Markov Bases Research Articles

Exploring the space of graphs with fixed discrete curvatures

Abstract Discrete curvatures are quantities associated to the nodes and edges of a graph that reflect the local geometry around them. These curvatures have a rich mathematical theory and they have recently found success as a tool to analyze networks across a wide range of domains. In this work, we consider the problem of constructing graphs with a prescribed set of discrete edge curvatures, and explore the space of such graphs. We address this problem in two ways: first, we develop an evolutionary algorithm to sample graphs with discrete curvatures close to a given set. We use this algorithm to explore how other network statistics vary when constrained by the discrete curvatures in the network. Second, we solve the exact reconstruction problem for the specific case of Forman–Ricci curvature. By leveraging the theory of Markov bases, we obtain a finite set of rewiring moves that connects the space of all graphs with a fixed discrete curvature.

Equivariant lattice bases

We study lattices in free abelian groups of infinite rank that are invariant under the action of the infinite symmetric group, with emphasis on finiteness of their equivariant bases. Our framework provides a new method for proving finiteness results in algebraic statistics. As an illustration, we show that every invariant lattice in Z ( N × [ c ] ) \mathbb {Z}^{(\mathbb {N}\times [c])} , where c ∈ N c\in \mathbb {N} , has a finite equivariant Graver basis. This result generalizes and strengthens several finiteness results about Markov bases in the literature.

When lattice bases are Markov bases

Sampling the fibre comprising the solutions of a linear inverse problem for count data is an important practical problem. Connectivity of the sampler is guaranteed only if a Markov basis, defining a sufficiently rich variety of sampling directions, is available. Computation of a Markov basis is typically challenging, and the mixing properties of the resulting sampler can be poor. However, for some problems a suitably chosen lattice basis will be a Markov basis. We provide an easily checkable condition for the existence of such a lattice Markov basis, and demonstrate that associated hit-and-run samplers will mix rapidly for uniform target distributions.

Haplotype frequency inference from pooled genetic data with a latent multinomial model.

In genetic association studies, haplotype data provide more refined information than data about separate genetic markers. However, large-scale studies that genotype hundreds to thousands of individuals may only provide results of pooled data. Methods for inferring haplotype frequencies from pooled genetic data that scale well with pool size rely on a normal approximation, which we observe to produce unreliable inference when applied to real data. We illustrate cases where the approximation fails, due to the normal covariance matrix being nearsingular. As an alternative to approximate methods, in this paper we propose two exact methods to infer haplotype frequencies from pooled genetic data based on a latent multinomial model, where the pooled results are considered integer combinations of latent, unobserved haplotype counts. One of our methods, latent count sampling via Markov bases, achieves approximately linear runtime with respect to pool size. Our exact methods produce more accurate inference over existing approximate methods for synthetic data and for haplotype data from the 1000 Genomes Project. We also demonstrate how our methods can be applied to time-series of pooled genetic data, as a proof of concept of how our methods are relevant to more complex hierarchical settings, such as spatiotemporal models.

Universally free numerical semigroups

A numerical semigroup is said to be universally free if it is free for any possible arrangement of its minimal generating set. In this work, we establish that toric ideals associated with universally free numerical semigroups can be generated by their set of circuits. Additionally, we provide a characterization of universally free numerical semigroups in terms of Gröbner bases. Specifically, a numerical semigroup is universally free if and only if all initial ideals of its corresponding toric ideal are complete intersections. Furthermore, we establish several equalities among the toric bases of a universally free numerical semigroup.We provide a complete characterization of 3-generated universally free numerical semigroups in terms of their minimal generating sets, and by proving the equality of certain toric bases. We compute exactly all the toric bases of a toric ideal defined by a 3-generated universally free numerical semigroup. Notably, we answer some questions posed by Tatakis and Thoma by demonstrating that toric ideals defined by 3-generated universally free numerical semigroups have a set of circuits and a universal Gröbner basis of size 3, while the universal Markov basis and the Graver basis can be arbitrarily large. We present partial results and propose several conjectures regarding universally free numerical semigroups with more than three generators.

Monte Carlo goodness-of-fit tests for degree corrected and related stochastic blockmodels

Abstract We construct Bayesian and frequentist finite-sample goodness-of-fit tests for three different variants of the stochastic blockmodel for network data. Since all of the stochastic blockmodel variants are log-linear in form when block assignments are known, the tests for the latent block model versions combine a block membership estimator with the algebraic statistics machinery for testing goodness-of-fit in log-linear models. We describe Markov bases and marginal polytopes of the variants of the stochastic blockmodel and discuss how both facilitate the development of goodness-of-fit tests and understanding of model behaviour. The general testing methodology developed here extends to any finite mixture of log-linear models on discrete data, and as such is the first application of the algebraic statistics machinery for latent-variable models.

