Reduction of Infinite Data Dimension via B Spline Smoothing

本研究项目开发了B样条平滑方法，用于：(1)机器学习降维；(2)非参数和半参数GARCH波动率模型，计算速度快，公式显式。给出了所有非参数估计的渐近同时置信带。该提案旨在发展基础理论，作为实际实施的关键指南。对于机器学习中的降维，重点是广义加性模型（GAM）和单指标模型（SIM），它们的维数趋于无穷大。对于从低到中等高（400-D）的维度，用于附加模型的样条背拟合核平滑程序和用于SIM的直接样条平滑程序在理论上是可靠的，直观地具有极快的计算吸引力。目前的项目将这些程序扩展到GAM和SIM，维度趋于无穷大，保留了理论，直观和计算的好处。研究者还研究了非参数和半参数GARCH模型的B样条平滑算法，实现了与核平滑相同的渐近性。由于GARCH模型的典型应用涉及从数千到数百万的样本量以及同样大量的滞后值，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了理论家的要求，也满足了金融分析师的要求。在信息过载的时代，几乎所有生物、医学、物理和社会科学领域的研究人员都经常面对大型数据集。这些大型数据集有成千上万个被称为变量或特征的特征，是有价值的科学信息的宝库。研究者开发的方法是强大的新工具，可以从大量数据集中提取这些有用的信息。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、州和联邦人口调查、联邦生物特征数据库等。用自由软件R编写的代码是公开的，可以广泛传播。来自行业和政府的从业者可以使用这些用户友好的模块实时、自信和精确地分析他们自己的大型数据集。该项目的一个显著特点是将前沿研究与研究生的教育和培训积极结合起来，特别是那些来自代表性不足群体的研究生。这与美国国家科学基金会的教育目标是一致的，也实现了美国国家科学基金会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.