Luis David Garc´Ia Puente
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A Computer Algebra System for R: Macaulay2 and the M2r Package
A computer algebra system for R: Macaulay2 and the m2r package David Kahle∗1, Christopher O'Neilly2, and Jeff Sommarsz3 1Department of Statistical Science, Baylor University 2Department of Mathematics, University of California, Davis 3Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago Abstract Algebraic methods have a long history in statistics. The most prominent manifestation of modern algebra in statistics can be seen in the field of algebraic statistics, which brings tools from commutative algebra and algebraic geometry to bear on statistical problems. Now over two decades old, algebraic statistics has applications in a wide range of theoretical and applied statistical domains. Nevertheless, algebraic statistical methods are still not mainstream, mostly due to a lack of easy off-the-shelf imple- mentations. In this article we debut m2r, an R package that connects R to Macaulay2 through a persistent back-end socket connection running locally or on a cloud server. Topics range from basic use of m2r to applications and design philosophy. 1 Introduction Algebra, a branch of mathematics concerned with abstraction, structure, and symmetry, has a long history of applications in statistics. For example, Pearson's early work on method of moments estimation in mixture models ultimately involved systems of polynomial equations that he painstakingly and remarkably solved by hand (Pearson, 1894; Am´endolaet al., 2016). Fisher's work in design was strongly algebraic and combinato- rial, focusing on topics such as Latin squares (Fisher, 1934). Invariance and equivariance continue to form a major pillar of mathematical statistics through the lens of location-scale families (Pitman, 1939; Bondesson, 1983; Lehmann and Romano, 2005). -
Algebraic Statistics Tutorial I Alleles Are in Hardy-Weinberg Equilibrium, the Genotype Frequencies Are
Example: Hardy-Weinberg Equilibrium Suppose a gene has two alleles, a and A. If allele a occurs in the population with frequency θ (and A with frequency 1 − θ) and these Algebraic Statistics Tutorial I alleles are in Hardy-Weinberg equilibrium, the genotype frequencies are P(X = aa)=θ2, P(X = aA)=2θ(1 − θ), P(X = AA)=(1 − θ)2 Seth Sullivant The model of Hardy-Weinberg equilibrium is the set North Carolina State University July 22, 2012 2 2 M = θ , 2θ(1 − θ), (1 − θ) | θ ∈ [0, 1] ⊂ ∆3 2 I(M)=paa+paA+pAA−1, paA−4paapAA Seth Sullivant (NCSU) Algebraic Statistics July 22, 2012 1 / 32 Seth Sullivant (NCSU) Algebraic Statistics July 22, 2012 2 / 32 Main Point of This Tutorial Phylogenetics Problem Given a collection of species, find the tree that explains their history. Many statistical models are described by (semi)-algebraic constraints on a natural parameter space. Generators of the vanishing ideal can be useful for constructing algorithms or analyzing properties of statistical model. Two Examples Phylogenetic Algebraic Geometry Sampling Contingency Tables Data consists of aligned DNA sequences from homologous genes Human: ...ACCGTGCAACGTGAACGA... Chimp: ...ACCTTGGAAGGTAAACGA... Gorilla: ...ACCGTGCAACGTAAACTA... Seth Sullivant (NCSU) Algebraic Statistics July 22, 2012 3 / 32 Seth Sullivant (NCSU) Algebraic Statistics July 22, 2012 4 / 32 Model-Based Phylogenetics Phylogenetic Models Use a probabilistic model of mutations Assuming site independence: Parameters for the model are the combinatorial tree T , and rate Phylogenetic Model is a latent class graphical model parameters for mutations on each edge Vertex v ∈ T gives a random variable Xv ∈ {A, C, G, T} Models give a probability for observing a particular aligned All random variables corresponding to internal nodes are latent collection of DNA sequences ACCGTGCAACGTGAACGA Human: X1 X2 X3 Chimp: ACGTTGCAAGGTAAACGA Gorilla: ACCGTGCAACGTAAACTA Y 2 Assuming site independence, data is summarized by empirical Y 1 distribution of columns in the alignment. -
Elect Your Council
