a stochastic approximation algorithm with markov chain monte carlo method for incomplete data estimation problems we propose a general procedure for solving incomplete data estimation problems the procedure can be used to find the maximum likelihood estimate or to solve estimating equations in difficult cases such as estimation with the censored or truncated regression model the nonlinear structural measurement error model and the random effects model the procedure is based on the general principle of stochastic approximation and the markov chain monte carlo method applying the theory on adaptive algorithms we derive conditions under which the proposed procedure converges simulation studies also indicate that the proposed procedure consistently converges to the maximum likelihood estimate for the structural measurement error logistic regression model

knottedness is in np modulo grh abstract given a tame knot k presented in the form of a knot diagram we show that the problem of determining whether k is knotted is in the complexity class np assuming the generalized riemann hypothesis grh in other words there exists a polynomial length certificate that can be verified in polynomial time to prove that k is non trivial grh is not needed to believe the certificate but only to find a short certificate this result complements the result of hass lagarias and pippenger that unknottedness is in np our proof is a corollary of major results of others in algebraic geometry and geometric topology

on the number of different permanents of some sparse 0 1 circulant matrices abstract starting from known results about the number of possible values for the permanents of 0 1 circulant matrices with three nonzero entries per row and whose dimension n is prime we prove corresponding results for n power of a prime n product of two distinct primes and n 2 3 h supported by some experimental results we also conjecture that the number of different permanents of n n 0 1 circulant matrices with k nonzero per row is asymptotically equal to n k 2 k o n k 3

on the permanent of certain 0 1 toeplitz matrices abstract we obtain convenient expressions and or efficient algorithms for the permanent of certain very sparse 0 1 toeplitz matrices the classes of matrices considered here include some nontrivial examples of circulants to which none of the previous approaches could be successfully applied

on prime factors of determinants of circulant matrices abstract we find the probability that the determinant of an integer circulant n n matrix is divisible by the prime p where p does not divide n

permanental compounds and permanents of 0 1 circulants abstract it was shown by the author in a recent paper that a recurrence relation for permanents of 0 1 circulants can be generated from the product of the characteristic polynomials of permanental compounds of the companion matrix of a polynomial associated with 0 1 circulants of the given type in the present paper general properties of permanental compounds of companion matrices are studied and in particular of convertible companion matrices i e matrices whose permanental compounds are equal to the determinantal compounds after changing the signs of some of their entries these results are used to obtain formulas for the limit of the n th root of the permanent of the n n 0 1 circulant of a given type as n tends to infinity the root squaring method is then used to evaluate this limit for a wide range of circulant types whose associated polynomials have convertible companion matrices

new algorithm for the ising problem partition function for finite lattice graphs we present a new efficient method to find the ising problem partition function for finite lattice graphs embeddable on an arbitrary orientable surface with integral coupling constants bounded in the absolute value by a polynomial of the size of the lattice graph the algorithm has been implemented for toroidal lattices using modular arithmetic and the generalized nested dissection method the implementation has substantially better performance than any other as far as we know

parameter estimation for a stationary process on a d dimensional lattice summary we study asymptotic properties of various estimation procedures for a general stationary regular process on a d dimensional lattice differences between d 1 time series and d 2 spatial processes are pointed out we suppose that the process is observed on a set pn with cn points which tends to infinity with the same speed in all directions the relative edge effect is of order cn lid increasing with d this effect is without statistical importance if d 1 but is important if d 2 we give a lcn consistent asymptotically normal estimator of the underlying parameter the procedure being constructed by a modification of whittle s approximation to the log likelihood in the gaussian case this procedure is asymptotically efficient

recurrence formulas for permanents of 0 1 circulants abstract a general method is developed for constructing linear recurrence formulas expressing the permanents of 0 1 circulants in terms of permanents of 0 1 circulants of the same type of lower orders the method extends to all 0 1 circulants a result of n metropolis m l stein and p r stein on permanents of 0 1 circulants of a special type

convergence of a stochastic approximation version of the em algorithm the expectation maximization em algorithm is a powerful computational technique for locating maxima of functions it is widely used in statistics for maximum likelihood or maximum a posteriori estimation in incomplete data models in certain situations however this method is not applicable because the expectation step cannot be performed in closed form to deal with these problems a novel method is introduced the stochastic approximation em saem which replaces the expectation step of the em algorithm by one iteration of a stochastic approximation procedure the convergence of the saem algorithm is established under conditions that are applicable to many practical situations moreover it is proved that under mild additional conditions the attractive stationary points of the saem algorithm correspond to the local maxima of the function presented to support our findings

