Efficient and robust inference for regularization with regular and functional data

许多广为人知的参数模型，包括某些线性多元回归模型、广义线性模型和大多数单指标模型，都是带有协变量的模型。通常研究中包含许多协变量，但由于稀疏性，这些观察到的协变量中只有一部分被认为与响应变量真正相关。例如，在医学实验中，将协变量与治疗效果联系起来的特定模型往往更多地是为了方便和简单地解释，而不是为了有效性。正则化方法对于识别与响应相关的变量子集和同时进行参数估计是有用的。有效的变量选择也可以产生简洁的模型，具有更好的预测精度和更容易解释。近年来，对这一领域进行了大量的研究，并研究了许多可靠的程序。（这里的“稳健”一词指的是程序在模型不规范和/或存在异常值时保持其有效性的能力。）这些方法在处理受污染的数据方面取得了不同程度的成功。现在，人们已经广泛认识到在统计推断中需要良好的稳健的程序。实际应用中的一个常见问题是数据中存在异常值。此外，统计模型只是对现实的近似，真实的数据永远不会完全来自指定的模型。提出的研究目标是开发同时高效和鲁棒的正则化方法。在许多科学研究中，越来越多地遇到功能数据。功能数据是由曲线、曲面或其他在连续体（如时间和空间）上变化的物体的重复测量组成的。近年来，功能数据分析在许多领域受到越来越多的关注，包括从脑成像数据和纵向数据研究中对神经系统疾病的临床诊断。例如，人脑弥散张量成像数据是弧长方面的功能数据。另一个例子是功能性磁共振成像数据，其中小脑血流动力学反应函数的平均变化被用来预测注意缺陷多动障碍指数。在函数数据中经常遇到离群点，包括整个离群曲线（全局离群点）以及具有局部离群特征的曲线，这些曲线可以在时域或频域进行局部化（局部离群点）。在这些研究中，稳健的方法很重要，因为它们代表了应用实验的结果。该领域的研究目标是研究功能回归模型的有效稳健估计和正则化方法，因为高效和稳健的过程对于有效的数据分析至关重要。********

Many widely known parametric models, including certain linear multivariate regression models, generalized linear models and most single-index models, are models with covariates. Often many covariates are included in studies, but only a part of these observed covariates is believed to be truly relevant to the response variable due to sparsity. For instance, in medical experiments particular models relating covariates to treatment effects are often adopted more for convenience and simplicity of interpretation than for validity. Regularization methods are useful for identifying a subset of variables that is associated with a response and for parameter estimation simultaneously. Effective variable selection can also lead to parsimonious models with better prediction accuracy and easier interpretation. In recent years, a considerable amount of research has been devoted to this area, and many robust procedures have also been studied. (Here the word ‘robust' refers to the ability of a procedure to retain its validity under a model misspecification and/or when outliers are present.) These methods have had varying degrees of success in dealing with contaminated data. The need for good robust procedures in statistical inference has been widely recognized now. A common problem in practical applications is the presence of outliers in the data. Furthermore, statistical models are just approximations to reality and that real data never come from the specified model exactly. A goal of the proposed research is to develop regularization methods that are simultaneously efficient and robust.***In many scientific studies functional data are increasingly encountered. Functional data are made up of repeated measurements taken as curves, surfaces or other objects varying over a continuum such as time and space. Functional data analysis has gained increasing attention during recent years in many areas, including in clinical diagnosis of neurological diseases from the brain imaging data and in longitudinal data studies. For example, the diffusion tensor imaging data of human brain are functional data in terms of arc-length. Another example is the functional magnetic resonance imaging data where the averaged changes of hemodynamic response functions for cerebellum are used to predict the attention deficit hyperactivity disorder index. Outliers are frequently encountered in functional data, including entire outlying curves (global outliers) as well as curves with local outlying features, which can be localized in either the time or frequency domain (local outliers). A robust methodology is important in these studies, as they represent outcomes of applied experiments. A goal of the proposed research in this area is to investigate efficient robust estimation and regularization methods for functional regression models, as efficient and robust procedures are vital for effective data analysis.********