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<ul><li><p>Nuclear Instruments and Methods in Physics Research A 502 (2003) 792794</p><p>What do we want, what do we have, what can we do?(unfolding in LHC era)</p><p>V.B. Anikeev</p><p>Institute for High Energy Physics, 1 Pobeda Street, Protvino, Moscow Region, Russia</p><p>Abstract</p><p>Nonparametric estimation of measured distribution is essential before extracting the parameters even if a model</p><p>exists. We consider the model of measurements in the general form Ks s0: High-precision measurements in the LHCera require that errors should be allowed for in both the operator K and the right-hand side s0: Several algorithms tosolve the problem are discussed and the existing software is reviewed.</p><p>r 2003 Elsevier Science B.V. All rights reserved.</p><p>PACS: 02.30.Zz; 02.60.Cb</p><p>Keywords: Inverse problem; Unfolding; Mathematical simulation</p><p>We study the situation in terms of the math-ematical simulation [1]. This method includes thefollowing three steps: (i) mathematical model, i.e.,the integral equation, which can be written to bethe operator equation Ks s0; (ii) the numericalmethod for solving the above equation; (iii) thecomputer program that estimates the solution andthe errors for given data. In the rst step, wechoose the model on the basis of the priorinformation we have. We can apply the powerand abstractions of mathematical analysis (innitevalues, convergence, etc.) and, basing on them, wecan study the properties of the solution, i.e.,existence of a solution in some sense or theinuence of the data and operator errors on thesolution. In the second step, we have to construct anumerical method because we can hardly nd theanalytical form of the solution. The numerical</p><p>methods are based on approximation (reductionof dimension, discretization, etc.) or on iterations(or superposition of both). This is the sourceof additional errors in the solution (errors ofthe method). In the last step, we must takeinto account that a nite set of numbers inthe computer is at our disposal. This is thesource of calculation difculties (over/underow)and of some additional errors in the solutioncaused by the computer arithmetics. Theabove three steps are interrelated and are initerative development for the problem underconsideration.In the initial phase at low luminosity, the</p><p>experiment will function as a factory for QCDprocesses, heavy avor and gauge bosons produc-tion. This will allow a large number of precisionmeasurements in the early stages of the experi-ment. [ATLAS TDR 15, CERN/LHCC 99-15]We want to extract from experimental data the</p><p>estimates of distributions in the form of the vectorE-mail address: anikeev@mx.ihep.su (V.B. Anikeev).</p><p>0168-9002/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved.</p><p>doi:10.1016/S0168-9002(03)00585-0</p></li><li><p>of values at some points #sxi and their statisticalerrors and bias. If we have the parametric modelsx; a of such a distribution, we want to estimate #aparameters of the model that t the data best insome sense and the errors of the parameters. Theprecision requirements depend on the physicalproblem, and the relative precision varies within awide range. In our opinion, the nonparametricestimation of the distribution is an essential step ofdata analysis even if a parametric model exists.This helps at least to choose a parametric modelwith open eyes and can reveal some new effects.The measurement model [2] (Fredholm integralequation) is</p><p>s0x0 Z NN</p><p>E0x0Kx0; xExsx dx;</p><p>where x is(are) some kinematical variable(s),continuous, discrete, or mixed, x xtrue; x0 xmeas; Kx0; xfix (kernel of equation)resolutionfunction (is unimodal, has properties of p.d.f., i.e.Kx0; xfixX0;</p><p>RNN Kx</p><p>0; xfix dx0 1:) Innitelimits mean an interval of possible variable values.This way, the true distribution sx is subsequentlyaffected by Ex effectiveness and acceptance, bynite resolution Kx0; x and by E0x0; the trigger,cuts, and effectiveness of the data processingroutine. Kx0; x is known from Monte Carlosimulation, and the high-precision measurementsrequire knowledge of the kernel and effectivenesserrors. We