Various models have been proposed in the literature to study non-negative integer-valued time series. In this paper, we study estimators for the generalized Poisson autoregressive process of order 1, a model developed...Various models have been proposed in the literature to study non-negative integer-valued time series. In this paper, we study estimators for the generalized Poisson autoregressive process of order 1, a model developed by Alzaid and Al-Osh [1]. We compare three estimation methods, the methods of moments, quasi-likelihood and conditional maximum likelihood and study their asymptotic properties. To compare the bias of the estimators in small samples, we perform a simulation study for various parameter values. Using the theory of estimating equations, we obtain expressions for the variance-covariance matrices of those three estimators, and we compare their asymptotic efficiency. Finally, we apply the methods derived in the paper to a real time series.展开更多
To study the Poisson theory of the generalized Birkhoff systems, the Lie algebra and the Poisson brackets were used to establish the Poisson theorem. The generalized Poisson condition for the first integral and the ge...To study the Poisson theory of the generalized Birkhoff systems, the Lie algebra and the Poisson brackets were used to establish the Poisson theorem. The generalized Poisson condition for the first integral and the generalized Poisson theorem of the generalized Birkhoff systems are obtained. An example is given to illustrate the application of the result.展开更多
A series of problems in mechanics and physics are governed by the ordinary Poisson equation which demands linearity,isotropy,and material homo- geneity.In this paper a generalization with respect to nonlinearity,aniso...A series of problems in mechanics and physics are governed by the ordinary Poisson equation which demands linearity,isotropy,and material homo- geneity.In this paper a generalization with respect to nonlinearity,anisotropy,and inhomogeneity is made.Starting from the canonical basic equations in the primal and dual formulation respectively we derive systematically the corresponding generalized variational principles;under certain conditions they can be extended to so called complementary extremum principles allowing for global bounds.For simplicity a restriction to two dimensional problems is made,including twice-connected domains.展开更多
Deep learning(DL)has proven to be important for computed tomography(CT)image denoising.However,such models are usually trained under supervision,requiring paired data that may be difficult to obtain in practice.Diffus...Deep learning(DL)has proven to be important for computed tomography(CT)image denoising.However,such models are usually trained under supervision,requiring paired data that may be difficult to obtain in practice.Diffusion models offer unsupervised means of solving a wide range of inverse problems via posterior sampling.In particular,using the estimated unconditional score function of the prior distribution,obtained via unsupervised learning,one can sample from the desired posterior via hijacking and regularization.However,due to the iterative solvers used,the number of function evaluations(NFE)required may be orders of magnitudes larger than for single-step samplers.In this paper,we present a novel image denoising technique for photon-counting CT by extending the unsupervised approach to inverse problem solving to the case of Poisson flow generative models(PFGM)++.By hijacking and regularizing the sampling process we obtain a single-step sampler,that is NFE=1.Our proposed method incorporates posterior sampling using diffusion models as a special case.We demonstrate that the added robustness afforded by the PFGM++framework yields significant performance gains.Our results indicate competitive performance compared to popular supervised,including state-of-the-art diffusion-style models with NFE=1(consistency models),unsupervised,and non-DL-based image denoising techniques,on clinical low-dose CT data and clinical images from a prototype photon-counting CT system developed by GE HealthCare.展开更多
Braverman and Kazhdan(2000)introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula.Their conjectures should imply that quite general Langlands L-functions have ...Braverman and Kazhdan(2000)introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula.Their conjectures should imply that quite general Langlands L-functions have meromorphic continuations and functional equations as predicted by Langlands'functoriality conjecture.As an evidence for their conjectures,Braverman and Kazhdan(2002)considered a setting related to the so-called doubling method in a later paper and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved.The connection between the two papers is made explicit in the work of Li(2018).In this paper,we consider a special case of the setting in Braverman and Kazhdan's later paper and prove a refined Poisson summation formula that eliminates the restrictive assumptions of that paper.Along the way we provide analytic control on the Schwartz space we construct;this analytic control was conjectured to hold(in a slightly different setting)in the work of Braverman and Kazhdan(2002).展开更多
文摘Various models have been proposed in the literature to study non-negative integer-valued time series. In this paper, we study estimators for the generalized Poisson autoregressive process of order 1, a model developed by Alzaid and Al-Osh [1]. We compare three estimation methods, the methods of moments, quasi-likelihood and conditional maximum likelihood and study their asymptotic properties. To compare the bias of the estimators in small samples, we perform a simulation study for various parameter values. Using the theory of estimating equations, we obtain expressions for the variance-covariance matrices of those three estimators, and we compare their asymptotic efficiency. Finally, we apply the methods derived in the paper to a real time series.
