Dynamic Markov Bases

This article presents a computational approach for generating Markov bases for multiway contingency tables whose cell counts might be constrained by fixed marginals and by lower and upper bounds. Our framework includes tables with structural zeros as a particular case. Instead of computing the entire Markov bases in an initial step, our framework finds sets of local moves that connect each table in the reference set with a set of neighbor tables. We construct a Markov chain on the reference set of tables that requires only a set of local moves at each iteration. The union of these sets of local moves forms a dynamic Markov basis. We illustrate the practicality of our algorithms in the estimation of exact p-values for a three-way table with structural zeros and a sparse eight-way table. This article has online supplementary materials.     

Markov Bases of Binary Graph Models

This paper is concerned with the topological invariant of a graph given by the maximum degree of a Markov basis element for the corresponding graph model for binary contingency tables. We describe a degree four Markov basis for the model when the underlying graph is a cycle and generalize this result to the complete bipartite graph K_2,n. We also give a combinatorial classification of degree two and three Markov basis moves as well as a Buchberger-free algorithm to compute moves of arbitrary given degree. Finally, we compute the algebraic degree of the model when the underlying graph is a forest.AMS Subject Classification: 05C99, 13P10, 62Q05.     

There are significantly more nonegative polynomials than sums of squares

We study the quantitative relationship between the cones of nonnegative polynomials, cones of sums of squares and cones of sums of even powers of linear forms. We derive bounds on the volumes (raised to the power reciprocal to the ambient dimension) of compact sections of the three cones. We show that the bounds are asymptotically exact if the degree is fixed and number of variables tends to infinity. When the degree is larger than two, it follows that there are significantly more nonnegative polynomials than sums of squares and there are significantly more sums of squares than sums of even powers of linear forms. Moreover, we quantify the exact discrepancy between the cones; from our bounds it follows that the discrepancy grows as the number of variables increases.     

Square-free integers as ideal norms

Elaborating on Katzel＇s result, Zhai obtained a sharp short interval result for the number of ways of expressing a square-free integer as sums of two squares. In this note we view Zhai＇s decom- position as one by degree 1 prime ideals in the Gaussian field and extend Zhai＇s approach to a wider range of quadratic fields etc.     

Bases and nonbases of square-free integers

A basis is a set A of nonnegative integers such that every sufficiently large integer n can be represented in the form n = a_i + a_j with a_i, a_i ∈ A. If A is a basis, but no proper subset of A is a basis, then A is a minimal basis. A nonbasis is a set of nonnegative integers that is not a basis, and a nonbasis A is maximal if every proper superset of A is a basis. In this paper, minimal bases consisting of square-free numbers are constructed, and also bases of square-free numbers no subset of which is minimal. Maximal nonbases of square-free numbers do not exist. However, nonbases of square-free numbers that are maximal with respect to the set of square-free numbers are constructed, and also nonbases of square-free numbers that are not contained in any nonbasis of square-free numbers maximal with respect to the square-free numbers.     

Estimating the number of zero-one multi-way tables via sequential importance sampling

In 2005, Chen et al. introduced a sequential importance sampling (SIS) procedure to analyze zero-one two-way tables with given fixed marginal sums (row and column sums) via the conditional Poisson (CP) distribution. They showed that compared with Monte Carlo Markov chain (MCMC)-based approaches, their importance sampling method is more efficient in terms of running time and also provides an easy and accurate estimate of the total number of contingency tables with fixed marginal sums. In this paper, we extend their result to zero-one multi-way ( $d$ -way, $d \ge 2$ ) contingency tables under the no $d$ -way interaction model, i.e., with fixed $d-1$ marginal sums. Also, we show by simulations that the SIS procedure with CP distribution to estimate the number of zero-one three-way tables under the no three-way interaction model given marginal sums works very well even with some rejections. We also applied our method to Samson’s monks data set.     

Monte Carlo Algorithms for Identifying Densely Connected Subgraphs

The problem of finding densely connected subgraphs in a network has attracted a lot of recent interest. Such subgraphs are sometimes referred to as communities in social networks or molecular modules in protein networks. In this article, we propose two Monte Carlo optimization algorithms for identifying the densest subgraphs with a fixed size or with size in a given range. The new algorithms combine the idea of simulated annealing and efficient moves for the Markov chain, and both algorithms are shown to converge to the set of optimal states (densest subgraphs) with probability 1. When applied to a yeast protein interaction network and a stock market graph, the algorithms identify interesting new densely connected subgraphs. Supplementary materials for the article are available online.     

