Theory and simulations of covariance mapping in multiple dimensions for data analysis in high-event-rate experiments

V. Zhaunerchyk, L. J. Frasinski, J. H. D. Eland, and R. Feifel

Phys. Rev. A 89, 053418 – Published 22 May 2014

Abstract

Multidimensional covariance analysis and its validity for correlation of processes leading to multiple products are investigated from a theoretical point of view. The need to correct for false correlations induced by experimental parameters which fluctuate from shot to shot, such as the intensity of self-amplified spontaneous emission x-ray free-electron laser pulses, is emphasized. Threefold covariance analysis based on simple extension of the two-variable formulation is shown to be valid for variables exhibiting Poisson statistics. In this case, false correlations arising from fluctuations in an unstable experimental parameter that scale linearly with signals can be eliminated by threefold partial covariance analysis, as defined here. Fourfold covariance based on the same simple extension is found to be invalid in general. Where fluctuations in an unstable parameter induce nonlinear signal variations, a technique of contingent covariance analysis is proposed here to suppress false correlations. In this paper we also show a method to eliminate false correlations associated with fluctuations of several unstable experimental parameters.

Received 8 November 2013
Revised 10 April 2014

DOI:https://doi.org/10.1103/PhysRevA.89.053418

Authors & Affiliations

V. Zhaunerchyk^1,2,*, L. J. Frasinski³, J. H. D. Eland^1,4, and R. Feifel^1,2

¹Department of Physics and Astronomy, Uppsala University, 751 20 Uppsala, Sweden
²Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden
³Department of Physics, Imperial College London SW7 2AZ, United Kingdom
⁴Department of Chemistry, Oxford University, Oxford OX1 3QZ, United Kingdom

^*Corresponding author: vitali.zhaunerchyk@physics.gu.se

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Vol. 89, Iss. 5 — May 2014

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Images

Figure 1
Simulated one-dimensional spectrum of the products according to the reaction scheme discussed [Eq. (5)], obtained for an ionization rate of 1 shot $^{- 1}$ .
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Figure 2
Simulated two-fold coincidence (left panels) and covariance (right panels) maps for $2 \times 10^{6}$ laser shots at the ionization rates of 0.01 shot $^{- 1}$ [(a) and (b)], 1 shot $^{- 1}$ [(c) and (d)], and 10 shot $^{- 1}$ [(e) and (f)], respectively. The values in the covariance maps are multiplied by $M_{shots}$ . The product channels considered are summarized in Eq. (5); it is assumed that products A–E are detected at channel positions $E$ (of arbitrary units) of 10, 15, 20, 25, and 30, respectively. In the maps (c)–(f) the maximum value of the color bar corresponds to the peak value of the island formed by products D and E. False coincidences clearly visible in map (e) are removed from covariance map (f).
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Figure 3
Simulated total (a) and partial (b) covariance maps multiplied by $M_{shots}$ for a process similar to the reaction scheme depicted in Figs. 1 and 2 but with two fluctuating parameters, $I$ and $J$ , that induce false correlations in map (a). Equation (13) removes these false correlations as shown in map (b).
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Figure 4
Slices of simulated 3D covariance maps for the reaction presented in Eq. (19), obtained for an ionization rate of 1 shot $^{- 1}$ . The covariance values are multiplied by $M_{shots}$ . The slices shown in panels (a), (b), (c), and (d) correspond to the products A, B, D, and E, respectively. They are expected to be registered at the channel peak positions $E$ (of arbitrary units) 10, 15, 25, and 30, respectively. The maps correctly show correlations among products A, B, D, and E, and their lack with uncorrelated product C (no islands at $E$ = 20 and the unshown slice at this position is blank).
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Figure 5
Simulation results of 3D total (left panels) and partial (right panels) covariance mapping for the reaction scheme summarized in Eq. (19). The covariance values are multiplied by $M_{shots}$ . The laser pulse intensity $I$ was assumed to fluctuate from shot to shot by 100% with an average ionization rate of 5 shot $^{- 1}$ , and the number of detected products was assumed to depend linearly on the intensity variations. Maps (a) and (c) present slices of total 3D covariance mapping corresponding to products A and B, respectively, and maps (b) and (d) present similar slices but obtained with the 3D partial covariance given by Eq. (28).
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Figure 6
Simulated 2D partial (a) and contingent (b) covariance maps for the reaction scheme presented in Eq. (5), assuming that the ionization cross section is proportional to the fourth power of the light intensity. The light intensity fluctuations are assumed to correspond to an ionization rate which is uniformly distributed between 0 and 10 shots $^{- 1}$ . The covariance values are multiplied by $M_{shots}$ .
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