Estimation of Flat‐topped Gaussian distribution with application in system identification

Uniformly distributed uncertainty exists in industrial process; additive error introduced by quantization is an example. To be able to handle additive uniform and Gaussian measurement uncertainty simultaneously in system identification, the Flat‐topped Gaussian distribution is considered in this paper as an alternative to the Gaussian distribution. To incorporate this type of uncertainty in the maximum likelihood estimation framework, the explicit form of its density function is of necessity. This work proposes an approach for obtaining both the functional structure and corresponding parameter estimation of Flat‐topped Gaussian distribution by a moment fitting strategy. The performance of the proposed approximation function is verified by comparison to the Flat‐topped Gaussian distributed random variable with different Gaussian and uniform components. Results of numerical simulations and industrial applications in system identification are presented to verify the effectiveness of the Flat‐topped Gaussian distribution for noise distribution in handling additional uniform uncertainty.     

Using an r-dependent Gaussian width in calculations of the globally uniform semiclassical wave function

The globally uniform semiclassical wave function expresses the solution to the time independent Schrodinger equation in terms of fixed width Gaussian wave packets traveling along a set of trajectories. There is a globally uniform wave function (GUWF) for each value of the Gaussian width parameter gamma. Numerical data show that a small Gaussian width is needed in some regions to obtain accurate results, while a broad Gaussian width provides better results in other regions. Since there is a semiclassically valid GUWF for every positive value of gamma, it is reasonable to employ the GUWF corresponding to a Gaussian width that provides good results at each value of r. A criterion for the r dependent choice of gamma is proposed and tested on one and two dimensional model problems. The results show that the use of an r dependent gamma in the GUWF results in improved accuracy for the model problems considered.     

Quantum statistical mechanics with Gaussians: equilibrium properties of van der Waals clusters

The variational Gaussian wave-packet method for computation of equilibrium density matrices of quantum many-body systems is further developed. The density matrix is expressed in terms of Gaussian resolution, in which each Gaussian is propagated independently in imaginary time beta=(k(B)T)(-1) starting at the classical limit beta=0. For an N-particle system a Gaussian exp[(r-q)(T)G(r-q)+gamma] is represented by its center qinR(3N), the width matrix GinR(3Nx3N), and the scale gammainR, all treated as dynamical variables. Evaluation of observables is done by Monte Carlo sampling of the initial Gaussian positions. As demonstrated previously at not-very-low temperatures the method is surprisingly accurate for a range of model systems including the case of double-well potential. Ideally, a single Gaussian propagation requires numerical effort comparable to the propagation of a single classical trajectory for a system with 9(N(2)+N)/2 degrees of freedom. Furthermore, an approximation based on a direct product of single-particle Gaussians, rather than a fully coupled Gaussian, reduces the number of dynamical variables to 9N. The success of the methodology depends on whether various Gaussian integrals needed for calculation of, e.g., the potential matrix elements or pair correlation functions could be evaluated efficiently. We present techniques to accomplish these goals and apply the method to compute the heat capacity and radial pair correlation function of Ne(13) Lennard-Jones cluster. Our results agree very well with the available path-integral Monte Carlo calculations.     

Gaussian induced dipole polarization model

A new induced dipole polarization model based on interacting Gaussian charge densities is presented. In contrast to the original induced point dipole model, the Gaussian polarization model is capable of finite interactions at short distances. Aspects of convergence related to the Gaussian model will be explored. The Gaussian polarization model is compared with the damped Thole-induced dipole model and the point dipole model. It will be shown that the Gaussian polarization model performs slightly better than the Thole model in terms of fitting to molecular polarizability tensors. An advantage of the model based on Gaussian charge distribution is that it can be easily generalized to other multipole moments and provide effective damping for both permanent electrostatic and polarization models. Finally, a method of parameterizing polarizabilities is presented. This method is based on probing a molecule with point charges and fitting polarizabilities to electrostatic potential. In contrast to the generic atom type polarizabilities fit to molecular polarizability tensors, probed polarizabilities are significantly more accurate in terms of reproducing molecular polarizability tensors and electrostatic potential, while retaining conformational transferability.     