efficient recursions for general factorisable models let n s valued categorical variables be jointly distributed according to a distribution known only up to an unknown normalising constant for an unnormalised joint likelihood expressible as a product of factors we give an algebraic recursion which can be used for computing the normalising constant and other summations a saving in computation is achieved when each factor contains a lagged subset of the components combining in the joint distribution with maximum computational efficiency as the subsets attain their minimum size if each subset contains at most r 1 of the n components in the joint distribution we term this a lag r model whose normalising constant can be computed using a forward recursion in o sr 1 computations as opposed to o sn for the direct computation we show how a lag r model represents a markov random field and allows a neighbourhood structure to be related to the unnormalised joint likelihood we illustrate the method by showing how the normalising constant of the ising or autologistic model can be computed

recursive computing and simulation free inference for general factorizable models summary we illustrate how the recursive algorithm of reeves pettitt 2004 for general factorizable models can be extended to allow exact sampling maximization of distributions and computation of marginal distributions all of the methods we describe apply to discrete valued markov random fields with nearest neighbour integrations defined on regular lattices in particular we illustrate that exact inference can be performed for hidden autologistic models defined on moderately sized lattices in this context we offer an extension of this methodology which allows approximate inference to be carried out for larger lattices without resorting to simulation techniques such as markov chain monte carlo in particular our work offers the basis for an automatic inference machine for such models

fitting nonlinear ordinary differential equation models with random effects and unknown initial conditions using the stochastic approximation expectation maximization saem algorithm the past decade has evidenced the increased prevalence of irregularly spaced longitudinal data in social sciences clearly lacking however are modeling tools that allow researchers to fit dynamic models to irregularly spaced data particularly data that show nonlinearity and heterogeneity in dynamical structures we consider the issue of fitting multivariate nonlinear differential equation models with random effects and unknown initial conditions to irregularly spaced data a stochastic approximation expectation maximization algorithm is proposed and its performance is evaluated using a benchmark nonlinear dynamical systems model namely the van der pol oscillator equations the empirical utility of the proposed technique is illustrated using a set of 24 h ambulatory cardiovascular data from 168 men and women pertinent methodological challenges and unresolved issues are discussed

an adaptive exchange algorithm for sampling from distributions with intractable normalizing constants sampling from the posterior distribution for a model whose normalizing constant is intractable is a long standing problem in statistical research we propose a new algorithm adaptive auxiliary variable exchange algorithm or in short adaptive exchange aex algorithm to tackle this problem the new algorithm can be viewed as a mcmc extension of the exchange algorithm which generates auxiliary variables via an importance sampling procedure from a markov chain running in parallel the convergence of the algorithm is established under mild conditions compared to the exchange algorithm the new algorithm removes the requirement that the auxiliary variables must be drawn using a perfect sampler and thus can be applied to many models for which the perfect sampler is not available or very expensive compared to the approximate exchange algorithms such as the double metropolis hastings sampler the new algorithm overcomes their theoretical difficulty in convergence the new algorithm is tested on the spat

efficient calculation of the normalizing constant of the autologistic and related models on the cylinder and lattice motivated by the autologistic model for the analysis of spatial binary data on the two dimensional lattice we develop efficient computational methods for calculating the normalizing constant for models for discrete data defined on the cylinder and lattice because the normalizing constant is generally unknown analytically statisticians have developed various ad hoc methods to overcome this difficulty our aim is to provide computationally and statistically efficient methods for calculating the normalizing constant so that efficient likelihood based statistical methods are then available for inference we extend the so called transition method to find a feasible computational method of obtaining the normalizing constant for the cylinder boundary condition to extend the result to the free boundary condition on the lattice we use an efficient path sampling markov chain monte carlo scheme the methods are generally applicable to association patterns other than spatial such as clustered binary data and to variables taking three or more values described by for example potts models

an update on minc s survey of open problems involving permanents abstract we summarise the progress which has been made since 1986 on the conjectures and open problems listed in h minc s survey articles on the theory of permanents

