbounded
contingency table under a certain model differs from a Markov basis for bounded
tables. Rapallo, (2007) applied Lawrence lifting to compute a Markov basis for
contingency tables whose cell entries are bounded. However, in the process, one
has to compute the universal Gr\"obner basis of the ideal associated with the
design matrix for a model which is, in general, larger than any reduced
Gr\"obner basis. Thus, this is also infeasible in small- and medium-sized
problems. In this paper we focus on bounded two-way contingency tables under
independence model and show that if these bounds on cells are positive, i.e.,
they are not structural zeros, the set of basic moves of all
minors connects all tables with given margins. We end this paper with an open
problem that if we know the given margins are positive, we want to find the
necessary and sufficient condition on the set of structural zeros so that the
set of basic moves of all minors connects all incomplete
contingency tables with given margins.Comment: 22 pages. It will appear in the Annals of the Institution of
Statistical Mathematic
Minimal and minimal invariant Markov bases of decomposable models for contingency tables
We study Markov bases of decomposable graphical models consisting of
primitive moves (i.e., square-free moves of degree two) by determining the
structure of fibers of sample size two. We show that the number of elements of
fibers of sample size two are powers of two and we characterize primitive moves
in Markov bases in terms of connected components of induced subgraphs of the
independence graph of a hierarchical model. This allows us to derive a complete
description of minimal Markov bases and minimal invariant Markov bases for
decomposable models.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ207 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Algebraic method for independence model of two-way contingency tables
The main purpose of the study is to propose an algebraic method to obtain the set of all independence models of I×J two-way contingency tables with the same row sums and column sums which is called fiber in algebraic statistics. This method involves solving a system of linear algebraic equations that only rely on row sums and column sums of the I×J two-way contingency table. The MATLAB software was used to solve this system. The effectiveness of the purposed method is illustrated by applying to a contingency table of agriculture teachers’ perception of secondary school agriculture
Blurring and deblurring digital images using the dihedral group
A new method of blurring and deblurring digital images is presented. The approach is based on using new filters generating from average filter and H-filters using the action of the dihedral group. These filters are called HB-filters; used to cause a motion blur and then deblurring affected images. Also, enhancing images using HB-filters is presented as compared to other methods like Average, Gaussian, and Motion. Results and analysis show that the HB-filters are better in peak signal to noise ratio (PSNR) and RMSE
Algebraic and Geometric Properties of Hierarchical Models
In this dissertation filtrations of ideals arising from hierarchical models in statistics related by a group action are are studied. These filtrations lead to ideals in polynomial rings in infinitely many variables, which require innovative tools. Regular languages and finite automata are used to prove and explicitly compute the rationality of some multivariate power series that record important quantitative information about the ideals. Some work regarding Markov bases for non-reducible models is shown, together with advances in the polyhedral geometry of binary hierarchical models
Noetherianity up to symmetry
These lecture notes for the 2013 CIME/CIRM summer school Combinatorial
Algebraic Geometry deal with manifestly infinite-dimensional algebraic
varieties with large symmetry groups. So large, in fact, that subvarieties
stable under those symmetry groups are defined by finitely many orbits of
equations---whence the title Noetherianity up to symmetry. It is not the
purpose of these notes to give a systematic, exhaustive treatment of such
varieties, but rather to discuss a few "personal favourites": exciting examples
drawn from applications in algebraic statistics and multilinear algebra. My
hope is that these notes will attract other mathematicians to this vibrant area
at the crossroads of combinatorics, commutative algebra, algebraic geometry,
statistics, and other applications.Comment: To appear in Springer's LNM C.I.M.E. series; several typos fixe
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