Predicting long pendant edges in model phylogenies, with applications to biodiversity and tree inference

In the simplest phylodynamic model (the pure-birth Yule process), lineages split independently at a constant rate λ 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 1/(2λ) as t grows, and thus is essentially independent of t. However, the behaviour of the length L of the longest pendant edge reveals remarkably different behaviour: L converges to t/2 as the expected number of leaves grows. Extending this model to allow an extinction rate μ (where μ<λ), we also establish a similar result for birth--death trees, except that t/2 is replaced by t/2 x (1-μ/λ). 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 questions concerning 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.

A STUDY OF SLANG WORDS IN SONG LYRICS USED BY BRUNO MARS

The study concerned with discussing about slang words used in Bruno Mars song lyrics. The research design of this study is descriptive qualitative method. In this study, the theory of word-formation process by George Yule process (1986) in his book “the study of language” is used to analyze the formation process of each slang word. The researcher only analyzes five in Bruno Mars song lyrics. They are count on me, show me, the lazy song, liquor store blues and just the way you are. There are consists of two research problem : (1) what is the type of slang word used by Bruno Mars in lyrics, (2) what is the dominant slang word found in Bruno Mars song lyrics. The data were document analysis, the research tried to find and identity the slang word in lyric by marking, finding and analyzing the slang word. The result shows that from 58 data taken, not all process of slang-word formation appear in the lyrics. Five process of slang-word formation found in the lyrics. The process of slang word formation that are used in the lyrics are clipping(160%), blending(120%), coinage(80%), borrowing(0%), and acronym(60%).

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

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.

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.