Yule Process Research Articles

Topology and inference for Yule trees with multiple states.

We introduce two models for random trees with multiple states motivated by studies of trait dependence in the evolution of species. Our discrete time model, the multiple state ERM tree, is a generalization of Markov propagation models on a random tree generated by a binary search or 'equal rates Markov' mechanism. Our continuous time model, the multiple state Yule tree, is a generalization of the tree generated by a pure birth or Yule process to the tree generated by multi-type branching processes. We study state dependent topological properties of these two random tree models. We derive asymptotic results that allow one to infer model parameters from data on states at the leaves and at branch-points that are one step away from the leaves.

Stochastic data mining tools for pipe blockage failure prediction

Failure prediction plays an important role in the management of urban water systems infrastructures. An accurate description of the deterioration of urban drainage systems is essential for optimal investment and rehabilitation planning. In the study presented in this paper, a new method to predict sewer pipe failure based on robust decision trees is proposed. Five other different stochastic failure prediction models – the non-homogeneous Poisson process, the zero-inflated non-homogeneous Poisson process, classical decision tress (CART and Random Forest algorithms), the Weibull accelerated lifetime model and the linear extended Yule process – are also implemented and explored in order to identify models that combine good failure prediction results with robustness. The six models were tested on the asset register and pipe failure register of a large US wastewater utility; only pipe blockage failures were considered in this study. The linear extended Yule process and the zero-inflated non-homogeneous Poisson process presented the overall best results throughout the models’ comparison, showing a good ability to detect pipes with high likelihood of blockage failure. Decision trees based on robust random forests only detected pipes with high likelihood of failure when considering a short-term prediction window; the accuracy of the predictions was one of the best when using the robust decision tree model. The Weibull accelerated lifetime model provided some of the best medium-term predictions but performed less well for shorter prediction windows.

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.

Phylogenetic confidence intervals for the optimal trait value

We consider a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny. The unknown tree is modelled by a Yule process conditioned on n contemporary nodes. The trait value is assumed to evolve along lineages as an Ornstein-Uhlenbeck process. As a result, the trait values of the n species form a sample with dependent observations. We establish three limit theorems for the sample mean corresponding to three domains for the adaptation rate. In the case of fast adaptation, we show that for large n the normalized sample mean is approximately normally distributed. Using these limit theorems, we develop novel confidence interval formulae for the optimal trait value.

Phylogenetic confidence intervals for the optimal trait value

We consider a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny. The unknown tree is modelled by a Yule process conditioned on n contemporary nodes. The trait value is assumed to evolve along lineages as an Ornstein-Uhlenbeck process. As a result, the trait values of the n species form a sample with dependent observations. We establish three limit theorems for the sample mean corresponding to three domains for the adaptation rate. In the case of fast adaptation, we show that for large n the normalized sample mean is approximately normally distributed. Using these limit theorems, we develop novel confidence interval formulae for the optimal trait value.

Test of two hypotheses explaining the size of populations in a system of cities

Two classical hypotheses are examined about the population growth in a system of cities: Hypothesis 1 pertains to Gibrat's and Zipf's theory which states that the city growth–decay process is size independent; Hypothesis 2 pertains to the so-called Yule process which states that the growth of populations in cities happens when (i) the distribution of the city population initial size obeys a log-normal function, (ii) the growth of the settlements follows a stochastic process. The basis for the test is some official data on Bulgarian cities at various times. This system was chosen because (i) Bulgaria is a country for which one does not expect biased theoretical conditions; (ii) the city populations were determined rather precisely. The present results show that: (i) the population size growth of the Bulgarian cities is size dependent, whence Hypothesis 1 is not confirmed for Bulgaria; (ii) the population size growth of Bulgarian cities can be described by a double Pareto log-normal distribution, whence Hypothesis 2 is valid for the Bulgarian city system. It is expected that this fine study brings some information and light on other usually considered to be more pertinent countries of city systems.

Statistical properties of pairwise distances between leaves on a random Yule tree.

