Statistical Modeling, Inference, and Analysis of Nanoscopic Phenomena

纳米现象的统计建模、推理和分析

• 批准号: RGPIN-2014-04225
• 负责人: Lysy, Martin
• 金额: $ 1.09万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=566381
• 关键词: Statistical; Modeling; Inference; Analysis; Nanoscopic

With the current state-of-the-art in nanoscopic measurement technology, researchers are able to observe the dynamics of molecules and atoms with unprecedented accuracy and reproducibility. These nanoscopic experiments present formidable challenges and opportunities for statistical modeling. On one hand, randomness naturally emerges from deterministic physical systems in which countless degrees of freedom are unobserved. On the other, departures from ideal experimental conditions require a detailed analysis of measurement error. Ultimately, the scientific goal is to describe a dynamical phenomenon with an explanatory model. This research program will address two important challenges to achieving this goal: efficient statistical inference for stochastic physical models, and effective evaluation of the agreement between theoretical models and experimental data. Specifically, the statistical models for nanoscopic phenomena are given in continuous time, whereas recorded data is almost always discrete. This poses a major hurdle to parametric inference, as the likelihood function induced by these data is generally unavailable. A common strategy known as "data augmentation" is to impute the missing continuous path of the data in order to calculate the likelihood, and then integrate it back out. Efficient imputation of continuous data in this context is one of the primary objectives of this research program, and will be performed by Gaussian Process Regression (GPR). This constitutes a novel application of this long-standing stochastic interpolation technique. Most of the statistical literature on inference for continuous-time models operates under a Markov assumption. While this is appropriate when dynamical phenomena have rapid decorrelation times, for many cutting-edge experiments with high frequency measurements, this is not the case. A particularly important non-Markov stochastic model is known as a Generalized Langevin Equation (GLE). This model has received considerable attention from the scientific community with applications in physics, chemistry, molecular biology, and even quantum mechanics. This research program will, for the very first time, address parametric inference for the GLE in the non-Gaussian setting. The key is an approximation to the continuous-time likelihood by a novel numerical discretization scheme, through which the path imputation framework described above can be directly applied. While the primary focus of the proposed research is on statistical inference, it is of critical importance to determine whether a given model -- possibly selected from competing alternatives -- in fact agrees with the experimental data. The validity of a scientific model lies, in part, is its ability to make predictions. In many nanoscopic experiments, however, the predictions of interest are not merely univariate quantities but entire functions, such as mean-squared displacement curves or frequency spectra. This research program will leverage the principles of functional data analysis to meaningfully reduce these functions to low-dimensional statistics, which can readily be used for both goodness-of-fit testing and model selection. In a broader statistical context, the inferential and analytical challenges encountered here arise when almost any continuous-time stochastic model is discretely observed. This is the case for a wide range of non-nanoscopic phenomena occurring in areas such as geology, climatology, neurology, and finance. As such, the proposed research program is highly relevant to a multitude of problems under active investigation by researchers in Canada and abroad.

利用目前最先进的纳米测量技术，研究人员能够以前所未有的准确性和再现性观察分子和原子的动力学。这些纳米级实验为统计建模带来了巨大的挑战和机遇。一方面，随机性自然地出现在确定性的物理系统中，在这些系统中，无数的自由度是不可观察的。另一方面，偏离理想的实验条件需要详细分析测量误差。最终，科学的目标是用一个解释性的模型来描述一个动力学现象。该研究计划将解决实现这一目标的两个重要挑战：随机物理模型的有效统计推断，以及理论模型和实验数据之间的一致性的有效评估。具体来说，纳米现象的统计模型是在连续的时间，而记录的数据几乎总是离散的。这对参数推断构成了一个主要障碍，因为由这些数据诱导的似然函数通常不可用。一种常见的策略称为“数据增强”，即估算数据的缺失连续路径，以计算可能性，然后将其整合回来。在这种情况下，连续数据的有效插补是本研究计划的主要目标之一，并将通过高斯过程回归（GPR）进行。这构成了一个新的应用，这个长期存在的随机插值技术。大多数关于连续时间模型推理的统计文献都是在马尔可夫假设下进行的。虽然当动力学现象具有快速去相关时间时，这是合适的，但对于许多具有高频测量的尖端实验，情况并非如此。一个特别重要的非马尔可夫随机模型被称为广义朗之万方程（GLE）。这个模型在物理学、化学、分子生物学甚至量子力学中得到了科学界的广泛关注。这项研究计划将首次解决非高斯环境下GLE的参数推断问题。关键是通过一种新的数值离散化方案来近似连续时间似然，通过该方案可以直接应用上述路径插补框架。虽然拟议研究的主要重点是统计推断，但至关重要的是确定一个给定的模型-可能是从竞争的替代品中选择的-实际上是否与实验数据一致。科学模型的有效性部分在于它做出预测的能力。然而，在许多纳米级实验中，感兴趣的预测不仅仅是单变量，而是整个函数，例如均方位移曲线或频谱。该研究计划将利用函数数据分析的原理，有意义地将这些函数减少到低维统计，可以很容易地用于拟合优度测试和模型选择。在更广泛的统计背景下，这里遇到的推理和分析的挑战时，几乎任何连续时间随机模型离散观察。在地质学、气候学、神经学和金融学等领域发生的广泛的非纳米现象就是这种情况。因此，拟议的研究计划与加拿大和国外研究人员正在积极调查的众多问题高度相关。