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18
The FourierSeries Method For Inverting Transforms Of Probability Distributions
, 1991
"... This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remar ..."
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Cited by 197 (52 self)
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This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy to use, requiring programs of less than fifty lines. The Fourierseries method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourierseries method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...
Practical Aspects of the MoreauYosida Regularization I: Theoretical Properties
, 1994
"... When computing the infimal convolution of a convex function f with the squared norm, one obtains the socalled MoreauYosida regularization of f . Among other things, this function has a Lipschitzian gradient. We investigate some more of its properties, relevant for optimization. Our main result co ..."
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Cited by 62 (2 self)
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When computing the infimal convolution of a convex function f with the squared norm, one obtains the socalled MoreauYosida regularization of f . Among other things, this function has a Lipschitzian gradient. We investigate some more of its properties, relevant for optimization. Our main result concerns secondorder differentiability and is as follows. Under assumptions that are quite reasonable in optimization, the MoreauYosida is twice diffferentiable if and only if f is twice differentiable as well. In the course of our development, we give some results of general interest in convex analysis. In particular, we establish primaldual relationship between the remainder terms in the firstorder development of a convex function and its conjugate.
Regularization methods for semidefinite programming
 SIAM JOURNAL ON OPTIMIZATION
, 2009
"... We introduce a new class of algorithms for solving linear semidefinite programming (SDP) problems. Our approach is based on classical tools from convex optimization such as quadratic regularization and augmented Lagrangian techniques. We study the theoretical properties and we show that practical im ..."
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Cited by 43 (5 self)
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We introduce a new class of algorithms for solving linear semidefinite programming (SDP) problems. Our approach is based on classical tools from convex optimization such as quadratic regularization and augmented Lagrangian techniques. We study the theoretical properties and we show that practical implementations behave very well on some instances of SDP having a large number of constraints. We also show that the “boundary point method” from [PRW06] is an instance of this class.
Petri Nets with k Simultaneously Enabled Generally Distributed Timed Transistions
 Performance Evaluation
, 1998
"... Stochastic Petri nets have been used to analyze the performance and reliability of complex systems comprising concurrency and synchronization. Various extensions have been proposed in literature in order to broaden their field of application to an increasingly larger range of real situations. In thi ..."
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Cited by 13 (3 self)
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Stochastic Petri nets have been used to analyze the performance and reliability of complex systems comprising concurrency and synchronization. Various extensions have been proposed in literature in order to broaden their field of application to an increasingly larger range of real situations. In this paper we extend the class of Markov Regenerative Stochastic Petri Nets* (MRSPN*s), removing the restriction that at most one generally distributed timed transition can be enabled in any marking. This new class of Petri Nets, which we call Concurrent Generalized Petri Nets (CGPNs) allows simultaneous enabling of immediate, exponentially and generally distributed timed transitions, under the hypothesis that the latter are all enabled at the same instant. The stochastic process underlying a CGPN is shown to be still an MRGP. We evaluate the kernel distribution of the underlying MRGP and define the steps required to generate it automatically. The methodology described is used to assess the beh...
Variance Estimation and Ranking of Gaussian Mixture Distributions in Target Tracking Applications
 Proc. IEEE Conf. Decision and Control 2002, Las Vegas, NV
, 2002
"... Variance estimation and ranking methods are developed for stochastic processes modeled by Gaussian mbcture distributions. It is shown that the variance estimate from a Gaussian mixture distribution has the same properties as a variance estimate from a single Gaussian distribution based on a reduced ..."
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Cited by 6 (2 self)
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Variance estimation and ranking methods are developed for stochastic processes modeled by Gaussian mbcture distributions. It is shown that the variance estimate from a Gaussian mixture distribution has the same properties as a variance estimate from a single Gaussian distribution based on a reduced number of samples. Hence, well known tools of variance estimation and ranking of single Gaussian distributions can be applied to Gaussian mixture distributions. As an application example, optimization of sensor processing order in the sequential multitarget multisensor joint probabilistic data associ ation (MSJPDA) algorithm is presented.
Variance Estimation and Ranking of Target Tracking Position Errors Modeled Using Gaussian Mixture Distributions
, 2002
"... In this pap e#) Gaussian mixture# are use# to mode#the distribution of positione#tio intarge# tracking algorithms. A gre#]6 e## e##]649) maximization (EM)algorithm isconstructe# to e#)83W46 parame#D44 of a kcompone# t Gaussianmixture base# on a sample se# produce# by a targe# tracking simulator. Fu ..."
