Complexity to Clarity: Nonparametric Procedures that Exploit Structured Data and Models

现代物理科学研究涉及使用非标准的集合数据对象(例如，图像、光谱或飓风轨迹)，这些对象不能直接服从传统的统计方法。自然，产生这些观测数据的物理过程也往往非常复杂。通常，唯一有意义的“模型”是高分辨率的理论模型或模拟模型。例如，在宇宙学中，科学家经常使用大型流体动力学模拟来了解宇宙是如何形成和演化的。该项目的目标是开发一种新的方法，将科学现象的仔细统计建模与可扩展的过程相结合，充分利用大量复杂数据的丰富性，而不会将数据简化为一组特征或模板。该项目基于调和分析和谱方法的思想，开发灵活和自适应的非参数方法，利用复杂数据中的稀疏(和潜在的非线性)结构进行高维推理。这些方法导出适用于底层数据分布的固有几何(例如子流形结构)的类傅立叶基函数，并使用经验基函数来估计高维聚集对象上的函数。这些方法在预测方面超越了点估计，而是对复杂、高维数据的条件密度、密度比和可能性进行了非参数估计。这些方法的一个关键应用是校准复杂的仿真模型：确定这些模型的输入值的设置以使其输出接近真实数据或更复杂且计算密集的仿真代码的输出的推理挑战。在更广泛的范围内，这项工作将为在广泛的科学应用中建立、解释和使用高维、复杂结构数据的概率模型做出关键的方法论贡献。

Modern research in the physical sciences involves the use of non-standard aggregate data objects (for example, images, spectra, or hurricane tracks), which are not directly amenable to traditional statistical methods. Naturally, the physical processes that generated these observed data are also often very complicated. Typically the only meaningful "model" is in the form of a high-resolution theoretical or simulation model. For example in cosmology, scientists regularly use large hydrodynamic simulations to understand how the universe formed and evolved. The goal of this project is to develop a new means of combining careful statistical modeling of scientific phenomena with scalable procedures that fully exploit the richness of large collections of complex data without reducing the data to a set of features or templates.Building on ideas from harmonic analysis and spectral methods, this project is to develop flexible and adaptive nonparametric methods for high-dimensional inference that exploit sparse (and potentially nonlinear) structure in complex data. These methods derive Fourier-like bases that adapt to the intrinsic geometry (e.g., submanifold structure) of the underlying data distribution, and use the empirical basis functions to estimate functions on high-dimensional aggregate objects. The methods go beyond point-estimates in prediction to nonparametric estimation of conditional densities, density ratios and likelihoods of complex, high-dimensional data. A key application of these methods is the calibration of complex simulation models: the inference challenge of determining the settings of input values to these models so that their output approximates either real data or the output of a more complex and computationally-intensive simulation code. On a broader scale, this work will make key methodological contributions to building, interpreting, and using probability models for high-dimensional, complexly-structured data in a wide range of scientific applications.