A Sufficient Condition for Reducing Recursions in Hidden Markov Models

In hidden Markov models, the probability of observing a set of strings can be computed using recursion relations. We construct a sufficient condition for simplifying the recursion relations for a certain class of hidden Markov models. If the condition is satisfied, then one can construct a reduced recursion where the dependence on Markov states completely disappears. We discuss a specific example—namely, statistical multiple alignment based on the TKF-model—in which the sufficient condition is satisfied.     

A rumor transmission model with various contact interactions

We consider a rumor transmission model with various contact interactions and explore what effect such interactions have on the spread of a rumor, in particular whether they can explain the rumor recursion. Through mathematical analysis and computer simulations, we conjecture that rumor recursion remains a major challenge to mathematical models of rumors beyond our model proposed here.     

Nets with distance bias

A distance bias is imposed on the probability of direct connection between every pair of points in a random net. The probability that there exists a path from a given point in the net to another point is now a function of both the axone density and the distance between the points. A recursion formula is derived in terms of which this probability can be computed. The rate of spread of an epidemic where probability of contact depends on the distance between the individuals can also be computed from the recursion formula.     

Distinctive signatures of recursion

Although recursion has been hypothesized to be a necessary capacity for the evolution of language, the multiplicity of definitions being used has undermined the broader interpretation of empirical results. I propose that only a definition focused on representational abilities allows the prediction of specific behavioural traits that enable us to distinguish recursion from non-recursive iteration and from hierarchical embedding: only subjects able to represent recursion, i.e. to represent different hierarchical dependencies (related by parenthood) with the same set of rules, are able to generalize and produce new levels of embedding beyond those specified a priori (in the algorithm or in the input). The ability to use such representations may be advantageous in several domains: action sequencing, problem-solving, spatial navigation, social navigation and for the emergence of conventionalized communication systems. The ability to represent contiguous hierarchical levels with the same rules may lead subjects to expect unknown levels and constituents to behave similarly, and this prior knowledge may bias learning positively. Finally, a new paradigm to test for recursion is presented. Preliminary results suggest that the ability to represent recursion in the spatial domain recruits both visual and verbal resources. Implications regarding language evolution are discussed.     

Inexpensive Computation of the Inverse of the Genomic Relationship Matrix in Populations with Small Effective Population Size

Many computations with SNP data including genomic evaluation, parameter estimation, and genome-wide association studies use an inverse of the genomic relationship matrix. The cost of a regular inversion is cubic and is prohibitively expensive for large matrices. Recent studies in cattle demonstrated that the inverse can be computed in almost linear time by recursion on any subset of ∼10,000 individuals. The purpose of this study is to present a theory of why such a recursion works and its implication for other populations. Assume that, because of a small effective population size, the additive information in a genotyped population has a small dimensionality, even with a very large number of SNP markers. That dimensionality is visible as a limited number of effective SNP effects, independent chromosome segments, or the rank of the genomic relationship matrix. Decompose a population arbitrarily into core and noncore individuals, with the number of core individuals equal to that dimensionality. Then, breeding values of noncore individuals can be derived by recursions on breeding values of core individuals, with coefficients of the recursion computed from the genomic relationship matrix. A resulting algorithm for the inversion called “algorithm for proven and young” (APY) has a linear computing and memory cost for noncore animals. Noninfinitesimal genetic architecture can be accommodated through a trait-specific genomic relationship matrix, possibly derived from Bayesian regressions. For populations with small effective population size, the inverse of the genomic relationship matrix can be computed inexpensively for a very large number of genotyped individuals.     

Statecharts for Gene Network Modeling

State diagrams (stategraphs) are suitable for describing the behavior of dynamic systems. However, when they are used to model large and complex systems, determining the states and transitions among them can be overwhelming, due to their flat, unstratified structure. In this article, we present the use of statecharts as a novel way of modeling complex gene networks. Statecharts extend conventional state diagrams with features such as nested hierarchy, recursion, and concurrency. These features are commonly utilized in engineering for designing complex systems and can enable us to model complex gene networks in an efficient and systematic way. We modeled five key gene network motifs, simple regulation, autoregulation, feed-forward loop, single-input module, and dense overlapping regulon, using statecharts. Specifically, utilizing nested hierarchy and recursion, we were able to model a complex interlocked feed-forward loop network in a highly structured way, demonstrating the potential of our approach for modeling large and complex gene networks.     

