Single Parameter Case Research Articles

Real analytic multi-parameter singular Radon transforms: Necessity of the Stein-Street condition

We study operators of the form T f ( x ) = ψ ( x ) ∫ f ( γ t ( x ) ) K ( t ) d t , \begin{equation*} Tf(x)= \psi (x) \int f(\gamma _t(x))K(t)\,dt, \end{equation*} where γ t ( x ) \gamma _t(x) is a real analytic function of ( t , x ) (t,x) mapping from a neighborhood of ( 0 , 0 ) (0,0) in R N × R n \mathbb {R}^N \times \mathbb {R}^n into R n \mathbb {R}^n satisfying γ 0 ( x ) ≡ x \gamma _0(x)\equiv x , ψ ( x ) ∈ C c ∞ ( R n ) \psi (x) \in C_c^\infty (\mathbb {R}^n) , and K ( t ) K(t) is a “multi-parameter singular kernel” with compact support in R N \mathbb {R}^N ; for example when K ( t ) K(t) is a product singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth γ t ( x ) \gamma _t(x) , in the single-parameter case when K ( t ) K(t) is a Calderón-Zygmund kernel. Street and Stein generalized their work to the multi-parameter case, and gave sufficient conditions for the L p L^p -boundedness of such operators. This paper shows that when γ t ( x ) \gamma _t(x) is real analytic, the sufficient conditions of Street and Stein are also necessary for the L p L^p -boundedness of T T , for all such kernels K K .

Optimal multi-parameter sampling for geostructural design

The common use of a minimal number of samples to determine geotechnical material parameters is a frequent source of difficulty for structural design engineers. One-parameter decision analysis has suggested that numbers of samples greater than that commonly used in current practice would be more risk-optimal. Building on this result, a multi-parameter theory is developed for the risk-optimal number of samples to be measured when characterising individual material parameters. The theory also considers the effect of coupled (measured in a single test) and correlated parameter pairs. As in the single-parameter case, the number of samples appropriate to a particular problem is dictated by the level of reliability used in the design, the cost of testing, and the damages for which the owner would be liable in the event of failure. By implementing the framework for a square footing and a slope embankment, it is shown that optimal sample sizes of 5-20 measurements per parameter are optimal, with the upper end of the spectrum commensurate with more severe failure consequences. It is suggested that the single-parameter end-case provides a conservative guideline to the risk-optimal number of sample measurements, appropriate for use in standards of practice.

The Sample Complexity of Up-to-ε Multi-dimensional Revenue Maximization

We consider the sample complexity of revenue maximization for multiple bidders in unrestricted multi-dimensional settings. Specifically, we study the standard model of additive bidders whose values for heterogeneous items are drawn independently. For any such instance and any , we show that it is possible to learn an -Bayesian Incentive Compatible auction whose expected revenue is within of the optimal -BIC auction from only polynomially many samples. Our fully nonparametric approach is based on ideas that hold quite generally and completely sidestep the difficulty of characterizing optimal (or near-optimal) auctions for these settings. Therefore, our results easily extend to general multi-dimensional settings, including valuations that are not necessarily even subadditive , and arbitrary allocation constraints. For the cases of a single bidder and many goods, or a single parameter (good) and many bidders, our analysis yields exact incentive compatibility (and for the latter also computational efficiency). Although the single-parameter case is already well understood, our corollary for this case extends slightly the state of the art.

Inverse Stable Prior for Exponential Models

We consider a class of non-conjugate priors as a mixing family of distributions for a parameter (e.g., Poisson or gamma rate, inverse scale or precision of an inverse-gamma, inverse variance of a normal distribution) of an exponential subclass of discrete and continuous data distributions. The prior class is proper, nonzero at the origin (unlike the gamma and inverted beta priors with shape parameter less than one and Jeffreys prior for a Poisson rate), and is easy to generate random numbers from. The prior class also provides flexibility in capturing a wide array of prior beliefs (right-skewed and left-skewed) as modulated by a bounded parameter $$\alpha \in (0, 1)$$ . The resulting posterior family in the single-parameter case can be expressed in closed form and is proper, making calibration unnecessary. The mixing induced by the inverse stable family results in a marginal prior distribution in the form of a generalized Mittag–Leffler function, which covers a broad array of distributional shapes. We derive closed-form expressions of some properties like the moment generating function and moments. We propose algorithms to generate samples from the posterior distribution and calculate the Bayes estimators for real data analysis. We formulate the predictive prior and posterior distributions. We test the proposed Bayes estimators using Monte Carlo simulations. The extension to hierarchical modeling and inverse variance components models is straightforward. We explore the global shrinkage model in some detail to show the potential value of the inverse stable prior. We show that the inverse stable prior has some better properties than the inverted beta prior corresponding to the half-Cauchy prior (commonly recommended for adoption in such cases). We illustrate the methodology using a real data set, introduce a hyperprior density for the hyperparameters, and extend the model to a heavy-tailed distribution.

