Collaborative Research: High-Dimensional Projection Tests and Related Topics

虽然高维数据分析已成为统计学中最活跃的研究领域，但仍有许多具有挑战性的问题尚未解决，需要新的方法和理论的发展。本项目旨在开发新的统计工具和软件，对高维数据进行统计建模和推理。预计拟议的研究将大大提高用于分析高维数据的统计工具和软件的可用性，这些数据经常在许多研究领域收集，包括基因组学，生物医学成像，功能性磁共振成像，断层扫描，肿瘤分类和金融。因此，预计拟议的工作将使各个领域的广泛科学家和研究人员受益。在过去的10年里，高维估计和稀疏恢复得到了相当多的关注，但对假设检验的了解却少得多。在这个项目中，PI首先计划为高维单样本和双样本均值问题开发新的投影Hotelling检验和卡方检验。这些测试与现有的测试的区别在于，它们基于最佳投影方向，这些方向是为了实现最佳功率性能而导出的。PI进一步提出了一种有效的数据驱动的方法来估计最佳投影方向的样本分裂策略。建议的程序可以很容易地执行。他们计划通过正则化方法研究稀疏最佳投影方向的估计。线性判别分析在分类中取得了巨大的成功，但大多数现有的程序不能处理不同数量的类。在这个项目中，他们还计划研究具有发散类数的多维线性判别分析，并开发新的程序，使研究人员能够将低维线性判别分析技术应用于超高维线性判别分析，并使具有发散类数的超高维线性判别分析在实际计算中可行。这种模型和相关的新方法在大数据分析方面具有很高的潜力。PI计划继续与工程师，气象学家，公共卫生科学研究人员和预防研究人员合作，并向统计学和生物统计学之外的科学家介绍拟议的方法。研究所计划通过出版物、会议介绍和软件分发来传播研究成果。

Although high-dimensional data analysis has become the most active research area in statistics, there are still many challenging unsolved problems which call for the development of new methods and theory. This project aims to develop new statistical tools and software to statistical modeling and inference on high-dimensional data. The proposed research is expected to significantly enhance the availability of statistical tools and software for analysis of high-dimensional data, which have frequently been collected in many research areas including genomics, biomedical imaging, functional magnetic resonance imaging, tomography, tumor classifications and finance. Hence, the proposed work is expected to benefit a broad range of scientists and researchers in various fields. Considerable attention has been devoted to high-dimensional estimation and sparsity recovery over the last 10 years, but much less is known about hypothesis testing. In this project, the PIs first plan to develop new projection Hotelling's test and chi-squares tests for high-dimensional one-sample and two-sample mean problems. The tests are distinguished from the existing ones in that they are based on optimal projection directions that are derived to achieve optimal power performance. The PIs further propose an effective data-driven method to estimate the optimal projection direction by a sample-splitting strategy. The proposed procedure can be easily carried out. They plan to investigate the estimation of the sparsity optimal projection direction via regularization methods. Linear discriminant analysis has been hugely successful in classification, but most of the existing procedures cannot handle diverging number of classes. In this project, they also plan to study ultrahigh dimensional linear discriminant analysis with a diverging number of classes and develop new procedures enable researchers to apply low-dimensional linear discriminant analysis techniques for ultrahigh-dimensional linear discriminant analysis, and make ultrahigh-dimensional linear discriminant analysis with a diverging number of classes computationally feasible in practice. This model and associated new methodology have high potential for big data analysis. The PIs plan to continue collaborating with engineers, meteorologists, public health science researchers and prevention researchers and introduce the proposed methodology to scientists beyond statistics and biostatistics. The PIs plan to disseminate the research results through publications, conference presentations and software distribution.