Yule Processes Research Articles

Λ-coalescents arising in a population with dormancy

Consider a population evolving from year to year through three seasons: spring, summer and winter. Every spring starts with N dormant individuals waking up independently of each other according to a given distribution. Once an individual is awake, it starts reproducing at a constant rate. By the end of spring, all individuals are awake and continue reproducing independently as Yule processes during the whole summer. In the winter, N individuals chosen uniformly at random go to sleep until the next spring, and the other individuals die. We show that because an individual that wakes up unusually early can have a large number of surviving descendants, for some choices of model parameters the genealogy of the population will be described by a Λ-coalescent. In particular, the beta coalescent can describe the genealogy when the rate at which individuals wake up increases exponentially over time. We also characterize the set of all Λ-coalescents that can arise in this framework.

The random transposition dynamics on random regular graphs and the Gaussian free field

Une permutation donnée, vue comme réunion de cycles disjoints, représente un graphe régulier de degré $2$. Considérons $d$ permutations aléatoires indépendantes, en superposons leurs structures de graphes. Ceci est le modèle de permutation bien connu donnant un (multi-)graphe régulier aléatoire de degré $2d$. Nous considérons un champ $2$-dimensionnel de $d$ permutations indexé par la taille et le temps. La taille de chaque permutation croît selon des processus couplés de restaurants chinois, tandis que chaque permutation évolue dans le temps selon une chaîne de transpositions aléatoires. À travers le modèle de permutation, ceci se projette en une famille à deux paramètres de graphes qui croissent en taille (« dimension ») et qui évoluent en temps. Dans cet ensemble de graphes aléatoires, on observe asymptotiquement une évolution remarquable des petits cycles et des statistiques linéaires des valeurs propres en dimension et en temps. En dimension, il avait été montré par Johnson et Pal (Ann. Probab. 42 (2014) 1396–1437) que les nombres de cycles sont décrits par un champ poissonnien de processus de Yule. Ici, nous donnons une description en dimension et en temps du processus des cycles en termes en termes d’un surface aléatoire poissonnienne, pour tout $d$. Lorsque $d$ tend vers l’infini, les fluctuations des nombres de cycles convergent vers un processus gaussien indexé par la dimension et le temps. Les marginales en dimension se trouvent être le champ libre gaussien, et le processus est stationnaire en temps. Une structure similaire de covariance apparaît dans les fluctuations des valeurs propres du processus des mineurs d’une matrice de Wigner réelle symétrique dont les coordonnées évoluent selon des processus stochastiques stationnaires i.i.d.. Ainsi, cet article décrit un analogue poissonnien d’une dynamique markovienne naturelle sur le champ libre gaussien, et étudie ses propriétés trajectorielles.

WORD FORMATION OF SLANG WORD IN SONG ALBUMS CREATED BY INDONESIAN RAPPER, YOUNG LEX

This research is aimed to identify (1) What are the word formation process of the slang word found in the lyrics of “Young Lex” songs (2) What are the common characteristics of slangs found in the lyrics of “Young Lex” songs. The method used in this study is descriptive qualitative method.This research paper usedGeorge Yule (2006) processes of word formation. The researcher found 6 types word formation in slang word in the lyrics of Young Lex among others blending, compounding, clipping, borrowing, coinage and acronym and 1 theory of Elissa Matiello, Variation.The data were collected by listening to 9 songs by Young Lex and reading the lyrics of the songs. There were 27 slang of variation, 6 slang of blending, 11 slang of clipping, 4 slang of compounding, 4 slang coinage, 17 slang borrowing and 6 slang of acronym.

A spatio-temporal point process model for particle growth

AbstractA spatio-temporal model of particle or star growth is defined, whereby new unit masses arrive sequentially in discrete time. These unit masses are referred to as candidate stars, which tend to arrive in mass-dense regions and then either form a new star or are absorbed by some neighbouring star of high mass. We analyse the system as time increases, and derive the asymptotic growth rate of the number of stars as well as the size of a randomly chosen star. We also prove that the size-biased mass distribution converges to a Poisson–Dirichlet distribution. This is achieved by embedding our model into a continuous-time Markov process, so that new stars arrive according to a marked Poisson process, with locations as marks, whereas existing stars grow as independent Yule processes. Our approach can be interpreted as a Hoppe-type urn scheme with a spatial structure. We discuss its relevance for and connection to models of population genetics, particle aggregation, image segmentation, epidemic spread, and random graphs with preferential attachment.

