Mathematical Sciences: Flexible Regression and Classification

摘要：本研究涉及多个研究方向，但有一个共同的主题：将被广泛接受但有限的统计工具推向更具冒险性的方向，同时保留其一些有吸引力的特征，如模型可解释性。具体而言，研究涉及到：a)多类响应的逻辑回归的非参数扩展，包括加性、投影追踪和基展开技术，以及类似于Fisher LDA的降阶模型；b)一种新的自适应基选择算法，类似于Friedman的MARS模型，该算法使用自然惩罚准则同时选择变量并缩小其系数；C)一种局部适应最近邻距离度量来对抗维度诅咒的技术。数据分析和建模中的许多重要问题都集中在预测上。一些重要的例子包括计算机辅助疾病诊断（例如阅读数字乳房x光片）、心脏病风险评估、自动读取手写数字（例如信封上的邮政编码）、语音识别等等。这项研究是关于以一种自然的方式丰富现有的良好建立的统计模型工具箱，以解决一些更复杂的情况。通常，新的外来技术，如神经网络，是“黑盒子”，似乎产生了良好的结果，但并没有向分析师提供一个可解释的模型、诊断或类似的反馈，让他们相信盒子已经产生了合理的结果。通过开发具有竞争力和可辩护性的模型，统计学可以在这些重要的预测和数据分析问题中发挥积极作用。这项研究正是通过在经典技术和允许模型探索的新技术之间创建混合技术来实现的。

Proposal: DMS 9504495 PI: Trevor Hastie Institution: Stanford University Title: Flexible Regression and Classification Abstract: The research concerns several research directions with a common theme: to push widely accepted but limited statistical tools in more adventurous directions, while retaining some of their attractive features, such as model interpretability. Specifically, the research involves the development of: a) nonparametric extensions of logistic regression for multiclass responses, including additive, projection pursuit and basis expansion techniques, as well as rank reduced models similar to Fisher's LDA; b) a new adaptive algorithm for basis selection, similar to Friedman's MARS model, which uses a natural penalized criterion to simultaneously select variables and shrinks their coefficients; c) a technique for locally adapting the nearest neighbor distance metric to combat the curse of dimensionality. Many important problems in data analysis and modeling focus on prediction. Some important examples include computer assisted diagnosis of disease (e.g. reading digital mammograms), heart disease risk assessment, automatic reading of handwritten digits (e.g. zip-codes on envelopes), speech recognition, to name a few. This research is about enriching the current toolbox of well established statistical models in a natural way to address some of these more complex scenarios. Often new exotic techniques, such as neural networks, are ``black boxes'' that appear to produce good results, but do not provide the analyst with an interpretable model, diagnostics or similar feedback to give them confidence that the box has produced sensible results. Statistics can play an active role in these important prediction and data analysis problems through the development competitive and defensible models. This research does just that by creating a blend between the well understood classical techniques and the new techniques that allow for model exploration.