On spreading speeds and traveling waves for growth and migration models in a periodic habitat

 It is shown that the methods previously used by the author [Wei82] and by R. Lui [Lui89] to obtain asymptotic spreading results and sometimes the existence of traveling waves for a discrete-time recursion with a translation invariant order preserving operator can be extended to a recursion with a periodic order preserving operator. The operator can be taken to be the time-one map of a continuous time reaction-diffusion model, or it can be a more general model of time evolution in population genetics or population ecology in a periodic habitat. Methods of estimating the speeds of spreading in various directions will also be presented. Received: 12 July 2001 / Revised version: 19 July 2002 / Published online: 17 October 2002 Mathematics Subject Classification (2000): 92D40, 92D25, 35K55, 35K57, 35B40 Keywords or phrases: Periodic – Spreading speed – Traveling wave     

Analysis of linear determinacy for spread in cooperative models

 The discrete-time recursion system $\u_{n+1}=Q[\u_n]$ with $\u_n(x)$ a vector of population distributions of species and $Q$ an operator which models the growth, interaction, and migration of the species is considered. Previously known results are extended so that one can treat the local invasion of an equilibrium of cooperating species by a new species or mutant. It is found that, in general, the resulting change in the equilibrium density of each species spreads at its own asymptotic speed, with the speed of the invader the slowest of the speeds. Conditions on $Q$ are given which insure that all species spread at the same asymptotic speed, and that this speed agrees with the more easily calculated speed of a linearized problem for the invader alone. If this is true we say that the recursion has a single speed and is linearly determinate. The conditions are such that they can be verified for a class of reaction-diffusion models. Received: 7 August 2000 / Revised version: 5 January 2002 / Published online: 17 July 2002     

Interrelationship Between Fractal Ornament and Multilevel Selection Theory

Interdisciplinarity is one of the features of modern science, defined as blurring the boundaries of disciplines and overcoming their limitations or excessive specialization by borrowing methods from one discipline into another, integrating different theoretical assumptions, and using the same concepts and terms. Often, theoretical knowledge of one discipline and technological advances of another are combined within an interdisciplinary science, and new branches or disciplines may also emerge. Biosemiotics, a field that arose at the crossroads of biology, semiotics, linguistics, and philosophy, enables scientists to borrow theoretical assumptions from semiotics and extend them to different biological theories. The latter applies especially to extended synthesis, wherein culture is viewed as one of the factors influencing evolution. In the present research, the semiotic system of Ukrainian folk ornament is analyzed through the theory of fractals, key features of which are recursion and self-similarity. As a result, an assumption is made about the fractal structure of culture and social life on a conceptual level. What follows is a discussion of how this assumption can contribute to the multilevel selection theory, one of the foundations of extended synthesis, which employs the concept of self-similarity at all levels of the biological hierarchy.     

RNA二级子结构的计数

多核苷酸的二级结构可视为一类顶点标号平面图,通常通过枚举每类RNA二级结构图的各种子图来计算其递推公式．本文给出了若干限制端环长度的RNA二级子结构的递推公式及渐近值。     

Modeling Critical Infrastructure Interdependency: The Case of the Mexico City Metro Transport System

ABSTRACT

“Critical infrastructures” are exceedingly complex and highly interconnected. In order to gain a full understanding and comprehensive awareness of the risks associated with critical infrastructures it becomes essential to consider in a coherent way all the aspects that may cause a failure of such systems. That is, there is a need for a systemic approach to interdependencies among critical infrastructures. The article presents the application of a systemic safety management system (SSMS) model to interdependency modeling for the case of the Mexico City Metro transport network. The model has highlighted that interdependencies occur vertically and horizontally. Horizontal interdependency occurs at every level of recursion and can be: operational, managerial, and environmental. Vertical interdependency, on the other hand, occurs between two levels of recursion only. The SSMS model has shown the potential to be used to model interdependencies among critical infrastructures. It is hoped that the approach presented may help to gain a better understanding of critical infrastructure interdependency.     

农作物产量预报模型研究与实践

选取玉米、大豆、小麦三种作物,建立产量的定性和定量预报模型,即年景趋势预报模型、逐步回归周期分量预报模型和多层递阶预报模型。经过检验和预报实践检验,表明所得到的预测模型具有一定的实用价值。     

A Likelihood Approach to Estimate the Number of Co-Infections

The number of co-infections of a pathogen (multiplicity of infection or MOI) is a relevant parameter in epidemiology as it relates to transmission intensity. Notably, such quantities can be built into a metric in the context of disease control and prevention. Having applications to malaria in mind, we develop here a maximum-likelihood (ML) framework to estimate the quantities of interest at low computational and no additional costs to study designs or data collection. We show how the ML estimate for the quantities of interest and corresponding confidence-regions are obtained from multiple genetic loci. Assuming specifically that infections are rare and independent events, the number of infections per host follows a conditional Poisson distribution. Under this assumption, we show that a unique ML estimate for the parameter () describing MOI exists which is found by a simple recursion. Moreover, we provide explicit formulas for asymptotic confidence intervals, and show that profile-likelihood-based confidence intervals exist, which are found by a simple two-dimensional recursion. Based on the confidence intervals we provide alternative statistical tests for the MOI parameter. Finally, we illustrate the methods on three malaria data sets. The statistical framework however is not limited to malaria.     

