# 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 

```{math}
\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[^metho_pca_purpose]. By doing so, the goal is to be able to reduce the 
dimension of our problem while loosing as few information as possible.

[^metho_pca_purpose]: 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.