Statistical Inference for Complex Dynamical Models

动力学模型描述一个动力系统的变化率。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。虽然动力模型中的参数通常有科学解释，但它们的值往往是未知的，有时难以直接估计。这使得通过展示其解与实际数据的拟合程度来验证动态模型变得困难。我提出的研究计划的长期目标是开发统计方法，用于在存在测量误差的情况下从动态系统的测量中估计动态模型中的参数。我将广泛研究动态模型的参数估计。我的研究计划突出了一些由重要应用驱动的有趣的研究问题。例如，当动力系统对多个对象进行复制测量时，每个对象可能具有不同但相关的模型参数值。因此，在动力学模型中同时考虑总体参数（固定效应）和个体参数（随机效应）是非常重要的，这种模型被称为混合效应动力学模型。受试者可能缺少群体信息，在这种情况下，动力系统的测量可以合理地假设遵循混合分布。我提出了一种半参数方法来估计混合分布数据的混合效应动力学模型。动态模型中的参数依赖于被试的某些特征（协变量），而这些参数的值是未知的，这是另一个研究问题。同时估计模型参数及其与这些协变量的关系可能是可取的。我提出了一种方差分析（ANOVA）方法来检测治疗对动力系统的影响。我提出的研究可以潜在地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.