Bayesian superposition of pure-birth destructive cure processes for tumor latency

In this paper, a new Bayesian stochastic cure rate model, based on the individual frailty as the total number of descendent cells of each damaged cell (known as the volume of the tumor) up to specific time and the repair mechanism (known as the destructive mechanism), is formulated as a superposition of pure-birth stochastic processes (Yule processes). The proposed model deals with combined modeling of long-term effects (cure rate) and short-term effects (tumor growth) which can be regarded as a Bayesian alternative to the Cox regression and promotion cure rate models when the proportional hazards assumption is violated, such as in the case of crossing survival functions. A simulation study and an application to a melanoma data, using the Bayesian RStan package, are provided to illustrate the usefulness of the proposed model and also to evaluate the impact of cancer treatment over the long-term effect and the progression stage of the tumor (tumor growth). Besides this interesting new biological interpretation of the tumor growth, the model can be viewed as an extended Bayesian version of the destructive cure rate model of Rodrigues et al. and an alternative to the two-sample semiparametric model of Yang and Prentice.

A Delayed Yule Process

In now classic work, David Kendall (1966) recognized that the Yule process and Poisson process could be related by a (random) time change. Furthermore, he showed that the Yule population size rescaled by its mean has an almost sure exponentially distributed limit as t → ∞ t\to \infty . In this note we introduce a class of coupled delayed continuous time Yule processes parameterized by 0 > α ≤ 1 0 > \alpha \le 1 and find a representation of the Poisson process as a delayed Yule process at delay rate α = 1 / 2 \alpha = {1/2} . Moreover we extend Kendall’s limit theorem to include a larger class of positive martingales derived from functionals that gauge the population genealogy. Specifically, the latter is exploited to uniquely characterize the moment generating functions of distributions of the limit martingales, generalizing Kendall’s mean one exponential limit. A connection with fixed points of the Holley-Liggett smoothing transformation also emerges in this context, about which much is known from general theory in terms of moments, tail decay, and so on.

Weak limits for the largest subpopulations in Yule processes with high mutation probabilities

Abstract We consider a Yule process until the total population reaches size n ≫ 1, and assume that neutral mutations occur with high probability 1 - p (in the sense that each child is a new mutant with probability 1 - p, independently of the other children), where p = pn ≪ 1. We establish a general strategy for obtaining Poisson limit laws and a weak law of large numbers for the number of subpopulations exceeding a given size and apply this to some mutation regimes of particular interest. Finally, we give an application to subcritical Bernoulli bond percolation on random recursive trees with percolation parameter pn tending to 0.

Estimating Divergence Times and Substitution Rates in Rhizobia.

Accurate estimation of divergence times of soil bacteria that form nitrogen-fixing associations with most leguminous plants is challenging because of a limited fossil record and complexities associated with molecular clocks and phylogenetic diversity of root nodule bacteria, collectively called rhizobia. To overcome the lack of fossil record in bacteria, divergence times of host legumes were used to calibrate molecular clocks and perform phylogenetic analyses in rhizobia. The 16S rRNA gene and intergenic spacer region remain among the favored molecular markers to reconstruct the timescale of rhizobia. We evaluate the performance of the random local clock model and the classical uncorrelated lognormal relaxed clock model, in combination with four tree models (coalescent constant size, birth–death, birth–death incomplete sampling, and Yule processes) on rhizobial divergence time estimates. Bayes factor tests based on the marginal likelihoods estimated from the stepping-stone sampling analyses strongly favored the random local clock model in combination with Yule process. Our results on the divergence time estimation from 16S rRNA gene and intergenic spacer region sequences are compatible with age estimates based on the conserved core genes but significantly older than those obtained from symbiotic genes, such as nodIJ genes. This difference may be due to the accelerated evolutionary rates of symbiotic genes compared to those of other genomic regions not directly implicated in nodulation processes.

Yule processes with rare mutation and their applications to percolation on $b$-ary trees

We consider supercritical Bernoulli bond percolation on a large $b$-ary tree, in the sense that with high probability, there exists a giant cluster. We show that the size of the giant cluster has non-gaussian fluctuations, which extends a result due to Schweinsberg in the case of random recursive trees. Using ideas in the recent work of Bertoin and Uribe Bravo, the approach developed in this work relies on the analysis of the sub-population with ancestral type in a system of branching processes with rare mutations, which may be of independent interest. This also allows us to establish the analogous result for scale-free trees.

