Principal Component Analysis

Abstract

Xuanye Chen Introduction Principal component analysis was first introduced by Karl Pearson for non-random variables, and then H. Hotelling extended this method to the case of random vectors. Principal component analysis (PCA) is a technique for reducing dimensionality, increasing interpretability, and at the same time minimizing information loss. Definition Principal Component Analysis (PCA) is a statistical method. Through orthogonal transformation, a group of variables that may be correlated is transformed into a group of linearly uncorrelated variables, which are called principal components. To be specific, it transforms the data to a new coordinate system such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. The Calculation of PCA F1 is used to represent the first linear combination selected, that is, the first comprehensive indicator. The larger the Var ( F1 ) is, the more information F1 contains. Therefore, F1 selected among all linear combinations has the largest variance, so F1 is called the first principal component. If the first principal component is not enough to represent the information of the original P indicators, then F2 is selected, that is, the second linear combination. In order to effectively reflect the original information, the existing information of F1 does not need to appear in F2. In other words, Cov(F1, F2) = 0, and F2 is called the second principal component. And so on, we can construct 3rd, 4th, ... , Pth principal component. Fp = a1i*ZX1 + a2i*ZX2 + ...... + api*ZXp