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

光谱学在包括天文学在内的许多科学领域都是必不可少的，在天文学中，将实验室的光谱与恒星的光谱进行比较可以揭示恒星的化学成分。时间分辨光谱学——测量随时间变化的光谱——已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异性，复杂化合物的分子动力学，以及生物系统中随时间变化的化学过程。该项目旨在开发一个框架，用于在相关物体上进行不完整测量的情况下分析时间分辨光谱数据，或者在一个系统中重复进行，在重复中看到随机变化。仔细汇集和分析重复的数据可以识别产生光谱变化的过程的特征。主要应用将是模拟时间分辨光谱的恒星，作为太阳系外行星系统的候选宿主，特别是从搜索类地行星围绕类太阳恒星（系外地球）的数据。小的系外行星是不能直接看到的，但是它们的存在是可以分辨出来的：行星对其恒星的牵引力会产生一种微小的摆动运动，这种运动可以通过测量光谱线极小的时变多普勒频移来探测到。目前，观测小行星的主要障碍是主星的光谱活动——黑子、亮斑和耀斑的来来去去可以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，从时间分辨光谱数据中分离恒星活动信号和行星信号。该项目还将支持培训各种各样的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，该函数表示摄谱仪测量的光的（相对）强度与波长和时间的关系。该项目的目标是开发一个框架来模拟测量单个动态光谱或许多相关动态光谱的数据，这些数据具有不完整的采样和噪声，例如，来自许多具有相似主恒星的候选系外行星系统的观测。该框架将整合来自近似理论和功能数据分析（FDA）的技术，功能数据分析是统计的一个分支，涉及分析包含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展开，将动态光谱写成波长（“speclets”）和时间（“modulators”）的成对单变量函数的乘积和。当波长和时间的二元函数给定时，近似理论利用非对称Hilbert-Schmidt分解（类似于奇异值分解（SVD）的过程）识别出最优的散斑和调制器。实际数据不能提供波长和时间二元函数的精确、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，以实现对噪声和不完整、不规则采样的speclet和调制器基的发现。该框架将应用于恒星种群的模拟光谱，以建立一个随机恒星变化的模型，这将使在多普勒径向速度搜索系外行星时发现系外地球成为可能。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.