Conditions on optimal support recovery in unmixing problems by means of multi-penalty regularization

Inspired by several real-life applications in audio processing and medical image analysis, where the quantity of interest is generated by several sources to be accurately modeled and separated, as well as by recent advances in regularization theory and optimization, we study the conditions on optimal support recovery in inverse problems of unmixing type by means of multi-penalty regularization. We consider and analyze a regularization functional composed of a data-fidelity term, where signal and noise are additively mixed, a non-smooth, convex, sparsity promoting term, and a quadratic penalty term to model the noise. We prove not only that the well-established theory for sparse recovery in the single parameter case can be translated to the multi-penalty settings, but we also demonstrate the enhanced properties of multi-penalty regularization in terms of support identification compared to sole -minimization. We additionally confirm and support the theoretical results by extensive numerical simulations, which give a statistics of robustness of the multi-penalty regularization scheme with respect to the single-parameter counterpart. Eventually, we confirm a significant improvement in performance compared to standard -regularization for compressive sensing problems considered in our experiments.

Stabilization of Nonlinear Systems Nonlinearly Depending on Fast Time-Varying Parameters: An Immersion and Invariance Approach

The problem of stabilization of nonlinear systems, which depend nonlinearly on fast time-varying parameters, is considered in the technical note. It is assumed that, if the plant parameters were known, a static state-feedback controller that achieves the stabilization objective with a Lyapunov-like function that is independent of the parameters is known. A constructive procedure to update the unknown parameters of the controller, based on the immersion and invariance approach, is proposed. The main contribution of the paper is to show that the proposed controller guarantees global convergence to zero of the systems state for arbitrary time variations of the plant parameters provided the controller parameters are bounded. To ensure the latter condition, an assumption, that in the single parameter case is strictly weaker than the monotonicity condition invoked in previous studies, is imposed. Stabilization is achieved via a, rather unique, combination of gradient-like parameter estimation and the construction of a monotonic signal that counters the deleterious effect of the parameter variations. Several simulation examples illustrate the applicability of the suggested method.

Derivation of quantum Chernoff metric with perturbation expansion method

We investigate a measure of distinguishability defined by the quantum Chernoff bound, which naturally induces the quantum Chernoff metric over a manifold of quantum states. Based on a quantum statistical model, we alternatively derive this metric by means of perturbation expansion. Moreover, we show that the quantum Chernoff metric coincides with the infinitesimal form of the quantum Hellinger distance, and reduces to the variant version of the quantum Fisher information for the single-parameter case. We also give the exact form of the quantum Chernoff metric for a qubit system containing a single parameter.

ON THE STATISTICAL ANALYSIS OF X-RAY POLARIZATION MEASUREMENTS

In many polarimetry applications, including observations in the X-ray band, the measurement of a polarization signal can be reduced to the detection and quantification of a deviation from uniformity of a distribution of measured angles. We explore the statistics of such polarization measurements using Monte Carlo simulations and chi-squared fitting methods. We compare our results to those derived using the traditional probability density used to characterize polarization measurements and quantify how they deviate as the intrinsic modulation amplitude grows. We derive relations for the number of counts required to reach a given detection level (parameterized by beta, the "number of sigma's" of the measurement) appropriate for measuring the modulation amplitude by itself (single interesting parameter case) or jointly with the position angle (two interesting parameters case). We show that for the former case when the intrinsic amplitude is equal to the well known minimum detectable polarization (MDP) it is, on average, detected at the 3-sigma level. For the latter case, when one requires a joint measurement at the same confidence level, then more counts are needed than that required to achieve the MDP level. This additional factor is amplitude-dependent, but is approximately 2.2 for intrinsic amplitudes less than about 20%. It decreases slowly with amplitude and is 1.8 when the amplitude is 50%. We find that the position angle uncertainty at 1-sigma confidence is well described by the relation 28.5 (deg) / beta.

Solving Inverse Structural Reliability Problem Using Artificial Neural Networks and Small-Sample Simulation

A new general inverse reliability analysis approach based on artificial neural networks is proposed. An inverse reliability analysis is a problem of obtaining design parameters corresponding to a specified reliability (reliability index or theoretical failure probability). Design parameters can be deterministic or they can be associated with random variables. The aim is to generally solve not only single design parameter cases but also multiple parameter problems with given multiple reliability constraints. Inverse analysis is based on the coupling of a stochastic simulation of the Monte Carlo type (the small-sample simulation method Latin hypercube sampling) and an artificial neural network. The validity and efficiency of this approach is shown using numerical examples with single as well as multiple reliability constraints and with single as well as multiple design parameters, and with independent basic random variables as well as random variables with prescribed statistical correlations.

