The truncated Hausdorff moment problem solved by using kernel density functions

In this work, the problem of an efficient representation and its exploitation to the approximate determination of a compactly supported, continuous probability density function (pdf) from a finite number of its moments is addressed. The representation used is a finite superposition of kernel density functions. This representation preserves positivity and can approximate any continuous pdf as closely as it is required. The classical theory of the Hausdorff moment problem is reviewed in order to make clear how the theoretical results as, e.g. the moment bounds, can be exploited in the numerical procedure. Various difficulties arising from the well-known ill-posedness of the numerical moment problem have been identified and solved. The kernel coefficients of the pdf expansion are calculated by solving a constrained, non-negative least-square problem. The consistency, numerical convergence and robustness of the solution algorithm have been illustrated by numerical examples with unimodal and bimodal pdfs. Although this paper is restricted to univariate, compactly supported pdfs, the method can be extended to general pdfs either univariate or multivariate, with finite or infinite support.     

Moment data can be analytically completed

When an unknown (target) probability density function (pdf) is to be reconstructed from its moments with the aid of a multiparametric (or semi-parametric) representation model, a large number of constraints (moment equations) is needed, usually larger than the unknown parameters. However, there is a strong belief, expressed by various authors, see Fasino D, Inglese G (Rendiconti dell' Istituto di Matematica dell' Università di Trieste XXVIII (1996) 183], and French JB [In: Dalton BJ, et al., Theory and application of moments problems in many-fermion systems (1979) 1], that most of the information defining a compactly supported pdf is usually contained in its first few moments. In addition, numerical experimentation has shown [Probab Engng Mech 17 (2002) 273] that the variation of the reconstructed pdf when higher-order moments vary within the recursively defined moment bounds, is weak, becoming insignificant as the moment order N increases. Based on these observations, we propose a method for extending a set of given moments, in cases that they are insufficient for solving the Hausdorff moment problem using a multiparametric model. Additional moments, compatible with the given ones, are obtained by exploiting upper and lower moment bounds. Then, two sequences of ‘dual’ (upper and lower) approximant pdfs are constructed, along with an easily realizable criterion of mutual convergence. The numerical performance and consistency of the solution algorithm have been illustrated for unimodal and bimodal pdfs.     

Efficient evaluation of the pdf of a random variable through the kernel density maximum entropy approach

In this paper, a new approach for the evaluation of the probability density function (pdf) of a random variable from the knowledge of its lower moments is presented. At first the classical moment problem (MP) is revisited, which gives the conditions such that the assigned sequence of sample moments represent really a sequence of moments of any distribution. Then an alternative approach is presented, termed as the kernel density maximum entropy (MaxEnt) method by the authors, which approximates the target pdf as a convex linear combination of kernel densities, transforming the original MP into a discrete MP, which is solved through a MaxEnt approach. In this way, simply solving a discrete MaxEnt problem, not requiring the evaluation of numerical integrals, an approximating pdf converging toward the MaxEnt pdf is obtained. The method is first demonstrated by approximating some known analytical pdfs (the chi‐square and the Gumbel pdfs) and then it is applied to some experimental engineering problems, namely for modelling the pdf of concrete strength, the circular frequency and the damping ratio of strong ground motions, the extreme wind speed in Messina's Strait region. All the developed numerical applications show the goodness and efficacy of the proposed procedure. Copyright © 2008 John Wiley & Sons, Ltd.     

The Principle of Maximum Entropy Applied in the Evaluation of the Measurement Uncertainty

The maximum entropy approach is a flexible and powerful tool for assigning a probability distribution to a measurable quantity treated as a random variable subjected to known moment constraints. The aim of this paper is to describe how the principle of maximum entropy may be used to transform information about the value of a quantity into a probability density function (pdf) that reflects exactly that information and nothing else. This principle will be applied to common cases of metrological interest, where different kinds of information are available. The derivation of the pdf is given in each case, and two practical examples with numerical results are reported to demonstrate the efficiency of the maximum entropy method     

Distribution system state estimation through Gaussian mixture model of the load as pseudo-measurement

This study presents an approach to utilise the loads as pseudo-measurements for the purpose of distribution system state estimation (DSSE). The load probability density function (pdf) in the distribution network shows a number of variations at different nodes and cannot be represented by any specific distribution. The approach presented in this study represents all the load pdfs through the Gaussian mixture model (GMM). The expectation maximisation (EM) algorithm is used to obtain the parameters of the mixture components. The standard weighted least squares (WLS) algorithm utilises these load models as pseudo-measurements. The effectiveness of WLS is assessed through some statistical measures such as bias, consistency and quality of the estimates in a 95-bus generic distribution network model.     