metrics and embeddings of generalizations of thompson s group f the distance from the origin in the word metric for generalizations f p of thompson s group f is quasi isometric to the number of carets in the reduced rooted tree diagrams representing the elements of f p this interpretation of the metric is used to prove that every f p admits a quasiisometric embedding into every f q and also to study the behavior of the shift maps under these embeddings the first example of a finitely presented infinite simple group was discovered by r j thompson in 1965 this group g arose in the study of logic and it has many interesting descriptions in 1974 higman 10 used a description as the group of automorphisms of the jonsson tarski algebra to generalize g to an infinite family of groups g has another description as a group of homeomorphisms of the cantor set it has subgroups which may be described as homeomorphism groups of different objects for instance the circle and the unit interval among the people studying these generalizations are brown 3 and bieri and strebel in a set of unpublished notes 1 in this paper we will concentrate on the groups f p which correspond to the groups fp in 3 these families of groups have also been extensively studied by brin and guzman 2 the group f 2 commonly known as thompson s group f can be regarded as the group of piecewise linear orientation preserving homeomorphisms of the interval 0 1 which have breakpoints only at dyadic points and whose slopes where defined are powers of two in 1984 brown and geoghegan 4 found it to be the first example of a finitely presented infinite dimensional torsion free fp group this fact has been extended to all f p in 3 and studied by stein 11 where generalizations to more general groups of homeomorphisms with general rational breakpoint sets are considered cleary 7 has studied these properties for groups of piecewise linear homeomorphisms with irrational breakpoints and slopes in 5 burillo established the relationship between the word metric of thompson s group f and an estimate of the distance derived from the unique normal form of the elements this algebraic estimate is quasi isometric to the word metric and was used to prove that some subgroups are nondistorted in f in this paper we find a geometric estimate of the word metric in terms of rooted tree diagrams see 6 show that this is quasi isometric to the word metric and use this interpretation to prove that some embeddings of f p into f q including some of those outlined in 2 are quasi isometric the interpretations of these embeddings in terms of received by the editors september 25 1998 and in revised form august 11 1999 2000 mathematics subject classification primary 20f65 secondary 20f05 20f38 20e99 05c25 c 2000 american mathematical society

computation of sparse circulant permanents via determinants we consider the problem of computing the permanent of circulant matrices we apply some recent results on the computation of the number of perfect matchings of small genus graphs in order to show that the permanent of a circulant matrix with three nonzero entries per row is the linear combination of just four determinants of circulant matrices with the same structure as the original matrix we also show that the same result holds true for a class of circulant matrices with four nonzero entries per row related to the dimer problem with periodic boundary conditions conversely we give hints at the fact that more general circulants do not share similar properties and thus should be dealt with by means of radically different approaches

an efficient markov chain monte carlo method for distributions with intractable normalising constants maximum likelihood parameter estimation and sampling from bayesian posterior distributions are problematic when the probability density for the parameter of interest involves an intractable normalising constant which is also a function of that parameter in this paper an auxiliary variable method is presented which requires only that independent samples can be drawn from the unnormalised density at any particular parameter value the proposal distribution is constructed so that the normalising constant cancels from the metropolis hastings ratio the method is illustrated by producing posterior samples for parameters of the ising model given a particular lattice realisation

congruences for permanents and determinants of circulants abstract starting from particular congruences concerning permanents of some 0 1 circulant matrices we derive more general congruences for permanents and determinants we also analyze the relation of such results with some congruences satisfied by the norms of algebraic integers in cyclotomic fields

exact solution of the ising model on group lattices of genus g 1 we discuss how to apply the dimer method to ising models on group lattices having nontrivial topological genus g we find that the use of group extension and the existence of both external and internal group isomorphisms greatly reduces the number of distinct pfaffians and leads to explicit topological formulas for their sign and weight in the expansion of the partition function the complete solution for the ising model on the klein lattice group l 2 7 with g 3 is given

how fast can one compute the permanent of circulant matrices abstract in this paper we address the problem of computing the permanent of 0 1 circulant matrices we investigate structural properties of circulant matrices showing that i if they are dense enough then they contain large arbitrary submatrices and ii if they are very sparse then they are not too far from convertible matrices building upon ii we then develop an efficient algorithm which allows us to compute permanents of very sparse circulants of size up to 200

fast sampling of gaussian markov random fields this paper demonstrates how gaussian markov random fields conditional autoregressions can be sampled quickly by using numerical techniques for sparse matrices the algorithm is general and efficient and expands easily to various forms for conditional simulation and evaluation of normalization constants we demonstrate its use by constructing efficient block updates in markov chain monte carlo algorithms for disease mapping

