formal group theoretic generalizations of the necklace algebra including aq deformation abstract n metropolis and g c rota adv math 50 1983 95 125 studied thenecklace polynomials and were lead to define thenecklace algebraas a combinatorial model for the classical ring ofwitt vectors which corresponds to the multiplicative formal group lawx y xy in this paper we define and study a generalized necklace algebra which is associated with an arbitrary formal group lawfover a torsion free ringa the map from the ring of witt vectors associated withfto the necklace algebra is constructed in terms of certain generalizations of the necklace polynomials we present a combinatorial interpretation for these polynomials in terms of words on a given alphabet the actions of theverschiebungandfrobeniusoperators as well as of thep typification idempotentare described and interpreted combinatorially aq analogue and other generalizations of thecyclotomic identityare also presented in general the necklace algebra can only be defined over the rationalizationa q nevertheless we show that for the family of formal group laws over the integersfq x y x y qxy q z we can define the corresponding necklace algebras over z we classify these algebras and define isomorphic ring structures on the groups of witt vectors and the groups of curves associated with the formal group lawsfq theq necklace polynomials which turn out to benumerical polynomialsin two variables can be interpreted combinatorially in terms of so calledq words and they satisfy an identity generalizing a classical one

convergence of best phi entropy estimates minimization problems involving phi entropy functionals a generalization of boltzmann shannon entropy are studied over a given set a and a sequence of sets a sub n and the properties of their optimal solutions x sub phi x sub n under certain conditions on the objective functional and the sets a and a sub n it is proven that as n increases to infinity the optimal solution x sub n converges in l sub 1 norm to the test phi entropy estimate x sub phi

spectrum estimation using multirate observations in this paper we are interested in estimating the power spectral density of a stationary random signal x n when the signal itself is not available but some low resolution measurements derived from it are observed we consider a model where x n is being measured using a set of linear multirate sensors each sensor outputs a measurement signal v sub i n whose sampling rate is only a fraction of the sampling rate assumed for the original signal based on this model we pose the following problem given certain autocorrelation coefficients of the observable signals v sub i n estimate the power spectral density of the original signal x n it turns out that this problem is ill posed we suggest to resolve this issue by using the principle of maximum entropy me we address technical difficulties associated with the me solution and then devise a practical algorithm for its approximate computation we demonstrate the viability of this algorithm through simulation examples

scalable estimation and regularization for the logistic normal multinomial model clustered multinomial data are prevalent in a variety of applications such as microbiome studies where metagenomic sequencing data are summarized as multinomial counts for a large number of bacterial taxa per subject count normalization with ad hoc zero adjustment tends to result in poor estimates of abundances for taxa with zero or small counts to account for heterogeneity and overdispersion in such data we suggest using the logistic normal multinomial lnm model with an arbitrary correlation structure to simultaneously estimate the taxa compositions by borrowing information across subjects we overcome the computational difficulties in high dimensions by developing a stochastic approximation em algorithm with hamiltonian monte carlo sampling for scalable parameter estimation in the lnm model the ill conditioning problem due to unstructured covariance is further mitigated by a covariance regularized estimator with a condition number constraint the advantages of the proposed methods are illustrated through simulations and an application to human gut microbiome data

the isometric log ratio transform for probabilistic multi label anatomical shape representation sources of uncertainty in the boundaries of structures in medical images have motivated the use of probabilistic labels in segmentation applications an important component in many medical image segmentation tasks is the use of a shape model often generated by applying statistical techniques to training data standard statistical techniques e g principal component analysis often assume data lies in an unconstrained vector space but probabilistic labels are constrained to the unit simplex if these statistical techniques are used directly on probabilistic labels relative uncertainty information can be sacrificed a standard method for facilitating analysis of probabilistic labels is to map them to a vector space using the logodds transform however the logodds transform is asymmetric in one of the labels which skews results in some applications the isometric log ratio ilr transform is a symmetrized version of the logodds transform and is so named as it is an isometry between the aitchison geometry the inherent geometry of the simplex and standard euclidean geometry we explore how to interpret the aitchison geometry when applied to probabilistic labels in medical image segmentation applications we demonstrate the differences when applying the logodds transform or the ilr transform to probabilistic labels prior to statistical analysis specifically we show that statistical analysis of ilr transformed data better captures the variability of anatomical shapes in cases where multiple different foreground regions share boundaries as opposed to foreground background boundaries

