Phylogenetic comparative methods use random processes, such as the Brownian Motion, to model the evolution of continuous traits on phylogenetic trees. Growing evidence for non-gradual evolution motivated the development of complex models, often based on Lévy processes. However, their statistical inference is computationally intensive, and currently relies on approximations, high dimensional sampling, or numerical integration. We consider here the Cauchy Process (CP), a particular pure-jump Lévy process in which the trait increment along each branch follows a centered Cauchy distribution with a dispersion proportional to its length. In this work, we derive an exact algorithm to compute both the joint probability density of the tip trait values of a phylogeny under a CP, and the ancestral trait values and branch increments posterior densities in quadratic time. A simulation study shows that the CP generates patterns in comparative data that are distinct from any Gaussian process, and that Restricted Maximum Likelihood (REML) parameter estimates and root trait reconstruction are unbiased and accurate for trees with 200 tips or less. The CP has only two parameters but is rich enough to capture complex pulsed evolution. It can reconstruct posterior ancestral trait distributions that are multimodal, reflecting the uncertainty associated with the inference of the evolutionary history of a trait from extant taxa only. Applied on empirical datasets taken from the Evolutionary Ecology and Virology literature, the CP suggests nuanced scenarios for the body size evolution of Greater Antilles Lizards and for the geographical spread of the West Nile Virus epidemics in North America, both consistent with previous studies using more complex models. The method is efficiently implemented in C with an R interface in package cauphy, that is open source and freely available online.

中文翻译：

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系统发育的柯西过程：脉冲进化的易处理模型


系统发育比较方法使用随机过程（例如布朗运动）来模拟系统发育树上连续性状的演化。越来越多的非渐进进化的证据推动了复杂模型的发展，这些模型通常基于莱维过程。然而，他们的统计推断是计算密集型的，目前依赖于近似、高维采样或数值积分。我们在这里考虑柯西过程 (CP)，这是一种特殊的纯跳跃 Lévy 过程，其中沿每个分支的特征增量遵循中心柯西分布，其离散度与其长度成比例。在这项工作中，我们推导了一种精确的算法来计算 CP 下系统发育的尖端特征值的联合概率密度，以及二次时间中祖先特征值和分支增量后验密度。模拟研究表明，CP 在比较数据中生成的模式与任何高斯过程都不同，并且限制最大似然 (REML) 参数估计和根性状重建对于尖端数为 200 或更少的树木来说是无偏且准确的。 CP 只有两个参数，但足够丰富以捕获复杂的脉冲演化。它可以重建多峰的后祖先性状分布，反映了仅从现有类群推断性状进化历史相关的不确定性。应用进化生态学和病毒学文献中的经验数据集，CP 提出了大安的列斯群岛蜥蜴的体型进化和西尼罗河病毒在北美的地理传播的细致情景，两者都与之前使用更复杂的研究相一致模型。 该方法通过 C 语言有效实现，并在 cauphy 包中使用 R 接口，该包是开源的并且可以在线免费获得。