A predator–prey SIR type dynamics on large complete graphs with three phase transitions

We study a variation of the SIR (Susceptible/Infected/Recovered) dynamics on the complete graph, in which infected individuals may only spread to neighboring susceptible individuals at fixed rate λ>0 while recovered individuals may only spread to neighboring infected individuals at fixed rate 1. This is also a variant of the so-called chase–escape process introduced by Kordzakhia and then Bordenave. Our work is the first study of this dynamics on complete graphs. Starting with one infected and one recovered individuals on the complete graph with N+2 vertices, and stopping the process when one type of individuals disappears, we study the asymptotic behavior of the probability that the infection spreads to the whole graph as N→∞ and show that for λ∈(0,1) (resp. for λ>1), the infection dies out (resp. does not die out) with probability tending to one as N→∞, and that the probability that the infection dies out tends to 1/2 for λ=1. We also establish limit theorems concerning the final state of the system in all regimes and show that two additional phase transitions occur in the subcritical phase λ∈(0,1): at λ=1/2 the behavior of the expected number of remaining infected individuals changes, while at λ=(5−1)/2 the behavior of the expected number of remaining recovered individuals changes. We also study the outbreak sizes of the infection, and show that the outbreak sizes are small (or self-limiting) if λ∈(0,1/2], exhibit a power-law behavior for 1/2<λ<1, and are pandemic for λ⩾1. Our method relies on different couplings: we first couple the dynamics with two independent Yule processes by using an Athreya–Karlin embedding, and then we couple the Yule processes with Poisson processes thanks to Kendall’s representation of Yule processes.

Cycles and eigenvalues of sequentially growing random regular graphs

Consider the sum of $d$ many i.i.d. random permutation matrices on $n$ labels along with their transposes. The resulting matrix is the adjacency matrix of a random regular (multi)-graph of degree $2d$ on $n$ vertices. It is known that the distribution of smooth linear eigenvalue statistics of this matrix is given asymptotically by sums of Poisson random variables. This is in contrast with Gaussian fluctuation of similar quantities in the case of Wigner matrices. It is also known that for Wigner matrices the joint fluctuation of linear eigenvalue statistics across minors of growing sizes can be expressed in terms of the Gaussian Free Field (GFF). In this article, we explore joint asymptotic (in $n$) fluctuation for a coupling of all random regular graphs of various degrees obtained by growing each component permutation according to the Chinese Restaurant Process. Our primary result is that the corresponding eigenvalue statistics can be expressed in terms of a family of independent Yule processes with immigration. These processes track the evolution of short cycles in the graph. If we now take $d$ to infinity, certain GFF-like properties emerge.

Probability distributions of ancestries and genealogical distances on stochastically generated rooted binary trees

The stationary birth-only, or Yule–Furry, process for rooted binary trees has been analysed with a view to developing explicit expressions for two fundamental statistical distributions: the probability that a randomly selected leaf is preceded by N nodes, or “ancestors”, and the probability that two randomly selected leaves are separated by N nodes. For continuous-time Yule processes, the first of these distributions is presented in closed analytical form as a function of time, with time being measured with respect to the moment of “birth” of the common ancestor (which is essentially inaccessible to phylogenetic analysis), or with respect to the instant at which the first bifurcation occurred. The second distribution is shown to follow in an iterative manner from a hierarchy of second-order ordinary differential equations. For Yule trees of a given number n of tips, expressions have been derived for the mean and variance for each of these distributions as functions of n, as well as for the distributions themselves. In addition, it is shown how the methods developed to obtain these distributions can be employed to find, with minor effort, expressions for the expectation values of two statistics on Yule trees, the Sackin index (sum over all root-to-leaf distances), and the sum over all leaf-to-leaf distances.

THE IMPACT OF GENE-TREE/SPECIES-TREE DISCORDANCE ON DIVERSIFICATION-RATE ESTIMATION

Molecular phylogenies are often used to test hypotheses about the tempo and mode of speciation and extinction. One commonly used statistic is Pybus and Harvey's γ, which measures the density of ordered internode distances on an ultrametric tree to infer earlier (negative γ) or later (positive γ) bursts of diversification. However, coalescent theory predicts that γ might be biased toward negative values (inferring early bursts of diversification) when using gene trees rather than species trees. Gene divergences predate species divergences, increasingly so at higher effective population sizes (N(e)), and proportionally more so toward the tips of the tree. Thus, gene trees will have a higher density of older nodes in many cases (particularly at higher N(e)), due to the disproportionate lengthening of terminal branches. This will yield an artifactual signature of early bursts of diversification when estimating γ from gene trees. We simulate gene trees within species trees under both Yule (pure-birth) and birth-death processes, and demonstrate support for these predictions. However, for most realistic estimates of θ in natural populations, gene trees provide relatively good estimates of γ, despite the disproportionate overestimation of younger node ages. This is corroborated with an empirical dataset of North American fence lizards (Sceloporus).

