Yule Process Research Articles

Application of semiclassical approximation to stochastic differential equations

Abstract A method for calculating the characteristics of stochastic differential equations using semiclassical approximation is proposed. For a stochastic differential equation arising in the study of the pure birth process (Yule process) and the Cox Ingersoll Ross (CIR) model, an analysis of the accuracy of the semiclassical approximation was carried out. This analysis is based on a comparison of approximate values with exact values for the mathematical expectation of a solution to the equation.

Error-induced extinction in a multi-type critical birth–death process

Extreme mutation rates in microbes and cancer cells can result in error-induced extinction (EEX), where every descendant cell eventually acquires a lethal mutation. In this work, we investigate critical birth–death processes with n distinct types as a birth–death model of EEX in a growing population. Each type-i cell divides independently (i)→(i)+(i)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(i)\\rightarrow (i)+(i)$$\\end{document} or mutates (i)→(i+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(i)\\rightarrow (i+1)$$\\end{document} at the same rate. The total number of cells grows exponentially as a Yule process until a cell of type-n appears, which cell type can only divide or die at rate one. This makes the whole process critical and hence after the exponentially growing phase eventually all cells die with probability one. We present large-time asymptotic results for the general n-type critical birth–death process. We find that the mass function of the number of cells of type-k has algebraic and stationary tail (size)-1-χk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\ ext {size})^{-1-\\chi _k}$$\\end{document}, with χk=21-k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi _k=2^{1-k}$$\\end{document}, for k=2,⋯,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=2,\\dots ,n$$\\end{document}, in sharp contrast to the exponential tail of the first type. The same exponents describe the tail of the asymptotic survival probability (time)-ξk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\ ext {time})^{-\\xi _k}$$\\end{document}. We present applications of the results for studying extinction due to intolerable mutation rates in biological populations.

Emergence of Power-Law Particle Size Distribution in Microfluidic Calcium Carbonate Precipitation: An Extended Yule Process with a Ripening Effect.

Precipitation of calcium carbonate in bulk solutions is well known to result in a bell-shaped or bimodal particle size distribution. However, it is unclear how the distribution behaves if precipitation occurs in a small, confined volume. In this Letter, we conduct microfluidic experiments where sodium carbonate and calcium chloride solutions are continuously injected into a microchannel to precipitate calcium carbonate particles. Results show that, regardless of the variations in reagent concentrations, mixing schemes, flow rates, and precipitation time, sizes of precipitated particles in the channel are power law distributed, with an exponent of 1.4. The data are described by an extended Yule process with the introduction of a ripening term. Since the Yule process is a general mechanism for power law generation, the extended Yule process proposed here provides a general model for systems where growth and ripening simultaneously present.

Predicting Long Pendant Edges in Model Phylogenies, with Applications to Biodiversity and Tree Inference.

In the simplest phylogenetic diversification model (the pure-birth Yule process), lineages split independently at a constant rate $\lambda$ for time $t$. The length of a randomly chosen edge (either interior or pendant) in the resulting tree has an expected value that rapidly converges to $\frac{1}{2\lambda}$ as $t$ grows and thus is essentially independent of $t$. However, the behavior of the length $L$ of the longest pendant edge reveals remarkably different behavior: $L$ converges to $t/2$ as the expected number of leaves grows. Extending this model to allow an extinction rate $\mu$ (where $\mu<\lambda$), we also establish a similar result for birth-death trees, except that $t/2$ is replaced by $t/2 \cdot (1-\mu/\lambda)$. This "complete" tree may contain subtrees that have died out before time $t$; for the "reduced tree" that just involves the leaves present at time $t$ and their direct ancestors, the longest pendant edge length $L$ again converges to $t/2$. Thus, there is likely to be at least one extant species whose associated pendant branch attaches to the tree approximately half-way back in time to the origin of the entire clade. We also briefly consider the length of the shortest edges. Our results are relevant to phylogenetic diversity indices in biodiversity conservation, and to quantifying the length of aligned sequences required to correctly infer a tree. We compare our theoretical results with simulations and with the branch lengths from a recent phylogenetic tree of all mammals. [Birth-death process; phylogenetic diversification models; phylogenetic diversity.].

Model Transfer and Universal Patterns: Lessons from the Yule Process

Model transfer refers to the observation that particular model structures are used across multiple distinct scientific domains. This paper puts forward an account to explain the inter-domain transfer of model structures. Central in the account is the role of validation criteria in determining whether a model is considered to be useful by practitioners. Validation criteria are points of reference to which model correctness for a particular purpose is assessed. I argue that validation criteria can be categorized as being mathematical, theoretical or phenomenological in nature. Model transfer is explained by overlap in validation criteria between scientific domains. Particular emphasis is placed on overlap between phenomenological criteria. Overlap in phenomenological criteria can be explained through the notion of universal patterns. Universal patterns are abstract structures that can be made to refer to multiple distinct phenomena when coupled with phenomena-specific empirical content. I present the case study of the Yule Process, in which universal patterns play a crucial role in explaining model transfer. This paper provides an account of model transfer that stays close to modelling practice and expands existing accounts by introducing the notion of universal patterns.

