# Chapter 5 Principal component analysis

Principal Component Analysis (PCA) is a powerful multivariate technique designed to summarize the most important features and relations of \(k\) numerical random variables \(X_1,\ldots,X_k\). PCA does *dimension reduction* of the original dataset by computing a new set of variables, the principal components \(\text{PC}_1,\ldots \text{PC}_k\), which explain the same information as \(X_1,\ldots,X_k\) but in an *ordered* way: \(\text{PC}_1\) explains the most of the information and \(\text{PC}_k\) the least.

There is *no response* \(Y\) or particular variable in PCA that deserves a particular attention – all variables are treated equally.