An Exact Paired Rank Test

An exact rank test for two dependent samples based on overall mid‐ranks is discussed which can be applied to metric as well as to ordinal data. The exact conditional distribution of the test statistic given the observed vector of rank differences is determined. A recursion formula is given as well as a fast shift algorithm in SAS/IML code. Moreover, it is demonstrated that the paired rank test can be more powerful than other tests for paired samples by means of a simulation study. Finally, the test is applied to a psychiatric trial with longitudinal ordinal data.     

Existence of traveling waves for integral recursions with nonmonotone growth functions

A class of integral recursion models for the growth and spread of a synchronized single-species population is studied. It is well known that if there is no overcompensation in the fecundity function, the recursion has an asymptotic spreading speed c*, and that this speed can be characterized as the speed of the slowest non-constant traveling wave solution. A class of integral recursions with overcompensation which still have asymptotic spreading speeds can be found by using the ideas introduced by Thieme (J Reine Angew Math 306:94–121, 1979) for the study of space-time integral equation models for epidemics. The present work gives a large subclass of these models with overcompensation for which the spreading speed can still be characterized as the slowest speed of a non-constant traveling wave. To illustrate our results, we numerically simulate a series of traveling waves. The simulations indicate that, depending on the properties of the fecundity function, the tails of the waves may approach the carrying capacity monotonically, may approach the carrying capacity in an oscillatory manner, or may oscillate continually about the carrying capacity, with its values bounded above and below by computable positive numbers. B. Li’s research was partially supported by the National Science Foundation under Grant DMS-616445. M. A. Lewis research was supported by “The Canada Research Chairs program,” and a grant from the Natural Sciences and Engineering Research Council of Canada.     

The kinetics of diffused substances in some biological systems

The purpose of this paper is to present a method for the determination of a solution for a coupled system of first order initial boundary-value problems arising from some biological systems. The physical problem is to determine the suspended and the superficial molecular concentrations of a traced substance passing through an organ containing a tangle of vessels, such as the kidney-ureter system. The approach to the problem is by successive approximation which leads to a recursion formula for the determination of the solution as well as error estimates for the approximations. The recursion formula involves only direct integration which indicates a promising possibility in obtaining numerical results by using a computer. In addition to the determination of a solution, some qualitative analysis of the solution is given. This includes the existence of a unique solution, the continuous dependency of the solution on the data, and the stability problem of a steady-state solution.     

A recursion model for cellular production/assembly systems

This article presents an efficient algorithm for applying the recursion modeling approach to describe the transient operation of cellular production/assembly systems that incorporate features such as finite buffers, job-shop routing, lot sequencing, and material handling. Tests evaluate the approximation method relative to number of machines at a station, capacity of input/output buffers, degree of balance among station processing times, and sequencing rule. Furthermore, the method is demonstrated in application to a hypothetical industrial setting that involves the assembly of electronic circuit cards in a facility composed of several cells. All tests indicate that the method gives accurate estimates of transient performance within reasonable runtime. In comparison with earlier recursion models, this research incorporates a number of new features (see list above), improves the accuracy of approximation, and facilitates implementation with a new, more efficient algorithm.     

Neutral evolution in spatially continuous populations

We introduce a general recursion for the probability of identity in state of two individuals sampled from a population subject to mutation, migration, and random drift in a two-dimensional continuum. The recursion allows for the interactions induced by density-dependent regulation of the population, which are inevitable in a continuous population. We give explicit series expansions for large neighbourhood size and for low mutation rates respectively and investigate the accuracy of the classical Malécot formula for these general models. When neighbourhood size is small, this formula does not give the identity even over large scales. However, for large neighbourhood size, it is an accurate approximation which summarises the local population structure in terms of three quantities: the effective dispersal rate, sigma(e); the effective population density, rho(e); and a local scale, kappa, at which local interactions become significant. The results are illustrated by simulations.     

A nonparametric empirical Bayes framework for large-scale multiple testing

We propose a flexible and identifiable version of the 2-groups model, motivated by hierarchical Bayes considerations, that features an empirical null and a semiparametric mixture model for the nonnull cases. We use a computationally efficient predictive recursion (PR) marginal likelihood procedure to estimate the model parameters, even the nonparametric mixing distribution. This leads to a nonparametric empirical Bayes testing procedure, which we call PRtest, based on thresholding the estimated local false discovery rates. Simulations and real data examples demonstrate that, compared to existing approaches, PRtest's careful handling of the nonnull density can give a much better fit in the tails of the mixture distribution which, in turn, can lead to more realistic conclusions.