Fractional Ridge Regression

技术进步使收集大量数据成为可能。对于企业如何运作（在线零售、精密制造、社交媒体）、科学如何开展（环境科学、气候建模、化学信息学、生物技术、工程）以及政府如何运作（医疗保健、公共安全、国土安全、国防、农业生产）的影响也相应巨大。对于大量数据集的许多用途，并非所有可用的信息都是相关的。例如，在估计的10万个人类基因中，通常只有一小部分与了解特定疾病和开发治疗方法有关（挑战在于确定这一小部分相关基因）。许多大数据探索的一个关键特征是识别和淡化多余的信息，并相应地识别和强调最相关的信息（去芜取草）。相对于目前使用的统计方法，分数岭回归（FRR）旨在提高大数据集统计分析的预测和可解释性。FRR改进了对大型数据集相关信息的识别和提取，从而改进了依赖于对大型数据集的分析和理解的商业、科学和政府政策的许多领域。随着几乎无限的应用，FRR研究是理想的从事不同的统计学生的研究项目。计算算法和统计软件将使所有学科的研究人员都可以使用FRR，从而使其对数据科学众多领域的教育和多样性的潜在好处成倍增加。研究者将为本科生和研究生确定子项目，并关注来自代表性不足群体的学生招募。分数脊回归加入了脊回归和统计学家回归建模工具箱中的套索。岭回归是由Hoerl和Kennard在1970年提出的，26年后，Tibshirani又引入了套索。从这些开创性的论文中产生的研究成果是惊人的，并且极大地促进了我们对收缩和选择方法的理解，以及在许多科学领域中回归模型的实践。在回归建模的一些应用中，目标仅仅是实现对未来响应值的最佳预测。在其他应用中，解释作为一种指导理解所调查过程的方法是很重要的。脊回归在预测方面非常好，尽管在预测和解释方面它经常被套索所掩盖，因为套索也允许选择。相对于套索，分数岭回归（FRR）提高了预测能力（以均方误差衡量）和可解释性（以变量选择特异性衡量）。FRR通过一个独特而聪明的惩罚函数实现了这两个目标，该函数自适应地降低了回归模型系数的数据驱动子集（一个分数）的权重，同时允许回归系数的互补子集自由变化，以获得最佳拟合模型。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Technological advances make it possible to collect enormous amounts of data. Implications for how businesses run (online retailing, precision manufacturing, social media), how science is conducted (environmental science, climate modeling, chemoinformatics, biotechnology, engineering), and how governments operate (health care, public safety, homeland security, national defense, agriculture production) are correspondingly enormous. For many uses of massive data sets, not all of the available information is relevant. For example, of the estimated 100,000 human genes, often only a handful are relevant to understanding a particular disease and developing a cure (the challenge is identifying the handful of relevant genes). A key feature in many big-data explorations is the identification and deemphasis of superfluous information with the corresponding identification and accentuation of the most relevant information (separating the wheat from the chaff). Fractional Ridge Regression (FRR) is designed to improve both prediction and interpretability of statistical analyses of large data sets relative to statistical methods currently in use. FRR improves the identification and extraction of relevant information from large data sets thereby improving the many areas of business, science, and government policy that rely on the analysis and understanding of large data sets. With nearly limitless applications, FRR research is ideal for engaging diverse statistics students in research projects. Computing algorithms and statistical software will make FRR available to researchers in all disciplines, thereby multiplying its potential benefits to education and diversity in numerous areas of data science. The investigator will identify sub-projects for undergraduate and graduate students with attention to student recruitment from under-represented groups.Fractional ridge regression joins ridge regression and the lasso in the statistician's regression modeling toolbox. Ridge regression was introduced by Hoerl and Kennard in 1970 and twenty-six years later was followed by the introduction of the lasso by Tibshirani. The body of research ensuing from these seminal papers is staggering, and has contributed immensely to our understanding of shrinkage and selection methodology and to the practice of regression modeling in many areas of science. In some applications of regression modeling the goal is simply to achieve the best possible predictions of future response values. In other applications, interpretation is important as a way to guide understanding of the process under investigation. Ridge regression is very good at prediction, although it is often eclipsed by the lasso in terms of both prediction and interpretation because the lasso also allows for selection. Fractional ridge regression (FRR) improves both prediction (measured by mean square error) and interpretability (measured by variable selection specificity) relative to the lasso. FRR accomplishes these twin goals via a unique and clever penalty function that adaptively downweighs only a data-driven subset of regression model coefficients (a fraction), while allowing for the complementary subset of regression coefficients to vary freely in order to obtain an optimal fitted model.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.