Efficient Recursive Methods for Partial Fraction Expansion of General Rational Functions

Ma, Youneng; Yu, Jinhua; Wang, Yuanyuan

doi:10.1155/2014/895036

Cited by 11 publications

(11 citation statements)

References 17 publications

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“…We first change the contour integral to a form which calculates the residue at infinity (eq. 32), which we subsequently write as a finite sum D j using the algorithm in [24]. Because we evaluate the integral with the residue at infinity, we can again allow for a Prior α = 0, which generalizes k mc → k mc + α = k * mc .…”

Section: General Weightsmentioning

confidence: 99%

Probabilistic treatment of the uncertainty from the finite size of weighted Monte Carlo data

Glüsenkamp¹

2018

Eur. Phys. J. Plus

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Parameter estimation in HEP experiments often involves Monte-Carlo simulation to model the experimental response function. A typical application are forward-folding likelihood analyses with reweighting, or time-consuming minimization schemes with a new simulation set for each parameter value. Problematically, the finite size of such Monte Carlo samples carries intrinsic uncertainty that can lead to a substantial bias in parameter estimation if it is neglected and the sample size is small. We introduce a probabilistic treatment of this problem by replacing the usual likelihood functions with novel generalized probability distributions that incorporate the finite statistics via suitable marginalization. These new PDFs are analytic, and can be used to replace the Poisson, multinomial, and sample-based unbinned likelihoods, which covers many use cases in high-energy physics. In the limit of infinite statistics, they reduce to the respective standard probability distributions. In the general case of arbitrary Monte Carlo weights, the expressions involve the fourth Lauricella function FD, for which we find a new finite-sum representation in a certain parameter setting. The result also represents an exact form for Carlson's Dirichlet average Rn with n > 0, and thereby an efficient way to calculate the probability generating function of the Dirichlet-multinomial distribution, the extended divided difference of a monomial, or arbitrary moments of univariate B-splines. We demonstrate the bias reduction of our approach with a typical toy Monte Carlo problem, estimating the normalization of a peak in a falling energy spectrum, and compare the results with previously published methods from the literature.

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Section: General Weightsmentioning

confidence: 99%

Probabilistic treatment of the uncertainty from the finite size of weighted Monte Carlo data

Glüsenkamp¹

2018

Eur. Phys. J. Plus

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“…A more suitable approach for the computation of A i,j and C i,j is the algorithm proposed in [27], which provides recursive expressions for the partial fraction residues of both proper and improper rational functions. According to [27, eq.…”

Section: Discussion On the Computation Of Partial Fraction Expansmentioning

confidence: 99%

A New Approach to the Statistical Analysis of Non-Central Complex Gaussian Quadratic Forms With Applications

Ramírez-Espinosa

Moreno-Pozas

Paris

et al. 2019

IEEE Trans. Veh. Technol.

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This paper proposes a novel approach to the statistical characterization of non-central complex Gaussian quadratic forms (CGQFs). Its key strategy is the generation of an auxiliary random variable (RV) that replaces the original CGQF and converges in distribution to it. The technique is valid for both definite and indefinite CGQFs and yields simple expressions of the probability density function (PDF) and the cumulative distribution function (CDF) that only involve elementary functions. This overcomes a major limitation of previous approaches, where the complexity of the resulting PDF and CDF does 3 or equivalently, the characteristic function, to obtain an approximation of the PDF of CGQFs [13][14][15]. Some works apply different series expansions to the characteristic function to allow such inversion [13,14], while the work by Biyari and Lindsey considers a specific non-central CGQF and inverts its MGF by solving some convolution integrals [15]. All these works present approximations for the PDF of non-central CGQFs in terms of double infinite sum of special functions. In particular, the PDF of positive-definite non-central CGQFs is given in terms of a double infinite sum of modified Bessel functions in [13], while the PDF of indefinite non-central CGQFs is expressed in terms of a double infinite sum of incomplete gamma functions [14] and of a double infinite sum of Laguerre polynomials [15]. Taking into account the limitations of direct inversion methods, since the solutions provided for the PDF and the CDF of non-central CGQFs are difficult to compute and not suitable for any further insightful analysis, very recently, Al-Naffouri et al. presented a different approach. They applied a transformation to the inequality that defines the CDF of non-central CGQFs, yielding a problem in which the well-known saddle point technique allows expressing the CDF as the solution of a differential equation [10].This paper proposes a completely different approach to the statistical analysis of indefinite noncentral CGQFs, which leads to simple expressions that approximate both the PDF and CDF of CGQFs. It is based on appropriately perturbing the non-zero mean components of the Gaussian vectors that build the quadratic form. This yields an auxiliary CGQF, denoted as confluent CGQF, which converges in distribution to the original quadratic form and whose analysis is surprisingly simpler. Specifically, this novel approach offers the following advantages over the recently proposed work in [10] and the other approaches given in the literature [13][14][15]:• The probability functions, namely PDF and CDF, are given as a linear combination of elementary functions (exponentials and powers) in a very tractable form.• Simple closed-form expression for the mean squared error (MSE) between the CGQF and the auxiliary one is provided, allowing the particularization of the auxiliary variable in order

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“…The terms we call f and ϕ are highlighted to indicate the structure used for the main theorem described in [17] (p. 19, 1.4.2). The term ∆ k represents an efficient solution [18] of the contour integral in eq. 74, and is written as…”

Section: A2 General Weightsmentioning

confidence: 99%

A unified perspective on modified Poisson likelihoods for limited Monte Carlo data

Glüsenkamp¹

2020

J. Inst.

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Counting experiments often rely on Monte Carlo simulations for predictions of Poisson expectations. The accompanying uncertainty from the finite Monte Carlo sample size can be incorporated into parameter estimation by modifying the Poisson likelihood. We first review previous Frequentist methods of this type by Barlow et al, Bohm et al, and Chirkin, as well as recently proposed probabilistic methods by the author and Argüelles et al. We show that all these approaches can be understood in a unified way: they all approximate the underlying probability distribution of the sum of weights in a given bin, the compound Poisson distribution (CPD). The Probabilistic methods marginalize the Poisson mean with a distribution that approximates the CPD, while the Frequentist counterparts optimize the same integrand treating the mean as a nuisance parameter. With this viewpoint we can motivate three new probabilistic likelihoods based on generalized gamma-Poisson mixture distributions which we derive in analytic form. Afterwards, we test old and new formulas in different parameter estimation settings consisting of a "background" and "signal" dataset. The probablistic counterpart to the Ansatz by Barlow et al. outperforms all other existing approaches in various scenarios. We further find a surprising outcome: usage of the exact CPD is actually bad for parameter estimation. A continuous approximation performs much better and in principle allows to perform bias-free inference at any level of simulated livetime if the first two moments of the CPD of each dataset are known exactly. Finally, we also discuss the situation where new Monte Carlo simulation is produced for a given parameter choice which leads to fluctuations in the computed likelihood value. Two of the new formulas allow to include this Poisson uncertainty directly into the likelihood which substantially decreases these fluctuations.

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Efficient Recursive Methods for Partial Fraction Expansion of General Rational Functions

Cited by 11 publications

References 17 publications

Probabilistic treatment of the uncertainty from the finite size of weighted Monte Carlo data

Probabilistic treatment of the uncertainty from the finite size of weighted Monte Carlo data

A New Approach to the Statistical Analysis of Non-Central Complex Gaussian Quadratic Forms With Applications

A unified perspective on modified Poisson likelihoods for limited Monte Carlo data

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