The role of risk-propensity in the risky driving of younger drivers

Young drivers are over-represented in road injury statistics, partly because they engage in more risky driving than older people. Although it is assumed that younger people have greater risk-propensity, defined as a positive attitude to risk, relevant theory is imprecise and relevant research is clouded by inappropriate measures. 89 participants aged 16-25 and 110 participants aged over 35 were recruited outside motor registries. Participants completed a battery of questionnaires including Rohrmann’s [Rohrmann, B. 2004. Risk attitude scales: concepts and questionnaires. Project report. Available at http://www.rohrmannresearch.net/pdfs/rohrmann-racreport.pdf (last accessed 12th February 2008)] measures of risk-aversion, risk-propensity, and risk-related motives for risky driving, as well as measures of risk-perception and risky driving. Compared to older drivers, younger drivers demonstrated lower risk-aversion, and higher propensity for taking accident risks, as well as stronger motives for risky driving in relation to experience-seeking, excitement, sensation-seeking, social influence, prestige-seeking, confidence/familiarity, underestimation of risk, irrelevance of risk, “letting off steam”, and “getting there quicker”. Further, these variables were associated with risky driving. Some evidence was observed for the possibility that risk propensity moderates the relationship between perceived risk and risky behaviour. These results suggest approaches to targeting the “young driver problem”.     

Screening of antibiotic susceptibility to β-lactam-induced elongation of Gram-negative bacteria based on dielectrophoresis

We demonstrate a rapid antibiotic susceptibility test (AST) based on the changes in dielectrophoretic (DEP) behaviors related to the β-lactam-induced elongation of Gram-negative bacteria (GNB) on a quadruple electrode array (QEA). The minimum inhibitory concentration (MIC) can be determined within 2 h by observing the changes in the positive-DEP frequency (pdf) and cell length of GNB under the cefazolin (CEZ) treatment. Escherichia coli and Klebsiella pneumoniae and the CEZ are used as the sample bacteria and antibiotic respectively. The bacteria became filamentous due to the inhibition of cell wall synthesis and cell division and cell lysis occurred for the higher antibiotic dose. According to the results, the pdfs of wild type bacteria decrease to hundreds of kHz and the cell length is more than 10 μm when the bacterial growth is inhibited by the CEZ treatment. In addition, the growth of wild type bacteria and drug resistant bacteria differ significantly. There is an obvious decrease in the number of wild type bacteria but not in the number of drug resistant bacteria. Thus, the drug resistance of GNB to β-lactam antibiotics can be rapidly assessed. Furthermore, the MIC determined using dielectrophoresis-based AST (d-AST) was consistent with the results of the broth dilution method. Utilizing this approach could reduce the time needed for bacteria growth from days to hours, help physicians to administer appropriate antibiotic dosages, and reduce the possibility of the occurrence of multidrug resistant (MDR) bacteria.     

The inverted beta loss function: properties and applications

In this paper, a general class of loss functions based on the inversion of the standard beta probability density function (pdf) is examined. The extension of this loss function from a standard beta pdf ranging from (0, 1) to the general beta pdf ranging from (p, q) is examined through the scale invariance property under a linear transformation. An industrial application in quality assurance is used to demonstrate this general class of loss functions. Mathematical derivations are attached in the Appendices.     

Evaluation by Monte Carlo simulations of the power limits and bit-error rate degradation in wavelength-division multiplexing networks caused by four-wave mixing

Fiber nonlinearities can degrade the performance of a wavelength-division multiplexing optical network. For high input power, a low chromatic dispersion coefficient, or low channel spacing, the most severe penalties are due to four-wave mixing (FWM). To compute the bit-error rate that is due to FWM noise, one must evaluate accurately the probability-density functions (pdf) of both the space and the mark states. An accurate evaluation of the pdf of the FWM noise in the space state is given, for the first time to the authors' knowledge, by use of Monte Carlo simulations. Additionally, it is shown that the pdf in the mark state is not symmetric as had been assumed in previous studies. Diagrams are presented that permit estimation of the pdf, given the number of channels in the system. The accuracy of the previous models is also investigated, and finally the results of this study are used to estimate the power limits of a wavelength-division multiplexing system.     

