Convergence and error propagation results on a linear iterative unfolding method. (English) Zbl 1369.47012

László, András

SIAM/ASA J. Uncertain. Quantif. 4, 1345-1371 (2016).

Summary: Unfolding problems often arise in the context of statistical data analysis. Such problematics occur when the probability distribution of a physical quantity is to be measured, but it is randomized (smeared) by some well-understood process, such as a nonideal detector response or a well-described physical phenomenon. In such case it is said that the original probability distribution of interest is folded by a known response function. The reconstruction of the original probability distribution from the measured one is called unfolding. That technically involves evaluation of the nonbounded inverse of an integral operator over the space of \(L^1\) functions, which is known to be an ill-posed problem. For the pertinent regularized operator inversion, we propose a linear iterative formula and provide proof of convergence in a probability theory context. Furthermore, we provide formulae for error estimates at finite iteration stopping order which are of utmost importance in practical applications: the approximation error, the propagated statistical error, and the propagated systematic error can be quantified. The arguments are based on the Riesz-Thorin theorem mapping the original \(L^1\) problem to \(L^2\) space, and subsequent application of ordinary \(L^2\) spectral theory of operators. A library implementation in C of the algorithm along with corresponding error propagation is also provided. A numerical example also illustrates the method in operation.

MSC:

47A52	Linear operators and ill-posed problems, regularization
47N30	Applications of operator theory in probability theory and statistics
65J10	Numerical solutions to equations with linear operators
62-07	Data analysis (statistics) (MSC2010)
65C60	Computational problems in statistics (MSC2010)

Keywords:

unfolding; convergence; error propagation; probability theory; statistics; functional analysis; Riesz-Thorin theorem

Software:

RooUnfold; GSL; Libunfold

References:

