Machine Learning and Dimension Reduction Methods for High-Dimensional Datasets

在当今计算机处理器功能强大而计算机内存便宜的环境中，研究人员能够收集和存储大量数据。分析这些数据需要复杂的统计和计算方法，因为大多数经典的统计方法是在数据收集不那么容易的时代开发的，数据集的数量级要小得多。充分降维（SDR）是一类用于回归和分类问题中的特征提取的方法，其目的是将多维数据集的大小减少到几个重要特征。该项目的重点是改进现有的方法，以获得更准确和计算速度更快的估计算法，以实现SDR。文献中最有趣的建议之一是使用机器学习算法，更具体地说是支持向量机（SVM）。该方法虽然强大，但可以在不同的方向上进行改进，因此学生可以在这个项目上采取一些方向。以下是一些例子：导出新的SDR方法对离群值的鲁棒性;导出稀疏SDR方法;当我们有缺失的预测因子时导出SDR方法;导出功能数据的SDR方法等等。

In today's environment where computer processors are powerful and computer memory cheap, researchers are able to collect and store huge amounts of data. Analysing that data needs sophisticated statistical and computational methods as most classic statistical methodology was developed at an era where data collection was not as easy and datasets where a lot of orders of magnitude smaller. Sufficient dimension reduction (SDR) is a class of methods for feature extraction in regression and classification problems with the purpose of reducing the size of a multidimensional dataset to a few important features. This has the potential of improving visualization of the most important relationships between the variables.This project focuses on the improvement of existing methodology for more accurate and computationally faster estimation algorithms to achieve SDR. Among the most interesting suggestions in the literature uses machine learning algorithms and more specifically Support Vector Machines (SVM). The method although powerful can be improved in different directions and therefore there are a number of directions that a student can take on this project. A few examples are: to derive new SDR methodology robust to outliers; to derive Sparse SDR methodology; to derive SDR methodology when we have missing predictors; to derive SDR methodology for functional data and many more.