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

摘要Wahba， Grace g.提案ID: DMS - 0505636原标题：统计模型构建和数据分析中的再现核希尔伯特空间方法新标题：统计模型构建和数据分析中的正定核方法正定函数（又称“核”）在统计模型构建、分类、聚类和数据挖掘中起着关键作用。这些核为其领域内的元素提供了距离度量，这些元素可以是函数（如在再现核希尔伯特空间中），或者最近的树、图形、图像、声音、DNA和蛋白质序列、微阵列基因表达数据、文本信息和其他对象。合理的距离度量是预测、分类和聚类的先决条件，获得这种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，用于在任意集合中与对象对之间的不相似性相关的有噪声的、粗糙的、不完整的信息可用的情况下获得核。这些方法被称为正则化核估计，因为它们涉及拟合可用的原始信息和核上的惩罚或复杂性函数之间的权衡，类似于经典正则化和偏差-方差权衡，但不相同。将提出最优调谐和降维程序，并研究它们的性质。所提出的方法被认为具有新的和重要的计算和理论优势，这些将被证明，通过有效的计算算法的发展，通过模拟，通过理论的发展，并通过应用于各种科学问题。这项工作的动机是通过获得描述对象之间“距离”的改进方法，获得上面提到的更好的对象聚类和分类方法。例如，微阵列基因芯片可能包含有关正在研究DMA的肿瘤类型的信息，预计本研究将为难以识别肿瘤类型的情况下提取这些信息提供改进的方法，最终将导致更好的诊断和治疗结果。同样，将蛋白质序列聚类到功能类中也很有趣，目的是通过关联“附近”的序列来识别功能。预计本研究将提供一种更有效的方法从原始或不完整的不相似数据中提取信息，并有助于了解蛋白质功能的长期技术。其他潜在的应用包括改进天气状态的分类，目标是对具有相似结果的局部情况进行聚类和分类，大型中微子探测器的信号检测，天体的分类，以及各种其他科学领域的分类和聚类问题。

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.