Statistical Inference for Complex Dynamical Models

动态模型描述动态系统的变化率。它们被广泛用于阐明许多领域的复杂系统，包括神经科学、生物学、医学、遗传学、生态学、工程学、物理学和金融学。虽然动态模型中的参数通常有科学解释，但它们的值通常是未知的，有时很难直接估计。这使得通过显示动力学模型的解与真实的数据的拟合程度来验证动力学模型变得困难。我提出的研究计划的长期目标是发展统计方法，估计参数的动态模型从测量的动态系统中存在的测量误差。我将广泛地研究动态模型的参数估计。我的研究计划突出了一些有趣的研究问题的重要应用的动机。例如，当动态系统在多个对象上具有重复测量时，每个对象可以具有不同但相关的模型参数值。然后，在动态模型中同时允许群体参数（固定效应）和个体参数（随机效应）是非常有趣的，然后将其称为混合效应动态模型。受试者可能缺少组信息，在这种情况下，可以合理地假设动力系统的测量值遵循混合分布。我提出了一个半参数方法来估计混合分布数据的混合效应动力学模型。另一个研究问题出现时，动力学模型中的参数依赖于一些字符（协变量）的主题，而这些参数有未知的值。可能需要同时估计模型参数及其与这些协变量的关系。我提出了一个方差分析（ANOVA）的方法来检测动态系统的治疗效果。我提出的研究可能应用于许多科学领域，如农业，生物学和遗传学。

Dynamical models describe the rate of change of a dynamical system. They are widely used to elucidate complex systems in many areas including neuroscience, biology, medicine, genetics, ecology, engineering, physics, and finance. While the parameters in dynamical models usually have scientific interpretations, their values are often unknown and can sometimes be difficult to estimate directly. This makes it hard to verify dynamical models by showing how well their solutions fit real data. My long-term objective of the proposed research program is to develop statistical methodologies for estimating parameters in dynamical models from measurements of the dynamical system in the presence of measurement errors. I will work broadly in parameter estimation for dynamical models. My research proposal highlights some interesting research problems motivated by important applications. For example, when the dynamical system has replicative measurements over multiple subjects, each subject may have different but correlated values of model parameters. Then it is of great interest to allow both population parameters (fixed effects) and individual parameters (random effects) in the dynamical model, which is then called a mixed-effects dynamical model. Subjects may have missing group information, in which case the measurements for the dynamical system can be reasonably assumed to follow a mixture distribution. I propose a semiparametric method to estimate mixed-effects dynamical models from mixture-distributed data. Another research problem arises when parameters in dynamical models depend on some characters (covariates) of subjects while these parameters have unknown values. It may be desirable to estimate model parameters and their relationships to those covariates simultaneously. I propose an analysis of variance (ANOVA) approach to detect treatment effects on dynamical systems. My proposed research can potentially be applied to many scientific areas such as agriculture, biology, and genetics.