Journal of Algebraic Statistics

Letter from the Editors

2022-03-06T07:11:27-05:00

“The present moment seems a very appropriate one to launch a new journal on Algebraic Statistics”
Fabrizio Catanese, Editor of the Journal of Algebraic Geometry

Many classical statistical models, such as Gaussian models from multivariate statistics and models for discrete random variables have algebraic structure in their parameter spaces. Algebraic statistics is a young, vibrant, quickly growing, and active discipline focused on the applications of algebraic geometry and its computational tools in statistical models. It has been applied in disclosure limitation, design of experiments, hypothesis testing for log-linear models, maximum likelihood estimation, and approximations to Bayesian integrals, and in the last decade it finds its applications in several other areas, such as computational biology, chemical networks, and finance. For example, the algebraic approach to phylogenetics is one of the subdisciplines of algebraic statistics, which has seen many applications in computational biology. Besides phylogenetics, this approach has seen biological applications in inferring the progression to drug resistance in HIV, determining the parametric behaviors of sequence alignments, and studying the geometry of fitness landscapes.

Example applications include mathematical modeling, operations research, model identification, system analysis and design, system verification, and system synthesis. Application areas include systems biology, genomics, proteomics, phylogenetics and evolutionary biology, chemical networks, finance, engineering, to name a few. Areas in algebra include polynomial methods, commutative algebra and algebraic geometry, group theory, string rewriting, automated reasoning, automata theory, formal languages, combinatorics, graph theory, and artificial intelligence, among others.

We, the editorial staff, believe this is an important new field, which has the potential to provide immeasurable benefits to all of mankind. However, as with all fields, a common intersection forum is needed. One in which researchers can share their visions, challenges, and victories. A forum which makes it easy for the neophytes - as well as the simply curious - to explore the field, and the experts to discuss how best to help inspire and teach them. Stemming from the mission and great success of the European Journal of Pure and Applied (EJPAM), we believe we can serve this research community by providing this forum, Journal of Algebraic Statistics (JAlgStat).

It is our great pleasure to invite original research papers and critical survey articles for submission to the Journal of Algebraic Statistics. The JAlgStat is intended to foster communication and interaction between researchers who apply algebraic methods and symbolic computation to various problems in statistics and the life sciences. Algebraists, biologists, statisticians, computer scientists, and scientists who are interested in algebraic methods to solve their problems need to develop a common language to discuss problems of common interest. While several of mathematicians, statisticians, computer scientists, and life scientists involved in this research area already have close collaborations with researchers in other fields, many others do not. Journal of Algebraic Statistics provides a much needed opportunity to further these connections. We will exert every effort to handle submissions swiftly and fairly to provide rapid reviews, taking advantage of the sophisticated electronic infrastructure and facilities provided by the Article Management System.

We all wish Journal of Algebraic Statistics -the first journal in the field- every success in the contemporary scientific world. Please join us in thanking those who have chosen to honor our inaugural issue with their research; with all your help, we can make this a truly valuable research forum. We are also especially grateful to our all Advisory and Editorial Board members, editorial message senders and all authors who have contributed to this inaugural volume.

Unsal Tekir, Eyup Cetin Editors-in-Chief

Ruriko Yoshida, Sonja Petrovic Associate Editors

J. Andrew Howe, Baris Kiremitci Production Editors

Editorial Messages Journal of Algebraic Statistics

2022-03-06T07:52:40-05:00

Just as it has been continually happening in the world of mathematical sciences, the group of mathematical scientists led by (for example) Professor Eyup Cetin and his colleagues (who are responsible for the remarkably successful journal, The European Journal of Pure and Applied Mathematics) have apparently broken the boundaries between pure and applied mathematics by establishing a new journal, the Journal of Algebraic Statistics. I am sure that both the mathematical as well as statistical communities at large will heartily welcome such an interesting and potentially useful addition to the list of broadbased journals in the mathematical sciences.

I do sincerely wish the Journal of Algebraic Statistics every success in its endeavor to attract and publish high-quality papers which are aimed essentially and substantially at significantly bridging the gaps between the various areas within the disciplines of the mathematical and statistical sciences.

