摘要翻译：
一组相似形状的曲线或图像是科学中越来越常见的功能数据集。主成分分析(PCA)是目前应用最广泛的功能数据变异分解技术。然而，主成分分析发现的线性变化模式并不总是能被实验者解释。此外，实验者感兴趣的变化模式并不总是线性的。本文提出了一种新的函数数据方差分析方法。我们的方法是通过将数据中的变化分解为预定的和可解释的方向（即模式）来推动的。由于其中一些模态可能是非线性的，我们发展了一个新的定义平方和比，它考虑了变分空间的曲率。在一般情况下，我们使用微分几何和形状统计的数学工具讨论了我们对变差估计的一致性。我们成功地将我们的方法应用到一个生物数据的激励示例中。这种分解使生物学家能够比较群体中不同遗传权衡的普遍程度，并量化选择对进化的影响。
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英文标题：
《Analysis of nonlinear modes of variation for functional data》
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作者：
Rima Izem, J.S. Marron
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最新提交年份：
2007
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分类信息：

一级分类：Statistics        统计学
二级分类：Methodology        方法论
分类描述：Design, Surveys, Model Selection, Multiple Testing, Multivariate Methods, Signal and Image Processing, Time Series, Smoothing, Spatial Statistics, Survival Analysis, Nonparametric and Semiparametric Methods
设计，调查，模型选择，多重检验，多元方法，信号和图像处理，时间序列，平滑，空间统计，生存分析，非参数和半参数方法
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英文摘要：
  A set of curves or images of similar shape is an increasingly common functional data set collected in the sciences. Principal Component Analysis (PCA) is the most widely used technique to decompose variation in functional data. However, the linear modes of variation found by PCA are not always interpretable by the experimenters. In addition, the modes of variation of interest to the experimenter are not always linear. We present in this paper a new analysis of variance for Functional Data. Our method was motivated by decomposing the variation in the data into predetermined and interpretable directions (i.e. modes) of interest. Since some of these modes could be nonlinear, we develop a new defined ratio of sums of squares which takes into account the curvature of the space of variation. We discuss, in the general case, consistency of our estimates of variation, using mathematical tools from differential geometry and shape statistics. We successfully applied our method to a motivating example of biological data. This decomposition allows biologists to compare the prevalence of different genetic tradeoffs in a population and to quantify the effect of selection on evolution.
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PDF链接：
https://arxiv.org/pdf/712.2657