Sequential importance sampling for multiway tables

We describe an algorithm for the sequential sampling of entries in multiway contingency tables with given constraints. The algorithm can be used for computations in exact conditional inference. To justify the algorithm, a theory relates sampling values at each step to properties of the associated toric ideal using computational commutative algebra. In particular, the property of interval cell counts at each step is related to exponents on lead indeterminates of a lexicographic Gr\"{o}bner basis. Also, the approximation of integer programming by linear programming for sampling is related to initial terms of a toric ideal. We apply the algorithm to examples of contingency tables which appear in the social and medical sciences. The numerical results demonstrate that the theory is applicable and that the algorithm performs well.Comment: Published at http://dx.doi.org/10.1214/009053605000000822 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Boolean Network Control with Ideals

A method is given for finding controls to transition an initial state x0 to a target set in deterministic or stochastic Boolean network control models. The algorithms use multivariate polynomial algebra. Examples illustrate the application

Monomials and Basin Cylinders for Network Dynamics

We describe methods to identify cylinder sets inside a basin of attraction for Boolean dynamics of biological networks. Such sets are used for designing regulatory interventions that make the system evolve towards a chosen attractor, for example initiating apoptosis in a cancer cell. We describe two algebraic methods for identifying cylinders inside a basin of attraction, one based on the Groebner fan that finds monomials that define cylinders and the other on primary decomposition. Both methods are applied to current examples of gene networks

Sampling large tables with constraints

Abstract: We describe a new sequential sampling method for constrained multi-way tables, with foundations in linear programming and sequential normal sampling. The method builds on techniques from other sequential algorithms in a way that scales well and can handle more challenging data sets. We apply the new algorithm to data to demonstrate its efficiency

Exact Tests for Singular Network Data

Abstract We propose methodology for exact statistical tests of hypotheses for models of network dynamics. The methodology formulates Markovian exponential families, then uses sequential importance sampling to compute expectations within basins of attraction and within level sets of a sufficient statistic for an overdispersion model. Comparisons of hypotheses can be done conditional on basins of attraction. Examples are presented

Computational Methods for Asynchronous Basins

For a Boolean network we consider asynchronous updates and define the exclusive asynchronous basin of attraction for any steady state or cyclic attractor. An algorithm based on commutative algebra is presented to compute the exclusive basin. Finally its use for targeting desirable attractors by selective intervention on network nodes is illustrated with two examples, one cell signalling network and one sensor network measuring human mobility

Conditional Tests on Basins of Attraction with Finite Fields

An iterative method is given for computing the polynomials that vanish on the basin of attraction of a steady state in discrete polynomial dynamics with finite field coefficients. The algorithm is applied to dynamics of a T cell survival network where it is used to compare transition maps conditional on a basin of attraction

Ideals, Big Varieties, and Dynamic Networks

The advantage of using algebraic geometry over enumeration for describing sets related to attractors in large dynamic networks from biology is advocated. Examples illustrate the gains

Exact Tests for Singular Network Data

We propose methodology for exact statistical tests of hypotheses for models of network dynamics. The methodology formulates Markovian exponential families, then uses sequential importance sampling to compute expectations within basins of attraction and within level sets of a sufficient statistic for an over-dispersion model. Comparisons of hypotheses can be done conditional on basins of attraction. Examples are presented