2.3.1.1. Purpose

The principle of this kind of analysis is to analyse a provided ensemble, called hereafter \(\mathcal{D}\), whose size is \(n_S\), and which can be written as

\[\mathcal{D} = \{\mathbf{x}^{i}\}_{i=1,\ldots,n_S}\]

where \(\mathbf{x}^{i}\) is the i-Th input vector, written as \(\mathbf{x}^{i}=(x^{i}_1,\ldots,x^{i}_{n_X})\) where \(n_X\) is the number of quantitative variable. It is basically a set of realisation of \(n_X\) random variables whose properties are completely unknown.

The aim is then to summarise (project/reduce) this sample into a smaller dimension space \(q\) (with \(1 \leq q \leq n_X\) ) these \(q\) factors being chosen in order to maximise the inertia and being orthogonal one to another[1]. By doing so, the goal is to be able to reduce the dimension of our problem while loosing as few information as possible.

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[1]

As a reminder, the dispersion of a quantitative variable is usually represented with its variance (or standard deviation), the inertia criteria is, for multi-dimension problems, the sum of all the variable’s variance.