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Cited by 4 (0 self)
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In this pap e#) Gaussian mixture# are use# to mode#the distribution of positione#tio intarge# tracking algorithms. A gre#]6 e## e##]649) maximization (EM)algorithm isconstructe# to e#)83W46 parame#D44 of a kcompone# t Gaussianmixture base# on a sample se# produce# by a targe# tracking simulator. Furthe#])D3fi it is shown thatthe distribution ofthe variance e#ance#) base# on n datasample# from a ze#9fiWfi)D Gaussianmixture distribution hasthe same prope#op)4 asthe distribution ofthe variance e#ance#) base# on are#fi62[ numb e# #nofsample# from aze#94]])D single Gaussian distribution. As are#32]8 the we ll known tools of variance e#nce#)D3] and ranking base# on prope#p)23 F distributions can be applie# to data originating from a ze#[W8)D] Gaussianmixture simply by taking into accountthe re##483)D #.The re#e#fi9fi) factor is found inclose# form for large n.The de# e lope d approach can be applie# to pe#44fi)D]34 e valuation and ranking of di#e#66 t targe# tracking algorithms usingthe rootme#8 square positione#tio asthe pe#[W69)D]2 me#]626 As an applicatione#pplica optimization ofse#8[4 proce#)94] orde# inthe se##][ tial multitarge# multise#D29 joint probabilistic data association (MSJPDA)algorithm ispre#9] te## Portions of this paper were submitted to the Conference on Decision and Control, to be held in Las Vegas, NV, December 2002. 1
Projection methods in conic optimization
"... Projection onto semidefinite positive matrices Consider the space Sn of symmetric nbyn matrices, equipped with the norm associated to the usual inner product 〈X,Y 〉 = ..."
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Cited by 4 (0 self)
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Projection onto semidefinite positive matrices Consider the space Sn of symmetric nbyn matrices, equipped with the norm associated to the usual inner product 〈X,Y 〉 =
Ranking and Optimization of Target Tracking Algorithms
, 2002
"... this dissertation isdev eloping a ranking and selection technique fore#ectiv ely comparing tracking algorithms that maintain target tracks after they hav e been initialized ..."
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Cited by 3 (1 self)
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this dissertation isdev eloping a ranking and selection technique fore#ectiv ely comparing tracking algorithms that maintain target tracks after they hav e been initialized
Data Fitting and Experimental Design in Dynamical Systems with User’s Guide
, 2009
"... EASYFITModelDesign is an interactive software system to identify parameters in explicit model functions, dynamical systems of equations, Laplace transformations, systems of ordinary differential equations, differential algebraic equations, or systems of onedimensional timedependent partial diffe ..."
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EASYFITModelDesign is an interactive software system to identify parameters in explicit model functions, dynamical systems of equations, Laplace transformations, systems of ordinary differential equations, differential algebraic equations, or systems of onedimensional timedependent partial differential equations with or without algebraic equations. Proceeding from given experimental data, i.e., observation times and measurements, the minimum least squares distance of measured data from a fitting criterion is computed, that depends on the solution of the dynamical system. Moreover, it is possible to predetermine an optimal experimental design by fixing the model parameters. Additional design parameters, for example initial concentrations or input feeds, are used to minimize the size of confidence intervals. Weight optimization helps to identify relevant time values where experiments can be taken. The mathematical background of the numerical algorithms is described in Schittkowski [438] in form of a comprehensive textbook. Also, the outcome of numerical comparative performance evaluations is found there, together with a chapter about numerical pitfalls, testing the validity of models, and a collection of 12 reallife case studies. Most of the case
Appendix 3 Ill  conditioned linear systems. Numerical solution approach A.3.1 Ill  conditioned linear systems
"... a linear system is much more valuable if it is a accompanied by an estimate of the error. The most often used approach in the error analysis of linear systems is inverse error analysis, which has been developed mainly by Wilkinson. Another approach is interval analysis, as developed by Hansen [ 5, 6 ..."
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a linear system is much more valuable if it is a accompanied by an estimate of the error. The most often used approach in the error analysis of linear systems is inverse error analysis, which has been developed mainly by Wilkinson. Another approach is interval analysis, as developed by Hansen [ 5, 6 ] and Moore [ 7 ]. In the latter method, the solution consists of an interval for each of the element of x, and the algorithm is designed so that the interval is guaranteed to contain the true value of that element. In this Appendix the solution of an ill  conditioned systems will be treated also as a problem of interval estimation and incorporate the a priori information about the solution into the problem as constraints. Then, it will be shown how the resulting constrained estimation problem can be reduced to a problem in mathematical programming which, in turn, can be solved to give confidence intervals for the true solution of the system. A.3.2 Numerical solutions An excellent surve