Clariant sets new 2025 financial targets and announces step-up in growth and profitability

Gibrat’s Law – also called law of proportionate effect – states that the size of a company does not influence its economic growth. The present article aims at testing the validity of this law on all companies based in the autonomous region of Andalusia (Spain). The study uses data on employment compiled by the Instituto de Estadística y Cartografía de Andalucía for every company belonging to the two-digit NACE code, with yearly data for the 2000 to 2015 period: 7,046,539 firms in total. The results obtained allow to identify by sector the impact of size on business growth, to check compliance in this relevant region of the European Union and analyse the influence of the recent economic/financial crisis in terms of Gibrat’s Law validation. In general terms, the degree of compliance with the law is limited in the analysed period, even more so in the years of economic crisis. We have fitted the double Pareto log-normal distribution to our data. That distribution fits well the firm’ size distribution in many sectors of the Andalusian economy. Thus, these sectors would follow a Yule process. At the aggregate level, the sectors of the Andalusian economy that comply with Gibrat’s law also follow a Yule process.

Moderate deviations of density-dependent Markov chains

A density dependent Markov chain (DDMC) introduced in Kurtz (1978) is a special continuous time Markov process. Examples are considered in fields like epidemics and processes which describe chemical reactions. Moreover the Yule process is a further example. In this paper we prove a moderate deviation principle for the paths of a certain class of DDMC. The proofs of the bounds utilize an exponential martingale as well as a generalized version of Girsanov’s theorem. The exponential martingale is defined according to the generator of the DDMC.

Firm growth in the 21st century: Does the Andalusian economy comply with Gibrat’s Law?

Gibrat’s Law – also called law of proportionate effect – states that the size of a company does not influence its economic growth. The present article aims at testing the validity of this law on all companies based in the autonomous region of Andalusia (Spain). The study uses data on employment compiled by the Instituto de Estadística y Cartografía de Andalucía for every company belonging to the two-digit NACE code, with yearly data for the 2000 to 2015 period: 7,046,539 firms in total. The results obtained allow to identify by sector the impact of size on business growth, to check compliance in this relevant region of the European Union and analyse the influence of the recent economic/financial crisis in terms of Gibrat’s Law validation. In general terms, the degree of compliance with the law is limited in the analysed period, even more so in the years of economic crisis. We have fitted the double Pareto log-normal distribution to our data. That distribution fits well the firm’ size distribution in many sectors of the Andalusian economy. Thus, these sectors would follow a Yule process. At the aggregate level, the sectors of the Andalusian economy that comply with Gibrat’s law also follow a Yule process.

Integro‐differential equations linked to compound birth processes with infinitely divisible addends

Stochastic modelling of fatigue (and other material's deterioration), as well as of cumulative damage in risk theory, are often based on compound sums of independent random variables, where the number of addends is represented by an independent counting process. We consider here a cumulative model where, instead of a renewal process (as in the Poisson case), a linear birth (or Yule) process is used. This corresponds to the assumption that the frequency of “damage” increments accelerates according to the increasing number of “damages”. We start from the partial differential equation satisfied by its transition density, in the case of exponentially distributed addends, and then we generalize it by introducing a space derivative of convolution type (i.e., defined in terms of the Laplace exponent of a subordinator). Then we are concerned with the solution of integro‐differential equations, under proper initial conditions, which, in a special case, reduce to a fractional one. Correspondingly, we analyze the related cumulative jump processes under a general infinitely divisible distribution of the (positive) jumps. Some special cases (such as the stable, tempered stable, gamma, and Poisson) are presented.

On the continuous-time limit of the Barabási–Albert random graph

We prove that, via an appropriate scaling, the degree of a fixed vertex in the Barabási–Albert model appeared at a large enough time converges in distribution to a Yule process. Using this relation we explain why the limit degree distribution of a vertex chosen uniformly at random (as the number of vertices goes to infinity), coincides with the limit distribution of the number of species in a genus selected uniformly at random in a Yule model (as time goes to infinity). To prove this result we do not assume that the number of vertices increases exponentially over time (linear rates). On the contrary, we retain their natural growth with a constant rate superimposing to the overall graph structure a suitable set of processes that we call the planted model and introducing an ad-hoc sampling procedure.

The Yule Process Revisited: Exploring its Potential in the Engineering Analysis of Certain Chemical Reactions

The Yule Process Revisited: Exploring its Potential in the Engineering Analysis of Certain Chemical Reactions

On the convergence of the maximum likelihood estimator for the transition rate under a 2-state symmetric model.

