A pair-formation model with a discrete set of offsprings and child care

We present a discrete newborns set-based deterministic model for a two-sex population structured by age and marital status. The model includes the spatial migration, a weighted harmonic mean-type pair formation function, and strong parental care and neglects the separation of pairs. Each sex has pre-reproductive and reproductive age intervals. All adult (of reproductive age) individuals are divided into single males, single females, and permanent pairs. All pairs are of two types: pairs without offsprings under parental care at the given time and pairs taking care of their young offsprings. All individuals of pre-reproductive age are divided into young (under parental care) and juvenile (offsprings who can live without parental care but cannot produce offsprings) groups. It is assumed that births can only occur from couples and after the death of any of the pair partner all young offsprings of this pair die. The model consists of integro-partial differential equations subject to the conditions of integral type. The number of these equations depends on the biologically possible maximal newborns number of the same generation produced by a pair. A class of separable solutions is studied for this model. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 93–129, January–March, 2007.     

An Age-Structured Population Dynamics Model with Females' Pregnancy and Child Care

We present a deterministic model for an age-structured population dynamics taking into account females' pregnancy, maternal care of offsprings, and environmental pressure with or without spatial migration. The model is based on the age-density notion for a group formed by a mother and her offsprings under maternal care. A harmonic-mean-type mating function of sexes without formation of permanent pairs is used. It is assumed that each sex has the pre-reproductive, reproductive, and post-reproductive age intervals. All adult individuals are divided into males, single females, fertilized females, and females taking child care. Individuals of post-reproductive age belong to the group of singles. All individuals of pre-reproductive age are divided into the young and juvenile groups. Only young offsprings are assumed to be under maternal care. Juvenile individuals can live without maternal care. The model consists of integro-PDEs subject to the conditions of integral type. The existence and uniqueness theorem is proved in the case of unlimited population. Separable solutions and their long-time behavior are studied for the limited nondispersing population. In the case of random migration two types of separable solutions and their long-time behavior for the homogeneous Dirichlet and Neumann boundary conditions are studied. In the case of directed migration in one-dimensional domain with special initial and Dirichlet boundary conditions, the unlimited invasive population dynamics is studied. In particular, an explicit formula for the migration front is given.     

A population dynamics model with strong maternal care

A two-sex age-structured nondispersing population dynamics deterministic model is presented taking into account strong maternal and weak paternal care of offspring. The model includes a weighted harmonic-mean type pair formation function and neglects the spatial dispersal and separation of pairs. It is assumed that each sex has pre-reproductive and reproductive age intervals. All adult individuals are divided into single males, single females, permanent pairs, and female-widows taking care of their offsprings after the death of their partners. All pairs are of two types: pairs without offspring under parental care at the given time and pairs taking child care. All individuals of pre-reproductive age are divided into young and juvenile groups. The young offspring are assumed to be under parental or maternal (after the death of their father) care. Juveniles can live without parental or maternal care but they cannot reproduce offsprings. It is assumed that births can only occur from couples. The model consists of nine integro-PDEs subject to the conditions of integral type. A class of separable solutions is studied, and a system for macro-moments evolving in time is derived in the case of age-independent vital ones. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 215–255, April–June, 2006.     

Nonparametric Bayesian Modeling for Multivariate Ordinal Data

This article proposes a probability model for k-dimensional ordinal outcomes, that is, it considers inference for data recorded in k-dimensional contingency tables with ordinal factors. The proposed approach is based on full posterior inference, assuming a flexible underlying prior probability model for the contingency table cell probabilities. We use a variation of the traditional multivariate probit model, with latent scores that determine the observed data. In our model, a mixture of normals prior replaces the usual single multivariate normal model for the latent variables. By augmenting the prior model to a mixture of normals we generalize inference in two important ways. First, we allow for varying local dependence structure across the contingency table. Second, inference in ordinal multivariate probit models is plagued by problems related to the choice and resampling of cutoffs defined for these latent variables. We show how the proposed mixture model approach entirely removes these problems. We illustrate the methodology with two examples, one simulated dataset and one dataset of interrater agreement.     

Computing p-values in conditional independence models for a contingency table

We present a Markov chain Monte Carlo (MCMC) method for generating Markov chains using Markov bases for conditional independence models for a four-way contingency table. We then describe a Markov basis characterized by Markov properties associated with a given conditional independence model and show how to use the Markov basis to generate random tables of a Markov chain. The estimates of exact p-values can be obtained from random tables generated by the MCMC method. Numerical experiments examine the performance of the proposed MCMC method in comparison with the χ ² approximation using large sparse contingency tables.     