illustration of bayesian inference in normal data models using gibbs sampling abstract the use of the gibbs sampler as a method for calculating bayesian marginal posterior and predictive densities is reviewed and illustrated with a range of normal data models including variance components unordered and ordered means hierarchical growth curves and missing data in a crossover trial in all cases the approach is straightforward to specify distributionally and to implement computationally with output readily adapted for required inference summaries

a logistic normal multinomial regression model for microbiome compositional data analysis changes in human microbiome are associated with many human diseases next generation sequencing technologies make it possible to quantify the microbial composition without the need for laboratory cultivation one important problem of microbiome data analysis is to identify the environmental biological covariates that are associated with different bacterial taxa taxa count data in microbiome studies are often over dispersed and include many zeros to account for such an over dispersion we propose to use an additive logistic normal multinomial regression model to associate the covariates to bacterial composition the model can naturally account for sampling variabilities and zero observations and also allow for a flexible covariance structure among the bacterial taxa in order to select the relevant covariates and to estimate the corresponding regression coefficients we propose a group l1 penalized likelihood estimation method for variable selection and estimation we develop a monte carlo expectation maximization algorithm to implement the penalized likelihood estimation our simulation results show that the proposed method outperforms the group l1 penalized multinomial logistic regression and the dirichlet multinomial regression models in variable selection we demonstrate the methods using a data set that links human gut microbiome to micro nutrients in order to identify the nutrients that are associated with the human gut microbiome enterotype

partially finite programming in l_1 and the existence of maximum entropy estimates best entropy estimation is a technique that has been widely applied in many areas of science it consists of estimating an unknown density from some of its moments by maximizing some measure of the entropy of the estimate this problem can be modelled as a partially finite convex program with an integrable function as the variable a complete duality and existence theory is developed for this problem and for an associated extended problem which allows singular measure theoretic solutions this theory explains the appearance of singular components observed in the literature when the burg entropy is used it also provides a unified treatment of existence conditions when the burg boltzmann shannon or some other entropy is used as the objective some examples are discussed

witt vectors and the algebra of necklaces where u is the classical mobius function m a n is called a necklace polynomial we shall be concerned with some remarkable identities satisfied by the polynomials m a n which apparently have not been previously noticed specifically we prove that m a n can be expressed as a quadratic polynomial in m a i and mj that m a n can be expressed linearly and integrally in terms of m a i the proofs given below of these identities are entirely combinatorial theorems l 6

markov chain monte carlo convergence diagnostics a comparative review abstract a critical issue for users of markov chain monte carlo mcmc methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest research into methods of computing theoretical convergence bounds holds promise for the future but to date has yielded relatively little of practical use in applied work consequently most mcmc users address the convergence problem by applying diagnostic tools to the output produced by running their samplers after giving a brief overview of the area we provide an expository review of 13 convergence diagnostics describing the theoretical basis and practical implementation of each we then compare their performance in two simple models and conclude that all of the methods can fail to detect the sorts of convergence failure that they were designed to identify we thus recommend a combination of strategies aimed at evaluating and accelerating mcmc sampler convergence including ap

symmetric functions formal group laws and lazard s theorem lazard s theorem is a central result in formal group theory it states that the ring over which the universal formal group law is defined known as the lazard ring is a polynomial algebra over the integers with infinitely many generators this ring also shows up in algebraic topology as the complex cobordism ring the main aim of this paper is to show that the polynomial structure of the lazard ring follows from the polynomial structure of a certain subalgebra of symmetric functions with integer coefficients the connection between symmetric functions and the lazard ring is provided by a certain hopf algebra map from symmetric functions to the covariant bialgebra of a formal group law we study this map by deriving formulas for the images of certain symmetric functions in passing we use this map to prove some symmetric function and catalan number identities based on the above results we prove lazard s theorem and present an application to the construction of certainp typical formal group laws over the integers combinatorial methods play a major role throughout this paper

