General formulation of HDMR component functions with independent and correlated variables

The High Dimensional Model Representation (HDMR) technique decomposes an n-variate function f (x) into a finite hierarchical expansion of component functions in terms of the input variables x = (x ₁, x ₂, . . . , x _n ). The uniqueness of the HDMR component functions is crucial for performing global sensitivity analysis and other applications. When x ₁, x ₂, . . . , x _n are independent variables, the HDMR component functions are uniquely defined under a specific so called vanishing condition. A new formulation for the HDMR component functions is presented including cases when x contains correlated variables. Under a relaxed vanishing condition, a general formulation for the component functions is derived providing a unique HDMR decomposition of f (x) for independent and/or correlated variables. The component functions with independent variables are special limiting cases of the general formulation. A novel numerical method is developed to efficiently and accurately determine the component functions. Thus, a unified framework for the HDMR decomposition of an n-variate function f (x) with independent and/or correlated variables is established. A simple three variable model with a correlated normal distribution of the variables is used to illustrate this new treatment.     

High dimensional model representation constructed by support vector regression. I. Independent variables with known probability distributions

High dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for capturing high dimensional input-output system behavior. In practice, the HDMR component functions are each approximated by an appropriate basis function expansion. This procedure often requires many input-output samples which can restrict the treatment of high dimensional systems. In order to address this problem we introduce svr-based HDMR to efficiently and effectively construct the HDMR expansion by support vector regression (SVR) for a function \(f(\mathbf{x})\). In this paper the results for independent variables sampled over known probability distributions are reported. The theoretical foundation of the new approach relies on the kernel used in SVR itself being an HDMR expansion (referred to as the HDMR kernel ), i.e., an ANOVA kernel whose component kernels are mutually orthogonal and all non-constant component kernels have zero expectation. Several HDMR kernels are constructed as illustrations. While preserving the characteristic properties of HDMR, the svr-based HDMR method enables efficient construction of high dimensional models with satisfactory prediction accuracy from a modest number of samples, which also permits accurate computation of the sensitivity indices. A genetic algorithm is employed to optimally determine all the parameters of the component HDMR kernels and in SVR. The svr-based HDMR introduces a new route to advance HDMR algorithms. Two examples are used to illustrate the capability of the method.     

General foundations of high‐dimensional model representations

A family of multivariate representations is introduced to capture the input–output relationships of highdimensional physical systems with many input variables. A systematic mapping procedure between the inputs and outputs is prescribed to reveal the hierarchy of correlations amongst the input variables. It is argued that for most welldefined physical systems, only relatively loworder correlations of the input variables are expected to have an impact upon the output. The highdimensional model representations (HDMR) utilize this property to present an exact hierarchical representation of the physical system. At each new level of HDMR, higherorder correlated effects of the input variables are introduced. Tests on several systems indicate that the few lowestorder terms are often sufficient to represent the model in equivalent form to good accuracy. The input variables may be either finitedimensional (i.e., a vector of parameters chosen from the Euclidean space ) or may be infinitedimensional as in the function space . Each hierarchical level of HDMR is obtained by applying a suitable projection operator to the output function and each of these levels are orthogonal to each other with respect to an appropriately defined inner product. A family of HDMRs may be generated with each having distinct character by the use of different choices of projection operators. Two types of HDMRs are illustrated in the paper: ANOVAHDMR is the same as the analysis of variance (ANOVA) decomposition used in statistics. Another cutHDMR will be shown to be computationally more efficient than the ANOVA decomposition. Application of the HDMR tools can dramatically reduce the computational effort needed in representing the input–output relationships of a physical system. In addition, the hierarchy of identified correlation functions can provide valuable insight into the model structure. The notion of a model in the paper also encompasses input–output relationships developed with laboratory experiments, and the HDMR concepts are equally applicable in this domain. HDMRs can be classified as nonregressive, nonparametric learning networks. Selected applications of the HDMR concept are presented along with a discussion of its general utility.     

