Collaborative Research: Model-Based and Model-Free Dimension Reduction with Applications to Bioinformatics

该提议集中于充分降维（SDR），其包括用于在回归或分类问题中参考响应Y来降低预测向量X的维度的方法。在过去的10到15年中，已经开发了各种SDR方法，这些方法不需要回归模型，并且利用了X给定Y的条件矩。这些方法已经积累了成功应用的惊人记录，并导致了各种各样的技术。研究人员建议引入逆还原模型来描述X给定Y的随机结构，而不是像传统回归中那样Y给定X。初步结果表明，这将导致在理论，方法和应用的重大进展。还原模型提供了一个统一的视角，连接传统的方法，如主成分和各种最近的无模型反演方法。此外，约简模型可以提供信息边界，这使得在可识别的上下文中评估和改进现有的无模型方法的性能成为可能。理解实验证据和探索科学假设比以往任何时候都更需要有意义地减少高维数据的方法。当代基因组科学尤其如此。测序技术、比对算法、微阵列和其他新兴的实验技术产生了关于基因组的信息，基因组中无数的新功能元件，它们所包含的数千个基因的同时表达模式，以及相关物种的进化模式。处理这些不断增长的信息的需求催生了一门全新的学科--生物信息学，其核心就是数据简化方法。在该提案中，研究人员计划研究一类逆归约模型，这些模型统一并改进了现有的降维方法，并且能够处理变量数量远远超过受试者数量的情况。这种情况是典型的基因组应用，很难或不可能使用现有的方法进行研究。

This proposal is focused on sufficient dimension reduction (SDR), which comprises methods for reducing the dimension of the predictor vector X in reference to the response Y in regression or classification problems. In the last 10 to15 years a variety of SDR methods have been developed that do not require a regression modeland that exploit the conditional moments of X given Y. These methods have accrued a striking record of successful applications and have led to a variety of techniques. The investigators propose to introduce inverse reductive models that describe the stochastic structure of X given Y, and not Y given X as in traditionalregression. Preliminary results indicate that this will lead to significant advances in theory, methods and applications. Reductive models provides a unified perspective linking traditional methods such as principal components and various recent model-free inverse methods. In addition, reductive models can provide information bounds, which make it possible to evaluate and improve upon the performance of existing model-free methods in recognizable contexts.High-throughput technologies produce massive amounts of complex and interconnected data. More than ever before, understanding experimental evidence and exploring scientific hypotheses require methods to meaningfully reduce high-dimensional data. This is particularly the case for contemporary genomic sciences. Sequencing techniques, alignment algorithms, microarrays and other emerging experimental technologies generate information on genomes, myriads of novel functional elements within them, patterns of simultaneous expression for the thousands of genes they contain, and patterns of evolution across related species. The need to handle this growing body of information has spun a whole new discipline, Bioinformatics,at the very heart of which are indeed data reduction methods. In this proposal the investigators plan to study a class of inverse reductive models that unify and improve on existing dimension reduction methods, and that are capable of handling situations where the number of variables far exceed the number of subjects. Suchsituations are typical for genomic applications, and are difficult or impossible to study using existing methods.