On a generalization of Regińska’s parameter choice rule and its numerical realization in large-scale multi-parameter Tikhonov regularization

A crucial problem concerning Tikhonov regularization is the proper choice of the regularization parameter. This paper deals with a generalization of a parameter choice rule due to Regińska (1996) [31], analyzed and algorithmically realized through a fast fixed-point method in Bazán (2008) [3], which results in a fixed-point method for multi-parameter Tikhonov regularization called MFP. Like the single-parameter case, the algorithm does not require any information on the noise level. Further, combining projection over the Krylov subspace generated by the Golub–Kahan bidiagonalization (GKB) algorithm and the MFP method at each iteration, we derive a new algorithm for large-scale multi-parameter Tikhonov regularization problems. The performance of MFP when applied to well known discrete ill-posed problems is evaluated and compared with results obtained by the discrepancy principle. The results indicate that MFP is efficient and competitive. The efficiency of the new algorithm on a super-resolution problem is also illustrated.

Uncertainty analysis of river flooding and dam failure risks using local sensitivity computations

The potential of Local Sensitivity Analysis (LSA) for analysis of uncertainty with respect to two major risks in river hydrodynamics – flash flood and dam failure – is assessed. LSA, implemented as an equation-based method, is compared to a Global Uncertainty Analysis (GUA) consisting in running Monte Carlo simulations with a hydrodynamic model. For a given statistical distribution of the model input parameters, the mean and standard deviation of the output variables are estimated with the two methods. In all single or multiple parameter cases investigated, including as much as±80% relative variation, LSA provides similar results to GUA, while requiring only one simulation instead of several hundreds or thousands. Only within a few meters of the shock (flow discontinuity) generated by the breaking of a dam do the two methods depart. This paper shows that despite the non-linearity of river flow processes, the first order, local approach remains generally valid for uncertainty analysis of hydrodynamic risks, even in the case of large parameter uncertainty. The contrast in importance of the various parameters on both sides of a shock is also highlighted.

Bayesian Robustness Modelling of Location and Scale Parameters

Abstract. The modelling process in Bayesian Statistics constitutes the fundamental stage of the analysis, since depending on the chosen probability laws the inferences may vary considerably. This is particularly true when conflicts arise between two or more sources of information. For instance, inference in the presence of an outlier (which conflicts with the information provided by the other observations) can be highly dependent on the assumed sampling distribution. When heavy‐tailed (e.g. t) distributions are used, outliers may be rejected whereas this kind of robust inference is not available when we use light‐tailed (e.g. normal) distributions. A long literature has established sufficient conditions on location‐parameter models to resolve conflict in various ways. In this work, we consider a location–scale parameter structure, which is more complex than the single parameter cases because conflicts can arise between three sources of information, namely the likelihood, the prior distribution for the location parameter and the prior for the scale parameter. We establish sufficient conditions on the distributions in a location–scale model to resolve conflicts in different ways as a single observation tends to infinity. In addition, for each case, we explicitly give the limiting posterior distributions as the conflict becomes more extreme.

On optimal damping of vibrating structures

AbstractThe paper deals with the problem of optimal damping of vibrating structures by means of collocated decentralized devices. We consider both H2 and H∞ criteria. We prove that in the H2 case the problem can be conduced to the minimization of a function which is the sum of a convex and a concave component. This structure is of help in the solution of the problem. In the H∞ case we show that there exists at most a local minimum for the problem, at least in the single–parameter case. We finally apply the results to the switching case, which is a promising approach in term of performance improvement.

Multi-parameter singular Radon transforms

The purpose of this announcement is to describe a development given in a series of forthcoming papers by the authors that concern operators of the form \[ f\mapsto \psi(x) \int f(\gamma_t(x)) K(t)\: dt, \] where $\gamma_t(x)=\gamma(t,x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in \mathbb{R}^N\times \mathbb{R}^n$ satisfying $\gamma_0(x)\equiv x$, $K(t)$ is a singular supported near $t=0$, and $\psi$ is a cutoff function supported near $x=0$. This note concerns the case when $K$ is a product kernel. The goal is to give conditions on $\gamma$ such that the above operator is bounded on $L^p$ for $1<p<\infty$. Associated maximal functions are also discussed. The single-parameter case when $K$ is a Calder\'on-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger. The theory here extends these results to the multi-parameter context and also deals effectively with the case when $\gamma$ is real-analytic.

