Editing Covariance matrix (section)

===Covariance mapping===
In '''covariance mapping''' the values of the <math> \operatorname{cov}(\mathbf{X}, \mathbf{Y}) </math> or <math> \operatorname{pcov}(\mathbf{X}, \mathbf{Y} \mid \mathbf{I}) </math> matrix are plotted as a 2-dimensional map. When vectors <math> \mathbf{X} </math> and <math> \mathbf{Y} </math> are discrete [[random function]]s, the map shows statistical relations between different regions of the random functions. Statistically independent regions of the functions show up on the map as zero-level flatland, while positive or negative correlations show up, respectively, as hills or valleys.

In practice the column vectors <math> \mathbf{X}, \mathbf{Y} </math>, and <math> \mathbf{I} </math> are acquired experimentally as rows of <math> n </math> samples, e.g.
<math display="block"> \left[\mathbf{X}_1, \mathbf{X}_2, \dots, \mathbf{X}_n\right] =
\begin{bmatrix}
  X_1(t_1) & X_2(t_1) & \cdots & X_n(t_1) \\ \\
  X_1(t_2) & X_2(t_2) & \cdots & X_n(t_2) \\ \\
  \vdots   &   \vdots & \ddots & \vdots   \\ \\
  X_1(t_m) & X_2(t_m) & \cdots & X_n(t_m)
\end{bmatrix} ,
</math>
where <math> X_j(t_i) </math> is the ''i''-th discrete value in sample ''j'' of the random function <math> X(t) </math>. The expected values needed in the covariance formula are estimated using the [[sample mean]], e.g.
<math display="block"> \langle \mathbf{X} \rangle = \frac{1}{n} \sum_{j=1}^{n} \mathbf{X}_j </math>
and the covariance matrix is estimated by the [[sample covariance]] matrix
<math display="block">
  \operatorname{cov}(\mathbf{X},\mathbf{Y})
  \approx \langle \mathbf{XY^\mathsf{T}} \rangle
  - \langle \mathbf{X} \rangle \langle \mathbf{Y}^\mathsf{T} \rangle ,
</math>
where the angular brackets denote sample averaging as before except that the [[Bessel's correction]] should be made to avoid [[bias of an estimator|bias]]. Using this estimation the partial covariance matrix can be calculated as
<math display="block">
  \operatorname{pcov}(\mathbf{X},\mathbf{Y} \mid \mathbf{I})
  = \operatorname{cov}(\mathbf{X},\mathbf{Y})
  - \operatorname{cov}(\mathbf{X},\mathbf{I})
  \left (    \operatorname{cov}(\mathbf{I},\mathbf{I})
  \backslash \operatorname{cov}(\mathbf{I},\mathbf{Y}) \right ),
</math>
where the backslash denotes the [[Division (mathematics)#Left and right division|left matrix division]] operator, which bypasses the requirement to invert a matrix and is available in some computational packages such as [[Matlab]].<ref name="LJF16">L J Frasinski "Covariance mapping techniques" ''J. Phys. B: At. Mol. Opt. Phys.'' '''49''' 152004 (2016), {{doi|10.1088/0953-4075/49/15/152004}}</ref>

[[Image:Stages of partial covariance mapping.png|thumb|600px|'''Figure 1: Construction of a partial covariance map of N<sub>2</sub> molecules undergoing Coulomb explosion induced by a free-electron laser.'''<ref name="OK13"/> Panels '''a''' and '''b''' map the two terms of the covariance matrix, which is shown in panel '''c'''. Panel '''d''' maps common-mode correlations via intensity fluctuations of the laser. Panel '''e''' maps the partial covariance matrix that is corrected for the intensity fluctuations. Panel '''f''' shows that 10% overcorrection improves the map and makes ion-ion correlations clearly visible. 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).]]
Fig. 1 illustrates how a partial covariance map is constructed on an example of an experiment performed at the [[DESY#FLASH|FLASH]] [[free-electron laser]] in Hamburg.<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), {{doi|10.1088/0953-4075/46/16/164028}}</ref> The random function <math> X(t) </math> is the [[Time-of-flight_mass_spectrometry|time-of-flight]] spectrum of ions from a [[Coulomb explosion]] of nitrogen molecules multiply ionised by a laser pulse. Since only a few hundreds of molecules are ionised at each laser pulse, the single-shot spectra are highly fluctuating. However, collecting typically <math> m=10^4 </math> such spectra, <math> \mathbf{X}_j(t) </math>, and averaging them over <math> j </math> produces a smooth spectrum <math> \langle \mathbf{X}(t) \rangle </math>, which is shown in red at the bottom of Fig. 1. The average spectrum <math> \langle \mathbf{X} \rangle </math> reveals several nitrogen ions in a form of peaks broadened by their kinetic energy, but to find the correlations between the ionisation stages and the ion momenta requires calculating a covariance map.

In the example of Fig. 1 spectra <math> \mathbf{X}_j(t) </math> and <math> \mathbf{Y}_j(t) </math> are the same, except that the range of the time-of-flight <math> t </math> differs. Panel '''a''' shows <math> \langle \mathbf{XY^\mathsf{T}} \rangle </math>, panel '''b''' shows <math> \langle \mathbf{X} \rangle \langle \mathbf{Y}^\mathsf{T} \rangle </math> and panel '''c''' shows their difference, which is <math> \operatorname{cov}(\mathbf{X},\mathbf{Y}) </math> (note a change in the colour scale). Unfortunately, this map is overwhelmed by uninteresting, common-mode correlations induced by laser intensity fluctuating from shot to shot. To suppress such correlations the laser intensity <math> I_j </math> is recorded at every shot, put into <math> \mathbf{I} </math> and <math> \operatorname{pcov}(\mathbf{X},\mathbf{Y} \mid \mathbf{I}) </math> is calculated as panels '''d''' and '''e''' show. The suppression of the uninteresting correlations is, however, imperfect because there are other sources of common-mode fluctuations than the laser intensity and in principle all these sources should be monitored in vector <math> \mathbf{I} </math>. Yet in practice it is often sufficient to overcompensate the partial covariance correction as panel '''f''' shows, where interesting correlations of ion momenta are now clearly visible as straight lines centred on ionisation stages of atomic nitrogen.