Collaborative Research: Capturing Salient Features in Point Process Models via Stochastic Process Discrepancies

在21世纪，数学建模已经成为科学研究的主要工具。科学家构建数学和计算机模型，旨在捕捉复杂物理现象的重要特征，并通过探索任何不确定模型参数的哪些值导致模型预测，从而更深入地了解该现象，这些模型预测与自然界中观察到的情况非常相似。尽管如此，这些模型最多也不能捕捉到自然界的所有复杂性——或者，正如乔治·博克斯（George Box）的名言所说，“所有的模型都是错误的，但有些模型是有用的。”这对于发现来说确实是一件好事——科学家通常可以通过精确地研究他们的模型与现实不符的原因来获得新的见解和更深入的理解。在这项研究中，pi将开发一种新的方法来量化数学模型和观测之间的“差异”，专门用于数据是事件或对象的数值计数的问题。几乎每个科学领域都有这样的例子——火山、地震或疾病病例的计数；星系、恒星或系外行星的；光子或伽马射线爆发脉冲或中微子。这个项目将专门解决天文学中的两类问题。一类是关于天文学家如何将测量来自天体物理物体（如恒星和星系）的光的原始数据转换为具有精确量化不确定性的光源属性（如亮度和颜色）的估计，即使在物体的精确形状未知且它们的图像重叠的情况下。第二类是利用天文调查目录来了解恒星、星系或小行星（如小行星）的主要人口统计学特性，如它们的光度或质量分布。由美国国家科学基金会资助的维拉鲁宾天文台将为这两类问题提供数据。这项研究将包括开发快速、开源的计算算法来实现这些新方法。项目的动机是应用领域的显著特征发现受到模型错误规范的威胁。在具有加性高斯误差的实值量级数据的应用中，统计学家通过将加性差异过程引入模型（通常使用高斯过程）来解决错误规范。这里讨论的两个问题领域需要分析离散计数或点处理数据：光子计数数据包括来自宇宙来源的图像和时间序列，或天文调查目录中的人口统计数据。这两个领域都依赖泊松点过程模型，用强度函数描述光子到达率（每单位面积）作为方向和时间的函数，或者星系密度作为空间位置和亮度的函数。加性差异模型不适用于这种离散数据设置。该团队将开发新的半参数方法，以非参数差异过程补充参数显著特征模型，灵活地模拟显著模型与真实数据生成过程的偏离。作为起点的方法表示真正的潜在强度函数，作为显著特征模型和随机乘法差异过程的乘积。显著特征模型将是一个参数强度函数（例如，具有位置、幅度、尺度和形状参数），有时是多个分量的叠加（例如，图像中的恒星、瞬态爆发中的脉冲或种群分量）。为了从显著模型中对差异进行建模，具有吸引人的理论性质的自然选择是乘法gamma差异过程；与泊松点过程的组合导致观测的总体负二项点过程。团队将实现这种方法，并将其推广到几个方向。对于人口统计模型，差异模型将嵌入到考虑测量误差和选择效应的分层模型中。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。

In the 21st century, mathematical modeling has emerged as a primary tool for scientific investigation. Scientists construct mathematical and computer models intended to capture the important features of a complex physical phenomenon, and learn more about the phenomenon by exploring what values of any uncertain model parameters lead to model predictions that mimic closely what is observed in nature. Still, at best the models typically cannot capture all of nature's complexity – or, as George Box famously put it, "All models are wrong, but some are useful." This is really a good thing for discovery – often scientists can get new insights and develop deeper understanding by studying precisely why their models fail to match reality. In this research the PIs will develop a new approach to quantifying this "discrepancy" between mathematical models and observations, intended specifically for problems in which the data are numerical counts of events or objects. Examples arise in nearly every scientific field – counts of volcanoes or earthquakes or disease cases; of galaxies or stars or exoplanets; of photons or gamma ray burst pulses or neutrinos. This project will specifically address two classes of problems in astronomy. One class concerns how astronomers can convert raw data measuring light from astrophysical objects (such as stars and galaxies) into estimates of properties of the sources (such as brightness and color) with accurately quantified uncertainties, even when the precise shapes of the objects are not known, and their images overlap. A second class concerns using astronomical survey catalogs to learn the dominant demographic properties of stars, galaxies, or minor planets (such as asteroids), such as the distribution of their luminosities or masses. The NSF-funded Vera Rubin Observatory will produce data for both types of problems. The research will include developing fast, open-sourcecomputational algorithms implementing the new approaches.The project is motivated by application areas in which salient feature discovery is threatened by model misspecification. In applications with real-valued magnitude data with additive Gaussian errors, statisticians have addressed misspecification by introducing additive discrepancy processes into models, often using Gaussian processes. The two problem areas addressed here require analysis of discrete count or point process data: photon counting data comprising images and time series from cosmic sources, or demographic data in astronomical survey catalogs. Both areas rely on Poisson point process models, with an intensity function describing, say, the photon arrival rate (per unit area) as a function of direction and time, or the density of galaxies as a function of spatial location and luminosity. Additive discrepancy models are not applicable to such discrete-data settings. The team will develop new semiparametric methods that supplement parametric salient feature models with nonparametric discrepancy processes that flexibly model the departure of salient models from the true data generating process. An approach serving as a starting point represents the true underlying intensity function as the product of the salient feature model and a stochastic multiplicative discrepancy process. The salient feature model will be a parametricintensity function (e.g., with location, amplitude, scale, and shape parameters), sometimes in a superposition of multiple components (e.g., stars in an image, pulses in a transient burst, or population components). To model discrepancy from the salient model, a natural choice with appealing theoretical properties is a multiplicative gamma discrepancy process; composition with the Poisson point process leads to an overall negative binomial point process for the observations. The team will implement this approach, and generalize it in several directions. For demographic models, the discrepancy models will be embedded in a hierarchical model that accounts for measurement error and selection effects.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.