Nonnegative Parameter Research Articles

PHD and CPHD Filtering With Unknown Detection Probability

A priori knowledge of target detection probability is of critical importance in the Gaussian mixture probability hypothesis density (PHD) and cardinalized PHD (CPHD) filters. In addition, these two filters require that the process noise and measurement noise of the state propagated in the recursion be Gaussian. These limitations may restrict the two filters application in real problems. To accommodate unknown target detection probability and nonnegative non-Gaussian parameters, this paper proposes a new implementation based on inverse gamma Gaussian mixtures, introducing a location independent feature whose posterior probability density and likelihood function are nonnegative non-Gaussian inverse gamma and gamma functions to determine detection probability incorporated into the recursions. The derivation of the merging inverse gamma components is also presented to prevent the unbounded increase of mixture components by minimizing the Kullback–Leibler divergence. First, a real heavy-clutter scenario is used to validate the effectiveness of the proposed filters in track initiation and target tracking without known detection probability. Then, simulations are presented to demonstrate that the proposed CPHD and PHD filters can achieve multitarget tracking performance similar to the standard counterparts with known target detection probability, and that they outperform the standard counterparts in scenarios with unknown and dynamically changing detection probability. The robustness of the proposed filters is tested in both real and simulation scenarios. It is also shown that the analytical and empirical computational complexities of the proposed filters are similar to those of their standard counterparts.

A Bayesian finite element model updating with combined normal and lognormal probability distributions using modal measurements

• A Bayesian approach combining normal and lognormal distributions is formulated in details based on maximum a posteriori . • Estimation of uncertainty and probabilistic change/damage is also developed. • Compared with maximum a posteriori based Bayesian updating using normal distribution. • Computational efficiency is compared with Markov Chain Monte Carlo (MCMC) based updating techniques. • With no need of mode matching, such updating remains possible for large structures. The present work is associated with Bayesian finite element (FE) model updating using modal measurements based on maximizing the posterior probability instead of any sampling based approach. Such Bayesian updating framework usually employs normal distribution in updating of parameters, although normal distribution has usual statistical issues while using non-negative parameters. These issues are proposed to be dealt with incorporating lognormal distribution for non-negative parameters. Detailed formulations are carried out for model updating, uncertainty-estimation and probabilistic detection of changes/damages of structural parameters using combined normal-lognormal probability distribution in this Bayesian framework. Normal and lognormal distributions are considered for eigen-system equation and structural (mass and stiffness) parameters respectively, while these two distributions are jointly considered for likelihood function. Important advantages in FE model updating (e.g. utilization of incomplete measured modal data, non-requirement of mode-matching) are also retained in this combined normal-lognormal distribution based proposed FE model updating approach. For demonstrating the efficiency of this proposed approach, a two dimensional truss structure is considered with multiple damage cases. Satisfactory performances are observed in model updating and subsequent probabilistic estimations, however level of performances are found to be weakened with increasing levels in damage scenario (as usual). Moreover, performances of this proposed FE model updating approach are compared with the typical normal distribution based updating approach for those damage cases demonstrating quite similar level of performances. The proposed approach also demonstrates better computational efficiency (achieving higher accuracy in lesser computation time) in comparison with two prominent Markov Chain Monte Carlo (MCMC) techniques (viz. Metropolis-Hastings algorithm and Gibbs sampling).

Degree of Approximation for Bivariate Generalized Bernstein Type Operators

In this paper we study an extension of the bivariate generalized Bernstein operators based on a non-negative real parameters. For these operators we obtain the order of approximation using Peetre’s K-functional, a Voronovskaja type theorem and the degree of approximation by means of the Lipschitz class. Further, we consider the Generalized Boolean Sum operators of generalized Bernstein type and we study the degree of approximation in terms of the mixed modulus of continuity. Finally, we show the comparisons by some illustrative graphics in Maple for the convergence of the operators to certain functions.

Counting all self-avoiding walks on a finite lattice strip of width one and two

In this paper, a closed-form expression for counting all SAWs, irrespective of length, but restricted to the finite lattice strip $\{ -a,\ldots ,0,\ldots ,b\}\times \{0,1\}$, shall be obtained in terms of the non-negative integer parameters $a$ and $b$. In addition, the argument used to prove this result will be extended to establish an enumerating formula for counting all SAWs, irrespective of length, but restricted to the half-finite lattice strip of width two $\{ 0,1,\ldots ,n\}\times \{ 0,1,2\}$, in terms of $n$.