An efficient and accurate implementation of centroid molecular dynamics using a Gaussian approximation

An approximate method for Centroid Molecular Dynamics (CMD) is presented which uses a Gaussian approximation. The resulting method, called Gaussian CMD (GCMD), is 100-1000 times faster than CMD because it replaces explicit path-integral sampling, which is the most time-consuming part of CMD, with a Gaussian averaging, which can be done analytically. Several methods for computing the Gaussian width parameter in the GCMD approach are also presented. This new method is shown to give satisfactory results for the position correlation function in simple one-dimensional systems when CMD itself is consistent with the exact result. The GCMD and CMD results are also compared for the case of 1-dimensional systems coupled to harmonic baths, with good success. GCMD is further compared to CMD with good success for liquid para-hydrogen at two different temperatures, 14 K and 25 K, and for ortho-deuterium at 20.7 K.     

On the measure of non-gaussianity of a nearly Gaussian noise

A procedure for the determining of the measure of non-gaussianity of nearly Gaussian noise is suggested, which used trimmed interval of concentration. The procedure is based on the Parseval finite-interval theorem. The following notions have been introduced: the scalar measure of non-gaussianity, the vector measure of non-gaussianity, and multicomponent partial measure of non-gaussianity. The method allows decreasing the probability of the mistaking of a purely Gaussian noise for nearly Gaussian one. The procedure was validated using a mixed normal distribution. The testing showed the finite-interval analog of the Gram-Charlier expansion to have only a few main (meaningful) components. The procedure for the determining of the measure of non-gaussianity of nearly Gaussian noise can be used in the noise diagnostics of various electrochemical systems and for noise monitoring of objects and processes in the electrochemical power engineering.     

Gaussian and finite-element Coulomb method for the fast evaluation of Coulomb integrals

The authors propose a new linear-scaling method for the fast evaluation of Coulomb integrals with Gaussian basis functions called the Gaussian and finite-element Coulomb (GFC) method. In this method, the Coulomb potential is expanded in a basis of mixed Gaussian and finite-element auxiliary functions that express the core and smooth Coulomb potentials, respectively. Coulomb integrals can be evaluated by three-center one-electron overlap integrals among two Gaussian basis functions and one mixed auxiliary function. Thus, the computational cost and scaling for large molecules are drastically reduced. Several applications to molecular systems show that the GFC method is more efficient than the analytical integration approach that requires four-center two-electron repulsion integrals. The GFC method realizes a near linear scaling for both one-dimensional alanine alpha-helix chains and three-dimensional diamond pieces.     

Matrix elements for Ĵ2 and Ĵz operators over explicitly correlated Cartesian Gaussian functions

General formalism for evaluation of multiparticle integrals involving J?² and J?_z operators over explicitly correlated Cartesian Gaussian functions is presented. The integrals are expressed in terms of the general overlap integrals. An explicitly correlated Cartesian Gaussian function is a product of spherical orbital Gaussian functions, powers of the Cartesian coordinates of the particle, and exponential Gaussian factors, which depend on interparticular distances. This development is relevant to both adiabatic and nonadiabatic calculations of energy and properties of multiparticle systems. © 1995 John Wiley & Sons, Inc.     