A Yule tree is the result of a branching process with constant birth and death rates. Such a process serves as an instructive null model of many empirical systems, for instance, the evolution of species leading to a phylogenetic tree. However, often in phylogeny the only available information is the pairwise distances between a small fraction of extant species representing the leaves of the tree. In this article we study statistical properties of the pairwise distances in a Yule tree. Using a method based on a recursion, we derive an exact, analytic and compact formula for the expected number of pairs separated by a certain time distance. This number turns out to follow a increasing exponential function. This property of a Yule tree can serve as a simple test for empirical data to be well described by a Yule process. We further use this recursive method to calculate the expected number of the n-most closely related pairs of leaves and the number of cherries separated by a certain time distance. To make our results more useful for realistic scenarios, we explicitly take into account that the leaves of a tree may be incompletely sampled and derive a criterion for poorly sampled phylogenies. We show that our result can account for empirical data, using two families of birds species.

A consistent estimator of the evolutionary rate.

We consider a branching particle system where particles reproduce according to the pure birth Yule process with the birth rate λ, conditioned on the observed number of particles to be equal to n. Particles are assumed to move independently on the real line according to the Brownian motion with the local variance σ(2). In this paper we treat n particles as a sample of related species. The spatial Brownian motion of a particle describes the development of a trait value of interest (e.g. log-body-size). We propose an unbiased estimator Rn(2) of the evolutionary rate ρ(2)=σ(2)/λ. The estimator Rn(2) is proportional to the sample variance Sn(2) computed from n trait values. We find an approximate formula for the standard error of Rn(2) based on a neat asymptotic relation for the variance of Sn(2).

Self-Similarity in Population Dynamics: Surname Distributions and Genealogical Trees

The frequency distribution of surnames turns out to be a relevant issue not only in historical demography but also in population biology, and especially in genetics, since surnames tend to behave like neutral genes and propagate like Y chromosomes. The stochastic dynamics leading to the observed scale-invariant distributions has been studied as a Yule process, as a branching phenomenon and also by field-theoretical renormalization group techniques. In the absence of mutations the theoretical models are in good agreement with empirical evidence, but when mutations are present a discrepancy between the theoretical and the experimental exponents is observed. Hints for the possible origin of the mismatch are discussed, with some emphasis on the difference between the asymptotic frequency distribution of a full population and the frequency distributions observed in its samples. A precise connection is established between surname distributions and the statistical properties of genealogical trees. Ancestors tables, being obviously self-similar, may be investigated theoretically by renormalization group techniques, but they can also be studied empirically by exploiting the large online genealogical databases concerning European nobility.

Average reductions between random tree pairs

There are a number of measures of degrees of similarity between rooted binary trees. Many of these ignore sections of the trees which are in complete agreement. We use computational experiments to investigate the statistical characteristics of such a measure of tree similarity for ordered, rooted, binary trees. We generate the trees used in the experiments iteratively, using the Yule process modeled upon speciation. Rooted binary trees arise in a wide range of settings, from biological evolutionary trees to efficient structures for searching datasets. There are a number of measures of tree similarity which arise in these settings. Here we investigate a measure which is relevant for ordered, rooted, binary trees of the same size. Examples of trees satisfying such conditions include some binary search trees. Our approach is to consider pairs of such trees of increasing size n, selected via a random process, and investigate the degree of commonality given by a natural measure of the degree to which they agree completely on peripheral subtrees. Using experimental evidence, we find that the degree of commonality appears to grown linearly with tree size, and we estimate the average behavior. There are a number of processes for selecting trees randomly. One method that is commonly studied is the uniform distribution on trees, where each tree is equally likely to be selected. Some properties of the reduction behavior of trees selected uniformly at random have been investigated by Cleary, Elder, Rechnitzer and Taback [Cleary et al. 2010] while studying statistical properties of Thompson’s group F, showing that a tree pair selected from the uniform distribution on tree pairs is almost surely unreduced in the sense described below. The common subtrees investigated here via reduction are a particular case of common edges, where in the

On the distribution of interspecies correlation for Markov models of character evolution on Yule trees

Efforts to reconstruct phylogenetic trees and understand evolutionary processes depend fundamentally on stochastic models of speciation and mutation. The simplest continuous-time model for speciation in phylogenetic trees is the Yule process, in which new species are “born” from existing lineages at a constant rate. Recent work has illuminated some of the structural properties of Yule trees, but it remains mostly unknown how these properties affect sequence and trait patterns observed at the tips of the phylogenetic tree. Understanding the interplay between speciation and mutation under simple models of evolution is essential for deriving valid phylogenetic inference methods and gives insight into the optimal design of phylogenetic studies. In this work, we derive the probability distribution of interspecies covariance under Brownian motion and Ornstein–Uhlenbeck models of phenotypic change on a Yule tree. We compute the probability distribution of the number of mutations shared between two randomly chosen taxa in a Yule tree under discrete Markov mutation models. Our results suggest summary measures of phylogenetic information content, illuminate the correlation between site patterns in sequences or traits of related organisms, and provide heuristics for experimental design and reconstruction of phylogenetic trees.

