Mathematical Programming and the Numerical Solution of Linear Equations.

Relles, Daniel A.; Rust, Burt W.; Burrus, W.R.

doi:10.2307/2285796

Cited by 24 publications

(7 citation statements)

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“…We adapt the nomenclature of [15] to fit the terms we have already defined. As a historical note, these intervals have also been described by Rust and O'Leary [18], Rust and Burrus [19], and Tenorio et al [20] for a simpler version of the problem with non-negativity constraints. For ease of description, we consider a statistical model described per eq.…”

Section: One-at-a-time Strict-bounds Intervals (Osb)mentioning

confidence: 89%

“…To address these concerns we explore the use of OSB intervals, described in [15,[18][19][20], and expand on these intervals with PO intervals. The OSB intervals can leverage physical constraints and elegantly handle rank-deficient response matrices, the use of which provides a mitigation technique for handling systematic error.…”

Section: Proposed Methods -One-at-a-time Strict Bounds and Prior-opti...mentioning

confidence: 99%

“…In this situation, 𝑲 cannot be inverted as it always has a non-trivial null space leading to non-identifiability in inferring 𝝀. The approach described in [15,18,19] is designed to handle both of these complications. Briefly, the approach is based on using constrained optimization to directly construct confidence intervals for 𝜃 in a way that allows one to handle the null space of 𝑲 and constraints on 𝝀.…”

Section: Jinst 17 P10013mentioning

confidence: 99%

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Uncertainty quantification for wide-bin unfolding: one-at-a-time strict bounds and prior-optimized confidence intervals

2022

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Unfolding is an ill-posed inverse problem in particle physics aiming to infer a true particle-level spectrum from smeared detector-level data. For computational and practical reasons, these spaces are typically discretized using histograms, and the smearing is modeled through a response matrix corresponding to a discretized smearing kernel of the particle detector. This response matrix depends on the unknown shape of the true spectrum, leading to a fundamental systematic uncertainty in the unfolding problem. To handle the ill-posed nature of the problem, common approaches regularize the problem either directly via methods such as Tikhonov regularization, or implicitly by using wide-bins in the true space that match the resolution of the detector. Unfortunately, both of these methods lead to a non-trivial bias in the unfolded estimator, thereby hampering frequentist coverage guarantees for confidence intervals constructed from these methods. We propose two new approaches to addressing the bias in the wide-bin setting through methods called One-at-a-time Strict Bounds (OSB) and Prior-Optimized (PO) intervals. The OSB intervals are a bin-wise modification of an existing guaranteed-coverage procedure, while the PO intervals are based on a decision-theoretic view of the problem. Importantly, both approaches provide well-calibrated frequentist confidence intervals even in constrained and rank-deficient settings. These methods are built upon a more general answer to the wide-bin bias problem, involving unfolding with fine bins first, followed by constructing confidence intervals for linear functionals of the fine-bin counts. We test and compare these methods to other available methodologies in a wide-bin deconvolution example and a realistic particle physics simulation of unfolding a steeply falling particle spectrum.

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Section: One-at-a-time Strict-bounds Intervals (Osb)mentioning

confidence: 89%

Section: Proposed Methods -One-at-a-time Strict Bounds and Prior-opti...mentioning

confidence: 99%

Section: Jinst 17 P10013mentioning

confidence: 99%

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Uncertainty quantification for wide-bin unfolding: one-at-a-time strict bounds and prior-optimized confidence intervals

2022

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“…In some important ill-posed problems, extra information is available about the solution; for example, we might know that x(ξ ) is non-negative or monotonic. Including such constraints can be important in achieving a good solution using techniques such as penalized maximum likelihood, Bayesian methods, maximum entropy, etc (see for example [2,4,19,24,29]), but we do not consider such constraints here. Neither do we consider the important question of robustness of our methods when our assumptions on the distribution of the error are violated.…”

Section: Systems Of First Kind Integral Equationsmentioning

confidence: 99%

Residual periodograms for choosing regularization parameters for ill-posed problems

Rust

O’Leary

2008

Inverse Problems

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Abstract. Consider an ill-posed problem transformed if necessary so that the errors in the data are independent identically normally distributed with mean zero and variance 1. We survey regularization and parameter selection from a linear algebra and statistics viewpoint and compare the statistical distributions of regularized estimates of the solution and the residual. We discuss methods for choosing a regularization parameter in order to assure that the residual for the model is statistically plausible. Ideally, as proposed by Rust (1998Rust ( , 2000, the results of candidate parameter choices should be evaluated by plotting the resulting residual along with its periodogram and its cumulative periodogram, but sometimes an automated choice is needed. We evaluate a method for choosing the regularization parameter that makes the residuals as close as possible to white noise, using a diagnostic test based on the periodogram. We compare this method with standard techniques such as the discrepancy principle, the L-curve, and generalized cross validation, showing that it performs better on two new test problems as well as a variety of standard problems.

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“…Let us return to the non-negativity constraint. A similar definition to (5) was suggested in [12] for the case where x is constrained to x 0. Define…”

Section: Introductionmentioning

confidence: 99%

Confidence intervals for linear discrete inverse problems with a non-negativity constraint

Fleck

Moses

2007

Inverse Problems

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We consider the problem of constructing confidence intervals for linear functionals β t x of a non-negative vector x given discrete indirect noisy observations y = Kx +. The intervals are defined by the range of the functional over constrained projected contours of a quadratic data misfit function. Rust and O'Leary (1994 J. Comput. Graph. Stat. 3 67) derived similar intervals but we show that some conditions are required for the intervals to have the correct coverage and provide valid intervals for the cases when the conditions are not met. The problem is reduced to two simple cases: for β > 0 a closed formula for the intervals can be used on reduced one-dimensional data without the need to solve an optimization problem. For β ≯ 0 the optimization is reduced to a two-dimensional problem with K = I but a non-diagonal covariance matrix.

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Mathematical Programming and the Numerical Solution of Linear Equations.

Cited by 24 publications

References 0 publications

Uncertainty quantification for wide-bin unfolding: one-at-a-time strict bounds and prior-optimized confidence intervals

Uncertainty quantification for wide-bin unfolding: one-at-a-time strict bounds and prior-optimized confidence intervals

Residual periodograms for choosing regularization parameters for ill-posed problems

Confidence intervals for linear discrete inverse problems with a non-negativity constraint

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