Minimization of transformed $$L_1$$ L 1 penalty: theory, difference of convex function algorithm, and robust application in compressed sensing

We study the minimization problem of a non-convex sparsity promoting penalty function, the transformed $$l_1$$ (TL1), and its application in compressed sensing (CS). The TL1 penalty interpolates $$l_0$$ and $$l_1$$ norms through a nonnegative parameter $$a \in (0,+\infty )$$ , similar to $$l_p$$ with $$p \in (0,1]$$ , and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem, and discuss the exact recovery of $$l_0$$ norm minimal solution based on the null space property (NSP). We then prove the stable recovery of $$l_0$$ norm minimal solution if the sensing matrix A satisfies a restricted isometry property (RIP). We formulated a normalized problem to overcome the lack of scaling property of the TL1 penalty function. For a general sensing matrix A, we show that the support set of a local minimizer corresponds to linearly independent columns of A. Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized constrained and unconstrained problems in CS. The DCATL1 algorithm involves outer and inner loops of iterations, one time matrix inversion, repeated shrinkage operations and matrix-vector multiplications. The inner loop concerns an $$l_1$$ minimization problem on which we employ the Alternating Direction Method of Multipliers. For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value $$a=1$$ , and compare DCATL1 with other CS algorithms on two classes of sensing matrices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). Among existing algorithms, the iterated reweighted least squares method based on $$l_{1/2}$$ norm is the best in sparse recovery for Gaussian matrices, and the DCA algorithm based on $$l_1$$ minus $$l_2$$ penalty is the best for over-sampled DCT matrices. We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with $$l_1$$ minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix A. DCATL1 is also competitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two hyperparameters.

Global Well-Posedness of the Incompressible Magnetohydrodynamics

This paper studies the Cauchy problem of the incompressible magnetohydro dynamic systems with or without viscosity ν. Under the assumption that the initial velocity field and the displacement of the initialmagnetic field froma non-zero constant are sufficiently small in certain weighted Sobolev spaces, the Cauchy problem is shown to be globally well-posed for all ν ≧ 0 and all spaces with dimension n ≧ 2. Such a result holds true uniformly in nonnegative viscosity parameters. The proof is based on the inherent strong null structure of the systems introduced by Lei (Commun Pure Appl Math 69(11):2072–2106, 2016) and the ghost weight technique introduced by Alinhac (Invent Math 145(3):597–618, 2001).

Blending Type Approximation by GBS Operators of Generalized Bernstein–Durrmeyer Type

In this article we study an extension of the bivariate generalized Bernstein–Durrmeyer operators based on a non-negative real parameter. For these operators we get a Voronovskaja type theorem, the order of approximation using Peetre’s K-functional and the degree of approximation by means of the Lipschitz class. Further, we introduce the generalized boolean sum operators of generalized Bernstein–Durrmeyer type and we estimate the degree of approximation in terms of the mixed modulus of smoothness.

Blending type approximation by bézier-summation-integral type operators

In this note we construct the B€zier variant of summation integral type operators based on a non-negative real parameter. We present a direct approximation theorem by means of the first order modulus of smoothness and the rate of convergence for absolutely continuous functions having a derivative equivalent to a function of bounded variation. In the last section, we study the quantitative Voronovska ja type theorem

Projections of spherical Brownian motion

We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R} ^{n+\ell }$. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of existence and pathwise uniqueness and show that they always hold. The square of the radial component is a Wright-Fisher diffusion with mutation and it features in a skew-product decomposition of the projected spherical Brownian motion. A more general SDE on the unit ball in $\mathbb{R} ^{n+\ell }$ allows us to geometrically realize the Wright-Fisher diffusion with general non-negative parameters as the radial component of its solution.

Dynamic thermal and IT resource management strategies for data center energy minimization

Energy-hungry data centers attract researchers’ attention to energy consumption minimization–a challenge confronted by both industry and academia. To assure the server reliability, its instantaneous temperature is generally controlled within a preset threshold. Nonetheless, field studies indicated that the occasional violation of extreme temperature constraints hardly affect the system reliability in practice. Therefore, strictly restraining the server temperature may contribute to meaningless energy consumption. As a response to this limitation, this paper presents a dynamic control algorithm without violating the average temperature constraint. We formulate a “soft” Server Temperature-Constrained Energy Minimization (STCEM) problem, where the object function consists of IT and Heat Ventilation & Air Conditioner (HVAC) energy. Based on the Lyapunov optimization, two algorithms, i.e., linear and quadratic control policies, are proposed to approximately solve the STCEM problem. The non-negative weight parameter V is introduced to trade off the energy consumption against server temperature constraint. Furthermore, extensive simulations have been carried out to evaluate the system performance for the proposed controlled algorithms. The experimental results demonstrate that the quadratic control policy outperforms the linear counterpart on the STCEM problem. Specifically, the energy consumption and server temperature constraint are well balanced when V≈5000.