Multicut-HDMR with an application to an ionospheric model

A new High Dimensional Model Representation (HDMR) tool, Multicut-HDMR, is introduced and applied to an ionospheric electron density model. HDMR is a general set of quantitative model assessment and analysis tools for improving the efficiency of deducing high-dimensional input-output system behavior. HDMR describes an output [f(x)] in terms of its input variables (x = [x(1), x(2), em leader, x(n)]) via a series of finite, hierarchical, correlated function expansions. Various forms of HDMR are constructed for different purposes such as modeling laboratory or field data, or reproducing a complicated mathematical model. The Cut-HDMR technique, which expresses f(x) with respect to a specified reference point x in the input space, is appropriate when the input space is sampled in an orderly fashion. However, if the desired domain of the input space is too large, the HDMR function expansion may not converge, and Cut-HDMR will be unable to accurately approximate f(x). The new Multicut-HDMR technique addresses this problem through the use of multiple reference points in the input space.     

On the fundamental conjecture of HDMR: a Fourier analysis approach

Although the HDMR decomposition has become an important tool for the understanding of high dimensional functions, the fundamental conjecture underlying its practical utility is still open for theoretical analysis. In this paper, we introduce the HDMR decomposition in conjunction with the Fourier-HDMR approximation leading to the following conclusions: (1) we suggest a type of Fourier-HDMR approximation for certain classes of differentiable functions; (2) utilizing the Fourier-HDMR method, we prove the fundamental conjecture about the dominance of low order terms in the HDMR expansion under relevant conditions, and we also obtain error estimates of the truncated HDMR expansion up to order u; (3) we prove the domain decomposition approximation theorem which shows that the global Fourier-HDMR approximation is not always optimal for a given accuracy order; (4) and finally, a piecewise Fourier-HDMR approach is discussed for high dimensional modeling. These results help to further understand how to efficiently represent the high dimensional functions.     

Regularized random-sampling high dimensional model representation (RS-HDMR)

High Dimensional Model Representation (HDMR) is under active development as a set of quantitative model assessment and analysis tools for capturing high-dimensional input–output system behavior. HDMR is based on a hierarchy of component functions of increasing dimensions. The Random-Sampling High Dimensional Model Representation (RS-HDMR) is a practical approach to HDMR utilizing random sampling of the input variables. To reduce the sampling effort, the RS-HDMR component functions are approximated in terms of a suitable set of basis functions, for instance, orthonormal polynomials. Oscillation of the outcome from the resultant orthonormal polynomial expansion can occur producing interpolation error, especially on the input domain boundary, when the sample size is not large. To reduce this error, a regularization method is introduced. After regularization, the resultant RS-HDMR component functions are smoother and have better prediction accuracy, especially for small sample sizes (e.g., often few hundred). The ignition time of a homogeneous H₂/air combustion system within the range of initial temperature, 1000 < T ₀ < 1500 K, pressure, 0.1 < P < 100 atm and equivalence ratio of H₂/O₂, 0.2 < R < 10 is used for testing the regularized RS-HDMR.      

Velocity measurement of dc plasma jets based on arc root fluctuations

The axial component of the radial velocity distribution of a plasma flow generated by a dc plasma spray torch was measured by using a nonintrusive optical method, based on the propagation of the plasma jet luminosity fluctuations. In contrast to the simplicity of the experimental set-up, a special effort was made in the data processing, namely by using numerical techniques defined in the context of signal theory. Both centerline and radial profiles of the axial velocity were obtained for pure Ar and Ar–H₂ plasma flows. These experimental results were satisfactorily validated by calculating enthalpy and mass balances.Notation 1 refers to the reference signal - 2 refers to the shifted signal - a dimensionless adjustment factor - A ₁,_{f 1} values used for the estimation ofs ₁(t) energy spectrum - B ₁ frequency cutoff (Hz) - f frequency (Hz) - F ₁(f),F ₂(f) optimum low-pass filters - h mass enthalpy (J/kg) - h° enthalpy flowrate (J/s) - I(y) wavelength integrated radiation intensity - n ₁(t),F ₂(t) white noise - N ₁(t),N ₂() n ₁(t),n ₂(t)Fourier transforms - P _i probability - r radial coordinate - R plasma radius - s ₁(t),s ₂(t) signal components to be correlated - S ₁(f),S ₂(f) s ₁(t),s ₂(t) Fourier transforms - t time - T temperature (K) - , _i velocity (m/s) - x ₁(t),x ₂(t) recorded signals - X ₁(f),X ₂(f) x ₁(t),x ₂(t) Fourier transforms - x horizontal distance from the jet axis - y vertical distance from the jet axis - average velocity - standard deviation - z axial distance from the nozzle exit Greek Symbols dimensionless constant - r radial position uncertainty - velocity uncertainty - x width of the emission coefficient profile - z distance between the two sampling points - (r) radiative emission coefficient - time delay betweens ₁(t) ands ₂(t) - ₁ ^dg () x₁(t) autocorrelation function = x ₁(t)x ₁(t+)dt - ₁ ^dg (f) x₁(t) energy spectrum - _s1(), _s2() s ₁(t),s ₂(t) autocorrelation functions - _xt(f) s ₁(t) energy spectrum - ₁ ² , ₂ ² energy spectrum of white noisen ₁(t),n ₂(t) - ₁ time of correlation - ₁₂() cross-correlation function - coherence factor     

