The exact distribution of indefinite quadratic forms in noncentral normal vectors

Provost, Serge B.; Rudiuk, Edmund M.

doi:10.1007/bf00054797

Cited by 43 publications

(18 citation statements)

References 16 publications

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“…is an indefinite quadratic form in Gaussian vectors, which can be expressed as a linear combination of independent noncentral -distributed variables [25]. which is just the conditional probability in (40).…”

Section: Discussionmentioning

confidence: 99%

Binary Symbol Recovery Via <formula formulatype="inline"> <tex Notation="TeX">$\ell_{\infty}$</tex></formula> Minimization in Faster-Than-Nyquist Signaling Systems

Jin

Zou

2014

IEEE Trans. Signal Process.

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The issue of binary symbol detection in fasterthan-Nyquist signaling systems (also known as overcomplete frame-modulated digital transmission systems) is addressed in this paper. Through convex relaxation, the original combinatorial optimization problem is transformed to an minimization problem. Following this idea, we further propose approximation algorithms to efficiently tackle such a convex optimization problem. For noiseless case, the recoverability of minimization is analyzed. It is shown that the binary symbol vector can be completely recovered via minimization if and only if there is a vector in the row space of the transmission matrix located in the same quadrant as . Otherwise, complete reconstruction via minimization is hopeless. At the same time, we give an upper bound to the reconstruction probability. For noisy case, the reconstruction probability is analyzed via the probability distribution function of indefinite quadratic form in Gaussian vectors. Numerical results are provided to study the detection performance of minimization. It is shown that, compared with semidefinite programming algorithm and linear minimum mean-squared error (LMMSE) method, the proposed minimization detection achieves a better performance-complexity trade-off.Index Terms-0-1 quadratic programming, binary, faster than Nyquist signaling, minimization, overcomplete frame.

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Section: Discussionmentioning

confidence: 99%

Binary Symbol Recovery Via <formula formulatype="inline"> <tex Notation="TeX">$\ell_{\infty}$</tex></formula> Minimization in Faster-Than-Nyquist Signaling Systems

Jin

Zou

2014

IEEE Trans. Signal Process.

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“…Provost and Rudiuk [30] derived an explicit formula for both the density (Theorem 2.1, pp. 386) and distribution function (Theorem 3.1, pp.…”

Section: Computation Of the Optimal Policymentioning

confidence: 99%

“…By definition G n := Y n − b, so that G n | X n = i ∼ N d i − b, i and P G T n+1 AG n+1 ≥ g W n (w) | X (n+1) = i in equation (5.5) can be computed explicitly using Theorem 3.1. of Provost and Rudiuk. [30] Using equations (5.4) and (5.5) we now can easily evaluate the quantities p mk , m , c m , m, k ∈ . Suppose at time n , the process is in state I m ∈ , then for M large, W n ≈ l m and we can approximate the transition probabilities…”

Section: Computation Of the Optimal Policymentioning

confidence: 99%

Optimal Control of a Partially Observable Failing System with Costly Multivariate Observations

Kim

Makiš

2012

Stochastic Models

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We model a partially observable deteriorating system subject to random failure. The state process follows an unobservable continuous time homogeneous Markov chain. At equidistant sampling times vector-valued observations having multivariate normal distribution with statedependent mean and covariance matrix are obtained at a positive cost. At each sampling epoch a decision is made either to run the system until the next sampling epoch or to carry out full preventive maintenance, which is assumed to be less costly than corrective maintenance carried out upon system failure. The objective is to determine the optimal control policy that minimizes the long-run expected average cost per unit time. We show that the optimal preventive maintenance region is a convex subset of Euclidean space. We also analyze the practical threestate version of this problem in detail and show that in this case the optimal policy is a control limit policy. An efficient computational algorithm is developed for the three-state problem, illustrated by a numerical example.

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“…Theorem 1 (Provost and Rudiuk 1996) Let Q = W − V = t j =1 l j T j − t+w j =t+1 l j T j where l j are positive real numbers and the T j are independent non-central chi-square variables with α j degrees of freedom and non-centrality parameter d j , j = 1, . .…”

Section: Appendixmentioning

confidence: 99%

“…(20) can be computed using Theorem 3.1 of Provost and Rudiuk (1996) who provided a closed-form expression for the cumulative distribution function of Z n |X n . For completeness, the Theorem 3.1 of Provost and Rudiuk (1996) is stated in Appendix 2.…”

Section: Computational Algorithm In the Smdp Frameworkmentioning

confidence: 99%

A Bayesian model and numerical algorithm for CBM availability maximization

2011

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In this paper, we consider an availability maximization problem for a partially observable system subject to random failure. System deterioration is described by a hidden, continuous-time homogeneous Markov process. While the system is operational, multivariate observations that are stochastically related to the system state are sampled through condition monitoring at discrete time points. The objective is to design an optimal multivariate Bayesian control chart that maximizes the long-run expected average availability per unit time. We have developed an efficient computational algorithm in the semi-Markov decision process (SMDP) framework and showed that the availability maximization problem is equivalent to solving a parameterized system of linear equations. A numerical example is presented to illustrate the effectiveness of our approach, and a comparison with the traditional age-based replacement policy is also provided.

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The exact distribution of indefinite quadratic forms in noncentral normal vectors

Abstract: Exact distribution function, exact density function, indefinite quadratic forms, noncentral chi-square variables, singular normal vectors, Whittaker's function,

Cited by 43 publications

References 16 publications

Binary Symbol Recovery Via <formula formulatype="inline"> <tex Notation="TeX">$\ell_{\infty}$</tex></formula> Minimization in Faster-Than-Nyquist Signaling Systems

Binary Symbol Recovery Via <formula formulatype="inline"> <tex Notation="TeX">$\ell_{\infty}$</tex></formula> Minimization in Faster-Than-Nyquist Signaling Systems

Optimal Control of a Partially Observable Failing System with Costly Multivariate Observations

A Bayesian model and numerical algorithm for CBM availability maximization

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