Avoiding abelian squares in partial words

Erd?s raised the question whether there exist infinite abelian square-free words over a given alphabet, that is, words in which no two adjacent subwords are permutations of each other. It can easily be checked that no such word exists over a three-letter alphabet. However, infinite abelian square-free words have been constructed over alphabets of sizes as small as four. In this paper, we investigate the problem of avoiding abelian squares in partial words, or sequences that may contain some holes. In particular, we give lower and upper bounds for the number of letters needed to construct infinite abelian square-free partial words with finitely or infinitely many holes. Several of our constructions are based on iterating morphisms. In the case of one hole, we prove that the minimal alphabet size is four, while in the case of more than one hole, we prove that it is five. We also investigate the number of partial words of length n with a fixed number of holes over a five-letter alphabet that avoid abelian squares and show that this number grows exponentially with n.     

Minimal invariant Markov basis for sampling contingency tables with fixed marginals

In this paper we define an invariant Markov basis for a connected Markov chain over the set of contingency tables with fixed marginals and derive some characterizations of minimality of the invariant basis. We also give a necessary and sufficient condition for uniqueness of minimal invariant Markov bases. By considering the invariance, Markov bases can be presented very concisely. As an example, we present minimal invariant Markov bases for all 2 × 2 × 2 × 2 hierarchical models. The invariance here refers to permutation of indices of each axis of the contingency tables. If the categories of each axis do not have any order relations among them, it is natural to consider the action of the symmetric group on each axis of the contingency table. A general algebraic algorithm for obtaining a Markov basis was given by Diaconis and Sturmfels (The Annals of Statistics, 26, 363–397, 1998). Their algorithm is based on computing Gröbner basis of a well-specified polynomial ideal. However, the reduced Gröbner basis depends on the particular term order and is not symmetric. Therefore, it is of interest to consider the properties of invariant Markov basis.     

Generating uniformly distributed random latin squares

By simulating an ergodic Markov chain whose stationary distribution is uniform over the space of n × n Latin squares, we can obtain squares that are (approximately) uniformly distributed; we offer two such chains. The central issue is the construction of “moves” that connect the squares. Our first approach uses the fact that an n × n Latin square is equivalent to an n × n × n contingency table in which each line sum equals 1. We relax the nonnegativity condition on the table's cells, allowing “improper” tables that have a single—1-cell. A simple set of moves connects this expanded space of tables [the diameter of the associated graph is bounded by 2(n − 1)³], and suggests a Markov chain whose subchain of proper tables has the desired uniform stationary distribution (with an average of approximately n steps between proper tables). By grouping these moves appropriately, we derive a class of moves that stay within the space of proper Latin squares [with graph diameter bounded by 4(n − 1)²]; these may also be used to form a suitable Markov chain. © 1996 John Wiley & Sons, Inc.     

Adaptive Component-Wise Multiple-Try Metropolis Sampling

One of the most widely used samplers in practice is the component-wise Metropolis–Hastings (CMH) sampler that updates in turn the components of a vector-valued Markov chain using accept–reject moves generated from a proposal distribution. When the target distribution of a Markov chain is irregularly shaped, a “good” proposal distribution for one region of the state–space might be a “poor” one for another region. We consider a component-wise multiple-try Metropolis (CMTM) algorithm that chooses from a set of candidate moves sampled from different distributions. The computational efficiency is increased using an adaptation rule for the CMTM algorithm that dynamically builds a better set of proposal distributions as the Markov chain runs. The ergodicity of the adaptive chain is demonstrated theoretically. The performance is studied via simulations and real data examples. Supplementary material for this article is available online.     

On Maxentropic Reconstruction of Countable Markov Chains and Matrix Scaling Problems

Using the Maximum Entropy Principle we consider maxentropic reconstruction of some countable Markov chains with given common steady-state probabilities. The probability transition matrix is obtained from the knowledge of the expected values of some random variables. For this, we consider the problem as an entropy optimization problem with linear constraints, which we solve by the geometric programming method. In the same way, we derive a method for solving a matrix scaling problem, namely maxentropic reconstruction of some nonnegative matrix with fixed row-column sums.     

Markov bases and subbases for bounded contingency tables

In this paper we study the computation of Markov bases for contingency tables whose cell entries have an upper bound. It is known that in this case one has to compute universal Gröbner bases, and this is often infeasible also in small- and medium-sized problems. Here we focus on bounded two-way contingency tables under independence model. We show that when these bounds on cells are positive the set of basic moves of all 2 × 2 minors connects all tables with given margins. We also give some results about bounded incomplete table and we conclude with an open problem on the necessary and sufficient condition on the set of structural zeros so that the set of basic moves of all 2 × 2 minors connects all incomplete contingency tables with given positive margins.     

