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===How to read the map===
The covariance map obtained in the FEL experiment<ref name="LJF13"/> is shown in Fig. 3. Along the ''x'' and ''y'' axes the averaged spectra <math>\langle\mathbf{X}\rangle</math> and <math>\langle\mathbf{Y}\rangle</math> are shown. These spectra are resolved on the map into pairwise correlations between energies of electrons coming from the same process. For example, if the process is the first process depicted in Fig. 2 (PP), then two low-energy electrons are ejected from the Ne core giving a positive island in the bottom-left corner of the map (one of the white ones). The island is positive because if one of the electrons is detected, there is higher than average probability of detecting also the other electron and the covariance of the signals at the two energies takes a positive value.
'''Figure 3: A covariance map revealing correlations between electrons emitted from neon (and from some N<sub>2</sub> and water vapour contamination).''' The map is constructed shot by shot from electron energy spectra, which are shown along the x and y axes after averaging over 480 000 FEL shots. Volumes of the features on the map give relative probabilities of various ionization sequences. (Note that the colour scale is non-linear to accommodate a large dynamic range of the map.) <ref name="LJF13"/>▼
The volumes of the islands are directly proportional to the relative probabilities of the ionisation processes.<ref name="LJF89"/> This useful quality of the map follows from a property of the [[Poisson distribution]], which governs the number of neon atoms in the focal volume and the number of electrons produced at a particular energy, <math>X_n(E_i)</math>. The property employed here is that the [[variance]] of a Poisson distribution is equal to its [[mean]] and this property is also inherited by covariance. Therefore the covariance plotted on the map is proportional to the number of neon atoms that produce pairs of electrons of particular energies. This is a big advantage of covariance in analysis of particle counting experiments over other bivariate estimators, such as [[Pearson's correlation coefficient]].
On the diagonal of the map there is an autocorrelation line. It is present because the same spectra are used for the ''x'' and ''y'' axes. Thus, if an electron pulse is present at a particular energy on one axis, it is also present on the other axis giving variance signal along the <math>E_x = E_y</math> line, which is usually stronger than the neighbouring covariance islands. The mirror symmetry of the map with respect to this line has the same origin. The autocorrelation line and the mirror symmetry would not be present if two different detectors were used for the ''x'' and ''y'' signals, for example to detect ions and electrons.<ref name="LJF92">L J Frasinski, M Stankiewicz, P A Hatherly, G M Cross and K Codling “Molecular
Much more information is present on the map than on the averaged, 1D spectrum. The single, often broad and indistinct peaks on the 1D spectrum, are resolved into several island on the map. It is particularly useful that impurities, such as water vapour or nitrogen, give islands usually away from the islands of the species studied (see Fig. 3).
▲'''Figure 3: A covariance map revealing correlations between electrons emitted from neon (and from some N<sub>2</sub> and water vapour contamination).''' The map is constructed shot by shot from electron energy spectra, which are shown along the x and y axes after averaging over 480 000 FEL shots. Volumes of the features on the map give relative probabilities of various ionization sequences. (Note that the colour scale is non-linear to accommodate a large dynamic range of the map.) <ref name="LJF13"/>
===Negative correlations===
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