文摘To study the Poisson theory of the generalized Birkhoff systems, the Lie algebra and the Poisson brackets were used to establish the Poisson theorem. The generalized Poisson condition for the first integral and the generalized Poisson theorem of the generalized Birkhoff systems are obtained. An example is given to illustrate the application of the result.
文摘A series of problems in mechanics and physics are governed by the ordinary Poisson equation which demands linearity,isotropy,and material homo- geneity.In this paper a generalization with respect to nonlinearity,anisotropy,and inhomogeneity is made.Starting from the canonical basic equations in the primal and dual formulation respectively we derive systematically the corresponding generalized variational principles;under certain conditions they can be extended to so called complementary extremum principles allowing for global bounds.For simplicity a restriction to two dimensional problems is made,including twice-connected domains.
基金supported by MedTechLabs,GE HealthCare,the Swedish Research council,No.2021-05103the Göran Gustafsson foundation,No.2114.
文摘Deep learning(DL)has proven to be important for computed tomography(CT)image denoising.However,such models are usually trained under supervision,requiring paired data that may be difficult to obtain in practice.Diffusion models offer unsupervised means of solving a wide range of inverse problems via posterior sampling.In particular,using the estimated unconditional score function of the prior distribution,obtained via unsupervised learning,one can sample from the desired posterior via hijacking and regularization.However,due to the iterative solvers used,the number of function evaluations(NFE)required may be orders of magnitudes larger than for single-step samplers.In this paper,we present a novel image denoising technique for photon-counting CT by extending the unsupervised approach to inverse problem solving to the case of Poisson flow generative models(PFGM)++.By hijacking and regularizing the sampling process we obtain a single-step sampler,that is NFE=1.Our proposed method incorporates posterior sampling using diffusion models as a special case.We demonstrate that the added robustness afforded by the PFGM++framework yields significant performance gains.Our results indicate competitive performance compared to popular supervised,including state-of-the-art diffusion-style models with NFE=1(consistency models),unsupervised,and non-DL-based image denoising techniques,on clinical low-dose CT data and clinical images from a prototype photon-counting CT system developed by GE HealthCare.
基金supported by the National Science Foundation of the USA(Grant Nos.DMS-1405708 and DMS-1901883)supported by the National Science Foundation of the USA(Grant Nos.DMS-1702218 and DMS-1848058)a start-up fund from the Department of Mathematics at Purdue University。
文摘Braverman and Kazhdan(2000)introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula.Their conjectures should imply that quite general Langlands L-functions have meromorphic continuations and functional equations as predicted by Langlands'functoriality conjecture.As an evidence for their conjectures,Braverman and Kazhdan(2002)considered a setting related to the so-called doubling method in a later paper and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved.The connection between the two papers is made explicit in the work of Li(2018).In this paper,we consider a special case of the setting in Braverman and Kazhdan's later paper and prove a refined Poisson summation formula that eliminates the restrictive assumptions of that paper.Along the way we provide analytic control on the Schwartz space we construct;this analytic control was conjectured to hold(in a slightly different setting)in the work of Braverman and Kazhdan(2002).