Maximum likelihood estimators are used extensively to estimate unknown parameters of stochastic trait evolution models on phylogenetic trees. Although the MLE has been proven to converge to the true value in the independent-sample case, we cannot appeal to this result because trait values of different species are correlated due to shared evolutionary history. In this paper, we consider a 2-state symmetric model for a single binary trait and investigate the theoretical properties of the MLE for the transition rate in the large-tree limit. Here, the large-tree limit is a theoretical scenario where the number of taxa increases to infinity and we can observe the trait values for all species. Specifically, we prove that the MLE converges to the true value under some regularity conditions. These conditions ensure that the tree shape is not too irregular, and holds for many practical scenarios such as trees with bounded edges, trees generated from the Yule (pure birth) process, and trees generated from the coalescent point process. Our result also provides an upper bound for the distance between the MLE and the true value.

선형 신장 율 과정을 활용한 무기체계 공용품목 수요예측에 관한 사례연구

선형 신장 율 과정을 활용한 무기체계 공용품목 수요예측에 관한 사례연구

Power Laws in Stochastic Processes for Social Phenomena: An Introductory Review

Many phenomena with power laws have been observed in various fields of the natural and social sciences, and these power laws are often interpreted as the macro behaviors of systems that consist of micro units. In this paper, we review some basic mathematical mechanisms that are known to generate power laws. In particular, we focus on stochastic processes including the Yule process and the Simon process as well as some recent models. The main purpose of this paper is to explain the mathematical details of their mechanisms in a self-contained manner.

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 &gt; α ≤ 1 0 &gt; \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.

Tempo and rates of diversification in the South American cichlid genus Apistogramma (Teleostei: Perciformes: Cichlidae).

Evaluating biodiversity and understanding the processes involved in diversification are noticeable conservation issues in fishes subject to large, sometimes illegal, ornamental trade purposes. Here, the diversity and evolutionary history of the Neotropical dwarf cichlid genus Apistogramma from several South American countries are investigated. Mitochondrial and nuclear markers are used to infer phylogenetic relationships between 31 genetically identified species. The monophyly of Apistogramma is suggested, and Apistogramma species are distributed into four clades, corresponding to three morphological lineages. Divergence times estimated with the Yule process and an uncorrelated lognormal clock dated the Apistogramma origin to the beginning of the Eocene (≈ 50 Myr) suggesting that diversification might be related to marine incursions. Our molecular dating also suggests that the Quaternary glacial cycles coincide with the phases leading to Apistogramma speciation. These past events did not influence diversification rates in the speciose genus Apistogramma, since diversification appeared low and constant through time. Further characterization of processes involved in recent Apistogramma diversity will be necessary.

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.

Dynamics of analyst forecasts and emergence of complexity: Role of information disparity.

We report complex phenomena arising among financial analysts, who gather information and generate investment advice, and elucidate them with the help of a theoretical model. Understanding how analysts form their forecasts is important in better understanding the financial market. Carrying out big-data analysis of the analyst forecast data from I/B/E/S for nearly thirty years, we find skew distributions as evidence for emergence of complexity, and show how information asymmetry or disparity affects financial analysts’ forming their forecasts. Here regulations, information dissemination throughout a fiscal year, and interactions among financial analysts are regarded as the proxy for a lower level of information disparity. It is found that financial analysts with better access to information display contrasting behaviors: a few analysts become bolder and issue forecasts independent of other forecasts while the majority of analysts issue more accurate forecasts and flock to each other. Main body of our sample of optimistic forecasts fits a log-normal distribution, with the tail displaying a power law. Based on the Yule process, we propose a model for the dynamics of issuing forecasts, incorporating interactions between analysts. Explaining nicely empirical data on analyst forecasts, this provides an appealing instance of understanding social phenomena in the perspective of complex systems.

Maximal displacement of critical branching symmetric stable processes

Nous considérons un processus de branchement en temps continu critique (processus de Yule) dont les individus suivent indépendamment un processus $\alpha$-stable symétrique réel issu de leur point de naissance. Le processus de branchement étant critique, il s’éteint presque surement et nous pouvons définir la valeur $M$ qui représente la position maximale jamais atteinte par un individu. Nous montrons que la distribution de $M$ satisfait l’estimée asymptotique suivante : $P\{M\ge x\}\sim (2/\alpha)^{1/2}x^{-\alpha/2}$.

Interstellar Travel and Galactic Colonization: Insights from Percolation Theory and the Yule Process.

In this paper, percolation theory is employed to place tentative bounds on the probability p of interstellar travel and the emergence of a civilization (or panspermia) that colonizes the entire Galaxy. The ensuing ramifications with regard to the Fermi paradox are also explored. In particular, it is suggested that the correlation function of inhabited exoplanets can be used to observationally constrain p in the near future. It is shown, by using a mathematical evolution model known as the Yule process, that the probability distribution for civilizations with a given number of colonized worlds is likely to exhibit a power-law tail. Some of the dynamical aspects of this issue, including the question of timescales and generalizing percolation theory, were also studied. The limitations of these models, and other avenues for future inquiry, are also outlined. Complex life-Extraterrestrial life-Panspermia-Life detection-SETI. Astrobiology 16, 418-426.