发表机构
School of Agriculture and Environment, College of Sciences, Massey University（梅西大学理学院农业与环境学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出类BME四重奏权重，推导其数学性质，将其扩展至多叉树并证明一致性，为系统发育树评估与应用提供基础以提升有限数据下的统计效率。
AI 中文摘要
与成对距离类似，四重奏在系统发育树上具有高度冗余性和相关性，其数量增长量级为n⁴而非n²。本文探索类BME权重，用于在对四重奏得分求和以评估完整树之前对其进行重加权。针对无根二叉树考虑三种权重：w_ext(q)=2^(-I_ext(q))、w_int(q)=2^(-I_int(q))和w_tot(q)=2^(-I_tot(q))=w_ext(q)w_int(q)，其中指数统计四重奏最小连接子树中指定的内部节点。针对6至10个分类单元的所有无标签无根二叉树形状，计算精确的树形状数量、总四重奏权重和内部边交叉和。对于w_ext，总四重奏权重具有树形状不变性，边交叉和仅取决于分裂大小；对于任意n叶树，证明sum_q w_ext(q)=(n-2)(n-3)/8，且穿过分裂为a|b的内部边的四重奏之和等于(a-1)(b-1)/4。还为w_int和w_tot推导了精确的树形状特异性归一化因子，针对多叉树给出了度数校正的硬多分支扩展，且条件一致性结果表明，当基础四重奏估计对真实诱导四重奏状态本身一致时，这些正权重可保持一致性。这些结果为评估和应用类BME四重奏权重以减少冗余、特别是提高有限数据下的统计效率提供了数学基础。
英文摘要
Like pairwise distances, quartets can be highly redundant and correlated on a phylogenetic tree, and their number grows on the order of n^4 rather than n^2. I explore BME-like weights for reweighting quartet scores before summing them to score a full tree. Three weights are considered on an unrooted binary tree: w_ext(q)=2^(-I_ext(q)), w_int(q)=2^(-I_int(q)), and w_tot(q)=2^(-I_tot(q))=w_ext(q)w_int(q), where the exponents count specified internal nodes in the minimal connecting subtree of a quartet. Exact tree-shape counts, total quartet-weight sums, and internal-edge crossing sums are calculated for all unlabeled unrooted binary tree shapes on 6-10 taxa. For w_ext, the total quartet weight is tree-shape-invariant and the edge-crossing sum depends only on split size. For any n-leaf tree, we prove sum_q w_ext(q)=(n-2)(n-3)/8, and the sum over quartets crossing an internal edge with split a|b equals (a-1)(b-1)/4. Exact tree-shape-specific normalizers are also derived for w_int and w_tot. A degree-corrected hard-polytomy extension is given for multifurcating trees, and a conditional consistency result shows that these positive weights preserve consistency when the underlying quartet estimates are themselves consistent for the true induced quartet states. These results provide a mathematical foundation for evaluating and applying BME-like quartet weights to reduce redundancy with the particular aim of improving statistical efficiency with finite data.
Comments31 pages. Lean 4 verification and table-reproduction materials are provided as ancillary files
