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157
NonUniform Random Variate Generation
, 1986
"... Abstract. This is a survey of the main methods in nonuniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various ..."
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Cited by 620 (21 self)
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Abstract. This is a survey of the main methods in nonuniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods.
Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization
, 2000
"... ..."
A note on the stochastic realization problem
 Hemisphere Publishing Corporation
, 1976
"... Abstract. Given a mean square continuous stochastic vector process y with stationary increments and a rational spectral density such that (oo) is finite and nonsingular, consider the problem of finding all minimal (wide sense) Markov representations (stochastic realizations) of y. All such realizati ..."
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Cited by 98 (23 self)
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Abstract. Given a mean square continuous stochastic vector process y with stationary increments and a rational spectral density such that (oo) is finite and nonsingular, consider the problem of finding all minimal (wide sense) Markov representations (stochastic realizations) of y. All such realizations are characterized and classified with respect to deterministic as well as probabilistic properties. It is shown that only certain realizations (internal stochastic realizations) can be determined from the given output process y. All others (external stochastic realizations)require that the probability space be extended with an exogeneous random component. A complete characterization of the sets of internal and external stochastic realizations is provided. It is shown that the state process of any internal stochastic realization can be expressed in terms of two steadystate KalmanBucy filters, one evolving forward in time over the infinite past and one backward over the infinite future. An algorithm is presented which generates families Of external realizations defined on the same probability space and totally ordered with respect to state covariances. 1. Introduction. One
The Classical Moment Problem as a SelfAdjoint Finite Difference Operator
, 1998
"... This is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators. Among the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix and convergence of Pade approximants appears as the strong r ..."
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Cited by 89 (7 self)
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This is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators. Among the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix and convergence of Pade approximants appears as the strong resolvent convergence of finite matrix approximations to a Jacobi matrix. As a bonus of this, we obtain new results on the convergence of certain Pade approximants for series of Hamburger.
Optimal inequalities in probability theory: A convex optimization approach
 SIAM Journal of Optimization
"... Abstract. We propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for P (X ∈ S), for a set S defined by polynomial inequalities and a random vector X defined on Ω ⊆Rn that has a given collection of up to kthorder moments. In the univariate case, we provi ..."
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Cited by 66 (10 self)
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Abstract. We propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for P (X ∈ S), for a set S defined by polynomial inequalities and a random vector X defined on Ω ⊆Rn that has a given collection of up to kthorder moments. In the univariate case, we provide optimal bounds on P (X ∈ S), when the first k moments of X are given, as the solution of a semidefinite optimization problem in k + 1 dimensions. In the multivariate case, if the sets S and Ω are given by polynomial inequalities, we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n, for fixed k. We characterize the complexity of the problem of deriving tight moment inequalities. We show that it is NPhard to find tight bounds for k ≥ 4 and Ω = Rn and for k ≥ 2 and Ω = Rn +, when the data in the problem is rational. For k =1andΩ=Rn + we show that we can find tight upper bounds by solving n convex optimization problems when the set S is convex, and we provide a polynomial time algorithm when S and Ω are unions of convex sets, over which linear functions can be optimized efficiently. For the case k =2andΩ=Rn, we present an efficient algorithm for finding tight bounds when S is a union of convex sets, over which convex quadratic functions can be optimized efficiently. Key words. optimization probability bounds, Chebyshev inequalities, semidefinite optimization, convex
Sums of squares, moment matrices and optimization over polynomials
, 2008
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NPhard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory ..."
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Cited by 62 (9 self)
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We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NPhard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main properties: asymptotic/finite convergence, optimality certificate, and extraction of global optimum solutions. We review the mathematical tools underlying these properties, in particular, some sums of squares representation results for positive polynomials, some results about moment matrices (in particular, of Curto and Fialkow), and the algebraic eigenvalue method for solving zerodimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.
Superfast solution of real positive definite Toeplitz systems
 SIAM J. Matrix Anal. Appl
, 1988
"... Abstract. We describe an implementation of the generalized Schur algorithm for the superfast solution of real positive definite Toeplitz systems of order n + 1, where n = 2ν. Our implementation uses the splitradix fast Fourier transform algorithms for real data of Duhamel. We are able to obtain the ..."
