Notes about the last principal component

摘要

This article gives results specifically related to the last principal component. This linear combination of variables plays an important role when one tries to fit a hyperplane to a cloud of points. In data analysis, variables are often subject to a linear transformation. The results that we provide here take account of such a transformation. Firstly, we define a `distance' between a variable and a set U of variables. Among the linear combinations of the variables in U, the last principal component is the most `distant' from U. Secondly, we clarify the link between the last principal component and (i) linear dependence, (ii) equation of an optimal hyperplane, and (iii) invariance of the optimal hyperplane under linear transformation of the variables. Finally, we show how the decomposition formula of inertia of a hyperplane in principal component analysis and the decomposition formula of the total sum of squares in linear regression both derive from a common principle.