Estimation of laser beam pointing parameters in the presence of atmospheric turbulence

The problem of estimating mechanical boresight and jitter performance of a laser pointing system in the presence of atmospheric turbulence is considered. A novel estimator based on maximizing an average probability density function (pdf) of the received signal is presented. The proposed estimator uses a Gaussian far-field mean irradiance profile, and the irradiance pdf is assumed to be lognormal. The estimates are obtained using a sequence of return signal values from the intended target. Alternatively, one can think of the estimates being made by a cooperative target using the received signal samples directly. The estimator does not require sample-to-sample atmospheric turbulence parameter information. The approach is evaluated using wave optics simulation for both weak and strong turbulence conditions. Our results show that very good boresight and jitter estimation performance can be obtained under the weak turbulence regime. We also propose a novel technique to include the effect of very low received intensity values that cannot be measured well by the receiving device. The proposed technique provides significant improvement over a conventional approach where such samples are simply ignored. Since our method is derived from the lognormal irradiance pdf, the performance under strong turbulence is degraded. However, the ideas can be extended with appropriate pdf models to obtain more accurate results under strong turbulence conditions.     

Accounting for residual stock in continuous review coordinated control of a family of items

Inventory models which permit coordinated control (or joint replenishment) of families of items are of considerable interest in practice. A major complexity in coordinated control is that, typically, one item triggers the family order and the other items have stock remaining above their reorder points. Account should be taken of this so-called residual stock when establishing the reorder points. In this paper, modelling the inventory position as a diffusion process permits us to develop expressions for the components of the residual stock probability density functions. Heuristics are presented for quick calculation of these expressions. Then we show how residual stock is taken into account in determining the reorder points of the individual items. The relationship of these residual stock pdf expressions to other components in a new class of coordinated control models is also discussed. This class of models has been shown to outperform existing models (IBM's IMPACT Inventory Control package(.     

Efficient determination of stability lobe diagrams by in-process varying of spindle speed and cutting depth

The experimental determination of stability lobe diagrams (SLDs) in milling can be realized by either continuously varying the spindle speed or by varying the depth of cut. In this paper, a method for combining both these methods along with an online chatter detection algorithm is proposed for efficient determination of SLDs. To accomplish this, communication between the machine control and chatter detection algorithm is established, and the machine axes are controlled to change the spindle speed or depth of cut. The efficiency of the proposed method is analyzed in this paper.The full text can be downloaded at https://link.springer.com/content/pdf/10.1007%2Fs40436-018-0225-x.pdf     

A simple method for including shear deformations in thin plate elements

A method is described for including the effect of shear deformations in existing thin plate finite elements, and thereby extending their range of application to include moderately thick plates. The method does not add extra degrees of freedom to the final element, so the thick and thin plate elements can be used interchangeably, and the thick plate solution is not appreciably more expensive than the thin plate solution. It is assumed that the shear deformations are constant over the element and, to account for this, two extra internal shear strain variables are added to the element. Various methods for eliminating these internal variables are examined but it is shown to be impossible to simultaneously satisfy both the constant bending moment and constant shear patch tests, except for parallelograms. However, one method gives elements which pass the constant shear patch test and, although failing the constant bending moment patch test for arbitrary geometries, gives errors which are small enough to be neglected in most engineering applications. This method has been applied to a triangular plate element and it is shown that the results obtained with this element converge (for all practical purposes) to the correct thick plate results.     

Receive maximal-ratio combining with outdated arbitrary transmit antenna selection in Nakagami-m fading

The use of transmit antenna selection algorithms in multiple-antenna systems enables significant reduction in implementation cost and complexity while maintaining acceptable performance. An attractive and quite flexible selection algorithm is to allow the receiver to pick any of the transmit antennas that can satisfy a predetermined performance target. Such an algorithm is referred to as the arbitrarily ordered transmit antenna selection algorithm. However, the effectiveness of transmit antenna selection is decreased by several propagation impairments over the feedback channel from the receiver to the transmitter. Of these impairments, the feedback channel time delay may impose a significant impact on the achieved performance. This paper aims to investigate the impact of this time delay on the performance of receive maximal-ratio combining (MRC) diversity employing the arbitrarily ordered transmit antenna selection algorithm. In order to obtain quantitative measures for this impact, new expressions for various performance criteria are obtained by using the new derived formulas for the probability density function (pdf) and the moment generating function (MGF) of the combined signal-to-noise ratio (SNR). Numerical and simulation results are presented to illustrate the effect of delayed (i.e. outdated) feedback information on the system performance for various transmit antenna selection scenarios.     

