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Title: Detecting epistasis via Markov bases
Creator: Malaspinas, Anna-Sapfo, Uhler, Caroline
Date: 2011, 2011
Description: Rapid research progress in genotyping techniques have allowed large genome-wide association studies. Existing methods often focus on...
Show moreRapid research progress in genotyping techniques have allowed large genome-wide 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 two-stage method for detecting epistasis by combining the traditionally used single-locus 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 two-stage 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 genome-wide 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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Collection: Journal of Algebraic Statistics

Title: Open Problems on Connectivity of Fibers with Positive Margins in Multi-dimensional Contingency Tables
Creator: Yoshida, Ruriko
Date: 2010, 2010
Description: Diaconis-Sturmfels developed an algorithm for sampling from conditional distributions for a statistical model of discrete exponential families...
Show moreDiaconis-Sturmfels developed an algorithm for sampling from conditional distributions for a statistical model of discrete exponential families, based on the algebraic theory of toric ideals. This algorithm is applied to categorical data analysis through the notion of Markov bases. Initiated with its application to Markov chain Monte Carlo approach for testing statistical fitting of the given model, many researchers have extensively studied the structure of Markov bases for models in computational algebraic statistics. In the Markov chain Monte Carlo approach for testing statistical fitting of the given model, a Markov basis is a set of moves connecting all contingency tables satisfying the given margins. Despite the computational advances, there are applied problems where one may never be able to compute a Markov basis. In general, the number of elements in a minimal Markov basis for a model can be exponentially many. Thus, it is important to compute a reduced number of moves which connect all tables instead of computing a Markov basis. In some cases, such as logistic regression, positive margins are shown to allow a set of Markov connecting moves that are much simpler than the full Markov basis. Such a set is called a Markov subbasis with assumption of positive margins. In this paper we summarize some computations of and open problems on Markov subbases for contingency tables with assumption of positive margins under specific models as well as develop algebraic methods for studying connectivity of Markov moves with margin positivity to develop Markov sampling methods for exact conditional inference in statistical models where the Markov basis is hard to compute.
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Collection: Journal of Algebraic Statistics

Title: An Euclidean norm based criterion to assess robots’ 2D path-following performance, AS2015 Special Issue articles: This issue includes a series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics Conference held in Genoa, Italy, in June 2015. Special issue guest editors: Piotr Zwiernik and Fabio Rapallo.
Creator: Saggini, Eleonora, Torrente, Maria-Laura
Date: 2016, 2016-07-12
Description: A current need in the robotics field is the definition of methodologies for quantitatively evaluating the results of experiments. This paper...
Show moreA current need in the robotics field is the definition of methodologies for quantitatively evaluating the results of experiments. This paper contributes to this by defining a new criterion for assessing path-following tasks in the planar case, that is, evaluating the performance of robots that are required to follow a desired reference path. Such criterion comes from the study of the local differential geometry of the problem. New conditions for deciding whether or not the zero locus of a given polynomial intersects the neighbourhood of a point are defined. Based on this, new algorithms are presented and tested on both simulated data and experiments conducted at sea employing an Unmanned Surface Vehicle.
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Collection: 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, 2015-11-09
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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Collection: Journal of Algebraic Statistics

Title: Markov degree of configurations defined by fibers of a configuration
Creator: Koyama, Takayuki, Ogawa, Mitsunori, Takemura, Akimichi
Date: 2015, 2015-11-09
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 two-way transportation polytopes is three.
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Collection: Journal of Algebraic Statistics

Title: The maximum likelihood degree of Fermat hypersurfaces
Creator: Agostini, Daniele, Alberelli, Davide, Grande, Francesco, Lella, Paolo
Date: 2015, 2015-11-09
Description: We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in...
Show moreWe study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section.
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Collection: Journal of Algebraic Statistics

Title: Moment Varieties of Gaussian Mixtures, AS2015 Special Issue articles: This issue includes a series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics Conference held in Genoa, Italy, in June 2015. Special issue guest editors: Piotr Zwiernik and Fabio Rapallo.
Creator: Améndola, Carlos, Faugère, Jean-Charles, Sturmfels, Bernd
Date: 2016, 2016-07-12
Description: The points of a moment variety are the vectors of all moments up to some order, for a given family of probability distributions. We study the...
Show moreThe points of a moment variety are the vectors of all moments up to some order, for a given family of probability distributions. We study the moment varieties for mixtures of multivariate Gaussians. Following up on Pearson’s classical work from 1894, we apply current tools from computational algebra to recover the parameters from the moments. Our moment varieties extend objects familiar to algebraic geometers. For instance, the secant varieties of Veronese varieties are the loci obtained by setting all covariance matrices to zero. We compute the ideals of the 5-dimensional moment varieties representing mixtures of two univariate Gaussians, and we offer a comparison to the maximum likelihood approach.
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Collection: Journal of Algebraic Statistics

Title: On multivariable cumulant polynomial sequences with applications, AS2015 Special Issue articles: This issue includes a series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics Conference held in Genoa, Italy, in June 2015. Special issue guest editors: Piotr Zwiernik and Fabio Rapallo.
Creator: Di Nardo, Elvira
Date: 2016, 2016-07-12
Description: A new family of polynomials, called cumulant polynomial sequence, and its extension to the multivariate case is introduced relying on a purely...
Show moreA new family of polynomials, called cumulant polynomial sequence, and its extension to the multivariate case is introduced relying on a purely symbolic combinatorial method. The coefficients are cumulants, but depending on what is plugged in the indeterminates, moment sequences can be recovered as well. The main tool is a formal generalization of random sums, when a not necessarily integer-valued multivariate random index is considered. Applications are given within parameter estimations, L ?evy processes and random matrices and, more generally, problems involving multivariate functions. The connection between exponential models and multivariable Sheffer polynomial sequences offers a different viewpoint in employing the method. Some open problems end the paper.
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Collection: Journal of Algebraic Statistics

