Fractional Ridge Regression

技术进步使收集大量数据成为可能。对企业如何运作（在线零售、精密制造、社交媒体）、科学如何运作（环境科学、气候建模、化学信息学、生物技术、工程学）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，通常只有少数与理解特定疾病和开发治疗方法相关（挑战在于识别少数相关基因）。 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（将小麦与谷壳分开）。分数岭回归（FRR）的目的是提高预测和解释性的统计分析的大型数据集相对于目前使用的统计方法。 FRR改进了从大型数据集中识别和提取相关信息的能力，从而改进了依赖于大型数据集分析和理解的许多商业、科学和政府政策领域。FRR研究具有几乎无限的应用，是吸引不同统计学专业学生参与研究项目的理想选择。 计算算法和统计软件将使所有学科的研究人员都能使用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。研究者将确定本科生和研究生的子项目，并注意从代表性不足的群体中招收学生。分数岭回归加入岭回归和统计学家回归建模工具箱中的套索。 岭回归是由Hoerl和肯纳德在1970年引入的，26年后，Tibshirani又引入了套索。从这些开创性的论文中得出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，以及回归建模在许多科学领域的实践。在回归建模的某些应用中，目标只是实现对未来响应值的最佳可能预测。在其他应用中，解释是重要的，作为一种方式来指导了解正在调查的过程。岭回归在预测方面非常出色，尽管它在预测和解释方面经常被套索所掩盖，因为套索也允许选择。分数岭回归（FRR）相对于套索提高了预测（通过均方误差测量）和可解释性（通过变量选择特异性测量）。 FRR通过一个独特而聪明的惩罚函数来实现这两个目标，该惩罚函数仅自适应地降低回归模型系数的数据驱动子集的权重（分数），该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的评估来支持。影响审查标准。

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.