[1]	A. László, A linear iterative unfolding method, J. Phys. Conf. Ser., 368 (2012), 012043.
[2]	A. László, A robust iterative unfolding method for signal processing, J. Phys. A, 39 (2006), 13621. · Zbl 1107.94004
[3]	G. Cowan, Proceedings of the Conference on Advanced Statistical Techniques in Particle Physics, Durham, UK, 2002.
[4]	V. Blobel, Unfolding for HEP experiments, presented at DESY Computing Seminar, (2008).
[5]	G. Bohm and G. Zech, Introduction to Statistics and Data Analysis for Physicists, Verlag Deutsches Elektronen-Synchrotron, Hamburg, 2010.
[6]	M. Kuusela and V. M. Panaretos, Statistical unfolding of elementary particle spectra: Empirical Bayes estimation and bias-corrected uncertainty quantification, Ann. Appl. Stat., 9 (2015), pp. 1671–1705. · Zbl 1454.62533
[7]	M. Kuusela and P. B. Stark, Shape-Constrained Uncertainty Quantification in Unfolding Steeply Falling Elementary Particle Spectra, preprint, , 2015. · Zbl 1380.62274
[8]	H. P. Dembinski and M. Roth, An algorithm for automatic unfolding of one-dimensional data distributions, Nucl. Instr. Meth. A, 729 (2013), pp. 410–416.
[9]	I. Dattner, A. Goldenshluger, and A. Juditsky, On deconvolution of distribution functions, Ann. Statist., 39 (2011), pp. 2477–2501, . · Zbl 1232.62056
[10]	I. Dattner, M. Reiß, and M. Trabs, Adaptive quantile estimation in deconvolution with unknown error distribution, Bernoulli, 22 (2016), pp. 143–192. · Zbl 1388.62095
[11]	J. Fan, On the optimal rates of convergence for nonparametric deconvolution problems, Ann. Statist., 19 (1991), pp. 1257–1272. · Zbl 0729.62033
[12]	C. H. Hesse, Iterative density estimation from contaminated observations, Metrika 64 (2006), pp. 151–165. · Zbl 1100.62042
[13]	F. Comte and C. Lacour, Anisotropic adaptive kernel deconvolution, Ann. Inst. H. Poincaré Probab. Stat., 49 (2013), pp. 569–609. · Zbl 1348.62121
[14]	M. C. Liu and R. L. Taylor, A consistent nonparametric density estimator for the deconvolution problem, Canad. J. Statist., 17 (1989), pp. 427–438. · Zbl 0694.62017
[15]	L. A. Stefanski and R. J. Carol, Deconvoluting kernel density estimators, Statistics, 21 (1990), pp. 169–184. · Zbl 0697.62035
[16]	J. Kalifa and B. Rouge, Deconvolution by thresholding in mirror wavelet bases, IEEE Trans. Image Process., 12 (2003), pp. 446–457.
[17]	A. Hoecker and V. Kartvelishvili, SVD Approach to data unfolding, Nuclear Instr. Meth. A, 372 (1996), pp. 469–481.
[18]	G. D’Agostini, A multidimensional unfolding method based on Bayes’ theorem, Nuclear Instr. Meth. A, 362 (1995), pp. 487–498.
[19]	G. Zech, Iterative unfolding with the Richardson-Lucy algorithm, Nuclear Instr. Meth. A, 716 (2013), pp. 1–9.
[20]	C. ALT et al., High transverse momentum Hadron spectra at \(\sqrt{s_{{}_{NN}}}=17.3\,\text{GeV}\), in Pb+Pb and p+p Collisions, Phys. Rev. C, 77 (2008), 034906.
[21]	W. H. Richardson, Bayesian-based iterative method of image restoration, J. Opt. Soc. Amer. A, 62 (1972), pp. 55–59.
[22]	L. B. Lucy, An iterative technique for the rectification of observed distributions, Astronomi. J., 79 (1974), p. 745.
[23]	L. A. Shepp and Y. Vardi, Maximum likelihood reconstruction for emission tomography, IEEE Trans. Med. Imag., 1 (1982), pp. 113-122.
[24]	A. Kondor, Method of convergent weights – An iterative procedure for solving Fredholm’s integral equations of the first kind, Nuclear Instr. Meth., 216 (1983), pp. 177–181.
[25]	H. N. Mülthei and B. Schorr, On an iterative method for a class of integral equations of the first kind, Math. Methods Appl. Sci., 9 (1987). · Zbl 0628.65130
[26]	H. N. Mülthei and B. Schorr, On an iterative method for the unfolding of spectra, Nuclear Instr. Meth. A, 257 (1987), pp. 371–377.
[27]	P. D. Lax, Functional Analysis, Chichester, Wiley-Interscience, 2002, · Zbl 1009.47001
[28]	W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973. · Zbl 0253.46001
[29]	G. Arfken, Fourier convolution theorem, in Mathematical Methods for Physicists, 7th ed., Elsevier, Amsterdam, 2013.
[30]	P. Bracewell, Convolution theorem, in The Fourier Transform and Its Applications, 3rd ed., McGraw-Hill, New York, 1999, pp. 108–112.
[31]	L. Landweber, An iteration formula for Fredholm integral equations of the first kind, Amer. J. Math., 73 (1951), pp. 615–624. · Zbl 0043.10602
[32]	G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Wiley-Interscience, New York, 1999. · Zbl 0924.28001
[33]	N. Dinculeanu, Vector Measures, Elsevier, New York, 1967.
[34]	A. László, The Libunfold Package, (2011).
[35]	V. Khachatryan et al., Transverse-momentum and pseudorapidity distributions of charged Hadrons in pp collisions at \(\sqrt{s}=7\) TeV, Phys. Rev. Lett., 105 (2010), 022002.
[36]	E. Yazgan, The CMS barrel calorimeter response to particle beams from 2 to 350 GeV/c, J. Phys. Conf. Ser., 160 (2009), 012056.
[37]	T. Adye et al., The ROOUnfold Package, .
[38]	GNU Scientific Library, .