正常拖尾色谱峰的塔板模型表达式

得到了描述正常拖尾色谱峰的塔板模型表达式,根据这一表达式,正常的色谱流出曲线应是非对称的拖尾峰,而对称的高斯型分布函数是对塔板模型进行近似处理的结果。和扩散模型的色谱流出曲线方程相比,二者在形式上完全相同,因此,尽管塔板模型和扩散模型的机理不同,但它们对于色谱流出曲线的数学描述是完全相同的。     

On the use of the gaussian chain as a monte carlo simulation model for the equilibrium properties of polymer solutions

We have explored the performance of a simulation model for Gaussian chains at different concentrations in a good solvent. The Gaussian statistics for the distances between contiguous beads in the model is directly implemented in the individual moves of a Monte Carlo algorithm. When the results of conformational properties for the Gaussian model are compared with those provided by a freely jointed model in the same conditions, significant differences arise at finite concentrations. The modeled Gaussian chain yields incorrect results for the quadratic average dimensions 〈R²〉 and 〈S²〉 at high concentrations, but correctly reproduces the results for the scaled end-to-end distance distribution function at any concentration, showing the effects of the screening of excluded volume when concentration increases. The reason for the wrong behavior of the simulated Gaussian model comes from a strong distortion of the “bond distance” distribution as a result of the concentration increase. We conclude that this model can only be safely applied to infinitely dilute solutions.     

Rapid characterization of anthocyanins in red raspberry fruit by high-performance liquid chromatography coupled to single quadrupole mass spectrometry

This paper describes the results of a comparison of four peak functions in describing real chromatographic peaks. They are the empirically transformed Gaussian, polynomial modified Gaussian, generalized exponentially modified Gaussian and hybrid function of Gaussian and truncated exponential functions. Real chromatographic peaks of different shapes (fronting. symmetric, and tailing) are obtained by various separation conditions of reversed-phase liquid chromatography. They are then fitted to the peak functions via the Marquardt-Levenberg algorithm, a nonlinear least-squares curve-fitting procedure, by Microsoft Solver. The qualities of the fits are evaluated by the sum of the squares of the residuals. It is concluded in the study that the empirically transformed Gaussian function offers the highest flexibility (best fits) to all shapes of chromatographic peaks, including extremely asymmetric tailing peaks with a peak asymmetry of up to 8. The flexibility of this function should improve our ability to process chromatographic peaks such as deconvolution of overlapped peaks and smoothing noisy peaks for the determination of statistical moments.     

Time propagation of constrained coupled Gaussian wave packets

The dynamics of quantum systems can be approximated by the time propagation of Gaussian wave packets. Applying a time dependent variational principle, the time evolution of the parameters of the coupled Gaussian wave packets can be calculated from a set of ordinary differential equations. Unfortunately, the set of equations is ill behaved in most practical applications, depending on the number of propagated Gaussian wave packets, and methods for regularization are needed. We present a general method for regularization based on applying adequate nonholonomic inequality constraints to the evolution of the parameters, keeping the equations of motion well behaved. The power of the method is demonstrated for a nonintegrable system with two degrees of freedom.     

Distributed basis sets of s-type Gaussian functions for molecular electronic structure calculations: Applications of the Gaussian cell model to one-electron polycentric linear molecular systems

A particular formulation of the distributed Gaussian basis-set approach, the extended Gaussian cell model, is applied to the simplest polycentric molecule, the linear H₃²⁺ ion. Calculations of the total energy using two extensions of the original Gaussian cell model are described and results are reported for the ground state and the first excited state. A comparison with recently reported finite element calculations is made for a number of nuclear geometries. © 1996 John Wiley & Sons, Inc.     

What role do surfaces play in GB models? A new‐generation of surface‐generalized born model based on a novel gaussian surface for biomolecules

We have developed a version of our surface generalized Born (SGB) model that employs a Gaussian surface, as opposed to the van der Waals surface used previously. The Gaussian surface is smooth and its properties are analytically differentiable with respect to the positions of atoms. A significant advantage of a solvent model based on this analytically differentiable surface is the availability of analytical gradients of the surface and solvation forces. An efficient and robust algorithm is designed to construct and triangulate the Gaussian surface for large biomolecules with arbitrary shapes, and to compute the various terms required for energy gradients. The Gaussian surface is shown to better mimic the boundary between the solute and solvent by properly addressing solvent accessibility, as is demonstrated by comparisons with standard Poisson-Boltzmann calculations for proteins of different sizes. These results also demonstrate that surface definition is a dominant contribution to differences between GB and PB calculations, especially if the system is large. Application of the new surface to prediction of long loop regions is presented, and significant improvement in the energetics is seen compared with results obtained using the van der Waals surface, even in the absence of optimized empirical correction terms that were used in the latter calculations.     