Title: Algebraic geometry of Poisson regression, AS2015 Special Issue articles: This issue includes a series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics Conference held in Genoa, Italy, in June 2015. Special issue guest editors: Piotr Zwiernik and Fabio Rapallo.
Creator: Kahle, Thomas, Oelbermann, Kai-Friederike, Schwabe, Rainer
Date: 2016, 2016-07-12
Description: Designing experiments for generalized linear models is difficult because optimal designs depend on unknown parameters. Here we investigate...
Show moreDesigning experiments for generalized linear models is difficult because optimal designs depend on unknown parameters. Here we investigate local optimality. We propose to study for a given design its region of optimality in parameter space. Often these regions are semi-algebraic and feature interesting symmetries. We demonstrate this with the Rasch Poisson counts model. For any given interaction order between the explanatory variables we give a characterization of the regions of optimality of a special saturated design. This extends known results from the case of no interaction. We also give an algebraic and geometric perspective on optimality of experimental designs for the Rasch Poisson counts model using polyhedral and spectrahedral geometry.
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Collection: Journal of Algebraic Statistics

Title: A linear-algebraic tool for conditional independence inference
Creator: Tanaka, Kentaro, Studeny, Milan, Takemura, Akimichi, Sei, Tomonari
Date: 2015, 2015-11-09
Description: In this note, we propose a new linear-algebraic method for the implication problem among conditional independence statements, which is...
Show moreIn this note, we propose a new linear-algebraic method for the implication problem among conditional independence statements, which is inspired by the factorization characterization of conditional independence. First, we give a criterion in the case of a discrete strictly positive density and relate it to an earlier linear-algebraic approach. Then, we extend the method to the case of a discrete density that need not be strictly positive. Finally, we provide a computational result in the case of six variables.
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Collection: Journal of Algebraic Statistics

Title: Mode Poset Probability Polytopes, AS2015 Special Issue articles: This issue includes a series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics Conference held in Genoa, Italy, in June 2015. Special issue guest editors: Piotr Zwiernik and Fabio Rapallo.
Creator: Montúfar, Guido, Rauh, Johannes
Date: 2016, 2016-07-12
Description: A mode of a probability distribution is an elementary event that has more probability mass than each of its direct neighbors, with respect to...
Show moreA mode of a probability distribution is an elementary event that has more probability mass than each of its direct neighbors, with respect to some vicinity structure on the set of elementary events. The mode inequalities cut out a polytope from the simplex of probability distributions. Related to this is the concept of strong modes. A strong mode is an elementary event that has more probability mass than all its direct neighbors together. The set of probability distributions with a given set of strong modes is again a polytope. We study the vertices, the facets, and the volume of such polytopes depending on the sets of (strong) modes and the vicinity structures.
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Collection: Journal of Algebraic Statistics

Title: Technology News, April 01, 1949
Creator: Illinois Institute of Technology
Date: 1949-04-01, 1949-04-01
Collection: Technology News Microfilm collection, 1928-1981

Title: Ken Isaacs with the Knowledge Box, Illinois Institute of Technology, Chicago, Illinois, 1962
Creator: Illinois Institute of Technology. Institute of Design
Date: 1962
Description: Photograph of Institute of Design visiting lecturer Ken Isaacs and the Knowledge Box.
Collection: Institute of Design records, 1937-ca. 1962

Title: Ken Isaacs with the Knowledge Box, Illinois Institute of Technology, Chicago, Illinois, 1962
Creator: Illinois Institute of Technology. Institute of Design
Date: 1962
Description: Photograph of Institute of Design visiting lecturer Ken Isaacs and the Knowledge Box.
Collection: Institute of Design records, 1937-ca. 1962

Title: Ken Isaacs with the Knowledge Box, Illinois Institute of Technology, Chicago, Illinois, 1962
Creator: Illinois Institute of Technology. Institute of Design
Date: 1962
Description: Photograph of Institute of Design visiting lecturer Ken Isaacs and the Knowledge Box.
Collection: Institute of Design records, 1937-ca. 1962

Title: Bed race, Illinois Institute of Technology, Chicago, Illinois, ca. 1980
Date: 1980
Description: Photograph of Illinois Tech fraternity and sorority members in a bed race during Greek Week. Photographer unknown. Date of photograph is...
Show morePhotograph of Illinois Tech fraternity and sorority members in a bed race during Greek Week. Photographer unknown. Date of photograph is unknown. Date listed is approximate.
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Collection: Office of Communications and Marketing photographs, 1905-1999

Title: New Bauhaus scrapbook, 1936-1938
Date: 1936-1938
Description: Thirty-seven page scrapbook with hand made fabric covered box with lower case letter b (for Bauhaus) on font in silver metal. Scrapbook (16 x...
Show moreThirty-seven page scrapbook with hand made fabric covered box with lower case letter b (for Bauhaus) on font in silver metal. Scrapbook (16 x 16”) has black plastic covers, spiral bound with manila card stock pages. Includes news clippings and tear sheets from periodicals about the opening of The New Bauhaus as well as the academic catalogue and program announcements, and a letter from Walter Gropius (5/18/1937) recommending Laszlo Moholy-Nagy as director. Creator unknown, assumed to be affiliated with the New Bauhaus.
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Collection: Institute of Design records, 1937-ca. 1962