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Cited by 54 (1 self)
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Abstract. We describe an implementation of the generalized Schur algorithm for the superfast solution of real positive definite Toeplitz systems of order n + 1, where n = 2ν. Our implementation uses the splitradix fast Fourier transform algorithms for real data of Duhamel. We are able to obtain the nth Szegő polynomial using fewer than 8n log2 2 n real arithmetic operations without explicit use of the bitreversal permutation. Since Levinson’s algorithm requires slightly more than 2n2 operations to obtain this polynomial, we achieve crossover with Levinson’s algorithm at n = 256. Key words. Toeplitz matrix, Schur’s algorithm, splitradix Fast Fourier Transform
A Convex Optimization Approach to the Rational Covariance Extension Problem
 SIAM J. Control Optim
, 1999
"... In this paper we present a convex optimization problem for solving the rational covariance extension problem. Given a partial covariance sequence and the desired zeros of the modeling filter, the poles are uniquely determined from the unique minimum of the corresponding optimization problem. In this ..."
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Cited by 50 (24 self)
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In this paper we present a convex optimization problem for solving the rational covariance extension problem. Given a partial covariance sequence and the desired zeros of the modeling filter, the poles are uniquely determined from the unique minimum of the corresponding optimization problem. In this way we obtain an algorithm for solving the covariance extension problem, as well as a constructive proof of Georgiou's seminal existence result and his conjecture, a stronger version of which we have resolved in [7]. K3 words. rational covariance extension, partial stochastic realization, trigonometric moment problem, spectral estimation, speech processing, stochastic modeling AMS subject classifications.30ERR 60G35, 62M15, 93A30,93E0 1.
Canonical Correlation Analysis, Approximate Covariance Extension, and Identification of Stationary Time Series
 Automatica
, 1996
"... In this paper we analyze a class of statespace identification algorithms for timeseries, based on canonical correlation analysis, in the light of recent results on stochastic systems theory. In principle, these so called "subspace methods" can be described as covariance estimation followed by stoc ..."
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Cited by 37 (17 self)
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In this paper we analyze a class of statespace identification algorithms for timeseries, based on canonical correlation analysis, in the light of recent results on stochastic systems theory. In principle, these so called "subspace methods" can be described as covariance estimation followed by stochastic realization. The methods o#er the major advantage of converting the nonlinear parameter estimation phase in traditional ARMA models identification into the solution of a Riccati equation but introduce at the same time some nontrivial mathematical problems related to positivity. The reason for this is that an essential part of the problem is equivalent to the wellknown rational covariance extension problem. Therefore the usual deterministic arguments based on factorization of a Hankel matrix are not valid for generic data, something that is habitually overlooked in the literature. We demonstrate that there is no guarantee that several popular identification procedures based on the same principle will not fail to produce a positive extension, unless some rather stringent assumptions are made which, in general, are not explicitly reported. In this paper the statistical problem of stochastic modeling from estimated covariances is phrased in the geometric language of stochastic realization theory. We review the basic ideas of stochastic realization theory in the context of identification, discuss the concept of stochastic balancing and of stochastic model reduction by principal subsystem truncation. The model reduction method of Desai and Pal, based on truncated balanced stochastic realizations, is partially justified, showing that the reduced system structure has a positive covariance sequence but is in general not balanced. As a byproduct of this analysis we obtain a t...
GloptiPoly 3: Moments, Optimization and Semidefinite Programming
 Soft. PROGRAMMING FOR NPLAYER GAMES 21
, 2007
"... We describe a major update of our Matlab freeware GloptiPoly for parsing generalized problems of moments and solving them numerically with semidefinite programming. 1 What is GloptiPoly? Gloptipoly 3 is intended to solve, or at least approximate, the Generalized Problem of Moments (GPM), an infinite ..."
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Cited by 34 (18 self)
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We describe a major update of our Matlab freeware GloptiPoly for parsing generalized problems of moments and solving them numerically with semidefinite programming. 1 What is GloptiPoly? Gloptipoly 3 is intended to solve, or at least approximate, the Generalized Problem of Moments (GPM), an infinitedimensional optimization problem which can be viewed as an extension of the classical problem of moments [8]. From a theoretical viewpoint, the GPM has developments and impact in various areas of mathematics such as algebra, Fourier analysis, functional analysis, operator theory, probability and statistics, to cite a few. In addition, and despite a rather simple and short formulation, the GPM has a large number of important applications in various fields such as optimization, probability, finance, control, signal processing, chemistry, cristallography, tomography, etc. For an account of various methodologies as well as some of potential applications, the interested reader is referred to [1, 2] and the nice collection of papers [5]. The present version of GloptiPoly 3 can handle moment problems with polynomial data. Many important applications in e.g. optimization, probability, financial economics and 1