ABSTRACT

We consider supervised dimension reduction problems, namely to identify a low dimensional projection of the predictors $x$ which can retain the statistical relationship between $x$ and the response variable y. We follow the idea of the sliced inverse regression (SIR) and the sliced average variance estimation (SAVE) type of methods, which is to use the statistical information of the conditional distribution $\pi (x|y)$ to identify the dimension reduction (DR) space. In particular we focus on the task of computing this conditional distribution without slicing the data. We propose a Bayesian framework to compute the conditional distribution where the likelihood function is constructed using the Gaussian process regression model. The conditional distribution $\pi (x|y)$ can then be computed directly via Monte Carlo sampling. We then can perform DR by considering certain moment functions (e.g. the first or the second moment) of the samples of the posterior distribution. With numerical examples, we demonstrate that the proposed method is especially effective for small data problems.

摘要

我们考虑监督降维问题，即识别预测变量的低维投影 $X$ 可以保留之间的统计关系 $X$和响应变量y。我们遵循切片逆回归（SIR）和切片平均方差估计（SAVE）类型方法的思想，即利用条件分布的统计信息$\pi (X|是)$识别降维 (DR) 空间。特别是，我们专注于在不切片数据的情况下计算这种条件分布的任务。我们提出了一个贝叶斯框架来计算条件分布，其中使用高斯过程回归模型构建似然函数。条件分布$\pi (X|是)$然后可以通过蒙特卡罗采样直接计算。然后我们可以通过考虑后验分布样本的某些矩函数（例如一阶或二阶矩）来执行 DR。通过数值例子，我们证明了所提出的方法对于小数据问题特别有效。