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.     

Combined small scale high dimensional model representation

Nowadays the utilization of High Dimensional Model Representation (HDMR), which is an algorithm for approximating multivariate functions, is becoming more pervasive in the applications of approximation theory. This extensive usage motivates new works on HDMR, to get better solutions while approximating to the multivariate functions. One of them is recently developed “Combined Small Scale High Dimensional Model Representation (CSSHDMR)". This new scheme not only optimises HDMR results but also provides good approximation with less terms than HDMR does. This paper presents the theory and the numerical results of the new method and shows that it is possible to apply approximation to multivariate functions by keeping only constant term of HDMR. From this aspect CSSHDMR can be used in any scientific problem which includes multivariate functions, from chemistry to statistics.     

Hybrid HDMR method with an optimized hybridity parameter in multivariate function representation

High Dimensional Model Representation (HDMR) based methods are used to generate an approximation for a given multivariate function in terms of less variate functions. This paper focuses on Hybrid HDMR which is composed of Plain HDMR and Logarithmic HDMR. The Plain HDMR method works well for representing multivariate functions having additive nature. If the function under consideration has a multiplicative nature, then the Logarithmic HDMR method produces better approximation. Hybrid HDMR method aims to successfully represent a multivariate function having neither purely additive nor purely multiplicative nature under a hybridity parameter. The performance of the Hybrid HDMR method strongly depends on the value of this hybridity parameter because this parameter manages the contribution level of Plain and Logarithmic HDMR expansions. The main purpose of this work is to optimize the hybridity parameter to get the best approximations. Fluctuationlessness Approximation Theorem is used in this optimization process and in evaluating the multiple integrals appearing in HDMR based methods. A number of numerical implementations are given at the end of the paper to show the performance of our proposed method.     

Random sampling-high dimensional model representation (RS-HDMR) and orthogonality of its different order component functions

High dimensional model representation is under active development as a set of quantitative model assessment and analysis tools for capturing high-dimensional input-output system behavior based on a hierarchy of functions of increasing dimensions. The HDMR component functions are optimally constructed from zeroth order to higher orders step-by-step. This paper extends the definitions of HDMR component functions to systems whose input variables may not be independent. The orthogonality of the higher order terms with respect to the lower order ones guarantees the best improvement in accuracy for the higher order approximations. Therefore, the HDMR component functions are constructed to be mutually orthogonal. The RS-HDMR component functions are efficiently constructed from randomly sampled input-output data. The previous introduction of polynomial approximations for the component functions violates the strictly desirable orthogonality properties. In this paper, new orthonormal polynomial approximation formulas for the RS-HDMR component functions are presented that preserve the orthogonality property. An integrated exposure and dose model as well as ionospheric electron density determined from measured ionosonde data are used as test cases, which show that the new method has better accuracy than the prior one.     

Local relativistic exact decoupling

We present a systematic hierarchy of approximations for local exact decoupling of four-component quantum chemical Hamiltonians based on the Dirac equation. Our ansatz reaches beyond the trivial local approximation that is based on a unitary transformation of only the atomic block-diagonal part of the Hamiltonian. Systematically, off-diagonal Hamiltonian matrix blocks can be subjected to a unitary transformation to yield relativistically corrected matrix elements. The full hierarchy is investigated with respect to the accuracy reached for the electronic energy and for selected molecular properties on a balanced test molecule set that comprises molecules with heavy elements in different bonding situations. Our atomic (local) assembly of the unitary exact-decoupling transformation--called local approximation to the unitary decoupling transformation (DLU)--provides an excellent local approximation for any relativistic exact-decoupling approach. Its order-N(2) scaling can be further reduced to linear scaling by employing a neighboring-atomic-blocks approximation. Therefore, DLU is an efficient relativistic method well suited for relativistic calculations on large molecules. If a large molecule contains many light atoms (typically hydrogen atoms), the computational costs can be further reduced by employing a well-defined nonrelativistic approximation for these light atoms without significant loss of accuracy. We also demonstrate that the standard and straightforward transformation of only the atomic block-diagonal entries in the Hamiltonian--denoted diagonal local approximation to the Hamiltonian (DLH) in this paper--introduces an error that is on the order of the error of second-order Douglas-Kroll-Hess (i.e., DKH2) when compared with exact-decoupling results. Hence, the local DLH approximation would be pointless in an exact-decoupling framework, but can be efficiently employed in combination with the fast to evaluate DKH2 Hamiltonian in order to speed up calculations for which ultimate accuracy is not the major concern.     

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.     