The Yule Approximation for the Site Frequency Spectrum after a Selective Sweep

In the area of evolutionary theory, a key question is which portions of the genome of a species are targets of natural selection. Genetic hitchhiking is a theoretical concept that has helped to identify various such targets in natural populations. In the presence of recombination, a severe reduction in sequence diversity is expected around a strongly beneficial allele. The site frequency spectrum is an important tool in genome scans for selection and is composed of the numbers , where is the number of single nucleotide polymorphisms (SNPs) present in from individuals. Previous work has shown that both the number of low- and high-frequency variants are elevated relative to neutral evolution when a strongly beneficial allele fixes. Here, we follow a recent investigation of genetic hitchhiking using a marked Yule process to obtain an analytical prediction of the site frequency spectrum in a panmictic population at the time of fixation of a highly beneficial mutation. We combine standard results from the neutral case with the effects of a selective sweep. As simulations show, the resulting formula produces predictions that are more accurate than previous approaches for the whole frequency spectrum. In particular, the formula correctly predicts the elevation of low- and high-frequency variants and is significantly more accurate than previously derived formulas for intermediate frequency variants.

Yule-generated trees constrained by node imbalance

The Yule process generates a class of binary trees which is fundamental to population genetic models and other applications in evolutionary biology. In this paper, we introduce a family of sub-classes of ranked trees, called Ω-trees, which are characterized by imbalance of internal nodes. The degree of imbalance is defined by an integer 0≤ω. For caterpillars, the extreme case of unbalanced trees, ω=0. Under models of neutral evolution, for instance the Yule model, trees with small ω are unlikely to occur by chance. Indeed, imbalance can be a signature of permanent selection pressure, such as observable in the genealogies of certain pathogens. From a mathematical point of view it is interesting to observe that the space of Ω-trees maintains several statistical invariants although it is drastically reduced in size compared to the space of unconstrained Yule trees. Using generating functions, we study here some basic combinatorial properties of Ω-trees. We focus on the distribution of the number of subtrees with two leaves. We show that expectation and variance of this distribution match those for unconstrained trees already for very small values of ω.

Extending the Yule process to model recurrent pipe failures in water supply networks

This paper shows how to construct, within the mathematical framework of counting process theory, a Linear Extension of the Yule Process (LEYP) to model the recurrent failures of pressure pipes in water supply networks. The choice of the counting process framework is motivated by an analysis of the advantages and shortcomings of the modelling approaches proposed in the literature over the last thirty years. The parametric nature of the LEYP model enables the prediction of future failures, accounting for the effect of previous failures, pipe ageing, and explanatory factors, assuming proportional hazards. The counting process is Markovian by definition and shown to follow a negative binomial distribution. This property leads to a useful and simple formula for computing conditional predictions given past observed failures. The likelihood of the parameters with respect to a sequence of observed failures is derived using product-integration. Maximum likelihood estimates of the model parameters can be calculated with left-truncated data, i.e. failure observations restricted to a time interval that possibly starts long after pipe commissioning. Procedures for testing the significance of the model parameters, for assessing the model goodness of fit, and for validating the model predictions are presented. The predictive performance of the LEYP model is finally illustrated with extensive failure data provided by a French water utility.

Diversity, disparity, and evolutionary rate estimation for unresolved Yule trees.