Efficient Implementation of High Dimensional Model Representations

Physical models of various phenomena are often represented by a mathematical model where the output(s) of interest have a multivariate dependence on the inputs. Frequently, the underlying laws governing this dependence are not known and one has to interpolate the mathematical model from a finite number of output samples. Multivariate approximation is normally viewed as suffering from the curse of dimensionality as the number of sample points needed to learn the function to a sufficient accuracy increases exponentially with the dimensionality of the function. However, the outputs of most physical systems are mathematically well behaved and the scarcity of the data is usually compensated for by additional assumptions on the function (i.e., imposition of smoothness conditions or confinement to a specific function space). High dimensional model representations (HDMR) are a particular family of representations where each term in the representation reflects the individual or cooperative contributions of the inputs upon the output. The main assumption of this paper is that for most well defined physical systems the output can be approximated by the sum of these hierarchical functions whose dimensionality is much smaller than the dimensionality of the output. This ansatz can dramatically reduce the sampling effort in representing the multivariate function. HDMR has a variety of applications where an efficient representation of multivariate functions arise with scarce data. The formulation of HDMR in this paper assumes that the data is randomly scattered throughout the domain of the output. Under these conditions and the assumptions underlying the HDMR it is argued that the number of samples needed for representation to a given tolerance is invariant to the dimensionality of the function, thereby providing for a very efficient means to perform high dimensional interpolation. Selected applications of HDMR's are presented from sensitivity analysis and time-series analysis.     

Use of the parametric representation method in revealing the root structure and hopf bifurcation

In practical applications of dynamical systems, it is often necessary to determine the number and the stability of the stationary states. The parameric respresentation method is a useful tool in such problems. Consider the two parameter families of functions:f(x) =u _o +u ₁ x +g(x), whereu _o andu ₁ are the parameters. We are interested in the number of zeros as well as in the stability. We want to determine the stable region on the parameter plane, where the real parts of the roots off are negative. The D-curve (along which the discriminant off is zero) helps us. We applied the method to the cases of cubic and quartic equation, giving pictorial meaning to the root structure. In this respect, the R-curves and the I-curves (along which the sum or difference, respectively, of two zeros is constant) also have a significance. Using these concepts, we established a relation between the (n - 1)th Routh-Hurwitz condition and the Hopf bifurcation.     

The convolution theorem and the Franck–Condon integral

The convolution theorem is used to evaluate the Franck–Condon integral. It is shown that this integral becomes the matrix element between two “squeezed” states. This enables one to evaluate the integral by using boson operators. In addition, a general method is developed to obtain integrals involving Hermite polynomials with a displaced argument. In particular, the two‐center matrix element _g〈m|f(x_e)|n〉_e, is obtained, where f(x_e)=exp(Dx+Fx_e). ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 11–15, 1999     

Exchange kernel of density functional response theory from the common energy denominator approximation (CEDA) for the Kohn–Sham Green's function

