Principal components regression (PCR) is a well‐known method to achieve dimension reduction and often improved prediction over the ordinary least squares. The conventional PCR retains the principal components with large variance and discards those with smaller variance. This operation can easily lead to poor prediction when the response variable is related to principal components with small variance. In this work, we propose a simple remedy named response‐guided principal components regression (RgPCR) that selects principal components for regression based on both the variance of principal components and the goodness of fit to the response. RgPCR is easy to implement without using any optimization and works naturally for both low dimensional and high dimensional data. We derive a C_p type statistic for selecting the tuning parameter in RgPCR. In our numerical experiments, RgPCR is shown to enjoy promising performance.

中文翻译：

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一种改善主成分回归的简单方法

主成分回归（PCR）是一种众所周知的方法，可以实现降维效果，并且通常可以改善对普通最小二乘法的预测。常规PCR保留了具有较大差异的主成分，并丢弃了具有较小差异的主成分。当响应变量与方差较小的主成分相关时，此操作很容易导致预测不佳。在这项工作中，我们提出了一种简单的补救方法，称为响应指导的主成分回归（RgPCR），该方法根据主成分的方差和对响应的拟合优度来选择要回归的主成分。RgPCR易于实现，无需使用任何优化，并且自然适用于低维和高维数据。我们得出C _p类型统计信息，以在RgPCR中选择调整参数。在我们的数值实验中，RgPCR被证明具有良好的性能。