Volume 41 • Issue 3 IMS Bulletin April/May 2012 Elect your Council Each year IMS holds elections so that its members can choose the next President-Elect CONTENTS of the Institute and people to represent them on IMS Council. The candidate for 1 IMS Elections President-Elect is Bin Yu, who is Chancellor’s Professor in the Department of Statistics and the Department of Electrical Engineering and Computer Science, at the University 2 Members’ News: Huixia of California at Berkeley. Wang, Ming Yuan, Allan Sly, The 12 candidates standing for election to the IMS Council are (in alphabetical Sebastien Roch, CR Rao order) Rosemary Bailey, Erwin Bolthausen, Alison Etheridge, Pablo Ferrari, Nancy 4 Author Identity and Open L. Garcia, Ed George, Haya Kaspi, Yves Le Jan, Xiao-Li Meng, Nancy Reid, Laurent Bibliography Saloff-Coste, and Richard Samworth. 6 Obituary: Franklin Graybill The elected Council members will join Arnoldo Frigessi, Steve Lalley, Ingrid Van Keilegom and Wing Wong, whose terms end in 2013; and Sandrine Dudoit, Steve 7 Statistical Issues in Assessing Hospital Evans, Sonia Petrone, Christian Robert and Qiwei Yao, whose terms end in 2014. Performance Read all about the candidates on pages 12–17, and cast your vote at http://imstat.org/elections/. Voting is open until May 29. 8 Anirban’s Angle: Learning from a Student Left: Bin Yu, candidate for IMS President-Elect. 9 Parzen Prize; Recent Below are the 12 Council candidates. papers: Probability Surveys Top row, l–r: R.A. Bailey, Erwin Bolthausen, Alison Etheridge, Pablo Ferrari Middle, l–r: Nancy L. Garcia, Ed George, Haya Kaspi, Yves Le Jan 11 COPSS Fisher Lecturer Bottom, l–r: Xiao-Li Meng, Nancy Reid, Laurent Saloff-Coste, Richard Samworth 12 Council Candidates 18 Awards nominations 19 Terence’s Stuff: Oscars for Statistics? 20 IMS meetings 24 Other meetings 27 Employment Opportunities 28 International Calendar of Statistical Events 31 Information for Advertisers IMS Bulletin 2 . -
Luis David Garcıa Puente
Luis David Garc´ıa Puente Department of Mathematics and Statistics (936) 294-1581 Sam Houston State University [email protected] Huntsville, TX 77341–2206 http://www.shsu.edu/ldg005/ Professional Preparation Universidad Nacional Autonoma´ de Mexico´ (UNAM) Mexico City, Mexico´ B.S. Mathematics (with Honors) 1999 Virginia Polytechnic Institute and State University Blacksburg, VA Ph.D. Mathematics 2004 – Advisor: Reinhard Laubenbacher – Dissertation: Algebraic Geometry of Bayesian Networks University of California, Berkeley Berkeley, CA Postdoctoral Fellow Summer 2004 – Mentor: Lior Pachter Mathematical Sciences Research Institute (MSRI) Berkeley, CA Postdoctoral Fellow Fall 2004 – Mentor: Bernd Sturmfels Texas A&M University College Station, TX Visiting Assistant Professor 2005 – 2007 – Mentor: Frank Sottile Appointments Colorado College Colorado Springs, CO Professor of Mathematics and Computer Science 2021 – Sam Houston State University Huntsville, TX Professor of Mathematics 2019 – 2021 Sam Houston State University Huntsville, TX Associate Department Chair Fall 2017 – 2021 Sam Houston State University Huntsville, TX Associate Professor of Mathematics 2013 – 2019 Statistical and Applied Mathematical Sciences Institute Research Triangle Park, NC SAMSI New Researcher fellowship Spring 2009 Sam Houston State University Huntsville, TX Assistant Professor of Mathematics 2007 – 2013 Virginia Bioinformatics Institute (Virginia Tech) Blacksburg, VA Graduate Research Assistant Spring 2004 Virginia Polytechnic Institute and State University Blacksburg, -
Algebraic Statistics of Design of Experiments
Thammasat University, Bangkok Algebraic Statistics of Design of Experiments Giovanni Pistone Bangkok, March 14 2012 The talk 1. A short history of AS from a personal perspective. See a longer account by E. Riccomagno in • E. Riccomagno, Metrika 69(2-3), 397 (2009), ISSN 0026-1335, http://dx.doi.org/10.1007/s00184-008-0222-3. 2. The algebraic description of a designed experiment. I use the tutorial by M.P. Rogantin: • http://www.dima.unige.it/~rogantin/AS-DOE.pdf 3. Topics from the theory, cfr. the course teached by E. Riccomagno, H. Wynn and Hugo Maruri-Aguilar in 2009 at the Second de Brun Workshop in Galway: • http://hamilton.nuigalway.ie/DeBrunCentre/ SecondWorkshop/online.html. CIMPA Ecole \Statistique", 1980 `aNice (France) Sumate Sompakdee, GP, and Chantaluck Na Pombejra First paper People • Roberto Fontana, DISMA Politecnico di Torino. • Fabio Rapallo, Universit`adel Piemonte Orientale. • Eva Riccomagno DIMA Universit`adi Genova. • Maria Piera Rogantin, DIMA, Universit`adi Genova. • Henry P. Wynn, LSE London. AS and DoE: old biblio and state of the art Beginning of AS in DoE • G. Pistone, H.P. Wynn, Biometrika 83(3), 653 (1996), ISSN 0006-3444 • L. Robbiano, Gr¨obnerBases and Statistics, in Gr¨obnerBases and Applications (Proc. of the Conf. 33 Years of Gr¨obnerBases), edited by B. Buchberger, F. Winkler (Cambridge University Press, 1998), Vol. 251 of London Mathematical Society Lecture Notes, pp. 179{204 • R. Fontana, G. Pistone, M. Rogantin, Journal of Statistical Planning and Inference 87(1), 149 (2000), ISSN 0378-3758 • G. Pistone, E. Riccomagno, H.P. Wynn, Algebraic statistics: Computational commutative algebra in statistics, Vol. -
