Principal Component Analysis (PCA) is a transform for finding the principal components (PCs) that represent features of random data. PCA also provides a reconstruction of the PCs to the original data. We consider an extension of PCA which allows us to improve the associated accuracy and diminish the numerical load, in comparison with known techniques. This is achieved due to the special structure of the proposed transform which contains two matrices $T_0$ and $T_1$, and a special transformation $\mathcal{f}$ of the so called auxiliary random vector $\mathbf w$. For this reason, we call it the three-term PCA. In particular, we show that the three-term PCA always exists, i.e. is applicable to the case of singular data. Both rigorous theoretical justification of the three-term PCA and simulations with real-world data are provided.
翻译:主要元件分析(PCA) 是用于寻找代表随机数据特征的主要组成部分(PCs)的一种变换。 CPA还提供个人电脑与原始数据的重建。 我们考虑扩展CPA, 以便与已知技术相比, 提高相关准确性并减少数字负荷。 之所以能够实现这一点,是因为拟议变换的特殊结构包含两个基体$0美元和$1美元, 以及所谓的辅助随机矢量( $\mathcal{f}) 的特殊变换 $\ mathcal{f} 。 为此, 我们称之为三期五氯苯甲醚。 我们特别表明,三期CPA始终存在, 即适用于单项数据的情况。 提供了三期五氯苯甲醚和模拟真实世界数据的严格理论依据。