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===The principle===
Consider a [[random function]] <math>X_n(E)</math>, where index <math>n</math> labels a particular instance of the function and <math>E</math> is the independent variable. In the context of the FEL experiment, <math>X_n(E)</math> is a digitized electron energy spectrum produced by laser shot <math>n</math>. As the electron energy <math>E</math> takes a range of discrete values <math>E_i</math> in places where the spectrum is sampled, the spectra can be regarded as [[row vector]]s of experimental data:
:<math> \mathbf{X} = \mathbf{X}_n = [X_n(E_1), X_n(E_2), X_n(E_3),
The simplest way to analyse the data is to average the spectra over <math>N</math> laser shots:
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