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#REDIRECT [[Covariance_matrix#Covariance_mapping]]
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In [[statistics]], covariance mapping is an extension of the [[covariance]] concept from [[random variables]] to [[random function|random functions]]. Normal covariance is a scalar (a single number) that measures statistical relation between two random variables. Covariance maps are matrices (arrays of numbers) that show statistical relations between different regions of 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.
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==Application to data analysis==
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Covariance mapping can be applied to any repetitive, fluctuating signal to reveal information hidden in the fluctuations. This technique was first used to analyse [[mass spectra]] of molecules ionised and fragmented by intense laser pulses.<ref name="LJF89">L J Frasinski, K Codling and P A Hatherly "Covariance Mapping: A Correlation Method Applied to Multiphoton Multiple Ionisation" ''Science'' '''246''' 1029–1031 (1989)</ref>
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Covariance mapping is particularly well suited to [[free-electron laser]] (FEL) research, where the x-ray intensity is so high that the large number of photoelectron and photoions produced at each pulse overwhelms simpler [[Photoelectron photoion coincidence spectroscopy|coincidence techniques]]. Figure 1 shows a typical experiment<ref name="LJF13">L J Frasinski, V Zhaunerchyk, M Mucke, R J Squibb, M Siano, J H D Eland, P Linusson, P v.d. Meulen, P Salén, R D Thomas, M Larsson, L Foucar, J Ullrich, K Motomura, S Mondal, K Ueda, T Osipov, L Fang, B F Murphy, N Berrah, C Bostedt, J D Bozek, S Schorb, M Messerschmidt, J M Glownia, J P Cryan, R Coffee, O Takahashi, S Wada, M N Piancastelli, R Richter, K C Prince, and R Feifel "Dynamics of Hollow Atom Formation in Intense X-ray Pulses Probed by Partial Covariance Mapping" ''Phys. Rev. Lett.'' '''111''' 073002 (2013), [http://hdl.handle.net/10044/1/11746 open access]</ref>. X-ray pulses are focused on neon atoms and ionize them. The kinetic energy spectra of the photoelectrons ejected from neon are recorded at each laser shot using a suitable spectrometer (here a [[Time-of-flight mass spectrometry|time-of-flight spectrometer]]). The single-shot spectra are sent to a computer, which calculates and displays the covariance map.
'''Figure 1: Schematics of a covariance mapping experiment.''' The experiment was performed at the [[LCLS#LCLS|LCLS FEL]] at [[Stanford University]]. <ref name="LJF13"/>
===The need to correlate photoelectrons===
Even in a relatively simple system, such as neon atom, intense x-rays induce a plethora of ionization processes (see Fig. 2). As the kinetic energies of the electrons ejected in different processes largely overlap, it is impossible to identify these processes using simple [[Photoemission spectroscopy|photoelectron spectrometry]]. To do so, one needs to correlate the kinetic energies of the electrons ejected in a given process. Covariance mapping is a method of revealing such correlations.
'''Figure 2: Examples of ionization processes in neon induced by intense x-ray photons of 1062 eV energy.''' When a photon is absorbed, it may eject a photoelectron from the atom core (P) or from its valence shell (P<sub>V</sub>). An Auger process fills any core hole ejecting an Auger electron (A). A core photoelectron on its way out may also kick out an additional valence electron giving double electron ejection (D<sub>KV</sub>) by a single photon. The x-ray intensity so high that several photons can be absorbed by a single atom producing a large variety of ionization sequences.
===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> where the spectrum is sampled, 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), \text{ ... up to the last sample}] </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 [[Matrix multiplication|their multiplication]] gives a matrix. It is convenient to display the matrix as a false-colour map.
===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 H<sub>2</sub> in intense laser fields probed by electron-electron electron-ion and ion-ion covariance techniques” ''Phys. Rev. A'' '''46''' R6789–R6792 (1992), [http://hdl.handle.net/10044/1/11612 open access]</ref>
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).
===Negative correlations===
... others' research
==Partial covariance mapping==
... N2 pcov stages
<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>
==Three-dimensional covariance mapping==
... scan N2O pictures
==See also==
* [[Photoelectron photoion coincidence spectroscopy]]
==References==
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