可识别性、活动子空间和参数空间降维之间的潜在联系
The underlying connections between identifiability, active subspaces, and parameter space dimension reduction
摘要：
参数、模型结构和输出之间的相互作用可以确定给定系统可能的推断、预测和控制策略。因此，参数空间简化和参数估计-以及更一般地理解具有观测结构的模型中所包含的信息的形状-对于数学建模和不确定性量化中的许多问题是必不可少的。因此，不同的学科已经开发出并行的方法来处理各自领域的问题。这些方法中的许多方法，包括可识别性、马虎性和活动子空间，使用相关的思想来解决参数降维、参数估计和推理和感兴趣的量的鲁棒性的问题。在本文中，我们表明，积极的子空间方法有内在的联系，从灵敏度分析和可识别性的方法，它确实是可能的框架，在一个统一的框架中的每种方法。Fisher信息矩阵的一种特殊形式，我们将其表示为灵敏度矩阵，它是所有三种方法的基础-活动子空间、可识别性和草率性。通过一系列的例子和案例研究，我们说明了在几种情况下的灵敏度函数的属性。这些初步的例子表明，局部和全局、线性和非线性之间的相互作用强烈影响着每种方法可以产生的见解。这些观察结果强调，一个人的参数降维的方法应该由科学问题驱动，也打开了大门，使用其他方法的工具，以产生有用的见解。
英文摘要：
The interactions between parameters, model structure, and outputs can determine what inferences, predictions, and control strategies are possible for a given system. Parameter space reduction and parameter estimation---and, more generally, understanding the shape of the information contained in models with observational structure---are thus essential for many questions in mathematical modeling and uncertainty quantification. As such, different disciplines have developed methods in parallel for approaching the questions in their field. Many of these approaches, including identifiability, sloppiness, and active subspaces, use related ideas to address questions of parameter dimension reduction, parameter estimation, and robustness of inferences and quantities of interest. In this paper, we show that active subspace methods have intrinsic connections to methods from sensitivity analysis and identifiability, and indeed that it is possible to frame each approach in a unified framework. A particular form of the Fisher information matrix (FIM), which we denote the sensitivity FIM, is fundamental to all three approaches---active subspaces, identifiability, and sloppiness. Through a series of examples and case studies, we illustrate the properties of the sensitivity FIM in several contexts. These initial examples show that the interplay between local and global and linear and non-linear strongly impact the insights each approach can generate. These observations underline that one's approach to parameter dimension reduction should be driven by the scientific question and also open the door to using tools from the other approaches to generate useful insights.
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