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Arxiv:2011.03572V2 [Math.CO] 17 Dec 2020
ORDER-FORCING IN NEURAL CODES R. AMZI JEFFS, CAITLIN LIENKAEMPER, AND NORA YOUNGS Abstract. Convex neural codes are subsets of the Boolean lattice that record the inter- section patterns of convex sets in Euclidean space. Much work in recent years has focused on finding combinatorial criteria on codes that can be used to classify whether or not a code is convex. In this paper we introduce order-forcing, a combinatorial tool which recognizes when certain regions in a realization of a code must appear along a line segment between other regions. We use order-forcing to construct novel examples of non-convex codes, and to expand existing families of examples. We also construct a family of codes which shows that a dimension bound of Cruz, Giusti, Itskov, and Kronholm (referred to as monotonicity of open convexity) is tight in all dimensions. 1. Introduction A combinatorial neural code or simply neural code is a subset of the Boolean lattice 2[n], where [n] := f1; 2; : : : ; ng. A neural code is called convex if it records the intersection pattern of a collection of convex sets in Rd. More specifically, a code C ⊆ 2[n] is open convex if there d exists a collection of convex open sets U = fU1;:::;Ung in R such that \ [ σ 2 C , Ui n Uj 6= ?: i2σ j2 =σ T S U The region i2σ Ui n j2 =σ Uj is called the atom of σ in U, and is denoted Aσ . The collection U is called an open realization of C, and the smallest dimension d in which one can find an open realization is called the open embedding dimension of C, denoted odim(C). -
Matrix Model Superpotentials and ADE Singularities
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications, Department of Mathematics Mathematics, Department of 2008 Matrix model superpotentials and ADE singularities Carina Curto University of Nebraska - Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/mathfacpub Part of the Mathematics Commons Curto, Carina, "Matrix model superpotentials and ADE singularities" (2008). Faculty Publications, Department of Mathematics. 34. https://digitalcommons.unl.edu/mathfacpub/34 This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications, Department of Mathematics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. c 2008 International Press Adv. Theor. Math. Phys. 12 (2008) 355–406 Matrix model superpotentials and ADE singularities Carina Curto Department of Mathematics, Duke University, Durham, NC, USA CMBN, Rutgers University, Newark, NJ, USA [email protected] Abstract We use F. Ferrari’s methods relating matrix models to Calabi–Yau spaces in order to explain much of Intriligator and Wecht’s ADE classifi- cation of N = 1 superconformal theories which arise as RG fixed points of N = 1 SQCD theories with adjoints. We find that ADE superpoten- tials in the Intriligator–Wecht classification exactly match matrix model superpotentials obtained from Calabi–Yau with corresponding ADE sin- gularities. Moreover, in the additional O, A, D and E cases we find new singular geometries. These “hat” geometries are closely related to their ADE counterparts, but feature non-isolated singularities. As a byproduct, we give simple descriptions for small resolutions of Gorenstein threefold singularities in terms of transition functions between just two co-ordinate charts. -
Duke University News May 10, 2005
