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==Partial covariance mapping==
Covariance maps expose all kinds of correlations, including indirect ones that are induced by a fluctuating common parameter. Such common-mode correlations are often uninteresting and they obscure the interesting ones. For example, in laser experiments the pulse intensity may fluctuate from shot to shot. These fluctuations correlate every electron with every other electron, simply because a more intense pulse produces more electrons of every energy.
The influence of such uninteresting correlations can be removed using partial covariance mapping. This method exposes only a part of the correlations, the part that is independent of the fluctuating parameter <math>I</math>, which has to be measured at every shot. The formula for partial covariance<ref>W J Krzanowski "Principles of Multivariate Analysis" (Oxford University Press, New York, 1988), Chap. 14.4; K V Mardia, J T Kent and J M Bibby "Multivariate Analysis (Academic Press, London, 1997), Chap. 6.5.3; T W Anderson "An Introduction to Multivariate Statistical Analysis" (Wiley, New York, 2003), 3rd ed., Chaps. 2.5.1 and 4.3.1.</ref> is
: <math>\mathbf{pcov}(\mathbf{Y},\mathbf{X};I) = \mathbf{cov}(\mathbf{Y},\mathbf{X}) - \mathbf{cov}(\mathbf{Y},I) \mathbf{cov}(I,\mathbf{X})/cov(I,I)</math>,
where <math>cov(I,I)</math> is the variance of the fluctuating parameter.
It is instructive to see how this formula works on an example of another experiment performed at the [[DESY#FLASH|FLASH]] FEL in Hamburg. (In fact this method was also used to analyse the LCLS experiment described above, but to keep the description simple the partial covariance was not mentioned.) The FLASH experimental setup was very similar to the LCLS setup shown in Fig. 1, except molecular nitrogen and iodine were studied and their ions rather than electrons were detected.<ref name="OK13">O Kornilov, M Eckstein, M Rosenblatt, C P Schulz, K Motomura, A Rouzée, J Klei, L Foucar, M Siano, A Lübcke, F. Schapper, P Johnsson, D M P Holland, T Schlatholter, T Marchenko, S Düsterer, K Ueda, M J J Vrakking and L J Frasinski "Coulomb explosion of diatomic molecules in intense XUV fields mapped by partial covariance" ''J. Phys. B: At. Mol. Opt. Phys.'' '''46''' 164028 (2013), [http://hdl.handle.net/10044/1/12267 open access]</ref> Fig. 5a shows the correlated product <math> \langle \mathbf{YX} \rangle </math> and Fig. 5b the uncorrelated product <math> \langle \mathbf{Y} \rangle \langle \mathbf{X} \rangle </math>. Their difference gives the normal covariance map (c). The momentum correlation lines start to be visible (note a change in the colour scale) but the map is overwhelmed by correlations induced by FEL intensity fluctuations. These correlations are calculated in panel (d) and the correction is subtracted from map (c) giving map (e). The momentum correlation lines are now clearly visible but some the common-mode background is still present, which is likely to be caused by other fluctuating parameters, such as the sample density or the FEL pulse duration. As these parameters were not monitored, simply an excess of the correction (d) was subtracted giving map (f). This crude, ''ad hoc'' method significantly suppresses the common-mode background in the region of interest but the overcorrection introduces negative regions (magenta) at long time of flights. The detailed algorithm of partial covariance mapping is given in <ref name="LJF13"/>.
'''Figure 5: Stages of partial covariance mapping to resolve the ion momentum correlations in Coulomb explosion of N<sub>2</sub> molecules.'''<ref name="OK13"/> Owing to momentum conservation these correlations appear as lines approximately perpendicular to the autocorrelation line (and to the periodic modulations which are caused by detector ringing).
==See also==
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