Essential and incidental measurement error: Bayesian estimation and inference when sample measurements are random-variable-valued

背景：这项发现基金将支持进一步发展随机变量值测量理论（RVVMs）的长期目标，这是我作为博士后研究员和新助理教授在过去5年中一直在开发的经典和伯克森测量误差模型的新推广，并应用该理论解决生态学和濒危物种保护的现代问题。传统的测量误差概念将问题视为错误分类或错误测量。然而，在应用数据收集系统中，当样本测量不能简单地生成实值随机变量的样本实例时，通常会出现这样的情况。相反，我们可能会发现自己处于这样一种情况，即测量过程生成的样本数据本身就是新的随机变量，这些随机变量描述了在每次样本绘制中观察到特定值或值范围的目标现象的机会。目标：在资助期间要解决的最相关的理论问题是：(1)将其他流行的测量误差模型推广到RVVM设置的不同校准条件的表征；(2)基于RVVM样本数据的极大似然估计的研究；(3)对算子理论框架的推广以及与Hilbert空间上可观测值作为线性算子的数学物理方法的联系。此外，RVVM框架的以下应用将进一步发展：(1)在鸟类带带操作中，容易地实现多项值测量和种群建模和数据收集协议；(2)通过eBird及相关平台，方便地实现公民科学来源的鸟类种群数据的多项值和正态值测量；(3) RVVM框架适应时空建模场景，应用于难以调查的城市筑巢鸟类的不确定鸟巢识别；(4)将RVVM框架集成到海鸟群航拍影像的自动目标分类中。影响：这项工作的理论发展可以提高对数学、统计和决策科学中测量的科学理解，而其应用可以在生态和环境科学中产生直接影响，特别是在解决与濒危物种保护相关的实际和困难问题方面。这项工作将产生各种理论和应用论文和技术演示，以及为应用从业者创建几个R软件包，以便在各种研究场景中实现这些方法。该项目将培养数学、统计和生态科学领域的2名本科生、4名硕士生和2名博士生，为他们从事数据科学、统计、生态学和保护领域的工作做好准备。

BACKGROUND: This Discovery Grant will support the long term goal of further developing a theory of random-variable-valued measurements (RVVMs), a novel generalization of classical and Berkson measurement error modelling that I have been developing for the past 5 years as a postdoctoral researcher and new Assistant Professor, and applying this theory to solve modern problems in ecology and conservation of species-at-risk. Traditional notions of measurement error treat the problem as one of misclassification or mismeasurement. However, there are often instances that arise in an applied data collection system when sample measurements do not generate simply sample instantiations of a real-valued random variable. Instead, we may find ourselves in a situation where the measurement process generates sample data that are themselves new random variables that describe the chance of observing a particular value, or range of values, for the target phenomenon in question on each sample draw. OBJECTIVES: The most pertinent theoretical matters to be addressed over the duration of the grant are: (1) A characterization of different calibration conditions that generalize other popular measurement error models to the RVVM setting; (2) A study of maximum likelihood estimators derived from RVVM sample data; (3) Generalization to an operator theoretic framework and connections to the mathematical physics approach of observables as linear operators on a Hilbert space. In addition, the following applications of the RVVM framework will be further developed: (1) Easy R implementation of multinomial-valued measurements and protocols for population modelling and data collection in bird-banding operations; (2) Easy R implementation of multinomial-valued and normal-valued measurements for citizen-science-sourced population data of avian species via eBird and related platforms; (3) Adaptation of RVVM framework to spatio-temporal modelling scenarios for application to unsure nest identification of difficult-to-survey urban-nesting avian species; (4) Integration of RVVM framework into automated object classification from aerial imagery of seabird colonies. IMPACT: This work's theoretical developments can improve the scientific understanding of measurement in the mathematical, statistical, and decision sciences, while its applications can have immediate impact in the ecological and environmental sciences, particularly with regards to solving real and difficult problems related to conservation of species-at-risk. A variety of theoretical and applied papers and technical presentations will result from this work, as well as the creation of several R software packages for applied practitioners to implement these methods in various research scenarios. This program will train 2 undergraduates, 4 MSc and 2 PhD students in the mathematical, statistical, and ecological sciences and prepare them for careers in data science, statistics, ecology, and conservation.