Artificial neural network based inverse reliability analysis

AbstractAn inverse reliability analysis is the problem to find design parameters corresponding to specified reliability levels expressed by reliability index or by theoretical failure probability. Design parameters can be deterministic or they can be associated to random variables described by statistical moments. The aim is to solve generally not only the single design parameter case but also the multiple parameter problems with given multiple reliability constraints. A new general approach of inverse reliability analysis is proposed. The inverse analysis is based on the coupling of a stochastic simulation of Monte Carlo type and an artificial neural network. A novelty of the approach is the utilization of the efficient small‐sample simulation method Latin Hypercube Sampling used for the stochastic preparation of the training set. (© 2010 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Truthful approximation mechanisms for restricted combinatorial auctions

When attempting to design a truthful mechanism for a computationally hard problem such as combinatorial auctions, one is faced with the problem that most efficiently computable heuristics can not be embedded in any truthful mechanism (e.g. VCG-like payment rules will not ensure truthfulness). We develop a set of techniques that allow constructing efficiently computable truthful mechanisms for combinatorial auctions in the special case where each bidder desires a specific known subset of items and only the valuation is unknown by the mechanism (the single parameter case). For this case we extend the work of Lehmann, O'Callaghan, and Shoham, who presented greedy heuristics. We show how to use If-Then-Else constructs, perform a partial search, and use the LP relaxation. We apply these techniques for several canonical types of combinatorial auctions, obtaining truthful mechanisms with provable approximation ratios.

Optimizing the asymptotic convergence rate of the Diaconis–Holmes–Neal sampler

We consider the problem of optimizing the asymptotic convergence rate of a parameter-dependent nonreversible Markov chain. We begin with a single-parameter case studied by Diaconis, Holmes and Neal and then introduce multiple parameters. We use nonsmooth analysis to investigate whether the presence of multiple parameters allows a faster asymptotic convergence rate, and argue that for a specific parameterization, it does not, at least locally.

An approach for computing the exact stability domain for a class of LTI parameter dependent systems

In this paper, the complete stability domain for both single- and multi-parameter dependent LTI systems is synthesized by extending existing results in the literature. This domain is calculated through a guardian map which involves the determinant of the Kronecker sum of a matrix with itself. The single parameter case is easily computable, whereas the multi-parameter case is more involved. The determinant of the bialternate sum of a matrix with itself is also exploited to reduce the computational complexity of the results presented in this paper.

ROBUST STABILITY OF DISCRETE-TIME ADAPTIVE NONLINEAR CONTROL

We consider adaptive control of discrete-time nonlinear systems with unknown parameters and with bounded noise. In the single parameter case, we demonstrate that the necessary and sufficient condition for the existence of a robust feedback stabilizer is that the nonlinear growth rate of the system dynamics is less than 4. This result further confirms the conclusion of (Guo, 1997) where Gaussian noise were considered. In the multiple parameter case, the necessary and sufficient condition turns out to be governed by a polynomial rule, which is identical to the one obtained in (Xie and Guo, 1999) where also Gaussian noise were considered but only the necessity part was proved. To the authors’ knowledge, the stabilizing controller constructed in this paper seems the first capable of dealing with any nontrivial noise in the multiple parameter case.

Prime and Primitive Ideals in Graded Deformations of Algebraic Quantum Groups at Roots of Unity

Let G be a connected, simply connected complex semisimple Lie group of rank n. The deformations employed by Artin, Schelter and Tate, and Hodges, Levasseur and Toro can be applied to the single parameter quantizations, at roots of unity, of the Hopf algebra of regular functions on G. Each of the resulting complex multiparameter quantum groups F∈,p[G] depends on both a suitable root of unity ∈ and an antisymmetric bicharacter p: Z n ×Z n →C×. These quantizations differ significantly from their single parameter (root-of-unity) counterparts, and, in particular, may have infinite-dimensional irreducible representations. Our approach to F∈,p[G] depends on a natural ℋ×ℋ-action thereon, where ℋ is an n-torus, and our main result offers a classification of the primitive ideals: We use a multiparameter quantum Frobenius map to provide a bijection from (Prim F∈,p[G])/ℋ×ℋ onto G/H×H, where H is a maximal torus of G. In the single parameter case, this bijection is a consequence of work by De Concini and Lyubashenko, and De Concini and Procesi; our results require their analysis. Our methods also exploit earlier work by Moeglin and Rentschler concerning actions of algebraic groups on complex Noetherian algebras. In contrast to generic quantizations of the coordinate ring of G, the primitive spectrum of F∈,p[G] is not finitely stratified by the torus action.