Characterizing optimal sampling of binary contingency tables via the configuration model

A binary contingency table is an m × n array of binary entries with row sums r = (r₁, …, r_m) and column sums c = (c₁, …, c_n). The configuration model generates a contingency table by considering r_i tokens of type 1 for each row i and c_j tokens of type 2 for each column j, and then taking a uniformly random pairing between type‐1 and type‐2 tokens. We give a necessary and sufficient condition so that the probability that the configuration model outputs a binary contingency table remains bounded away from 0 as \begin{align*}N=\sum_{i=1}^m r_i=\sum_{j=1}^n c_j\end{align*} goes to ∞. Our finding shows surprising differences from recent results for binary symmetric contingency tables. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Partnership formation with age-dependent preferences

We analyze a model of partnership formation in which players’ preferences are based on the age of a prospective partner. There are two classes of individuals, called for convenience here male and female. Males and females are fertile for the same length of time, normalized to one unit. A male enters the mating pool at age 0 and meets prospective partners according to a Poisson process. At equilibrium, he accepts a female if the utility from mating exceeds the expected utility from future search, which depends on the acceptance strategies of all males and females and the corresponding steady-state distribution of age in the pool of unmated individuals. Females face an analogous problem. Mating pairs are only formed by mutual consent and individuals leave the pool of unmated individuals on finding a mating partner or reaching the age of 1. A policy iteration algorithm is used to determine the equilibrium acceptance strategies and the corresponding steady-state distribution of the age of individuals in the mating pool. Two examples are presented.     

Equilibria of two-sided matching games with common preferences

Problems of matching have long been studied in the operations research literature (assignment problem, secretary problem, stable marriage problem). All of these consider a centralized mechanism whereby a single decision maker chooses a complete matching which optimizes some criterion. This paper analyzes a more realistic scenario in which members of the two groups (buyers–sellers, employers–workers, males–females) randomly meet each other in pairs (interviews, dates) over time and form couples if there is mutual agreement to do so. We assume members of each group have common preferences over members of the other group. Generalizing an earlier model of Alpern and Reyniers [Alpern, S., Reyniers, D.J., 2005. Strategic mating with common preferences. J. Theor. Biol. 237, 337–354], we assume that one group (called males) is r   times larger than the other, r?1$r?1$. Thus all females, but only 1/r$1/r$ of the males, end up matched. Unmatched males have negative utility -c$-c$. We analyze equilibria of this matching game, depending on the parameters r   and c$c$. In a region of (r,c)$(r\text{,}c)$ space with multiple equilibria, we compare these, and analyze their ‘efficiency’ in several respects. This analysis should prove useful for designers of matching mechanisms who have some control over the sex ratio (e.g. by capping numbers of males at a ‘singles event’or by having ‘ladies free’ nights) or the nonmating cost c (e.g. tax benefits to married couples).     

用于枪炮射表整体解析逼近的一列基函数

本文提出用于枪炮射表整体解析逼近的一列基函数.在一个具体型号的火炮的射表上作出的数值试验表明效果良好.这列基函数对于其它型号枪炮射表的逼近应该同样有参考价值.为得出这列基函数而提出的开拓与不变性的分析方法,在数学模型工作中对于其它许多表函数的整体解析逼近应该同样有参考价值.     

Exact Inference for Kendall's S and Spearman's ρ with Extension to Fisher's Exact Test in r × c Contingency Tables

Abstract

We formulate the problem of exact inference for Kendall's S and Spearman's D algebraically, using a general recursion formula developed by Smid for the score S with ties in both rankings. Analogous recursion formulas are shown to hold for the score D as well as for a log transform, F, of the score used in Fisher's exact test of independence in contingency tables. A new implementation of Mehta and Patel's network algorithm is then applied to obtain exact significance levels of either S or D for observations from both continuous and discrete distributions. A simple extension is made to obtain Fisher's exact test in r x c contingency tables. Observed CPU times for contingency table problems four to six of Mehta and Patel and problems four and five of Clarkson, Fan, and Joe are roughly 2/3 of those obtained using Clarkson's et al. implementation of the network algorithm. It is shown that a hierarchy, with F > S > D, holds regarding the rate of aggregation. An algorithm for rapid lexicographic enumeration of entries in a frequency table is also given.     

Ordering chaos and synchronization transitions by chemical delay and coupling on scale-free neuronal networks

Chemical synaptic connections are more common than electric ones in neurons, and information transmission delay is especially significant for the synapses of chemical type. In this paper, we report a phenomenon of ordering spatiotemporal chaos and synchronization transitions by the delays and coupling through chemical synapses of modified Hodgkin–Huxley (MHH) neurons on scale-free networks. As the delay τ is increased, the neurons exhibit transitions from bursting synchronization (BS) to intermittent multiple spiking synchronizations (SS). As the coupling g_syn is increased, the neurons exhibit different types of firing transitions, depending on the values of τ. For a smaller τ, there are transitions from spatiotemporal chaotic bursting (SCB) to BS or SS; while for a larger τ, there are transitions from SCB to intermittent multiple SS. These findings show that the delays and coupling through chemical synapses can tame the chaotic firings and repeatedly enhance the firing synchronization of neurons, and hence could play important roles in the firing activity of the neurons on scale-free networks.     