The branching structure of biological evolution confers statistical dependencies on phenotypic trait values in related organisms. For this reason, comparative macroevolutionary studies usually begin with an inferred phylogeny that describes the evolutionary relationships of the organisms of interest. The probability of the observed trait data can be computed by assuming a model for trait evolution, such as Brownian motion, over the branches of this fixed tree. However, the phylogenetic tree itself contributes statistical uncertainty to estimates of rates of phenotypic evolution, and many comparative evolutionary biologists regard the tree as a nuisance parameter. In this article, we present a framework for analytically integrating over unknown phylogenetic trees in comparative evolutionary studies by assuming that the tree arises from a continuous-time Markov branching model called the Yule process. To do this, we derive a closed-form expression for the distribution of phylogenetic diversity (PD), which is the sum of branch lengths connecting the species in a clade. We then present a generalization of PD which is equivalent to the expected trait disparity in a set of taxa whose evolutionary relationships are generated by a Yule process and whose traits evolve by Brownian motion. We find expressions for the distribution of expected trait disparity under a Yule tree. Given one or more observations of trait disparity in a clade, we perform fast likelihood-based estimation of the Brownian variance for unresolved clades. Our method does not require simulation or a fixed phylogenetic tree. We conclude with a brief example illustrating Brownian rate estimation for 12 families in the mammalian order Carnivora, in which the phylogenetic tree for each family is unresolved.

Comparative Study of Three Stochastic Models for Prediction of Pipe Failures in Water Supply Systems

The prediction of pipe failures in urban water systems is a complex process because the available failure records, originating in work orders, are often short and incomplete. To identify a robust and simple model with good failure prediction results using short data history, three existing models were compared in this study: the single-variate Poisson process, the Weibull accelerated lifetime model, and the linear-extended Yule process. This work also presents modifications to these models that enable them to produce more accurate predictions and overcome computational issues for practical software implementation. The three models, together with the improvements where applicable, were applied to water supply system data provided by a Portuguese water utility, and the results were comparatively analysed to assess the accuracy of each model. The Weibull accelerated lifetime model yielded the best results among the three models, accurately predicting failures and detecting pipes with high failure likelihood; however, it is based on Monte Carlo simulations, which can be time-consuming. The linear extended Yule process could also effectively detect pipes with higher failure likelihood; however, it presented a clear tendency to overestimate the number of future failures. The single-variate Poisson process is the simplest of the three models and produced failure prediction results of lower quality.

Time to a single hybridization event in a group of species with unknown ancestral history

We consider a stochastic process for the generation of species which combines a Yule process with a simple model for hybridization between pairs of co-existent species. We assume that the origin of the process, when there was one species, occurred at an unknown time in the past, and we condition the process on producing n species via the Yule process and a single hybridization event. We prove results about the distribution of the time of the hybridization event. In particular we calculate a formula for all moments and show that under various conditions, the distribution tends to an exponential with rate twice that of the birth rate for the Yule process.

A Strong Law for the Size of Yule M-Oriented Recursive Trees

Abstract Let Nt be the total number of nodes in a Yule m-oriented recursive tree at time t. Then {Nt : t ∈ [0;1)} is a Yule process with birth rates λn = (m(n - 1) + 1)λ for n ≥ 1, where N0 = 1. In this paper, we first give the exact distribution of Nt, then prove that , almost surely

Size Frequency Distribution of Japanese Given Names

The size frequency distribution of Japanese given names is analyzed. A power law behavior is observed in a small size range with an exponent smaller than 2, which is close to that of Japanese family names. A similar tendency is observed in the cases of names in other countries. A Galton–Watson-type model with a generalized Yule process combined with an Exclusion Principle in Family (EPiF) is suggested.

Using a break prediction model for drinking water networks asset management: From research to practice

Break prediction models can help water utility decision-makers to build pipe rehabilitation programs. For many years, using them has been a specialist matter. After more than 15 years of research into the ageing of water pipes, Irstea (formerly Cemagref) has developed the Linear Extension of the Yule Process (LEYP) model based on counting process theory, which relies not only on a pipe's characteristics and environment but also on its age and previous breaks. It was then decided to develop a break prediction tool usable by water utilities: the ‘Casses’ freeware. To make this possible, it was necessary to deal with several constraints. To cope with the diversity of available data for various water utilities, flexible input data formats were designed as well as an importation module which checks the conformity and coherence of data. Tools for data management and an advice module dedicated to model calibration were conceived for non-statistician users. The break prediction results can be used directly to compare break evolution with different rehabilitation strategies and they can also feed multicriteria decision tools. In this case, the ‘Casses’ freeware can work as a ‘slave’ of the integrated application.