Reduction of Infinite Data Dimension via B Spline Smoothing

本研究计画发展B样条平滑方法：（1）机器学习中的降维;（2）非参数及半参数Gestival波动率模型，具有快速的计算速度及明确的公式。对所有非参数估计给出了渐近同时置信带。该提案旨在发展基本理论，作为实际执行的重要指南。机器学习中的降维问题主要集中在维数趋于无穷大的广义加性模型（GAM）和单指标模型（SIM）。对于从低维到中高维（400-D）的数据，加性模型的样条拟合核平滑方法和SIM模型的直接样条平滑方法在理论上是可靠的，具有直观的吸引力和极快的计算速度。目前的项目将这些程序扩展到GAM和SIM，其维数将达到无穷大，保留了理论，直观和计算的优势。本文还研究了非参数和半参数Gestival模型的B样条光滑算法，获得了与核光滑相同的渐近性.由于GARCH模型的典型应用涉及数千到数百万的样本大小和同样大量的滞后值，B样条平滑可以在几秒钟内计算出核平滑需要几天的时间。因此，所提出的方法既满足理论家又满足金融分析师的需求。在信息过载的时代，生物、医学、物理和社会科学几乎所有领域的研究人员都经常面临大型数据集。这些大型数据集包含数万种称为变量或特征的特征，是有价值的科学信息的宝库。研究人员开发的方法是从大型数据集中提取此类有用信息的强大新工具。此类数据的典型示例包括但不限于环境和全球变化研究、高频金融数据、州和联邦人口统计调查、联邦生物统计数据库等。来自行业和政府的从业者可以使用这些用户友好的模块来分析他们自己的大型数据集，真实的时间，自信和精确。该项目的一个显著特点是积极将前沿研究与研究生的教育和培训相结合，特别是那些来自代表性不足群体的研究生。这与NSF的教育目标是一致的，并履行了NSF对促进科学多样性原则的承诺。

This research project develops B spline smoothing methods for: (1) reducing dimension in machine learning and (2) non- and semi parametric GARCH volatility model, with fast computing and explicit formulae. Asymptotically simultaneous confidence band are provided for all nonparametric estimation. The proposal aims to develop the underlying theory as a crucial guide to practical implementation. For dimension reduction in machine learning, the focuses are on the generalized additive model (GAM) and the single index model (SIM), with dimensions tending to infinity. For dimensions from low to moderately high (400-D), spline-backfitted kernel smoothing procedure for additive model and direct spline smoothing procedure for SIM are theoretically reliable, intuitively appealing with extremely fast computing. The current project extends these procedures to GAM and SIM with dimension going to infinity, preserving the theoretical, intuitive and computing benefits. The investigator also studies B spline smoothing algorithms for non- and semi- parametric GARCH model, achieving the same asymptotics as kernel smoothing. As typical applications of GARCH model involve sample sizes from thousands to millions and equally large number of lagged values, B spline smoothing can compute in seconds what kernel smoothing would need days. Thus the proposed methods satisfy both theoreticians and financial analysts.In the age of information overload, researchers in nearly all areas of biological, medical, physical and social sciences are routinely confronted with large data sets. With tens of thousands of characteristics called variables or features, these large data sets are treasure troughs of valuable scientific information. The methods developed by the investigator are powerful new tools for drawing such useful information out of large data sets. Typical examples of such data include but are not limited to, environmental and global change studies, high frequency financial data, state and federal demographic surveys, federal biometric database, etc. Codes written in free software R are made publicly available for wide dissemination. Practitioners from industry and government can analyze their own large data sets with these user-friendly modules, in real time, with confidence and precision. A distinctive feature of the project is the active integration of cutting-edge research with the education and training of graduate students, especially those from underrepresented groups. This is consistent with the education goal of NSF and fulfills NSF's commitment to the principle of fostering diversity in science.