Principal Manifolds of Middles: A Framework and Estimation Procedure Using Mixture Densities
share
Principal manifolds are used to represent high-dimensional data in a low-dimensional space. They are high-dimensional generalizations of principal curves and surfaces. The existing methods for fitting principal manifolds have several shortcomings: model bias, heavy computational burden, sensitivity to outliers, and difficulty of use in applications. We propose a novel method for modeling principal manifolds that addresses these limitations. It is based on minimization of penalized mean squared error functionals, providing a nonlinear summary of the data points in Euclidean spaces. We introduce the framework in the context of principal manifolds of middles and develop an estimate by proposing a high-dimensional mixture density estimation procedure. The Sobolev embedding theorem guarantees the regularity of the derived manifolds and analytical expressions of the embedding maps are obtained. The algorithm is computationally efficient and robust to outliers. We used simulation studies to illustrate the comparative performance of the proposed method in low-dimensions and found that it performs better than competitors. In addition, we analyze computed tomography images of lung cancer tumors focusing on two important clinical questions - estimation of the tumor surface and identification of tumor interior classifier. We used the obtained analytic expressions of embedding maps to construct a tumor interior classifier.
READ FULL TEXT