Correlation method for variance reduction of Monte Carlo integration in RS-HDMR

The High Dimensional Model Representation (HDMR) technique is a procedure for efficiently representing high-dimensional functions. A practical form of the technique, RS-HDMR, is based on randomly sampling the overall function and utilizing orthonormal polynomial expansions. The determination of expansion coefficients employs Monte Carlo integration, which controls the accuracy of RS-HDMR expansions. In this article, a correlation method is used to reduce the Monte Carlo integration error. The determination of the expansion coefficients becomes an iteration procedure, and the resultant RS-HDMR expansion has much better accuracy than that achieved by direct Monte Carlo integration. For an illustration in four dimensions a few hundred random samples are sufficient to construct an RS-HDMR expansion by the correlation method with an accuracy comparable to that obtained by direct Monte Carlo integration with thousands of samples.     

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.     

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.     

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.     

Ratio control variate method for efficiently determining high-dimensional model representations

The High-Dimensional Model Representation (HDMR) technique is a family of approaches to efficiently interpolate high-dimensional functions. RS(Random Sampling)-HDMR is a practical form of HDMR based on randomly sampling the overall function, and utilizing orthonormal polynomial expansions to approximate the RS-HDMR component functions. The determination of the expansion coefficients for the component functions employs Monte Carlo integration, which controls the accuracy of the RS-HDMR interpolation. The control variate method is an established approach to improve the accuracy of Monte Carlo integration. However, this method is often not practical for an arbitrary function f(x) because there is no general way to find the analytical control variate function h(x), which needs to be very similar to f(x). In this article, we show that truncated RS-HDMR expansions can be used as h(x) for arbitrary f(x), and a more stable iterative ratio control variate method was developed for the determination of the expansion coefficients for the RS-HDMR component functions. As the asymptotic error (standard deviation) of the estimator given by the ratio control variate method is proportional to 1/N(sample size), it is more efficient than the direct Monte Carlo integration, whose error is proportional to 1/square root(N). In an illustration of a four-dimensional atmospheric model a few hundred random samples are sufficient to construct an RS-HDMR expansion by the ratio control variate method with an accuracy comparable to that obtained by direct Monte Carlo integration with thousands of samples.     

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.     

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.     

The influence of the support functions on the quality of enhanced multivariance product representation

This paper presents recently developed Enhanced Multivariance Product Representation (EMPR) method for multivariate functions. EMPR disintegrates a multivariate function to components which are respectively constant, univariate, bivariate and so on in ascending multivariance. Although the EMPR method has the same philosophy with the High Dimensional Model Representation (HDMR) method, it has been proposed to get better quality than HDMR’s with the help of the support functions. For this purpose, we investigate the EMPR truncation qualities with respect to the selection of the support functions. The obtained results and a number of numerical implementations to show the efficiency of the method are also given in this paper.     

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.     

A high dimensional model representation based numerical method for solving ordinary differential equations

A new numerical method for solving ordinary differential equations by using High Dimensional Model Representation (HDMR) has been developed in this work. Higher order ordinary differential equations can be reduced to a set of first order ODEs. Although HDMR is generally used for multivariate functions, univariate functions are taken into account throughout the work because of the ODEs’ natures. Not the numerical solution but its image under an appropriately chosen linear ordinary differential operator is expressed as a linear combination of the positive deviation powers of independent variable from its initial value. The linear combination of these image functions are expected to form a basis set under consideration. The unknown constants in the linear combination are found by maximizing the constancy measurer formed in terms of the HDMR components after they are evaluated. Results are compared with well-known step size based numerical methods. A semi qualitative error analysis of the proposed method is also established.     

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.     

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.     

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.     

High-dimensional model representations generated from low order terms--lp-RS-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. RS-HDMR is a particular form of HDMR based on random sampling (RS) of the input variables. The component functions in an HDMR expansion are optimal choices tailored to the n-variate function f(x) being represented over the desired domain of the n-dimensional vector x. The high-order terms (usually larger than second order, or equivalently beyond cooperativity between pairs of variables) in the expansion are often negligible. When it is necessary to go beyond the first and the second order RS-HDMR, this article introduces a modified low-order term product (lp)-RS-HDMR method to approximately represent the high-order RS-HDMR component functions as products of low-order functions. Using this method the high-order truncated RS-HDMR expansions may be constructed without directly computing the original high-order terms. The mathematical foundations of lp-RS-HDMR are presented along with an illustration of its utility in an atmospheric chemical kinetics model.     

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.     

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.     

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.     

Use of Auxiliary Functions for Construction of Series Expansion Relations of Noninteger Slater Functions in Standard Convention

Using complete orthonormal sets of ψ ^(α*) ‐self‐frictional exponential type orbitals (ψ ^(α*) ‐SFETOs) and Q^q‐noninteger auxiliary functions (Q^q‐NIAFs) introduced by the author, the combined formulas for the one‐ and two‐center one‐range addition theorems of χ‐noninteger Slater type orbitals (χ‐NISTOs) with arbitrary values of distances between centers R_ab (for R_ab = 0 and R_ab ≠ 0), and of integer (for α* = α, –∞ < α ≤ 2) and noninteger (for α* ≠ α, –∞ < α* < 3) self‐frictional (SF) quantum numbers are suggested. The presented relations for the one‐range addition theorems can be useful tools especially in the electronic structure studies of atoms, molecules and solids when χ‐NISTOs are employed as basis functions.     

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.     

Fast Proton Exchange in Histidine: Measurement of Rate Constants through Indirect Detection by NMR Spectroscopy

Owing to its imidazole side chain, histidine participates in various processes such as enzyme catalysis, pH regulation, metal binding, and phosphorylation. The determination of exchange rates of labile protons for such a system is important for understanding its functions. However, these rates are too fast to be measured directly in an aqueous solution by using NMR spectroscopy. We have obtained the exchange rates of the NH₃⁺ amino protons and the labile NH^ε2 and NH^δ1 protons of the imidazole ring by indirect detection through nitrogen‐15 as a function of temperature (272 K<T<293 K) and pH (1.3<pH<4.9) of uniformly nitrogen‐15‐ and carbon‐13‐labeled L ‐histidine ? HCl ? H₂O. Exchange rates up to 8.5×10⁴ s^?1 could be determined (i.e., lifetimes as short as 12 μs). The three chemical shifts δ_Hi of the invisible exchanging protons H_i and the three one‐bond scalar coupling constants ¹J(N,H_i) could also be determined accurately.