Minimum requirements for the estimation of measurement uncertainty: Recommendations of the joint Working group for uncertainty of measurement of the CSMBLM and CCMB.

The International vocabulary of metrology – Basic and general concepts and associated terms (VIM3, 2.26 measurement uncertainty, JCGM 200:2012) defines uncertainty of measurement as a non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information obtained from performing the measurement. Clinical Laboratory Standards Institute (CLSI) has published a very detailed guideline with a description of sources contributing to measurement uncertainty as well as different approaches for the calculation (Expression of measurement uncertainty in laboratory medicine; Approved Guideline, CLSI C51-A 2012). Many other national and international recommendations and original scientific papers about measurement uncertainty estimation have been published. In Croatia, the estimation of measurement uncertainty is obligatory for accredited medical laboratories. However, since national recommendations are currently not available, each of these laboratories uses a different approach in measurement uncertainty estimation. The main purpose of this document is to describe the minimal requirements for measurement uncertainty estimation. In such way, it will contribute to the harmonization of measurement uncertainty estimation, evaluation and reporting across laboratories in Croatia. This recommendation is issued by the joint Working group for uncertainty of measurement of the Croatian Society for Medical Biochemistry and Laboratory Medicine and Croatian Chamber of Medical Biochemists. The document is based mainly on the recommendations of Australasian Association of Clinical Biochemists (AACB) Uncertainty of Measurement Working Group and is intended for all medical biochemistry laboratories in Croatia.

The Kantorovich variant of an operator defined by D. D. Stancu

In this note we introduce Kantorovich variant of the operators considered by Stancu (1998) based on two nonnegative parameters. Here, we prove an approximation theorem with the help of Bohman–Korovkin’s principle and find the estimate of the rate of convergence by means of modulus of smoothness and Lipschitz type function for these operators. In the last section of the paper, we show the convergence of the operators by illustrative graphics in Mathematica to certain functions.

Semiparametric estimation of duration models when the parameters are subject to inequality constraints and the error distribution is unknown

This paper proposes a semiparametric method for estimating duration models when there are inequality constraints on some parameters and the error distribution may be unknown. Thus, the setting considered here is particularly suitable for practical applications. The parameters in duration models are usually estimated by a quasi-MLE. Recent advances show that a semiparametrically efficient estimator[SPE] has better asymptotic optimality properties than the QMLE provided that the parameter space is unrestricted. However, in several important duration models, the parameter space is restricted, for example in the commonly used linear duration model some parameters are non-negative. In such cases, the SPE may turn out to be outside the allowed parameter space and hence are unsuitable for use. To overcome this difficulty, we propose a new constrained semiparametric estimator. In a simulation study involving duration models with inequality constraints on parameters, the new estimator proposed in this paper performed better than its competitors. An empirical example is provided to illustrate the application of the new constrained semiparametric estimator and to show how it overcomes difficulties encountered when the unconstrained estimator of nonnegative parameters turn out to be negative.

Stationary Schrödinger equations in ℝ2 with unbounded or vanishing potentials and involving concave nonlinearities

ABSTRACTWe study the existence and multiplicity of solutions for the following class of stationary nonlinear Schrödinger equations:where is a nonnegative parameter, V and Q are unbounded or decaying radial potentials, the nonlinearity f(s) may exhibit exponential growth and g(x, s) is a concave term. The approach used here is based on a version of the Trudinger–Moser inequality, mountain-pass theorem and the Ekeland’s variational principle in a suitable weighted Sobolev space.

Approximation of functions by a new family of generalized Bernstein operators

The main object of this paper is to construct a new generalization of the Bernstein operator, depending on a non-negative real parameter. We investigate some elementary properties of this operator, such as end point interpolation, linearity and positivity, etc. By using these generating operators, we provide another proof of the Weierstrass Approximation Theorem. We give the detailed proofs to the rate of convergence and Voronovskaja type asymptotic estimate formula for the operators. Moreover, an upper bound for the error is obtained in terms of the usual modulus of continuity. Shape preserving properties of the generalized Bernstein operators are also studied. It is proved that monotonic or convex functions produce monotonic or convex generalized Bernstein polynomials.

Identification of fixedness and loading of an end of the Euler–Bernoulli beam by the natural frequencies of its vibrations

Under consideration is the uniform Euler–Bernoulli beam whose left end is fixed, and some load elastically fixed by two springs is concentrated at the right end. If the beam is hit then it begins vibrating. The aim of the article is to determine the parameters of fixedness (rigidity coefficients of springs) and loading (the mass andmoment of inertia of the load) of the right end of the beam by natural frequencies of its flexural vibrations. It is shown that the four unknown parameters of the boundary conditions at the right end of the beam are uniquely determined from the five natural frequencies of its flexural vibrations. Some counterexample is presented showing that four natural frequencies are insufficient for the unique identification of these four nonnegative parameters.