Hari M. Srivastava, University of Victoria , Canada

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The present moment seems a very appropriate one to launch a new journal on algebraic statistics. In fact many fields of mathematics are considering with interest concrete applications of well developed theories towards the solution of problems coming from everyday science and technology. This applies in particular to certain branches of algebraic geometry. I wish to the new journal a good success

Fabrizio Catanese, University of Bayreuth, Germany

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Algebraic Statistics is a rapidly growing discipline, and presents many opportunities for research and applications. The newly launched Journal of Algebraic Statistics will bring together researchers working on problems in this area and as such is highly welcome. I congratulate the Editors for bringing it out and wish them and the journal success.

Arjun K. Gupta, Bowling Green State University, USA

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Many people think that Algebra and Statistics have really nothing in common, except some applications of Linear Algebra to Statistics. This is far away from the truth. A main purpose of this new journal is to uncover the numerous connections between these fields, and hence to advance both Statistics and Algebra.

Many of these connections were not intended in the beginning and came as pleasant surprises. The applications go in both directions and bring new ideas and method from one area to the other.

I want to congratulate the founders and Editors-in-Chief of this new journal for establishing it and for promoting the study of this fascinating interplay.

Gunter F. Pilz, Johannes Kepler University, Austria

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Modern Algebra is central to all fields of mathematics, and impacts engineering fields such as coding theory and cryptography. Likewise, Statistics touches on all aspects of modern science. The intersection of these two fields, Algebraic Statistics, is becoming important in a number of application areas in the form of random walks on groups, random matrix theory, multivariate statistical analysis, geometric probability, and topological analysis of large data sets. Though efforts in these different areas have been published over the past half century in a variety of venues, having one place to go where readers interested in the theory and application of both Algebra and Statistics will enable significant advances by providing a hub from which connections to the broader literature can be more easily made. The Journal of Algebraic Statistics has the potential to be such a forum, and I look forward to the success of this new journal.

Gregory S. Chirikjian, Johns Hopkins University, USA

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I would like to congratulate the editorial team for the inaugural issue of the Journal of Algebraic Statistics. Algebraic Statistics is the emerging new field focused on the applications of algebraic geometry and its computational tools in the study of statistical models.

Algebraic Statistics is built around the observation that many statistical models are (semi)-algebraic sets. The study of the geometry and equations of these algebraic sets can be useful for making statistical inferences, thus the areas of interest include categorical data analysis, experimental design, graphical models, maximum likelihood estimation, and Bayesian methods.

Also some work shows applications of Algebraic Statistics to problems in computational biology. Nearly all statistical models for discrete random variables fall into the category above, and many models for continuous random variables can be treated this way as well. Thus, it is likely that these algebraic statistical techniques will be useful in many more areas of computational and mathematical biology such as systems biology, evolutionary biology, functional genomics, bioinformatics, and epidemiology.

Algebraic Statistics is an exciting field and attracts many younger researchers. Thus I wish for the Journal of Algebraic Statistics to be very successful.

Ruriko Yoshida, University of Kentucky, USA

Abstract Algebra in Statistics

2022-03-06T08:03:56-05:00

In this note we show how some specific classes of algebraic structures (“planar nearrings”) give rise to efficient Balanced Incomplete Block Designs, which in turn can excellently be used in statistical experiments.

Open Problems on Connectivity of Fibers with Positive Margins in Multi-dimensional Contingency Tables

2022-03-06T06:20:59-05:00

Diaconis-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.

Position and Orientation Distributions for Non-Reversal Random Walks using Space-Group Fourier Transforms

2022-03-06T08:08:10-05:00

This paper presents an efficient group-theoretic approach for computing the statistics of non-reversal random walks (NRRW) on lattices. These framed walks evolve on proper crystallographic space groups. In a previous paper we introduced a convolution method for computing the statistics of NRRWs in which the convolution product is defined relative to the space-group operation. Here we use the corresponding concept of the fast Fourier transform for functions on crystallographic space groups together with a non-Abelian version of the convolution theorem. We develop the theory behind this technique and present numerical results for two-dimensional and three-dimensional lattices (square, cubic and diamond). In order to verify our results, the statistics of the end-to-end distance and the probability of ring closure are calculated and compared with results obtained in the literature for the random walks for which closed-form expressions exist.