用于多维标度的局部双标图及其在微生物组中的应用
Local biplots for multi-dimensional scaling, with application to the microbiome
摘要：
我们提出了局部双标图，一个经典的主成分双标图多维尺度的扩展。注意到主成分双标图具有从数据空间到主子空间的映射的雅可比矩阵的解释，我们将局部双标图定义为多维缩放的类似映射的雅可比矩阵。在这个过程中，我们展示了我们的本地双标图轴，广义欧氏距离，广义主成分之间的密切关系。在模拟和真实数据中，我们展示了局部双标图如何揭示哪些变量或变量组合对于多维缩放提供的低维嵌入是重要的。他们对微生物组数据分析中常用的一类遗传学信息距离进行了特别深入的研究，表明这些距离的不同变体可以被解释为隐含地平滑了系统发育树上的数据，并且这种平滑的程度是可变的。
英文摘要：
We present local biplots, a an extension of the classic principal components biplot to multi-dimensional scaling. Noticing that principal components biplots have an interpretation as the Jacobian of a map from data space to the principal subspace, we define local biplots as the Jacobian of the analogous map for multi-dimensional scaling. In the process, we show a close relationship between our local biplot axes, generalized Euclidean distances, and generalized principal components. In simulations and real data we show how local biplots can shed light on what variables or combinations of variables are important for the low-dimensional embedding provided by multi-dimensional scaling. They give particular insight into a class of phylogenetically-informed distances commonly used in the analysis of microbiome data, showing that different variants of these distances can be interpreted as implicitly smoothing the data along the phylogenetic tree and that the extent of this smoothing is variable.
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