Constancy maximization based weight optimization in high dimensional model representation for multivariate functions

High Dimensional Model Representation (HDMR) method is a technique that represents a multivariate function in terms of less-variate functions. Even though the method has a finite expansion, to determine the components of this expansion is very expensive due to integration based natures of the components. Hence, the HDMR expansion is generally truncated at certain multivariance level and such approximations are produced to represent the given multivariate function approximately. The weight function selection becomes an important issue for the HDMR based applications when it is desired to give different importances to function values at different points. An appropriately chosen weight function may increase the quality of the approximation incredibly. This work aims at a multivariate weight function optimization to obtain high quality approximations through the HDMR method to represent multivariate functions. The proposed optimization considers constancy measurer maximization which produces a quadratic vector equation to be solved. Another contribution of this work is to use a recently developed method, fluctuation free integration, with HDMR, to solve this equation easily. This work is an extension of a previous work about weight optimization in HDMR for univariate functions.     

Fluctuation free multivariate integration based logarithmic HDMR in multivariate function representation

This paper focuses on the Logarithmic High Dimensional Model Representation (Logarithmic HDMR) method which is a divide–and–conquer algorithm developed for multivariate function representation in terms of less-variate functions to reduce both the mathematical and the computational complexities. The main purpose of this work is to bypass the evaluation of N–tuple integrations appearing in Logarithmic HDMR by using the features of a new theorem named as Fluctuationlessness Approximation Theorem. This theorem can be used to evaluate the complicated integral structures of any scientific problem whose values can not be easily obtained analytically and it brings an approximation to the values of these integrals with the help of the matrix representation of functions. The Fluctuation Free Multivariate Integration Based Logarithmic HDMR method gives us the ability of reducing the complexity of the scientific problems of chemistry, physics, mathematics and engineering. A number of numerical implementations are also given at the end of the paper to show the performance of this new method.     

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.     

studies in the relationship between molecular structure and chromatographic behaviour-III Mechanism of the reversed-phase chromatography of some alkyl phenols

It can be shown that thin-layer chromatography by the reversed-phase technique gives results which are sufficiently reproducible for correlation of Chromatographie behaviour with chemical structure to be attempted. To a first approximation the Martin additivity principle is valid, but variations in the intermolecular forces between solute molecules and solvent molecules, caused by the easy transmission of electronic effects through an aromatic system, result in deviations from ideal conditions.     

Addition by subtraction in coupled-cluster theory: a reconsideration of the CC and CI interface and the nCC hierarchy

The nCC hierarchy of coupled-cluster approximations, where n guarantees exactness for n electrons and all products of n electrons are derived and applied to several illustrative problems. The condition of exactness for n=2 defines nCCSD=2CC, with nCCSDT=3CC and nCCSDTQ=4CC being exact for three and four electrons. To achieve this, the minimum number of diagrams is evaluated, which is less than in the corresponding CC model. For all practical purposes, nCC is also the proper definition of a size-extensive CI. 2CC is also an orbitally invariant coupled electron pair approximation. The numerical results of nCC are close to those for the full CC variant, and in some cases are closer to the full CI reference result. As 2CC is exact for separated electron pairs, it is the natural zeroth-order approximation for the correlation problem in molecules with other effects introduced as these units start to interact. The nCC hierarchy of approximations has all the attractive features of CC including its size extensivity, orbital invariance, and orbital insensitivity, but in a conceptually appealing form suited to bond breaking, while being computationally less demanding. Excited states from the equation of motion (EOM-2CC) are also reported, which show results frequently approaching those of EOM-CCSDT.     

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.      

High Dimensional Model Representations Generated from Low Dimensional Data Samples. I. mp-Cut-HDMR

High dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for improving the efficiency of deducing high dimensional input–output system behavior. For a high dimensional system, an output f(x) is commonly a function of many input variables x=|x ₁,x ₂,...,x _n } with n10² or larger. HDMR describes f(x) by a finite hierarchical correlated function expansion in terms of the input variables. Various forms of HDMR can be constructed for different purposes. Cut- and RS-HDMR are two particular HDMR expansions. Since the correlated functions in an HDMR expansion are optimal choices tailored to f(x) over the entire domain of x, the high order terms (usually larger than second order, or beyond pair cooperativity) in the expansion are often negligible. When the approximations given by the first and the second order Cut-HDMR correlated functions are not adequate, this paper presents a monomial based preconditioned HDMR method to represent the higher order terms of a Cut-HDMR expansion by expressions similar to the lower order ones with monomial multipliers. The accuracy of the Cut-HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input–output samples without directly invoking the determination of higher order terms. The mathematical foundations of monomial based preconditioned Cut-HDMR is presented along with an illustration of its applicability to an atmospheric chemical kinetics model.     

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.     

Multivariate data modelling through Piecewise generalized HDMR method

This work aims to develop a new High Dimensional Model Representation (HDMR) based method which can construct an analytical structure for a given multivariate data modelling problem. Modelling multivariate data through a divide-and-conquer method stands for multivariate data partitioning process in which we deal with a number of less variate data sets instead of a single N dimensional problem. Generalized HDMR is one of these methods used to model a multivariate data set which has a number of scattered nodes with associated function values. However, Generalized HDMR includes a linear equation system with huge number of unknowns and equations to be solved. This equation sometimes has linearly dependent equations in it and this is an undesirable situation. This work offers a new method named Piecewise Generalized HDMR method which bypasses this disadvantage as well as reducing the mathematical complexity and CPU time needed to complete the algorithm of the previous method. Our new method splits the given problem domain into subdomains, applies the Generalized HDMR philosophy to each subdomain and superpositions the information coming from these subdomains. The algorithm of this new method and a number of numerical implementations are given in this paper.     