Error detection and gap filling in cubes of data

We consider the family ZET of algorithms for detecting gross errors and filling the gaps in data tables (object-attribute) and data cubes (object-attribute-time). To work with each element of a table we take the information only from its competent subtable rather than from the whole table. We consider some methods for choosing the “competent” subtable that include tools for avoiding local extrema. An example is presented of the use of a ZET algorithm to solve an applied problem.     

The first moment of Salié sums

The main objective of this article is to study the asymptotic behavior of Salié sums over arithmetic progressions. We deduce from our asymptotic formula that Salié sums possess a bias towards being positive. The method we use is based on the Kuznetsov formula for modular forms of half-integral weight. Moreover, in order to develop an explicit formula, we are led to determine an explicit orthogonal basis of the space of modular forms of half-integral weight.     

Balancing attainability,desirability and promotion steadiness in manpower planning systems

A Markov manpower planning model with fixed internal transition probabilities, enables assessing the feasibility to attain the most desirable personnel structure. In case the desirable personnel structure is not attainable under control by recruitment, the internal personnel flows can be modified while not disrupting the career progression expectations. This paper introduces the promotion steadiness degree to quantify the personnel policy deviation from the career progression expectations. As a result, this paper focuses on a model that balances three criteria, that is, the desirability degree, the attainability degree and the promotion steadiness degree, formulated by fuzzy membership functions. A new set of instances is introduced, and the algorithms are evidenced in a set of experiments.     

Realization of multi-input/multi-output switched linear systems from Markov parameters

This paper presents a four-stage algorithm for the realization of multi-input/multi-output (MIMO) switched linear systems (SLSs) from Markov parameters. In the first stage, a linear time-varying (LTV) realization that is topologically equivalent to the true SLS is derived from the Markov parameters assuming that the discrete states have a common MacMillan degree and a mild condition on their dwell times holds. In the second stage, stationary point set of a Hankel matrix with fixed dimensions built from the Markov parameters is examined. Splitting of this set into disjoint intervals and complements reveals linear time-invariant dynamics prevailing on these intervals. Clustering over a feature space permits recovery of the discrete states up to similarity transformations which is complete if a unimodality assumption holds and the discrete states satisfy a residence requirement. In the third stage, the switching sequence is estimated by three schemes. The first scheme is non-iterative in time. The second scheme is based on matching the estimated and the true Markov parameters of the SLS system over segments. The third scheme works also on the same principle, but it is a discrete optimization/hypothesis testing algorithm. The three schemes operate on different dwell time and model structure requirements, but the dwell time requirements are weaker than that needed to recover the discrete states. In the fourth stage, the discrete state estimates are brought to a common basis by a novel basis transformation which is necessary for predicting outputs to prescribed inputs. Robustness of the four-stage algorithm to amplitude bounded noise is studied and it is shown that small perturbations may only produce small deviations in the estimates vanishing as noise amplitude diminishes. Time complexities of the stages are also studied. A numerical example illustrates the derived results.     

Coulomb Scattering of a Slow Quantum Particle in a Space of Arbitrary Dimension

We assume that a charged quantum particle moves in a space of dimension d = 2, 3,... and is scattered by a fixed Coulomb center. We derive and study expansions of the wave function and all radial functions of such a particle in integer powers of the wave number and in Bessel functions of a real order. We prove that finite sums of such expansions are asymptotic approximations of the wave functions in the low-energy limit.     

On primitively universal quadratic forms

In 1999, Manjul Bhargava proved the Fifteen Theorem and showed that there are exactly 204 universal positive definite integral quaternary quadratic forms. We consider primitive representations of quadratic forms and investigate a primitive counterpart to the Fifteen Theorem. In particular, we give an efficient method for deciding whether a positive definite integral quadratic form in four or more variables with odd square-free determinant is almost primitively universal.     

A discrete adapted hierarchical basis solver for radial basis function interpolation

In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial degree. The HB forms an orthogonal set and is adapted to the kernel seed function and the placement of the interpolation nodes. Moreover, this basis is orthogonal to a set of polynomials up to a given degree defined on the interpolating nodes. We are thus able to decouple the RBF interpolation problem for any degree of the polynomial interpolation and solve it in two steps: (1) The polynomial orthogonal RBF interpolation problem is efficiently solved in the transformed HB basis with a GMRES iteration and a diagonal (or block SSOR) preconditioner. (2) The residual is then projected onto an orthonormal polynomial basis. We apply our approach on several test cases to study its effectiveness.