We propose a framework of principal manifolds to model high‐dimensional data. This framework is based on Sobolev spaces and designed to model data of any intrinsic dimension. It includes principal component analysis and principal curve algorithm as special cases. We propose a novel method for model complexity selection to avoid overfitting, eliminate the effects of outliers and improve the computation speed. Additionally, we propose a method for identifying the interiors of circle‐like curves and cylinder/ball‐like surfaces. The proposed approach is compared to existing methods by simulations and applied to estimate tumour surfaces and interiors in a lung cancer study.

中文翻译：

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通过模型复杂度选择进行主流形估计


我们提出了一个主流形框架来对高维数据进行建模。该框架基于 Sobolev 空间，旨在对任何内在维度的数据进行建模。作为特例，它包括主成分分析和主曲线算法。我们提出了一种新的模型复杂度选择方法，以避免过度拟合，消除异常值的影响并提高计算速度。此外，我们提出了一种识别圆形曲线和圆柱/球形表面内部的方法。通过模拟将所提出的方法与现有方法进行比较，并应用于估计肺癌研究中的肿瘤表面和内部。