MathDuke University News May 10, 2005 Mathematics since 1978, she has been a major Events participant in Calculus reform, and has been very active in the Women in Science and Engineering Math Department Party (WISE) program.This program o ers academic, nancial and social support to female undergrad- A large group of undergraduate and gradu- uate students in those elds. ate students mixed with the faculty at the an- nual mathematics department party on April 28. Jordan Ellenberg On the day after classes ended, members of the Duke mathematical community packed the de- In a March 1 talk scheduled by DUMU, Jordan partment lounge to enjoy sandwiches and con- Ellenberg of Princeton University and the Uni- versation in an informal setting. This party is the versity of Wisconsin gave an enjoyable talk relat- traditional occasion for the faculty to honor the ing the card game Set to questions in combinato- graduating students, contest participants and re- rial geometry. A dozen DUMU students enjoyed search students for their hard work and accom- chatting with him at dinner after his talk. plishments. Many math majors received the Set is a simple but addictive card game played 2005 Duke Math shirt, some received certi cates with a special 81-card deck. A standard "folk- and a few took home generous cash prizes. A lore question" among players of this game is: good time was had by all. what is the largest number of cards that can be on the table which do not allow a legal play?He Graduation Luncheon explained how this question, which seems to be At noon, immediately after Graduation Exer- about cards, is actually about geometry over a - cises on Sunday May 15, senior math majors nite eld.He presented several results and ended and their families will meet in the LSRC dining with a number of open mathematical problems room. -
A Widely Applicable Bayesian Information Criterion
JournalofMachineLearningResearch14(2013)867-897 Submitted 8/12; Revised 2/13; Published 3/13 A Widely Applicable Bayesian Information Criterion Sumio Watanabe [email protected] Department of Computational Intelligence and Systems Science Tokyo Institute of Technology Mailbox G5-19, 4259 Nagatsuta, Midori-ku Yokohama, Japan 226-8502 Editor: Manfred Opper Abstract A statistical model or a learning machine is called regular if the map taking a parameter to a prob- ability distribution is one-to-one and if its Fisher information matrix is always positive definite. If otherwise, it is called singular. In regular statistical models, the Bayes free energy, which is defined by the minus logarithm of Bayes marginal likelihood, can be asymptotically approximated by the Schwarz Bayes information criterion (BIC), whereas in singular models such approximation does not hold. Recently, it was proved that the Bayes free energy of a singular model is asymptotically given by a generalized formula using a birational invariant, the real log canonical threshold (RLCT), instead of half the number of parameters in BIC. Theoretical values of RLCTs in several statistical models are now being discovered based on algebraic geometrical methodology. However, it has been difficult to estimate the Bayes free energy using only training samples, because an RLCT depends on an unknown true distribution. In the present paper, we define a widely applicable Bayesian information criterion (WBIC) by the average log likelihood function over the posterior distribution with the inverse temperature 1/logn, where n is the number of training samples. We mathematically prove that WBIC has the same asymptotic expansion as the Bayes free energy, even if a statistical model is singular for or unrealizable by a statistical model. -
Combinatorial Geometry of Threshold-Linear Networks
COMBINATORIAL GEOMETRY OF THRESHOLD-LINEAR NETWORKS CARINA CURTO, CHRISTOPHER LANGDON, AND KATHERINE MORRISON Abstract. The architecture of a neural network constrains the potential dynamics that can emerge. Some architectures may only allow for a single dynamic regime, while others display a great deal of flexibility with qualitatively different dynamics that can be reached by modulating connection strengths. In this work, we develop novel mathematical techniques to study the dynamic con- straints imposed by different network architectures in the context of competitive threshold-linear n networks (TLNs). Any given TLN is naturally characterized by a hyperplane arrangement in R , and the combinatorial properties of this arrangement determine the pattern of fixed points of the dynamics. This observation enables us to recast the question of network flexibility in the language of oriented matroids, allowing us to employ tools and results from this theory in order