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 Title
 Properties of semielementary imsets as sums of elementary imsets
 Creator
 Kashimura, Takuya, Sei, Tomonari, Takemura, Akimichi, Tanaka, Kentaro
 Date
 2011, 2011
 Description

We study properties of semielementary imsets and elementary imsets introduced by Studeny [10]. The rules of the semigraphoid axiom ...
Show moreWe study properties of semielementary imsets and elementary imsets introduced by Studeny [10]. The rules of the semigraphoid axiom (decomposition, weak union and contraction) for conditional independence statements can be translated into a simple identity among three semielementary imsets. By recursively applying the identity, any semielementary imset can be written as a sum of elementary imsets, which we call a representation of the semielementary imset. A semielementary imset has many representations. We study properties of the set of possible representations of a semielementary imset and prove that all representations are connected by relations among four elementary imsets.
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 Journal of Algebraic Statistics
 Title
 Markov bases for twoway changepoint models of ladder determinantal tables
 Creator
 Aoki, Satoshi, Hibi, Takayuki
 Date
 2017, 20170208
 Description

To evaluate the goodnessoffit of a statistical model to given data, calculating a conditional p value by a Markov chain Monte Carlo method...
Show moreTo evaluate the goodnessoffit of a statistical model to given data, calculating a conditional p value by a Markov chain Monte Carlo method is one of the effective approaches. For this purpose, a Markov basis plays an important role because it guarantees the connectivity of the chain, which is needed for unbiasedness of the estimation, and therefore is investigated in various settings such as incomplete tables or subtable sum constraints. In this paper, we consider the twoway changepoint model for the ladder determinantal table, which is an extension of these two previous works, i.e., works on incomplete tables by Aoki and Takemura (2005, J. Stat. Comput. Simulat.) and subtable some constraints by Hara, Takemura and Yoshida (2010, J. Pure Appl. Algebra). Our main result is based on the theory of Gr ?obner basis for the distributive lattice. We give a numerical example for actual data.
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 Journal of Algebraic Statistics
 Title
 Detecting epistasis via Markov bases
 Creator
 Malaspinas, AnnaSapfo, Uhler, Caroline
 Date
 2011, 2011
 Description

Rapid research progress in genotyping techniques have allowed large genomewide association studies. Existing methods often focus on...
Show moreRapid research progress in genotyping techniques have allowed large genomewide association studies. Existing methods often focus on determining associations between single loci and a specific phenotype. However, a particular phenotype is usually the result of complex relationships between multiple loci and the environment. In this paper, we describe a twostage method for detecting epistasis by combining the traditionally used singlelocus search with a search for multiway interactions. Our method is based on an extended version of Fisher’s exact test. To perform this test, a Markov chain is constructed on the space of multidimensional contingency tables using the elements of a Markov basis as moves. We test our method on simulated data and compare it to a twostage logistic regression method and to a fully Bayesian method, showing that we are able to detect the interacting loci when other methods fail to do so. Finally, we apply our method to a genomewide data set consisting of 685 dogs and identify epistasis associated with canine hair length for four pairs of single nucleotide polymorphisms (SNPs).
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 Journal of Algebraic Statistics
 Title
 Uncovering Proximity of Chromosome Territories using Classical Algebraic Statistics
 Creator
 Arsuaga, Javier, Heskia, Ido, Hosten, Serkan, Maskalevich, Tatiana
 Date
 2015, 20151109
 Description

Exchange type chromosome aberrations (ETCAs) are rearrangements of the genome that occur when chromosomes break and the resulting fragments...
Show moreExchange type chromosome aberrations (ETCAs) are rearrangements of the genome that occur when chromosomes break and the resulting fragments rejoin with fragments from other chromosomes or from other regions within the same chromosome. ETCAs are commonly observed in cancer cells and in cells exposed to radiation. The frequency of these chromosome rearrangements is correlated with their spatial proximity, therefore it can be used to infer the three dimensional organization of the genome. Extracting statistical significance of spatial proximity from cancer and radiation data has remained somewhat elusive because of the sparsity of the data. We here propose a new approach to study the three dimensional organization of the genome using algebraic statistics. We test our method on a published data set of irradiated human blood lymphocyte cells. We provide a rigorous method for testing the overall organization of the genome, and in agreement with previous results we find a random relative positioning of chromosomes with the exception of the chromosome pairs {1,22} and {13,14} that have a significantly larger number of ETCAs than the rest of the chromosome pairs suggesting their spatial proximity. We conclude that algebraic methods can successfully be used to analyze genetic data and have potential applications to larger and more complex data sets.
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 Journal of Algebraic Statistics
 Title
 Markov degree of configurations defined by fibers of a configuration
 Creator
 Koyama, Takayuki, Ogawa, Mitsunori, Takemura, Akimichi
 Date
 2015, 20151109
 Description

We consider a series of configurations defined by fibers of a given base configuration. We prove that Markov degree of the configurations is...
Show moreWe consider a series of configurations defined by fibers of a given base configuration. We prove that Markov degree of the configurations is bounded from above by the Markov complexity of the base configuration. As important examples of base configurations we consider incidence matrices of graphs and study the maximum Markov degree of configurations defined by fibers of the incidence matrices. In particular we give a proof that the Markov degree for twoway transportation polytopes is three.
Show less  Collection
 Journal of Algebraic Statistics