A complete and explicit expression for the exchange kernel f _x of density functional response theory (DFRT) is derived in terms of the occupied Kohn-Sham (KS) orbitals _i. It is based on the common energy denominator approximation (CEDA) for the KS Green's function (O. V. Gritsenko and E. J. Baerends, Phys. Rev. A 64, 042506 (2001)). The kernel f _x ^CEDA is naturally subdivided into the Slater f _S ^CEDA and the 'response' f _resp ^CEDA parts, which are the derivatives of the Slater _S and response _resp potentials, respectively. While f _S ^CEDA is obtained with a straightforward differentiation of _S , some terms of f _resp ^CEDA are obtained from the solution of linear equations for the corresponding derivatives. All components of f _x ^CEDA are explicitly expressed in terms of the products _i ^* _j of the occupied KS orbitals taken at the positions r ₁ and r ₂, as well as the potentials of these products at r ₃. The coefficients in these expressions are obtained by inversion of the matrix, associated with the overlap matrix of the products _i ^* _j and _k ^* _l . Terms are indicated, which generate in an external electric field an ultra-nonlocal potential _x , counteracting an external field, and possible approximations to f _x ^CEDA are considered.     

Symmetry-adapted cluster f-orbitals and a vector method for generally oriented f-orbital overlaps

The irreducible representations consisting of linear combinations of cluster atomic f-orbitals are obtained for f, f,f and f orbitals in M₃(D _3h ), M₄(D _4h ), M₄(T _d ), M₆(O _h ) and M₈(O _h ) clusters. The charge overlap of any pair of two atoms in a cluster is decomposed in terms of a set of coefficients for the , , , and overlaps, respectively. A vector method is devised for this decomposition which may be extended to any arbitrary orientation and to higher orbitals. The decomposition coefficients represent fundamental geometrical properties of the cluster and are applicable to clusters of arbitrary dimensions.Contribution No. 0004 ITI Basic Research Laboratory. Presented in part at the 187th ACS National Meeting in St. Louis, MO, USA     

Coefficients of Linear Thermal Expansion of Solid Electrolytes with Co-cation Conduction Based on γ-K4P2O7, γ-KFeO2, and γ-LiAlO2 Structures

Linear thermal expansion coefficients (K) for co-cation solid electrolytes of three types are measured. The electrolytes include solid solutions (K_{1 – x} Rb _x )_3.8Me_0.1P₂O₇ (Me = Ca, Ba) with the structure of -K₄P₂O₇ (I); fcc-solid solutions (A_{1 – x} A" _x )MO₂–EO₂ (A, A" = K, Rb, Cs; M = Al, Ga, Fe; E = Si, Ge, Ti) of -KFeO₂ type (II); and tetragonal solid solutions 0.8(Li_{1 – x} Na _x AlO₂) · 0.2TiO₂ with the LiAlO₂ structure (III). Dependences of K on the ratio of alkali cation amounts in I and II have a maximum, i.e. the polyalkali effect (PAE) of K is observed, while in III this dependence is practically linear (PAE of K is absent). The correlation is found between PAE of K and the electroconductivity of the electrolytes.     

Ruthenium Tris(pyrazolyl)borate Complexes. Part 20 [1]. Synthesis, Characterization, and Reactivity of Neutral Trispyrazolylborate Ruthenium Vinylidene Complexes

The reaction of RuTp(COD)Cl (1) with PPh₂Prⁱ and terminal alkynes HCCR (R=C₆H₅, C₄H₃S, C₆H₄OMe, Fc, C₆H₄–Fc, C₆H₉) affords the neutral vinylidene complexes RuTp(PPh₂Prⁱ) (Cl)(=C=CHR) (2a–2f) in high yields. These complexes do not react with MeOH to give methoxy carbene complexes of the type RuTp(PPh₂Prⁱ)(Cl)(=C(OMe)CH₂R), but react with oxygen to yield the CO complex RuTp(PPh₂R)(Cl)(CO) (3). The structures of 2b, 2f, and 3 have been determined by X-ray crystallography.     

Ruthenium Tris(pyrazolyl)borate Complexes. Part 20 [1]. Synthesis,Characterization, and Reactivity of Neutral Trispyrazolylborate Ruthenium Vinylidene Complexes

Summary. The reaction of RuTp(COD)Cl (1) with PPh₂Prⁱ and terminal alkynes HCCR (R=C₆H₅, C₄H₃S, C₆H₄OMe, Fc, C₆H₄–Fc, C₆H₉) affords the neutral vinylidene complexes RuTp(PPh₂Prⁱ) (Cl)(=C=CHR) (2a–2f) in high yields. These complexes do not react with MeOH to give methoxy carbene complexes of the type RuTp(PPh₂Prⁱ)(Cl)(=C(OMe)CH₂R), but react with oxygen to yield the CO complex RuTp(PPh₂R)(Cl)(CO) (3). The structures of 2b, 2f, and 3 have been determined by X-ray crystallography.     

