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Notes about the last principal component

Contributors

Rolle, Jean-Daniel

Collection

  • Cahiers de recherche; 2000.16

Publication date

2000

Abstract

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 components analysis and the decomposition formula of the total sum of squares in linear regression both derive from a common principle.

Keywords

  • Principal components
  • Linear dependence
  • Inertia
  • Hyperplane
  • Projections

UNIGE affiliation entities

Citation (ISO format)

ROLLE, Jean-Daniel. Notes about the last principal component. 2000

Main files (1)

Report

Identifiers

  • PID : unige:5853
588views
958downloads

Technical informations

Creation

15/04/2010 12:20:44

First validation

15/04/2010 12:20:44

Update

14/03/2023 15:26:51

Status update

14/03/2023 15:26:50

Last indexation

29/10/2024 14:24:35
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