Symmetrically Colored Gaussian Graphical Models with Toric Vanishing Ideals

A colored Gaussian graphical model is a linear concentration model in which equalities among the concentrations are specified by a coloring of an underlying graph. The model is called RCOP if this coloring is given by the edge and vertex orbits of a subgroup of the automorphism group of the graph. We show that RCOP Gaussian graphical models on block graphs are toric in the space of covariance matrices and we describe Markov bases for them. To this end, we learn more about the combinatorial structure of these models and their connection to Jordan algebras.

No-Reference Blur Assessment Based on Re-Blurring Using Markov Basis

Blur is produced in a digital image due to low pass filtering, moving objects or defocus of the camera lens during capture. Image viewers are annoyed by blur artefact and the image's perceived quality suffers as a result. The high-quality input is relevant to communication service providers and imaging product makers because it may help them improve their processes. Human-based blur assessment is time-consuming, expensive and must adhere to subjective evaluation standards. This paper presents a revolutionary no-reference blur assessment algorithm based on re-blurring blurred images using a special mask developed with a Markov basis and Laplace filter. The final blur score of blurred images has been calculated from the local variation in horizontal and vertical pixel intensity of blurred and re-blurred images. The objective scores are generated by applying proposed algorithm on the two image databases i.e., Laboratory for image and video engineering (LIVE) database and Tampere image database (TID 2013). Finally, on the basis of objective and subjective scores performance analysis is done in terms of Pearson linear correlation coefficient (PLCC), Spearman rank-order correlation coefficient (SROCC), Mean absolute error (MAE), Root mean square error (RMSE) and Outliers ratio (OR). The existing no-reference blur assessment algorithms have been used various methods for the evaluation of blur from no-reference image such as Just noticeable blur (JNB), Cumulative Probability Distribution of Blur Detection (CPBD) and Edge Model based Blur Metric (EMBM). The results illustrate that the proposed method was successful in predicting high blur scores with high accuracy as compared to existing no-reference blur assessment algorithms such as JNB, CPBD and EMBM algorithms.

On the strongly robust property of toric ideals

To every toric ideal one can associate an oriented matroid structure, consisting of a graph and another toric ideal, called bouquet ideal. The connected components of this graph are called bouquets. Bouquets are of three types; free, mixed and non-mixed. We prove that the cardinality of the following sets - the set of indispensable elements, minimal Markov bases, the Universal Markov basis and the Universal Gröbner basis of a toric ideal - depends only on the type of the bouquets and the bouquet ideal. These results enable us to introduce the strongly robust simplicial complex and show that it determines the strongly robust property. For codimension 2 toric ideals, we study the strongly robust simplicial complex and prove that robust implies strongly robust.

On robustness and related properties on toric ideals

A toric ideal is called robust if its universal Gröbner basis is a minimal set of generators, and is called generalized robust if its universal Gröbner basis equals its universal Markov basis (the union of all its minimal sets of binomial generators). Robust and generalized robust toric ideals are both interesting from both a commutative algebra and an algebraic statistics perspective. However, only a few nontrivial examples of such ideals are known. In this work, we study these properties for toric ideals of both graphs and numerical semigroups. For toric ideals of graphs, we characterize combinatorially the graphs giving rise to robust and to generalized robust toric ideals generated by quadratic binomials. As a by-product, we obtain families of Koszul rings. For toric ideals of numerical semigroups, we determine that one of its initial ideals is a complete intersection if and only if the semigroup belongs to the so-called family of free numerical semigroups. Hence, we characterize all complete intersection numerical semigroups which are minimally generated by one of its Gröbner basis and, as a consequence, all the Betti numbers of the toric ideal and its corresponding initial ideal coincide. Moreover, also for numerical semigroups, we prove that the ideal is generalized robust if and only if the semigroup has a unique Betti element and that there are only trivial examples of robust ideals. We finish the paper with some open questions.

On the relative size of toric bases

We consider the Graver basis, the universal Gröbner basis, a Markov basis and the set of the circuits of a toric ideal. Let [Formula: see text] be any two of these bases such that [Formula: see text] in general, we prove that there is no polynomial on the size or on the maximal degree of the elements of [Formula: see text] which bounds the size or the maximal degree of the elements of [Formula: see text] correspondingly.

