Phylogenetic data arising on two possibly different tree topologies might be mixed through several biological mechanisms, including incomplete lineage sorting or horizontal gene transfer in the case of different topologies, or simply different substitution processes on characters in the case of the same topology. Recent work on a 2-state symmetric model of character change showed such a mixture model has non-identifiable parameters, and thus it is theoretically impossible to determine the two tree topologies from any amount of data under such circumstances. Here the question of identifiability is investigated for 2-tree mixtures of the 4-state group-based models, which are more relevant to DNA sequence data. Using algebraic techniques, we show that the tree parameters are identifiable for the JC and K2P models. We also prove that generic substitution parameters for the JC mixture models are identifiable, and for the K2P and K3P models obtain generic identifiability results for mixtures on the same tree. This indicates that the full phylogenetic signal remains in such mixtures, and that the 2-state symmetric result is thus a misleading guide to the behavior of other models.
Identifiability of a Coalescent-based Population Tree Model
进化树模型的可识别性是最近的一个讨论话题，一些模型已被证明是不可识别的。基于合并的有根种群树模型最初由Nielsen et al. 1998 [2]提出，在过去几年中已被许多作者使用，并且是一种简单的工具，可以准确地模拟树中等位基因频率的变化。然而，这个模型的可识别性从未被证明。在这里，我们证明了这个模型是可识别的模型参数可以表示为子样本的概率分布的函数。这是证明基于该模型的总体树的最大似然估计的一致性的一步。
A Profile Mixture Model is a model of protein evolution, describing sequence data in which sites are assumed to follow many related substitution processes on a single evolutionary tree. The processes depend in part on different amino acid distributions, or profiles, varying over sites in aligned sequences. A fundamental question for any stochastic model, which must be answered positively to justify model-based inference, is whether the parameters are identifiable from the probability distribution they determine. Here we show that a Profile Mixture Model has identifiable parameters under circum…
Identifiability is a crucial property for a statistical model since distributions in the model uniquely determine the parameters that produce them. In phylogenetics, the identifiability of the tree parameter is of particular interest since it means that phylogenetic models can be used to infer evolutionary histories from data. In this paper we introduce a new computational strategy for proving the identifiability of discrete parameters in algebraic statistical models that uses algebraic matroids naturally associated to the models. We then use this algorithm to prove that the tree parameters ar…
Identifiability of phylogenetic networks and quintet concordance factors
University of Edinburgh（爱丁堡大学）；New Jersey Institute of Technology（新泽理工学院）；Texas State University（德克萨斯州立大学）；University of Hawai‘i at Mānoa（夏威夷大学马诺阿分校）；University of Alaska Fairbanks（阿拉斯加大学费尔班克斯分校）
Phylogenetic networks are an extension of phylogenetic trees which are used to represent evolutionary histories in which reticulation events (such as recombination and hybridization) have occurred. A central question for such networks is that of identifiability, which essentially asks under what circumstances can we reliably identify the phylogenetic network that gave rise to the observed data? Recently, identifiability results have appeared for networks relative to a model of sequence evolution that generalizes the standard Markov models used for phylogenetic trees. However, these results are…
Identifiability of the discrete tree parameter is a key property for phylogenetic models since it is necessary for statistically consistent estimation of the tree from sequence data. Algebraic methods have proven to be very effective at showing that tree and network parameters of phylogenetic models are identifiable, especially when the underlying models are group-based. However, since group-based models are time-reversible, only the unrooted tree topology is identifiable and the location of the root is not. In this note we show that the rooted tree parameter of the Cavender-Farris-Neyman Mode…
On Tree-Network Distinguishability and Full Identifiability of Phylogenetic Networks
Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特理工大学应用数学研究所）；European Molecular Biology Laboratory, European Bioinformatics Institute (EMBL-EBI)（欧洲分子生物学实验室欧洲生物信息学研究所）
Inference of species networks from genomic data under the Network Multispecies Coalescent Model is currently severely limited by heavy computational demands. It also remains unclear how complicated networks can be for consistent inference to be possible. As a step toward inferring a general species network, this work considers its tree of blobs, in which non-cut edges are contracted to nodes, so only tree-like relationships between the taxa are shown. An identifiability theorem, that most features of the unrooted tree of blobs can be determined from the distribution of gene quartet topologies,…
In this paper we study group-based Markov models of evolution and their mixtures. In the algebreo-geometric setting, group-based phylogenetic tree models correspond to toric varieties, while their mixtures correspond to secant and join varieties. Determining properties of these secant and join varieties can aid both in model selection and establishing parameter identifiability. Here we explore the first natural geometric property of these varieties: their dimension. The expected projective dimension of the join variety of a set of varieties is one more than the sum of their dimensions. A join …
Diversification models describe the random growth of evolutionary trees, modeling the historical relationships of species through speciation and extinction events. One class of such models allows for independently changing traits, or types, of the species within the tree, upon which speciation and extinction rates depend. Although identifiability of parameters is necessary to justify parameter estimation with a model, it has not been formally established for these models, despite their adoption for inference. This work establishes generic identifiability up to label swapping for the parameters…
Identifiability of the unrooted species tree topology under the coalescent model with time-reversible substitution processes, site-specific rate variation, and invariable sites
The Tree of Life is the graphical structure that represents the evolutionary process from single-cell organisms at the origin of life to the vast biodiversity we see today. Reconstructing this tree from genomic sequences is challenging due to the variety of biological forces that shape the signal in the data, and many of those processes like incomplete lineage sorting and hybridization can produce confounding information. Here, we present the mathematical version of the identifiability proofs of phylogenetic networks under the pseudolikelihood model in SNaQ. We establish that the ability to de…
Phylogenetic networks extend phylogenetic trees to model non-vertical inheritance, by which a lineage inherits material from multiple parents. The computational complexity of estimating phylogenetic networks from genome-wide data with likelihood-based methods limits the size of networks that can be handled. Methods based on pairwise distances could offer faster alternatives. We study here the information that average pairwise distances contain on the underlying phylogenetic network, by characterizing local and global features that can or cannot be identified. For general networks, we clarify t…
Inference of network-like evolutionary relationships between species from genomic data must address the interwoven signals from both gene flow and incomplete lineage sorting. The heavy computational demands of standard approaches to this problem severely limit the size of datasets that may be analyzed, in both the number of species and the number of genetic loci. Here we provide a theoretical pointer to more efficient methods, by showing that logDet distances computed from genomic-scale sequences retain sufficient information to recover network relationships in the level-1 ultrametric case. Th…
A model of genomic sequence evolution on a species tree should include not only a sequence substitution process, but also a coalescent process, since different sites may evolve on different gene trees due to incomplete lineage sorting. Chifman and Kubatko initiated the study of such models, leading to the development of the SVDquartets methods of species tree inference. A key observation was that symmetries in an ultrametric species tree led to symmetries in the joint distribution of bases at the taxa. In this work, we explore the implications of such symmetry more fully, defining new models i…