to charac- terize the different dynamic regimes a given architecture can support. In particular, fixed points of a TLN correspond to cocircuits of an associated oriented matroid; and mutations of the matroid correspond to bifurcations in the collection of fixed points. As an application, we provide a com- plete characterization of all possible sets of fixed points that can arise in networks through size n = 3, together with descriptions of how to modulate synaptic strengths of the network in order to access the different dynamic regimes. These results provide a framework for studying the possible computational roles of various motifs observed in real neural networks. 1. Introduction An n-dimensional threshold-linear network (TLN) is a continuous dynamical system given by the nonlinear equations: n X (1)x _ i = −xi + Wijxj + bi i = 1; : : : ; n j=i + wherex _ i = dxi=dt denotes the time derivative, bi and Wij are all real numbers, Wii = 0 for each i, and [x]+ = maxfx; 0g. -
Homotopy Theory and Neural Information Networks
Homotopy Theory and Neural Information Networks Matilde Marcolli (Caltech, University of Toronto & Perimeter Institute) Focus Program on New Geometric Methods in Neuroscience Fields Institute, Toronto, 2020 Matilde Marcolli (Caltech, University of Toronto & Perimeter Institute)Homotopy Theory and Neural Information Networks • based on ongoing joint work with Yuri I. Manin (Max Planck Institute for Mathematics) • related work: M. Marcolli, Gamma Spaces and Information, Journal of Geometry and Physics, 140 (2019), 26{55. Yu.I. Manin and M. Marcolli, Nori diagrams and persistent homology, arXiv:1901.10301, to appear in Mathematics of Computer Science. • building mathematical background for future joint work with Doris Tsao (Caltech neuroscience) • this work partially supported by FQXi FFF Grant number: FQXi-RFP-1804, SVCF grant number 2018-190467 Matilde Marcolli (Caltech, University of Toronto & Perimeter Institute)Homotopy Theory and Neural Information Networks Motivation N.1: Nontrivial Homology Kathryn Hess' applied topology group at EPFL: topological analysis of neocortical microcircuitry (Blue Brain Project) formation of large number of high dimensional cliques of neurons (complete graphs on N vertices with a directed structure) accompanying response to stimuli formation of these structures is responsible for an increasing amount of nontrivial Betti numbers and Euler characteristics, which reaches a peak of topological complexity and then fades proposed functional interpretation: this peak of non-trivial homology is necessary for the -
Algebraic Statistics 2015 8-11 June, Genoa, Italy
ALGEBRAIC STATISTICS 2015 8-11 JUNE, GENOA, ITALY INVITED SPEAKERS TUTORIALS Elvira Di Nardo - University of Basilicata Luis García-Puente - Sam Houston State University Thomas Kahle - OvGU Magdeburg Luigi Malagò - Shinshu University and INRIA Saclay Sonja Petrović - Illinois Institute of Technology Giovanni Pistone - Collegio Carlo Alberto Piotr Zwiernik - University of Genoa Jim Q. Smith - University of Warwick Bernd Sturmfels - University of California Berkeley IMPORTANT DATES 30 April - deadline for abstract submissions website: http://www.dima.unige.it/~rogantin/AS2015/ 31 May - registration closed 2 Monday, June 8, 2015 Invited talk B. Sturmfels Exponential varieties. Page 7 Talks C. Long, S. Sullivant Tying up loose strands: defining equations of the strand symmetric model. Page 8 L. Robbiano From factorial designs to Hilbert schemes. Page 10 E. Robeva Decomposing tensors into frames. Page 11 Tutorial L. García-Puente R package for algebraic statistics. Page 12 Tuesday, June 9, 2015 Invited talks T. Kahle Algebraic geometry of Poisson regression. Page 13 S. Petrovic´ What are shell structures of random networks telling us? Page 14 Talks M. Compagnoni, R. Notari, A. Ruggiu, F. Antonacci, A. Sarti The geometry of the statistical model for range-based localization. Page 15 R. H. Eggermont, E. Horobe¸t, K. Kubjas Matrices of nonnegative rank at most three. Page 17 A. Engström, P. Norén Algebraic graph limits. Page 18 E. Gross, K. Kubjas Hypergraph decompositions and toric ideals. Page 20 3 J. Rodriguez, B. Wang The maximum likelihood degree of rank 2 matrices via Euler characteristics. Page 21 M. Studený, T. Kroupa A linear-algebraic criterion for indecomposable generalized permutohedra.