Bayesian Inference for the RC(m) Association Model

Describing the structure in a two-way contingency table in terms of an RC(m) association model, we are concerned with the computation of posterior distributions of the model parameters using prior distributions which take into account the nonlinear restrictions of the model. We are further involved with the determination of the order of association m, based on Bayesian arguments. Using projection methods, a prior distribution over the parameters of the simpler RC(m) model is induced from a prior of the parameters of the saturated model. The fit of the assumed RC(m) model is evaluated using the posterior distribution of its distance from the full model. Our methods are illustrated with a popular dataset.     

A mathematical analysis of an age-space-structured autosomal diploid population dynamics model with random mating and females' pregnancy

We discuss a deterministic model of the age-structured autosomal polylocal multiallelic diploid population dynamics that takes into account random mating of sexes, females' pregnancy, and its dispersal in the whole space. This model generalizes the previous one by taking into account the spatial dispersal whose mechanism is described by the general linear elliptic differential operator of the second order. The population consists of male, single (nonfertilized) female, and fertilized female subclasses. Using the fundamental solution method for the uniformly parabolic second-order differential operator with bounded Hölder continuous coefficients, we prove the existence and uniqueness theorem for the classic solution of the Cauchy problem for this model. In the case where dispersal moduli of fertilized females do not depend on the age of mated males, we analyze the population growth and decay. Mutation is not consisdered in this paper.     

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??hen the distribution of random variable V_1,V_2 and V_3 composes an I\times J\times K contingency table, this paper research the collapsibility of entropy, and present necessary and sufficient conditions for simple collapsibility and strong collapsibility of mutual information by the research on collapsibility of entropy, which are considered under the condition of sub-independence.     

Collapsibility of Mutual Information

hen the distribution of random variable V_1,V_2 and V_3 composes an I\times J\times K contingency table, this paper research the collapsibility of entropy, and present necessary and sufficient conditions for simple collapsibility and strong collapsibility of mutual information by the research on collapsibility of entropy, which are considered under the condition of sub-independence.     

有缺失数据的2×2×2列联表的参数估计

本文讨论了缺失数据的 2× 2× 2列联表的参数估计问题 ,通过给出适当的模型和观测数据模式 ,我们给出了各列联表各细胞参数的估计 ,并且证明这些估计是G -Markov的     

Exact Conditional P Value Calculation for the Quasi-Symmetry Model

Earlier literature introduced a network algorithm for computing an exact test of independence in a two-way contingency table. This article adapts that algorithm to tests of quasi-symmetry in square tables. The algorithm is generally faster than competing Monte Carlo methods, and essentially eliminates the need for asymptotic approximation of P values for assessing goodness-of-fit of the quasi-symmetry model. A macro written for the R computing package is available for implementing the method.     

Confidence Circles For Correspondence Analysis Using Orthogonal Polynomials

An alternative approach to classical correspondence analysis was developed in [3] and involves decomposing the matrix of Pearson contingencies of a contingency table using orthogonal polynomials rather than via singular value decomposition. It is especially useful in analysing contingency tables which are of an ordinal nature. This short paper demonstrates that the confidence circles of Lebart, Morineau and Warwick (1984) for the classical approach can be applied to ordinal correspondence analysis. The advantage of the circles in analysing a contingency table is that the researcher can graphically identify the row and column categories that contribute or not to the hypothesis of independence.     

Relationship between strictly collapsible and perfect contingency tables

Whittemore (1978) conjectured that an N-dimensional contingency table p is strictly collapsible over each factor with respect to the set of remaining factors if and only if p has a certain factorization. I prove this conjecture for N = 3 and show by counterexamples that it is false for N > 3.     

A structure theorem on bivariate positive quadrant dependent distributions and tests for independence in two-way contingency tables

In this paper, the set of all bivariate positive quadrant dependent distributions with fixed marginals is shown to be compact and convex. Extreme points of this convex set are enumerated in some specific examples. Applications are given in testing the hypothesis of independence against strict positive quadrant dependence in the context of ordinal contingency tables. The performance of two tests, one of which is based on eigenvalues of a random matrix, is compared. Various procedures based upon certain functions of the eigenvalues of a random matrix are also proposed for testing for independence in a two-way contingency table when the marginals are random.