Reproducing Kernel Hilbert Space Methods in Statistical Model Building and Data Analysis

Wahba，Grace G.Proposal ID：DMS-0505636原始标题：统计建模和数据分析中的再生核Hilbert空间方法新标题：统计建模和数据分析中的正定核方法正定函数(又名“核”)在统计建模、分类、聚类和数据挖掘中起着关键作用。这种核为它们的域中的元素提供距离度量，该距离度量可以是函数(如在再现核的希尔伯特空间中)，或者最近是树、图形、图像、声音、DNA和蛋白质序列、微阵列基因表达数据、文本消息和其他对象。合理的距离度量是预测、分类和聚类的先决条件，而获取这些度量的方法是一个活跃的研究领域。作者将介绍、发展和研究一类新的非参数方法的性质，用于在噪声、粗糙、不完全信息与任意集合中的对象对之间的相似性相关的情况下获得核。这种方法被称为正则化核估计，因为它们涉及到在拟合可用原始信息和核上的惩罚或复杂性泛函之间的权衡，类似于但不同于经典的正则化和偏差-方差权衡。提出了最优的调谐和降维过程，并研究了它们的性质。提出的方法被认为具有新的和重要的计算和理论优势，这些优势将通过高效计算算法的开发、模拟、理论的发展和对各种科学问题的应用而得到体现。这项工作的目的是获得更好的方法来对上述对象进行聚类和分类，通过获得改进的方法来描述对象之间的“距离”。例如，微阵列基因芯片可能包含有关其DMA正在研究的肿瘤类型的信息，预计这项研究将提供在难以识别肿瘤类型的情况下提取该信息的改进方法，这将最终导致更好的诊断和治疗结果。类似地，将蛋白质序列聚类成功能类是有意义的，目的是通过关联“邻近”的序列来识别功能。预计本研究将提供一种从粗略或不完整的相异数据中提取信息的更有效的方法，并有助于理解蛋白质功能的长期技术。其他可能的应用包括改进的天气状况分类，目的是对具有类似结果的当地情况进行分类和分类，大型中微子探测器中的信号探测，天体分类，以及其他各种科学领域的分类和分类问题。

Abstract Wahba, Grace G.Proposal ID: DMS - 0505636Original Title: Reproducing Kernel Hilbert Space Methods in StatisticalModel Building and Data AnalysisNew Title: Positive Definite Kernel Methods in Statistical ModelBuilding and Data AnalysisPositive definite functions (a.k.a."kernels") play a key role instatistical model building, classification, clustering and data mining. Such kernels provide a distance metric for elements in their domain, which may be functions (as in Reproducing Kernel Hilbert Spaces), or, more recently, trees, graphs, images, sounds, DNA and protein sequences, microarray gene expression data, text messages and otherobjects. A reasonable distance metric is a prerequesite for prediction, classification, and clustering, and methods for obtaining such metricsare an area of active research. The proposer will introduce, develop and study the properties of a new class of nonparametric methods for obtaining kernels in situations where noisy, crude, incomplete information related to dissimilarity between pairs of objects in arbitrary sets is available. The methods are called regularized kernel estimation, since they involve a tradeoff betweenfitting the crude information available and a penalty or complexity functional on the kernel, analogous to, but not the same as classicalregularization and the bias-variance tradeoff. Optimal tuning and dimensionality reduction procedures will be proposed and their properties studied. The methods proposed are believed to have new and important computational and theoretical advantages, and these will bedemonstrated, by development of efficient computational algorithms, by simulation, by development of the theory, and by application to a variety of scientific problems.This work is motivated by the goal of obtaining better methods for clustering and classifying objects mentioned above, by obtaining improved ways to describe the "distance" betweein objects. For example, microarray gene chips may contain information concerning, e. g. the type of tumor whose DMA is being studied, and it is anticipated that this research will provideimproved methods for extracting this information in cases where it is difficult to identify the type of tumor, and this will ultimately result in better diagnostic and treatmentoutcomes. Similarly, it is of interest to cluster protein sequences into functional classes, with the goal of identifying function by associating sequences that are "nearby". It is anticipated that the present research will provide a more efficient way of extracting information from crude or incomplete dissimilarity data and, contribute to the long-term technology of understanding protein function. Other potential applications includeimproved classification of weather states, with the goal of clusteringand classifying local situations that have similar outcomes, signaldetection in large neutrino detectors, classification of astronomical bodies, and classification and clustering problems in a variety of other scientific fields.