Half-Metallic Ferromagnetism in Strontium-Doped III–V: Ab Initio Calculations

First-principle calculations are employed by means of an all-electron full-potential linearized augmented plane wave to investigate the electronic structure and magnetic properties of AlP, AlAs, GaP, GaAs, InP, and InAs-based dilute magnetic semiconductors (DMS) with Sr impurities. It is shown that, in all the cases the ferromagnetic phase is energetically favored with respect to the paramagnetic one. In addition, these alloys are found to be half-metallic ferromagnets with a net total magnetic moment of 1.00 μ _B when they assume the zinc-blende structure at the equilibrium lattice constant. Ferromagnetism is induced by the spin polarization of the p shells of anions; the magnetic moment mainly comes from anion atoms surrounding the dopant atom, which is different from conventional DMS. These theoretical results make these materials interesting candidates for spin injection in spintronic devices.     

Acoustic emission theory for moment tensor analysis

Acoustic emission (AE) is extensively applied to nondestructive evaluation of materials and structures. In the conventional AE measurement, several AE parameters are detected and analyzed to elucidate characteristics of microfracturing behaviors in materials.Although theoretical treatment of AE waveforms was proposed more than one decade ago, the quantitative analysis was neither practical nor applicable to general AE waveforms. The crack mechanisms associated with AE generation consist of crack kinetics and crack kinematics. It has already been demonstrated that deconvolution analysis is available for determining crack kinetics. As for crack kinematics, it is known that the moment tensor analysis is promising, but has only been applied to marginal cases. In this respect, a practical procedure for the moment tensor analysis is recently formulated, selecting P wave portion from the full-space Green's function of homogeneous and isotropic material. A multi-channel observation is utilized to locate AE sources, based on arrival time differences of AE waves. Furthermore, a simplified moment tensor analysis is performed by analyzing the amplitudes of the first motions.It is clarified that the eigenvalues of general moment tensor components are available for classifying crack types and for determining crack orientations. To implement these results into an analytical procedure, a unified decomposition of the eigenvalues is proposed.With the emphasis on the development of this practical procedure for the moment tensor analysis, the theory of AE for the source characterization is reviewed. The results of a geological application and a test of reinforced concrete are discussed. A post-analysis is attempted to screen out poor solutions by comparing with theoretical solutions on the synthetic waveforms.     

EXACT DYNAMIC STIFFNESS MATRIX FOR BEAMS OF ARBITRARILY VARYING CROSS SECTIONS

In this paper, the exact dynamic stiffness matrix is derived for the transverse vibration of beams whose cross-sectional area and moment of inertia vary in accordance to any two arbitrary real-number powers. This variation represents a very large class of arbitrary varying beams and thus, fills the void currently existing in this area of research. With this approach, most beams can be modelled by just one element, and for beams having abrupt profile changes or with very complex profiles, they can be divided into separate distinct parts, with each of the part modelled by just one element, and then assembled together. The method is exact; however, the accuracy of the results depends only on the solver used to solve the exact frequency equation. To demonstrate the procedure, beams of non-linearly varying circular and elliptical cross-sections, and a combination beam consisting of a linear-tapered section, a uniform section and a non-linearly varying-section are analysed for their natural frequencies. Since there are no known solutions for these structures, comparison with finite element results was made and very good agreement was observed. © 1997 by John Wiley & Sons, Ltd.     

A compound scattering pdf for the ultrasonic echo envelope and its relationship to K and Nakagami distributions

A compound probability density function (pdf) is presented to describe the envelope of the backscattered echo from tissue. This pdf allows local and global variation in scattering cross sections in tissue. The ultrasonic backscattering cross sections are assumed to be gamma distributed. The gamma distribution also is used to model the randomness in the average cross sections. This gamma-gamma model results in the compound scattering pdf for the envelope. The relationship of this compound pdf to the Rayleigh, K, and Nakagami distributions is explored through an analysis of the signal-to-noise ratio of the envelopes and random number simulations. The three parameter compound pdf appears to be flexible enough to represent envelope statistics giving rise to Rayleigh, K, and Nakagami distributions.     

The diffusion process and residual stock in periodic review,coordinated control of families of items

Inventory control models which provide coordinated control ( or joint replenish ment) of families of items are very useful in practice. These models require expressions for the residual stock levels for each item in the family when a replenishment is placed. In this paper expressions are developed for the periodic review residual stock probability density function, when inventory position is modelled as a diffusion process, and heuristics are presented for quick calculation of these expressions. The importance of residual stock in setting inventory control model parameters is illustrated by showing how residual stock is taken into account in establishing item reorder points. The relationship of these residual stock pdf expressions to other components in a new class of coordinated control models is also discussed. This class of models has been shown to outperform existing models (IBM's IMPACT Inventory Control package).