bayesian tests and model diagnostics in conditionally independent hierarchical models abstract consider the conditionally independent hierarchical model cihm in which observations yi are independently distributed from f yi i the parameters i are independently distributed from distributions g and the hyperparameters are distributed according to a distribution h the posterior distribution of all parameters of the cihm can be efficiently simulated by markov chain monte carlo mcmc algorithms although these simulation algorithms have facilitated the application of cihms they generally have not addressed the problem of computing quantities useful in model selection this article explores how mcmc simulation algorithms and other related computational algorithms can be used to compute bayes factors that are useful in criticizing a particular cihm in the case where the cihm models a belief that the parameters are exchangeable or lie on a regression surface the bayes factor can measure the consistency of the data with the structural prior belief bayes factors can also be u

modeling household purchase behavior with logistic normal regression the successful development of marketing strategies requires the accurate measurement of household preferences and their reaction to variables such as price and advertising manufacturers for examp

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

q deformed necklace rings and q mobius function abstract we introduce a q deformation of the classical mobius function and investigate its properties in connection with q deformed truncated necklace rings also we study the strictly natural isomorphism of q deformed necklace rings

a dirichlet tree multinomial regression model for associating dietary nutrients with gut microorganisms summary understanding the factors that alter the composition of the human microbiota may help personalized healthcare strategies and therapeutic drug targets in many sequencing studies microbial communities are characterized by a list of taxa their counts and their evolutionary relationships represented by a phylogenetic tree in this article we consider an extension of the dirichlet multinomial distribution called the dirichlet tree multinomial distribution for multivariate over dispersed and tree structured count data to address the relationships between these counts and a set of covariates we propose the dirichlet tree multinomial regression model for which we develop a penalized likelihood method for estimating parameters and selecting covariates for efficient optimization we adopt the accelerated proximal gradient approach simulation studies are presented to demonstrate the good performance of the proposed procedure an analysis of a data set relating dietary nutrients with bacterial counts is used to show that the incorporation of the tree structure into the model helps increase the prediction power

properties of cross entropy minimization the principle of minimum cross entropy minimum directed divergence minimum discrimination information is a general method of inference about an unknown probability density when there exists a prior estimate of the density and new information in the form of constraints on expected values various fundamental properties of cross entropy minimization are proven and collected in one place cross entropy s well known properties as an information measure are extended and strengthened when one of the densities involved is the result of cross entropy minimization the interplay between properties of cross entropy minimization as an inference procedure and properties of cross entropy as an information measure is pointed out examples are included and general analytic and computational methods of finding minimum cross entropy probability densities are discussed

necklace rings and logarithmic functions in this paper we develop the theory of the necklace ring and the logarithmic function regarding the necklace ring we introduce the necklace ring functor nr from the category of special rings into the category of special rings and then study the associated adams operators as far as the logarithmic function is concerned we generalize the results in bryant s paper free lie algebras and formal power series j algebra 253 1 2002 167 188 to the case of graded lie super algebras with a group action by applying the euler poincare principle

convergence of best entropy estimates given a finite number of moments of an unknown density on a finite measure space the best entropy estimate that nonnegative density x with the given moments which minimizes the boltzmann shannon entropy i x x log x is considered a direct proof is given that i has the kadec property in l if yn converges weakly to 37 and i yn converges to i 37 then yn converges to 37 in norm as a corollary it is obtained that as the number of given moments increases the best entropy estimates converge in li norm to the best entropy estimate of the limiting problem which is simply in the determined case furthermore for classical moment problems on intervals with strictly positive and sufficiently smooth error bounds and uniform convergence are actually obtained

