The Principal Component Transform of Parametrized Functions

• 作者: Arcady Ponosov Ilia Zabrodskii
• 来源期刊: 应用数学(英文)
• 学科: 数学
• 关键词: Principal COMPONENT Analysis DISCRETIZATION of FUNCTIONS METAMODELING LATENT Parameters
• 年，卷（期）: yysxyw_2017,（4）
• 页码范围: 453-475
• 页数: 23页
• 分类号: O1

摘要

Many advanced mathematical models of biochemical, biophysical and other processes in systems biology can be described by parametrized systems of nonlinear differential equations. Due to complexity of the models, a problem of their simplification has become of great importance. In particular, rather challengeable methods of estimation of parameters in these models may require such simplifications. The paper offers a practical way of constructing approximations of nonlinearly parametrized functions by linearly parametrized ones. As the idea of such approximations goes back to Principal Component Analysis, we call the corresponding transformation Principal Component Transform. We show that this transform possesses the best individual fit property, in the sense that the corresponding approximations preserve most information (in some sense) about the original function. It is also demonstrated how one can estimate the error between the given function and its approximations. In addition, we apply the theory of tensor products of compact operators in Hilbert spaces to justify our method for the case of the products of parametrized functions. Finally, we provide several examples, which are of relevance for systems biology.