multitype spatial point patterns with hierarchical interactions multitype spatial point patterns with hierarchical interactions are considered here hierarchical interaction means directionality points on a higher level of hierarchy affect the locations of points on the lower levels but not vice versa such relations are common for example in ecological communities interacting point patterns are often modeled by gibbs processes with pairwise interactions however these models are inherently symmetric and the hierarchy can be acknowledged only when interpreting the results we suggest the following in allowing the inclusion of the hierarchical structure in the model instead of regarding the pattern as a realization of a stationary multivariate point process we build the pattern one type at a time according to the order of the hierarchy by using nonstationary univariate processes as interactions connected to points x on a certain level are considered the effect of the higher levels is interpreted as heterogeneity of the pattern x and the points on the lower levels are neglected because of the hierarchical structure

maximizing generalized linear mixed model likelihoods with an automated monte carlo em algorithm summary two new implementations of the em algorithm are proposed for maximum likelihood fitting of generalized linear mixed models both methods use random independent and identically distributed sampling to construct monte carlo approximations at the e step one approach involves generating random samples from the exact conditional distribution of the random effects given the data by rejection sampling using the marginal distribution as a candidate the second method uses a multivariate t importance sampling approximation in many applications the two methods are complementary rejection sampling is more efficient when sample sizes are small whereas importance sampling is better with larger sample sizes monte carlo approximation using random samples allows the monte carlo error at each iteration to be assessed by using standard central limit theory combined with taylor series methods specifically we construct a sandwich variance estimate for the maximizer at each approximate e step this suggests a rule for automatically increasing the monte carlo sample size after iterations in which the true em step is swamped by monte carlo error in contrast techniques for assessing monte carlo error have not been developed for use with alternative implementations of monte carlo em algorithms utilizing markov chain monte carlo e step approximations three different data sets including the infamous salamander data of mccullagh and nelder are used to illustrate the techniques and to compare them with the alternatives the results show that the methods proposed can be considerably more efficient than those based on markov chain monte carlo algorithms however the methods proposed may break down when the intractable integrals in the likelihood function are of high dimension

on the values of permanents of 0 1 circulant matrices with three ones per row abstract starting from previous works concerning permanents of 0 1 circulant matrices with three nonzero entries per row and just using algebraic methods we find some congruences satisfied by the values of such permanents along with simple generalizations to circulants with k ones per row k n k 3 we also give an alternative proof for known relations of fibonacci numbers with some of these values of permanents

markov chain monte carlo for autologistic regression models with application to the distribution of plant species summary in this paper we explore using autologistic regression models for spatial binary data with covariates autologistic regression models can handle binary responses exhibiting both spatial correlation and dependence on covariates we use markov chain monte carlo mcmc to estimate the parameters in these models the distributional behavior of the mcmc maximum likelihood estimates mcmc mles is studied via simulation we find that the mcmc mles are approximately normally distributed and that the mcmc estimates of fisher information may be used to estimate the variance of the mcmc mles and to construct confidence intervals finally we illustrate by example how our studies may be applied to model the distribution of plant species

maximum likelihood estimation for spatial models by markov chain monte carlo stochastic approximation we propose a two stage algorithm for computing maximum likelihood estimates for a class of spatial models the algorithm combines markov chain monte carlo methods such as the metropolis hastings green algorithm and the gibbs sampler and stochastic approximation methods such as the off line average and adaptive search direction a new criterion is built into the algorithm so stopping is automatic once the desired precision has been set simulation studies and applications to some real data sets have been conducted with three spatial models we compared the algorithm proposed with a direct application of the classical robbins monro algorithm using wiebe s wheat data and found that our procedure is at least 15 times faster

stochastic approximation in monte carlo computation the wang landau wl algorithm is an adaptive markov chain monte carlo algorithm used to calculate the spectral density for a physical system a remarkable feature of the wl algorithm is that it is not trapped by local energy minima which is very important for systems with rugged energy landscapes this feature has led to many successful applications of the algorithm in statistical physics and biophysics however there does not exist rigorous theory to support its convergence and the estimates produced by the algorithm can reach only a limited statistical accuracy in this article we propose the stochastic approximation monte carlo samc algorithm which overcomes the shortcomings of the wl algorithm we establish a theorem concerning its convergence the estimates produced by samc can be improved continuously as the simulation proceeds samc also extends applications of the wl algorithm to continuum systems the potential uses of samc in statistics are discussed through two classes of applications i