The norm of the saturation of a binomial ideal, with applications to Markov bases

Given a pure binomial ideal I in variables x_i, we define a new measure of the complexity of the saturation of I with respect to the product of the variables x_i, which we call the norm. We give a bound on the norm in terms of easily-computed invariants of the ideal. We discuss statistical applications both practical and theoretical.

Geometrically aware dynamic Markov bases for statistical linear inverse problems

SummaryFor statistical linear inverse problems involving count data, inference typically requires sampling a latent variable with conditional support comprising of the lattice points in a convex polytope. Irreducibility of random walk samplers is guaranteed only if a sufficiently rich array of sampling directions is available. In principle, this can be achieved by finding a Markov basis of moves ab initio, but in practice doing so may be computationally infeasible. What is more, the use of a full Markov basis can lead to very poor mixing. It is far simpler to find a lattice basis of moves, which can be tailored to the overall geometry of the polytope. However, a single lattice basis generally does not connect all points in the polytope. In response, we propose a dynamic lattice basis sampler. This sampler can access a sufficient variety of sampling directions to guarantee irreducibility, but also prefers moves that are well aligned to the polytope geometry, hence promoting good mixing. The probability with which the sampler selects different bases can be tuned. We present an efficient algorithm for updating the lattice basis, obviating the need for repeated matrix inversion.

Generated Markov Basis from HHB

Generated Markov Basis from HHB

State-Dependent Kernel Selection for Conditional Sampling of Graphs

This article introduces new efficient algorithms for two problems: sampling conditional on vertex degrees in unweighted graphs, and conditional on vertex strengths in weighted graphs. The resulting conditional distributions provide the basis for exact tests on social networks and two-way contingency tables. The algorithms are able to sample conditional on the presence or absence of an arbitrary set of edges. Existing samplers based on MCMC or sequential importance sampling are generally not scalable; their efficiency can degrade in large graphs with complex patterns of known edges. MCMC methods usually require explicit computation of a Markov basis to navigate the state space; this is computationally intensive even for small graphs. Our samplers do not require a Markov basis, and are efficient both in sparse and dense settings. The key idea is to carefully select a Markov kernel on the basis of the current state of the chain. We demonstrate the utility of our methods on a real network and contingency table. Supplementary materials for this article are available online.

Analysis of an Agricluture Data Using Markov Basis for Independent Model

Algebra statistical is one of the recent topics that has seen rapid development, it is considering as the consequence of the convergence of ideas between statistics and algebra. Moreover, Algebraic statistic has many applications in many different fields, one of these important field is agriculture. In this paper, we analyse a 3x2 contingency table which contains agriculture teaching experience and perception of secondary school agriculture. The method of algebraic statistic they will be used for a mainly Markov Basis of independent model and the computation of Entropy for variance alternative contingency tables based on Tsallis Entropy. In addition, the usefulness of this paper is to illustrate agriculture data. Finally, we will find Toric ideal to generate the elements of Markov Basis.

Unboundedness of Markov complexity of monomial curves in [formula omitted] for n ≥ 4

Computing the complexity of Markov bases is an extremely challenging problem; no formula is known in general and there are very few classes of toric ideals for which the Markov complexity has been computed. A monomial curve C in A3 has Markov complexity m(C) two or three. Two if the monomial curve is complete intersection and three otherwise. Our main result shows that there is no d∈N such that m(C)≤d for all monomial curves C in A4. The same result is true even if we restrict to complete intersections. We extend this result to all monomial curves in An, where n≥4.

What is... a Markov Basis?

This short piece defines a Markov basis. The aim is to introduce the statistical concept to mathematicians.

The Action of some Subgroups of Symmetric Group S3n on An Invariant Markov basis

In this paper, we will find some subgroups H_1 and H_2 of symmetric group S_3n, n∈N and n≥2, such that the Markov basis B is H invariant for (25n^3-66n^2+41n) ×3×n - contingency tables with fixed two dimensional marginal, where B is the Markov basis. We will get another 3×n - contingency tables have same the Markov basis B by using the action of H on these contingency tables.

On the Grobner Basis of the Toric Ideal for 3 X n- Contingency Tables

In this paper, The Grobner basis of the Toric Ideal for - contingency tables related with the Markov basis B introduced by Hussein S. MH, Abdulrahman H. M in 2018 is found. Also, the Grobner basis is a reduced and universal Grobner basis are shown.