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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>, the spectra can be regarded as [[row vector|row vectors]] of experimental data:
:<math> \mathbf{X} = \mathbf{X}_n = [X_n(E_1), X_n(E_2), X_n(E_3), ... ] </math>.
The simplest way to analyse the data is to average the spectra over <math>N</math> laser shots:
:<math> \langle \mathbf{X} \rangle = \frac{1}{N} \sum^{N}_{n=1} \mathbf{X}_n </math>.
Such spectra show kinetic energies of individual electrons but the correlations between the electrons are lost in the process of averaging. To reveal the correlations we need to calculate the covariance map:
:<math>\mathbf{cov}(\mathbf{Y},\mathbf{X}) = \langle \mathbf{YX} \rangle - \langle \mathbf{Y} \rangle \langle \mathbf{X} \rangle </math>,
where vector <math>\mathbf{Y}</math> is the [[transpose]] of vector <math>\mathbf{X}</math> and the angular brackets denote averaging over many laser shots as before. Note that the ordering of the vectors (a column followed by a row) ensures that their multiplication gives a matrix. It is convenient to display the matrix as a false-colour map.
===How to read the map===
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