Modeling of Substituent Effects: Self-Parametrization, a Case of Interdependence of the Coefficients, Illustrated on the Linear Plus Cross-Term Models

The effect of one or several substituents X _i on a molecular property f (reaction rate, spectroscopic property, etc.) is often described in terms of substituent parameters _i (X) depending on the nature X of the substituent, and the position i of substitution. We call self-parametrization the case when the substituents are parameterized by the value _i (X)=f _i (X) of f itself in the molecule monosubstituted by X in position i. On the example of the linear plus cross-term models, we show that self-parametrization implies that the coefficients of the model are interdependent. The theoretical relations between them are given, for equivalent or nonequivalent positions of substitution, and tested on measured acid–base equilibrium constants, and NMR coupling constants.     

Correlation factor for ground state of helium

If a variational trial function of form exp(–ar ₁ –ar ₂)f(r ₁₂) is postulated for the ground-state of helium-like ions, then for a given a it is shown that the variation principle leads to an ordinary second order differential equation for f, the solution of which represents the optimum function f for use with a trial function of this type, in the sense that this solution minimises the expectation value of the Hamiltonian for the system. A solution to the differential equation may be found by the usual series expansion method.

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Zusammenfassung Wenn ein Variationsansatz der Form exp(–ar ₁ –ar ₂ f(r ₁₂) für den Grundzustand von Heartigen Ionen vorausgesetzt wird, so wird gezeigt, daß (bei gegebenem a) das Variationsprinzip zu einer gewöhnlichen Differentialgleichung 2. Ordnung für f führt. Ihre Lösung stellt eine optimale, den Erwartungswert des Hamiltonoperators des Systems minimisierende Funktion des angegebenen Typs dar. Eine Lösung der Differentialgleichung kann mit der gewöhnlichen Methode eines Reihenentwicklungs-Ansatzes gefunden werden.

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Résumé Si l'on prend comme fonction variationnelle d'essai pour l'état fondamental des ions de type hélium: exp(–ar₁ –ar ₂)f(r ₁₂) le principe variationnel mène pour a constant à une équation différentielle du second ordre pour f. La solution de cette équation représente la «meilleure» fonction f à utiliser avec une fonction d'essai de ce type, car elle minimise la valeur moyenne de l'hamiltonien. Cette solution peut être obtenue par la méthode ordinaire de développement en série.

     

Glass Transition of Low‐Dimensional Polystyrene

Summary: A unified model is developed for the finite size‐effect on the glass‐transition temperature of polymers, T_g(D), where D denotes the diameter of particles or thickness of films. In terms of this model, T_g depends on both the size and interface conditions. The predicted results are consistent with the experimental evidence for polystyrene (PS) particles and films with different interface situations.

T_g(D) function of free‐standing PS films.     

Synthesis,Characterization, Electrochemical and Spectroscopic Properties of [Ru(dppt)(bpy)Cl]+ and [Ru(pta)(bpy)Cl]+

Two novel Ru^II complexes [Ru(dppt)(bpy)Cl]ClO₄ (1) and [Ru(pta)(bpy)Cl]ClO₄ (2)[dppt, pta and bpy = 3-(1,10-phenanthrolin-2-yl)-5,6-diphenyl-as-triazine, 3-(1,10-phenanthrolin-2-yl)-as-triazino[5,6-f]acenaphthylene and 2,2-bipyridine, respectively] were synthesized and characterized by elemental analysis and electrospray mass spectrometry, ¹H-n.m.r., and u.v.–vis spectroscopy. The redox properties of the complexes were examined using cyclic voltammetry. Due to the strong -accepting character of asymmetric ligands, the MLCT bands of (1) and (2) are shifted significantly to lower energies by comparison with [Ru(tpy)(bpy)Cl]⁺.