Double‐step block division plant‐wide fault detection and diagnosis based on variable distributions and relevant features

Large‐scale process data in plant‐wide process monitoring are characterized by two features: complex distributions and complex relevance. This study proposes a double‐step block division plant‐wide process monitoring method based on variable distributions and relevant features to overcome this limitation. First, the data distribution is considered, and the normality test method called the D‐test is applied to classify the variables with the same distribution (i.e., Gaussian distribution or non‐Gaussian distribution) in a block. Thus, the second block division is implemented on both blocks obtained in the previous step. The mutual information shared between two variables is used to generate relevant matrixes of the Gaussian and non‐Gaussian blocks. The K‐means method clusters the vectors of the relevant matrix. Principal component analysis is conducted to monitor each Gaussian subblock, whereas independent component analysis is conducted to monitor each non‐Gaussian subblock. A composite statistic is eventually derived through Bayesian inference. The proposed method is applied to a numerical system and the Tennessee Eastman process data set. The monitoring performance shows the superiority of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Adaptive density partitioning technique in the auxiliary plane wave method

We have developed the adaptive density partitioning technique (ADPT) in the auxiliary plane wave method, in which a part of the density is expanded to plane waves, for the fast evaluation of Coulomb matrix. Our partitioning is based on the error estimations and allows us to control the accuracy and efficiency. Moreover, we can drastically reduce the core Gaussian products that are left in Gaussian representation (its analytical integrals is the bottleneck in this method). For the taxol molecule with 6-31G** basis, the core Gaussian products accounted only for 5% in submicrohartree error.     

A study of the performance of numerical basis sets in DFT calculations on sulfur-containing molecules

The performance of numerical basis sets in relation to Gaussian basis sets is examined, by studying 20 small sulfur-containing molecules. The results of geometry optimization calculations are reported for each molecule using both density functional and Hartee-Fock methods. In comparison with experimental data, it is shown that the use of numerical bases tend to overestimate structural parameters, particularly bond lengths, and, in most cases, more than Gaussian basis sets. It is also shown that the use of a larger Gaussian basis set in DFT calculations has the effect of reducing bond lengths. © 1996 John Wiley & Sons, Inc.     

Long-time and unitary properties of semiclassical initial value representations

We numerically compare the semiclassical "frozen Gaussian" Herman-Kluk propagator [Chem. Phys. 91, 27 (1984)] and the "thawed Gaussian" propagator put forward recently by Baranger et al. [J. Phys. A 34, 7227 (2001)] by studying the quantum dynamics in some nonlinear one-dimensional potentials. The reasons for the lack of long-time accuracy and norm conservation in the latter method are uncovered. We amend the thawed Gaussian propagator with a global harmonic approximation for the stability of the trajectories and demonstrate that this revised propagator is a true alternative to the Herman-Kluk propagator with similar accuracy.     

Choice of Spin–Orbit Correction Terms in Gaussian Model Chemistries

Spin–orbit correction terms for use in Gaussian‐2 theory and other model chemistries for third‐row atoms and molecules are calculated by several methods with the objective of finding a reliable method that can be applied in a routine and economical manner in the spirit of Gaussian model chemistries. The results are evaluated for the test set of molecules and ions used in the original extension of Gaussian‐2 theory to third‐row atoms. Further results are presented for systems where Gaussian‐2 results are reported in the literature without spin–orbit correction terms and for some larger molecules. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1552–1556, 2001