cannot guarantee such properties ofkernel as Kx0; x Kx0 x; K jx0 xj;Kx; x0:One can rewrite the model in the operator form:</p><p>s0 Ks: If we use binning technique, the expectednumber of events in the bin n is</p><p>%Yn Zbinn</p><p>s0x0 dx0 Z NN</p><p>Anxsx dx:</p><p>It follows from the model that our nonpara-metric #sx or parametric #sx; a estimate cancontain an invisible component, i.e., a compo-nent orthogonal to the kernel or to the set ofapparatus functions Anx (i.e., high-frequencyoscillations). Kx0; x; Ex are xed and denedby the existing setup. We can affect only E 0x0(cuts selection), thus we can control sharpness(close to d-function) of apparatus functions Anx:</p><p>After discretization</p><p>%YnEX</p><p>i</p><p>Anxi dxi X</p><p>i</p><p>Ani dxi;</p><p>but the quadrature formulae are preferablebecause they provide higher precision. Binningin x and x0 is subject to discussion (i.e., equicon-tent in x; and proportional to resolution in x0).Existence and uniqueness of solution sx forthe exact model is provided by physics, whichis not true for the model with errors. So, we haveto introduce some generalized solution as aminimum of some functional (residual s0 Ks).Choice of the norm jjs0 Ksjj is also subject todiscussion.Generally, this model is ill-posed, i.e., small</p><p>variations in the right-hand side s0 or in theoperator K can cause unpredictable variations inthe solution s: The LHC era specics are non-negligible errors in the kernel. There is a theory forsolution of ill-posed problems including the casewhen errors are in both operator and the right-hand side, see Ref. [3]. There are algorithms forsolving this kind of problems [4]. In our previouswork [2], we analyzed the statistical errors and bias(as a consequence of regularization) for thenonparametric estimation (1D problem) causedby errors in the right-hand side. We expandthis analysis to include errors in operator.Anyway, we have to work with matrices, namely,to dene eigenvalues and estimate the conditionnumber (is the problem ill-posed?), matrixmultiplication for parameter estimation or iter-ative regularization, invert the Gram matrix forlinear estimates, etc. Computer arithmetic is thesource of additional errors in matrix computa-tions. The works of Godunovs school [5] solvethis problem.This way, we have components for new software</p><p>for model-independent unfolding of distributionswhen the kernel has errors. Now we are studyingthe new software package. The software underconstruction will meet the following requirements:(i) correct estimate of measured distributionrequires that the errors in both the right side s0</p><p>and the operator K allowed for, (ii) we have tobear in mind the invisible component, (iii) we haveto see if the problem is ill-conditioned, (iv) we</p><p>V.B. Anikeev / Nuclear Instruments and Methods in Physics Research A 502 (2003) 792794 793</p></li><li><p>must have estimates of statistical errors and of biasfor the estimate obtained, (v) to obtain reliableresults, we implement several regularization algo-rithms (linear algorithms like spectral window,Tikhonov, iterations, etc., and nonlinear ones likemaximum likelihood, maximum entropy), (vi)several methods to choose the regularizationparameter will be used.</p><p>Our thanks to F. Gianotti (CERN, ATLAS) forher interest in the problem.</p><p>References</p><p>[1] A.A. Samarsky, A.P. Mikhailov, Mathematical Simulation:</p><p>Ideas. Methods. Examples, Nauka, Moscow, 1997.[2] V.B. Anikeev, V.P. Zhigunov, Phys. Part. Nucl. 24(4) (1993)</p><p>9891055; http://www1.jinr.ru/Pepan/Pepan index.html.</p><p>[3] A.N. Tikhonov, A.S. Leonov, A.G. Yagola, Nonlinear Ill-</p><p>posed Problems, Nauka, Moscow, 1995.</p><p>[4] A.F. Verlan, V.S. Sizikov, Integral Equations: Methods,</p><p>Algorithms, Programs, Naukova dumka, Kiev, 1986.</p><p>[5] E.A. Biberdorf, N.I. Popova, Solution of the Linear</p><p>Systems with the Guaranteed Estimate of the Results</p><p>Accuracy, Part 1, preprint BINP 99-49.</p><p>V.B. Anikeev / Nuclear Instruments and Methods in Physics Research A 502 (2003) 792794794</p><p>What do we want, what do we have, what can we do? (unfolding in LHC era)References</p></li></ul>