Data from: Low-parameter phylogenetic inference under the general Markov model

In their 2008 and 2009 papers, Sumner and colleagues introduced the "squangles" -- a small set of Markov invariants for phylogenetic quartets. The squangles are consistent with the general Markov model (GM) and can be used to infer quartets without the need to explicitly estimate all parameters. As GM is inhomogeneous and hence non-stationary, the squangles are expected to perform well compared to standard approaches when there are changes in base-composition amongst species. However, the GM model assumes constant rates across sites, so the squangles should be confounded by data generated with invariant sites or other forms of rate-variation across sites. Here we implement the squangles in a least-squares setting that returns quartets weighted by either confidence or internal edge lengths, and we show how these weighted quartets can be used as input into a variety of supertree and supernetwork methods. For the first time, we quantitatively investigate the robustness of the squangles to breaking of the constant rates-across-sites assumption on both simulated and real data sets; and we suggest a modification that improves the performance of the squangles in the presence of invariant sites. Our conclusion is that the squangles provide a novel tool for phylogenetic estimation that is complementary to methods that explicitly account for rate-variation across sites, but rely on homogeneous -- and hence stationary -- models.Sumner及其研究团队在2008年与2009年的论文中提出了「squangles」——一类用于系统发育四元组（phylogenetic quartets）的小型马尔可夫不变量（Markov invariants）集合。该方法与一般马尔可夫模型（general Markov model, GM）相符，且无需显式估计全部参数即可用于推断四元组。由于GM模型具有非齐性（因而非平稳）的特性，当物种间碱基组成存在差异时，squangles的表现预计将优于标准分析方法。然而，GM模型假设位点间速率恒定，因此当数据由不变位点或其他形式的位点间速率异质性生成时，squangles的分析结果会受到混淆。本文在最小二乘框架下实现了squangles方法，可输出以置信度或内部边长度加权的四元组，并展示了如何将这些加权四元组作为输入应用于多种超树和超网络分析方法。本文首次针对模拟数据集与真实数据集，定量研究了squangles在违反位点间速率恒定假设时的鲁棒性，并提出了一种改进方案，可提升squangles在存在不变位点场景下的性能。本文结论表明，squangles为系统发育推断提供了一种全新工具，可与那些显式考虑位点间速率异质性但依赖齐性（因而平稳）模型的系统发育分析方法形成互补。