Occupancy schemes associated to Yule processes

An occupancy problem with an infinite number of bins and a random probability vector for the locations of the balls is considered. The respective sizes of the bins are related to the split times of a Yule process. The asymptotic behavior of the landscape of the first empty bins, i.e. the set of corresponding indices represented by point processes, is analyzed and convergences in distribution to mixed Poisson processes are established. Additionally, the influence of the random environment, the random probability vector, is analyzed. It is represented by two main components: an independent, identically distributed sequence and a fixed random variable. Each of these components has a specific impact on the qualitative behavior of the stochastic model. It is shown in particular that, for some values of the parameters, some rare events, which are identified, determine the asymptotic behavior of the average values of the number of empty bins in some regions.

Occupancy schemes associated to Yule processes

An occupancy problem with an infinite number of bins and a random probability vector for the locations of the balls is considered. The respective sizes of the bins are related to the split times of a Yule process. The asymptotic behavior of the landscape of the first empty bins, i.e. the set of corresponding indices represented by point processes, is analyzed and convergences in distribution to mixed Poisson processes are established. Additionally, the influence of the random environment, the random probability vector, is analyzed. It is represented by two main components: an independent, identically distributed sequence and a fixed random variable. Each of these components has a specific impact on the qualitative behavior of the stochastic model. It is shown in particular that, for some values of the parameters, some rare events, which are identified, determine the asymptotic behavior of the average values of the number of empty bins in some regions.

SOJOURN TIME TAILS IN THE M/D/1 PROCESSOR SHARING QUEUE

We consider the sojourn time V in the M/D/1 processor sharing (PS) queue and show that P(V &gt; x) is of the form Ce−γx as x becomes large. The proof involves a geometric random sum representation of V and a connection with Yule processes, which also enables us to simplify Ott's [21] derivation of the Laplace transform of V. Numerical experiments show that the approximation P(V &gt; x) ≈ Ce−γx is excellent even for moderate values of x.

On the suitability of Yule process to stochastically model some properties of object-oriented systems

We present a study of three large object-oriented software systems—VisualWorks Smalltalk, Java JDK and Eclipse—searching for scaling laws in some of their properties. We study four system properties related to code production, namely the inheritance hierarchies, the naming of variables and methods, and the calls to methods. We systematically found power-law distributions in these properties, most of which have never been reported before. We were also able to statistically model the programming activities leading to the studied properties as Yule processes, with very good correspondence between empirical data and the prediction of Yule model. The fact that a design and optimization process like software development can be modeled on the large with the laws of statistical physics poses intriguing issues to software engineers, and could be exploited for finding new metrics and quality measures.

STOCHASTIC COMPARISONS OF NONHOMOGENEOUS PROCESSES

The purpose of this article is to describe various conditions on the parameters of pairs of nonhomogeneous Poisson or pure birth processes under which the corresponding epoch times or interepoch intervals are stochastically ordered in various senses. We derive results involving the usual stochastic order, the multivariate hazard rate order, the multivariate likelihood ratio order, as well as the dispersive and the mean residual life orders. A sample of applications involving generalized Yule processes, load-sharing models, and minimal repairs in reliability theory illustrate the usefulness of the new results.

On Asymptotic Ancillarity and Inference for Yule and Regular Nonergodic Processes

By extending the notion of ancillarity to asymptotic ancillarity a conditional inference approach to statistical inference for a class of regular nonergodic processes is developed. This approach is closely related to that of Efron & Hinkley (1978) for the independent identically distributed case. The Yule process is used to illustrate the main ideas.

On asymptotic ancillarity and inference for Yule and regular nonergodic processes

SUMMARY By extending the notion of ancillarity to asymptotic ancillarity a conditional inference approach to statistical inference for a class of regular nonergodic processes is developed. This approach is closely related to that of Efron & Hinkley (1978) for the independent identically distributed case. The Yule process is used to illustrate the main ideas.