Transformed Schatten-1 iterative thresholding algorithms for low rank matrix completion

c 2017 International Press C OMMUN. M ATH. S CI. Vol. 15, No. 3, pp. 839–862 TRANSFORMED SCHATTEN-1 ITERATIVE THRESHOLDING ALGORITHMS FOR LOW RANK MATRIX COMPLETION ∗ SHUAI ZHANG † , PENGHANG YIN ‡ , AND JACK XIN § Abstract. We study a non-convex low-rank promoting penalty function, the transformed Schatten- 1 (TS1), and its applications in matrix completion. The TS1 penalty, as a matrix quasi-norm defined on its singular values, interpolates the rank and the nuclear norm through a nonnegative parameter a ∈ (0,+∞). We consider the unconstrained TS1 regularized low-rank matrix recovery problem and develop a fixed point representation for its global minimizer. The TS1 thresholding functions are in closed analytical form for all parameter values. The TS1 threshold values differ in subcritical (supercritical) parameter regime where the TS1 threshold functions are continuous (discontinuous). We propose TS1 iterative thresholding algorithms and compare them with some state-of-the-art algorithms on matrix completion test problems. For problems with known rank, a fully adaptive TS1 iterative thresholding algorithm consistently performs the best under different conditions, where ground truth matrices are generated by multivariate Gaussian, (0,1) uniform and Chi-square distributions. For problems with unknown rank, TS1 algorithms with an additional rank estimation procedure approach the level of IRucL-q which is an iterative reweighted algorithm, non-convex in nature and best in performance. Keywords. transformed Schatten-1 penalty; fixed point representation; closed form thresholding function; iterative thresholding algorithms; matrix completion AMS subject classifications. 90C26; 90C46 1. Introduction Low rank matrix completion problems arise in many applications such as collabora- tive filtering in recommender systems [4,17], minimum order system and low-dimensional Euclidean embedding in control theory [14,15], network localization [18], and others [26]. The mathematical problem is: min rank(X) X∈ m×n s.t. X ∈ L where L is a convex set. In this paper, we are interested in methods for solving the affine rank minimization problem (ARMP) min rank(X) X∈ m×n s.t. A (X) = b in p , where the linear transformation A : m×n → p and vector b ∈ p are given. The matrix completion problem min rank(X) X∈ m×n s.t. X i,j = M i,j , (i,j) ∈ Ω is a special case of (1.2), where X and M are both m × n matrices and Ω is a subset of index pairs {(i,j)}. ∗ Received: September 25, 2016; accepted (in revised form): October 28, 2016. Communicated by Wotao Yin. The work was partially supported by NSF grant DMS-1222507 and DMS-1522383. † Department of Mathematics, University of California, Irvine, CA, 92697, USA (szhang3@uci.edu). Phone: (949)-824-5309. Fax: (949)-824-7993. ‡ Department of Mathematics, University of California, Irvine, CA, 92697, USA (penghany@uci.edu). § Department of Mathematics, University of California, Irvine, CA, 92697, USA (jxin@math.uci.edu).

Global behavior of a higher order difference equation

In this paper we study the global behavior of the nonnegative equilibrium points of the difference equation xn+1=Axn-[email protected]?i=02k+1xn-i,n=0,1,... where A,B,C are nonnegative parameters, initial conditions are nonnegative real numbers and k,m are nonnegative integers, [email protected]?2k+1. Also we derive solutions of some special cases of this equation.

Distribution-free properties of some asymptotic cumulants for the Mallows C p and its modifications in usual and ridge regression

In multivariate multiple linear regression with a non-negative ridge parameter, when a model is underspecified, the asymptotic biases of the Mallows Cp and its modifications are derived up to order O(1) under non-normality. For a not underspecified model, the asymptotic biases are of smaller order and are shown to be distribution free. Similarly, under the latter condition, the common asymptotic variance of order O(1) for the statistics is distribution free. It is shown that the above results hold irrespective of the ridge parameter. Numerical illustrations with simulations under normality and non-normality give similar simulated results. These results justify the robust correct model selection under non-normality shown by simulations.

EXISTENCE OF POSITIVE SOLUTIONS TO NONLOCAL BOUNDARY VALUE PROBLEMS WITH BOUNDARY PARAMETER

We establish the existence of positive solutions to nonlocal boundary value problems with integral boundary condition and non-negative real boundary parameter by mainly using the Schauder-Fixed point theorem.