Analytical HDMR formulas for functions expressed as quadratic polynomials with a multivariate normal distribution

High Dimensional Model Representation (HDMR) is a general set of quantitative model assessment and analysis tools for systems with many variables. A general formulation for the HDMR component functions with independent and correlated variables was obtained previously. Since the HDMR component functions generally are coupled to one another and involve multi-dimensional integrals, explicit formulas for the component functions are not available for an arbitrary function with an arbitrary probability distribution amongst their variables. This paper presents analytical formulas for the HDMR component functions and the corresponding sensitivity indexes for the common case of a function expressed as a quadratic polynomial with a multivariate normal distribution over its variables. This advance is important for practical applications of HDMR with correlated variables.     

Theoretical additivity in multipath 13C, 1H coupling constants

Different path contributions to the Fermi contact term of carbon-hydrogen coupling constants in several bicycloalkanes were calculated using the ‘inner projections of the polarization propagator’ (IPPP) approach. The ground-state wave functions were calculated at the INDO level of approximation. Three and four-bond contributions to the couplings, and the through-space transmission via the rear lobes of the C? X (X ? H, F) bonds attached to the bridgehead carbon atoms, were calculated. The additivity of these contributions compared with the total INDO couplings is good. Similar paths in different molecules give dissimilar contributions to the couplings.     

Quantum-chemical foundations of the topological substructural molecular design

The topological substructural molecular design (TOPS-MODE) approach is formulated as a tight-binding quantum-chemical method. The approach is based on certain postulates that permit to express any molecular property as a function of the spectral moments of certain types of molecular and environment-dependent energies. We use several empirical potentials to account for these intrinsic and external molecular energies. We prove that any molecular property expressed in terms of a quantitative structure-property and structure-activity relationships (QSPR/QSAR) model developed by using the TOPS-MODE method can be expressed as a bond additivity function. In addition, such a property can also be expressed as a substructural cluster expansion function. The conditions for such bond contributions being transferable are also analyzed here. Several new statistical-mechanical electronic functions are introduced as well as a bond-bond thermal Green's function or a propagator accounting for the electronic hopping between pairs of bonds. All these new concepts are applied to the development and application of a new QSAR model for describing the toxicity of polyhalogenated-dibenzo-1,4-dioxins. The QSAR model obtained displays a significant robustness and predictability. It permits an easy structural interpretation of the structure-activity relationship in terms of bond additivity functions, which display some resemblances with other theoretical parameters obtained from first principle quantum-chemical methods.     

A Matrix Based Indexing HDMR method for multivariate data modelling

Modelling multivariate data of real life problems from engineering, chemistry, physics, mathematics or other related sciences, in which function values are known only at arbitrarily distributed points of the problem domain, is an important and complicated issue since there exist mathematical and computational complexities in the analytical structure construction process coming from the multivariance. The Plain High Dimensional Model Representation (HDMR) method expresses a multivariate problem in terms of less-variate problems. In this work, a Matrix Based Indexing HDMR method is developed to make the Plain HDMR philosophy employable for the multivariate data partitioning process. This new method will have the ability of dealing with less-variate data sets by partitioning the given data set into univariate, bivariate and trivariate data sets. Interpolating these partitioned data sets will construct an approximate analytical structure as the model of the given multivariate data modelling problem.     

THERM: a computer code for estimating thermodynamic properties for species important to combustion and reaction modeling

A computer package has been developed called THERM, an acronym for THermodynamic property Estimation for Radicals and Molecules. THERM is a versatile computer code designed to automate the estimation of ideal gas phase thermodynamic properties for radicals and molecules important to combustion and reaction-modeling studies. Thermodynamic properties calculated include heat of formation and entropies at 298 K and heat capacities from 300 to 1500 K. Heat capacity estimates are then extrapolated to above 5000 K, and NASA format polynomial thermodynamic property representations valid from 298 to 5000 K are generated. This code is written in Microsoft Fortran version 5.0 for use on machines running under MSDOS. THERM uses group additivity principles of Benson and current best values for bond strengths, changes in entropy, and loss of vibrational degrees of freedom to estimate properties for radical species from parent molecules. This ensemble of computer programs can be used to input literature data, estimate data when not available, and review, update, and revise entries to reflect improvements and modifications to the group contribution and bond dissociation databases. All input and output files are ASCII so that they can be easily edited, updated, or expanded. In addition, heats of reaction, entropy changes, Gibbs free-energy changes, and equilibrium constants can be calculated as functions of temperature from a NASA format polynomial database.