Low-Rank Functional Data Analysis for Time-Resolved Spectroscopy and the Search for Earth-Like Exoplanets

光谱学在许多科学领域都是必不可少的，包括天文学，其中基于实验室的光谱与恒星光谱的比较揭示了恒星的化学成分。时间分辨光谱测量光谱随时间的变化已经成为探测许多系统动力学的有力工具。它被用来研究恒星的变化，复杂化合物的分子动力学，以及生物系统中与时间相关的化学过程。该项目旨在开发一个框架，用于在相关对象之间进行不完整测量的设置中分析时间分辨光谱数据，或者重复一个系统，在复制过程中观察到随机变化。仔细汇集和分析重复的数据可以识别产生光谱变异性的过程的特征。主要应用将是模拟恒星的时间分辨光谱，作为太阳系外行星系统的候选宿主，特别是搜索围绕类太阳恒星（系外地球）运行的类地球行星的数据。太阳系外的小行星不能直接看到，但它们的存在是可以辨别的：行星对它的星星的牵引产生了一个小的摆动运动，这可以通过测量光谱线的非常小的、随时间变化的多普勒频移来检测。目前，观测小行星的主要障碍是主星的光谱活动--暗黑子、亮斑和耀斑的忽隐忽现会掩盖或模仿行星随时间变化的信号。该项目将开发新的算法，以便在时间分辨光谱数据中将恒星活动信号与行星信号分开。该项目还将支持培训各种天文学学生和高级统计学博士后研究人员，其中许多人将继续从事涉及数据科学的非学术STEM职业。时间分辨（动态）光谱可以用波长和时间的二元函数来描述，表示由摄谱仪测量的光的（相对）强度与波长和时间的关系。该项目的目标是建立一个框架，对测量单一动态光谱或多个相关动态光谱的数据进行建模，这些动态光谱具有不完整的采样和噪声，例如，来自对具有类似主星的许多候选系外行星系统的观测。该框架将整合近似理论和功能数据分析（FDA）的技术，FDA是统计学的分支，涉及分析包括函数集合测量的数据。该框架的一个核心组成部分将是使用可分离的扩展，将动态光谱写为波长（“speclets”）和时间（“modulators”）的成对单变量函数的乘积之和。当给定波长和时间的二元函数时，近似理论使用非对称Hilbert-Schmidt分解（类似于奇异值分解（SVD）的过程）来识别最佳反射镜和调制器。真实的数据不提供波长和时间的二元函数的精确、密集采样。该项目将建立在FDA方法的基础上，包括功能主成分分析（FPCA）和分层贝叶斯随机过程模型，以实现解释噪声和不完整、不规则采样的speclet和调制器基础发现。该框架将被应用于模拟的恒星种群光谱，以建立一个随机恒星变化模型，这将有助于在多普勒径向速度搜索系外行星时发现系外地球。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Spectroscopy is essential in many scientific fields, including astronomy, where comparisons of lab-based spectra to stellar spectra reveal the chemical compositions of stars. Time-resolved spectroscopy—measuring spectra changing over time—has emerged as a powerful tool for probing the dynamics of many systems. It is used to study the variability of stars, the molecular dynamics of complex compounds, and time-dependent chemical processes in biological systems. This project aims to develop a framework for analyzing time-resolved spectroscopic data in settings where incomplete measurements are made across related objects, or repeatedly for one system, with random variations seen across the replications. Careful pooling and analysis of data across the replications can identify signatures of the processes producing spectral variability. The main application will be modeling time-resolved spectroscopy of stars as candidate hosts of extrasolar planetary systems, particularly data from searches for Earth-like planets orbiting Sun-like stars (exo-Earths). Small extrasolar planets are not directly visible, but their presence can be discerned: the tug of a planet on its star produces a small wobbling motion, which can be detected by measuring extremely small, time-varying Doppler shifts of spectral lines. Currently, the main obstacle to observing small planets is the spectral activity of the host star—the comings and goings of dark sunspots, bright plages, and flares can mask or mimic a planet's time-dependent signal. The project will develop new algorithms to disentangle stellar activity signals from planet signals in time-resolved spectral data. The project will also support training of a diverse population of astronomy students and postdoctoral researchers in advanced statistics, many of whom will go on to pursue non-academic STEM careers involving data science. A time-resolved (dynamic) spectrum can be described with a bivariate function of wavelength and time representing the (relative) intensity of light measured by a spectrograph versus wavelength and time. The goal of this project is to develop a framework to model data measuring a single dynamic spectrum, or many related dynamic spectra, with incomplete sampling and noise, for example, from observations of many candidate exoplanet systems with similar host stars. The framework will integrate techniques from approximation theory, and from functional data analysis (FDA), the branch of statistics concerned with analyzing data comprising measurements of ensembles of functions. A core component of the framework will be use of separable expansions, writing the dynamic spectrum as a sum of products of paired univariate functions of wavelength ("speclets") and time ("modulators"). When the bivariate function of wavelength and time is given, approximation theory identifies optimal speclets and modulators, using the asymmetric Hilbert-Schmidt decomposition, a procedure resembling singular value decomposition (SVD). Real data do not provide precise, dense sampling of the bivariate function of wavelength and time. The project will build on FDA approaches, including functional principal components analysis (FPCA) and hierarchical Bayesian stochastic process models, to enable speclet and modulator basis discovery that accounts for noise and incomplete, irregular sampling. The framework will be applied to simulated spectra of populations of stars to build a model for stochastic stellar variability that will enable discovery of exo-Earths in Doppler radial velocity searches for exoplanets.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.