on jones subgroup of r thompson group f abstract recently vaughan jones showed that the r thompson group f encodes in a natural way all knots and links in r 3 and a certain subgroup f of f encodes all oriented knots and links we answer several questions of jones about f in particular we prove that the subgroup f is generated by x 0 x 1 x 1 x 2 x 2 x 3 where x i i n are the standard generators of f and is isomorphic to f 3 the analog of f where all slopes are powers of 3 and break points are 3 adic rationals we also show that f coincides with its commensurator hence the linearization of the permutational representation of f on f f is irreducible we show how to replace 3 in the above results by an arbitrary n and to construct a series of irreducible representations of f defined in a similar way finally we analyze jones construction and deduce that the thompson index of a link is linearly bounded in terms of the number of crossings in a link diagram

combinatorial and topological approach to the 3d ising model we extend the planar pfaffian formalism for the evaluation of the ising partition function to lattices of high topological genus g the 3d ising model on a cubic lattice where g is proportional to the number of sites is discussed in detail the expansion of the partition function is given in terms of 22g pfaffians classified by the oriented homology cycles of the lattice i e by its spin structures correct counting is guaranteed by a signature term which depends on the topological intersection of the oriented cycles through a simple bilinear formula the role of a gauge symmetry arising in the above expansion is discussed the same formalism can be applied to the counting problem of perfect matchings over general lattices and provides a determinant expansion of the permanent of 0 1 matrices

acceleration of stochastic approximation by averaging a new recursive algorithm of stochastic approximation type with the averaging of trajectories is investigated convergence with probability one is proved for a variety of classical optimization and identification problems it is also demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence

the computational complexity of knot and link problems we consider the problem of deciding whether a polygonal knot in 3 dimensional euclidean space is unknotted ie capable of being continuously deformed without self intersection so that it lies in a plane we show that this problem unknotting problem is in np we also consider the problem splitting problem of determining whether two or more such polygons can be split or continuously deformed without self intersection so that they occupy both sides of a plane without intersecting it we show that it also is in np finally we show that the problem of determining the genus of a polygonal knot a generalization of the problem of determining whether it is unknotted is in pspace we also give exponential worst case running time bounds for deterministic algorithms to solve each of these problems these algorithms are based on the use of normal surfaces and decision procedures due to w haken with recent extensions by w jaco and j l tollefson

optimization via enumeration a new algorithm for the max cut problem we present a polynomial time algorithm to find the maximum weight of an edge cut in graphs embeddable on an arbitrary orientable surface with integral weights bounded in the absolute value by a polynomial of the size of the graph

high dimensional exploratory item factor analysis by a metropolis hastings robbins monro algorithm a metropolis hastings robbins monro mh rm algorithm for high dimensional maximum marginal likelihood exploratory item factor analysis is proposed the sequence of estimates from the mh rm algorithm converges with probability one to the maximum likelihood solution details on the computer implementation of this algorithm are provided the accuracy of the proposed algorithm is demonstrated with simulations as an illustration the proposed algorithm is applied to explore the factor structure underlying a new quality of life scale for children it is shown that when the dimensionality is high mh rm has advantages over existing methods such as numerical quadrature based em algorithm extensions of the algorithm to other modeling frameworks are discussed

constrained monte carlo maximum likelihood for dependent data maximum likelihood estimates mles in autologistic models and other exponential family models for dependent data can be calculated with markov chain monte carlo methods the metropolis algorithm or the gibbs sampler which simulate ergodic markov chains having equilibrium distributions in the model from one realization of such a markov chain a monte carlo approximant to the whole likelihood function can be constructed the parameter value if any maximizing this function approximates the mle

spatial statistics and bayesian computation on wednesday may 6th 1992 professor b w silverman in the chair summary markov chain monte carlo mcmc algorithms such as the gibbs sampler have provided a bayesian inference machine in image analysis and in other areas of spatial statistics for several years founded on the pioneering ideas of ulf grenander more recently the observation that hyperparameters can be included as part of the updating schedule and the fact that almost any multivariate distribution is equivalently a markov random field has opened the way to the use of mcmc in general bayesian computation in this paper we trace the early development of mcmc in bayesian inference review some recent computational progress in statistical physics based on the introduction of auxiliary variables and discuss its current and future relevance in bayesian applications we briefly describe a simple mcmc implementation for the bayesian analysis of agricultural field experiments with which we have some practical experience

likelihood analysis of spatial point patterns summary the likelihood procedure for estimating the pairwise interaction potential function is developed for statistical analysis of homogeneous spatial point patterns approximation methods of the normalizing factor of gibbs canonical distribution are discussed both to estimate a scale parameter and to measure the softness or hardness of repulsive interactions the approximations are useful up to a considerably high density the validity of our procedure is demonstrated by some computer experi ments some real data are analysed