convergence rates for maximum entropy regularization the authors prove results about stability convergence and convergence rates for the maximum entropy method by interpreting it as a nonlinear regularization method the results are applied to a widely studied problem from physical chemistry namely the reconstruction of fluorescence lifetime distributions from the distribution of the delays between absorption and emission of photons

the multivariate poisson log normal distribution summary the statistical analysis of multivariate counts has proved difficult because of the lack of a parametric class of distributions supporting a rich enough correlation structure with increasing availability of powerful computing facilities an obvious candidate for consideration is now the multivariate log normal mixture of independent poisson distributions the multivariate poisson log normal distribution the properties of this discrete multivariate distribution are studied and its uses in a variety of applications to multivariate count data are illustrated

use of the von mises distribution to analyse continuous proportions summary the von mises distribution in p dimensions is suggested for the analysis of data which are proportions of a continuum such as time or volume the one way analysis ofvariance technique developed by watson 1956 is extended to a multiway layout and the methods are illustrated on data giving proportions of time spent in various activities other examples are briefly mentioned

aperiodic rings necklace rings and witt vectors in n metropolis and gian carlo rota witt vectors and the algebra of necklaces adv in math 50 1983 95 125 metropolis and rota show that the cyclotomic identity provides a linkage between the necklace ring and the ring of witt vectors in this paper we study this linkage let a be the class of rings a with the property that the additive group of a is torsion free we will prove that the ring of witt vectors over a for any a a is isomorphic to a suitably defined subring of the necklace ring of the rationalization a q of a alternatively to a subring of the aperiodic ring of a a major part of this paper is concerned with a development of the properties and relationships involving the aperiodic ring which parallels the treatment of the necklace ring given in n metropolis and gian carlo rota witt vectors and the algebra of necklaces adv in math 50 1983 95 125 in addition we translate the combinatorial proof of the cyclotomic identity given by metropolis and rota n metropolis and gian carlo rota the cyclotomic identity contemp math 34 1984 19 27 into the language of species in the sense of joyal thereby making clear the naturality of the identity

statistical interpretation of species composition the relative abundance of different species characterizes the structure of a biological community we analyze an experiment addressing the relationship between omnivorous feeding linkages and community stability our goal is to determine whether communities with different predator compositions respond similarly to environmental disturbance to evaluate these data we develop a hierarchical statistical model that combines aitchison s logistic normal distribution with a conditional multinomial observation distribution in addition we present an algebra for compositions that includes addition scalar multiplication and a metric for differences in compositions the algebra aids interpretation of treatment effects treatment interactions and covariates markov chain monte carlo mcmc is used for inference in a bayesian framework our experimental results indicate that a high degree of omnivory can help to stabilize community dynamics and prevent radical shifts in community composition this result is at od

the burnside ring of profinite groups and the witt vector construction abstract the classical construction of witt vectors can be understood as a special case of a construction which can be defined relative to any given profinite group g in terms of its completed burnside ring lde g it specializes to the classical situation in case g is infinite cyclic

the burnside ring of the infinite cyclic group and its relations to the necklace algebra λ rings and the universal ring of witt vectors abstract it is shown that well known product decompositions of formal power series arise from combinatorially defined canonical isomorphisms between the burnside ring of the infinite cyclic group on the one hand and grothendieck s ring of formal power series with constant term 1 as well as the universal ring of witt vectors on the other hand

generalized burnside grothendieck ring functor and aperiodic ring functor associated with profinite groups abstract for every profinite group g we construct two covariant functors g and ap g which are equivalent to the functor w g introduced in a dress c siebeneicher the burnside ring of profinite groups and the witt vectors construction adv math 70 1988 87 132 we call g the generalized burnside grothendieck ring functor and ap g the aperiodic ring functor associated with g in case g is abelian we also construct another functor ap g from the category of commutative rings with identity to itself as a generalization of the functor ap introduced in k varadarajan k wehrhahn aperiodic rings necklace rings and witt vectors adv math 81 1990 1 29 finally it is shown that there exist q analogues of these functors i e w g g ap g and ap g in case g is the profinite completion of the multiplicative infinite cyclic group c