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Publication# Uncertainty quantification in unfolding elementary particle spectra at the Large Hadron Collider

Résumé

This thesis studies statistical inference in the high energy physics unfolding problem, which is an ill-posed inverse problem arising in data analysis at the Large Hadron Collider (LHC) at CERN. Any measurement made at the LHC is smeared by the finite resolution of the particle detectors and the goal in unfolding is to use these smeared measurements to make nonparametric inferences about the underlying particle spectrum. Mathematically the problem consists in inferring the intensity function of an indirectly observed Poisson point process. Rigorous uncertainty quantification of the unfolded spectrum is of central importance to particle physicists. The problem is typically solved by first forming a regularized point estimator in the unfolded space and then using the variability of this estimator to form frequentist confidence intervals. Such confidence intervals, however, underestimate the uncertainty, since they neglect the bias that is used to regularize the problem. We demonstrate that, as a result, conventional statistical techniques as well as the methods that are presently used at the LHC yield confidence intervals which may suffer from severe undercoverage in realistic unfolding scenarios. We propose two complementary ways of addressing this issue. The first approach applies to situations where the unfolded spectrum is expected to be a smooth function and consists in using an iterative bias-correction technique for debiasing the unfolded point estimator obtained using a roughness penalty. We demonstrate that basing the uncertainties on the variability of the bias-corrected point estimator provides significantly improved coverage with only a modest increase in the length of the confidence intervals, even when the amount of bias-correction is chosen in a data-driven way. We compare the iterative bias-correction to an alternative debiasing technique based on undersmoothing and find that, in several situations, bias-correction provides shorter confidence intervals than undersmoothing. The new methodology is applied to unfolding the Z boson invariant mass spectrum measured in the CMS experiment at the LHC. The second approach exploits the fact that a significant portion of LHC particle spectra are known to have a steeply falling shape. A physically justified way of regularizing such spectra is to impose shape constraints in the form of positivity, monotonicity and convexity. Moreover, when the shape constraints are applied to an unfolded confidence set, one can regularize the length of the confidence intervals without sacrificing coverage. More specifically, we form shape-constrained confidence intervals by considering all those spectra that satisfy the shape constraints and fit the smeared data within a given confidence level. This enables us to derive regularized unfolded uncertainties which have by construction guaranteed simultaneous finite-sample coverage, provided that the true spectrum satisfies the shape constraints. The uncertainties are conservative, but still usefully tight. The method is demonstrated using simulations designed to mimic unfolding the inclusive jet transverse momentum spectrum at the LHC.

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Mikael Johan Kuusela, Victor Panaretos

We consider the high energy physics unfolding problem where the goal is to estimate the spectrum of elementary particles given observations distorted by the limited resolution of a particle detector. This important statistical inverse problem arising in data analysis at the Large Hadron Collider at CERN consists in estimating the intensity function of an indirectly observed Poisson point process. Unfolding typically proceeds in two steps: one first produces a regularized point estimate of the unknown intensity and then uses the variability of this estimator to form frequentist confidence intervals that quantify the uncertainty of the solution. In this paper, we propose forming the point estimate using empirical Bayes estimation which enables a data-driven choice of the regularization strength through marginal maximum likelihood estimation. Observing that neither Bayesian credible intervals nor standard bootstrap confidence intervals succeed in achieving good frequentist coverage in this problem due to the inherent bias of the regularized point estimate, we introduce an iteratively bias-corrected bootstrap technique for constructing improved confidence intervals. We show using simulations that this enables us to achieve nearly nominal frequentist coverage with only a modest increase in interval length. The proposed methodology is applied to unfolding the Z boson invariant mass spectrum as measured in the CMS experiment at the Large Hadron Collider.

The high energy physics unfolding problem is an important statistical inverse problem in data analysis at the Large Hadron Collider (LHC) at CERN. The goal of unfolding is to make nonparametric inferences about a particle spectrum from measurements smeared by the finite resolution of the particle detectors. Previous unfolding methods use ad hoc discretization and regularization, resulting in confidence intervals that can have significantly lower coverage than their nominal level. Instead of regularizing using a roughness penalty or stopping iterative methods early, we impose physically motivated shape constraints: positivity, monotonicity, and convexity. We quantify the uncertainty by constructing a nonparametric confidence set for the true spectrum, consisting of all those spectra that satisfy the shape constraints and that predict the observations within an appropriately calibrated level of fit. Projecting that set produces simultaneous confidence intervals for all functionals of the spectrum, including averages within bins. The confidence intervals have guaranteed conservative frequentist finite-sample coverage in the important and challenging class of unfolding problems for steeply falling particle spectra. We demonstrate the method using simulations that mimic unfolding the inclusive jet transverse momentum spectrum at the LHC. The shape-constrained intervals provide usefully tight conservative inferences, while the conventional methods suffer from severe undercoverage.

Tomas Masák, Victor Panaretos, Tomas Rubin

Nonparametric inference for functional data over two-dimensional domains entails additional computational and statistical challenges, compared to the one-dimensional case. Separability of the covariance is commonly assumed to address these issues in the densely observed regime. Instead, we consider the sparse regime, where the latent surfaces are observed only at few irregular locations with additive measurement error, and propose an estimator of covariance based on local linear smoothers. Consequently, the assumption of separability reduces the intrinsically four-dimensional smoothing problem into several two-dimensional smoothers and allows the proposed estimator to retain the classical minimax-optimal convergence rate for two-dimensional smoothers. Even when separability fails to hold, imposing it can be still advantageous as a form of regularization. A simulation study reveals a favorable bias-variance tradeoff and massive speed-ups achieved by our approach. Finally, the proposed methodology is used for qualitative analysis of implied volatility surfaces corresponding to call options, and for prediction of the latent surfaces based on information from the entire dataset, allowing for uncertainty quantification. Our cross-validated out-of-sample quantitative results show that the proposed methodology outperforms the common approach of pre-smoothing every implied volatility surface separately. Supplementary materials for this article are available online.