Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. Sobol, On the distribution of points in a cube and the approximate evaluation of each integrals, Comput. Math. Math. Phys. 7 (1976) 86–112.
Y. Shreider, The Monte Carlo Method (Pergamon Press, Oxford, 1967).
G.G. Lorentz, M.V. Golitschek and Y. Makovoz, Constructive Approximation (Springer, New York, 1996).
F. Girosi and T. Poggio, Representation properties of networks: Kolmogorov's theorem is irrelevant, Neurocomputing 1 (1989) 465–469.
J. Friedman and W. Stuetzle, Projection pursuit regression, J. Amer. Statist.Assoc. 76 (1981) 817–823.
P. Diaconis and M. Shahshahani, On nonlinear functions of linear combinations, SIAM J. Sci. Statist. Comput. 5(1) (1984) 175–191.
P. Huber, Projection pursuit, Ann. Statist. 13(2) (1981) 435–525.
D. Parker, Learning logic, Working paper 47, Center for Computational Research in Economics and Management Science, Massachusetts Institute of Technology (1985).
T. Poggio and F. Girosi, Networks for approximation and learning, Proc. IEEE 78 (1990) 1481–1497.
Stone, Additive regression and other nonparametric models, Ann. Statist. 13 (1985) 689–705.
H. Scheffe, The Analysis of Variance (Wiley, New York, 1959).
B. Efron and C. Stein, The jackknife estimate of variance, Ann. Statist. 9(3) (1981) 586–596.
H. Rabitz and Ö.F. Alışs, General foundations of high dimensional model representations, J. Math. Chem. 25 (1999) 197–233.
J.A. Shorter, P.C. Ip and H. Rabitz, An efficient chemical kinetics solver using high dimensional model representation, J. Phys. Chem. A 103 (1999) 7192–7198.
J.A. Shorter, P.C. Ip and H. Rabitz, Radiation transport simulation by means of fully equivalent operational model, Geophys. Res. Lett. (2000) tin press.
H. Rabitz and K. Shim, Multicomponent semiconductor material discovery guided by a generalized correlated function expansion, J. Chem. Phys. 111 (1999) 10640–10651.
K. Shim and H. Rabitz, Independent and correlated composition behavior of material properties: application to energy band gaps for the Ga?In1??P?As1?? and Ga?In1??P?Sb? As1???? alloys, Phys. Rev. B 58 (1998) 1940–1946.
H. Rabitz, Ö.F. Al?¸s, J. Shorter and K. Shim, Efficient input–output model representations, Comput. Phys. Comm. 117 (1999) 11–20.
I. Sobol, Sensitivity estimates for nonlinear mathematical models, Math. Mod. Comp. Exp. 1 (1993) 407–414.
A. Saltelli and I. Sobol, About the use of rank transformation in sensitivity analysis of model output, Reliab. Eng. Sys. Safety 50 (1995) 225–239.
A. Saltelli and J. Hjorth, Uncertainty and sensitivity analyses of OH-initiated dimethylsulphide (DMS) oxidation kinetics, J. Atm. Chem. 21 (1995) 187–221.
T. Homma and A. Saltelli, Importance measures in global sensitivity analysis of nonlinear models, Reliab. Eng. Sys. Safety 52 (1996) 1–17.
M. Casdagli, Nonlinear prediction of chaotic time series, Phys. D 51 (1989) 52–98.
J.P. Crutchfield and B.S. McNamara, Equations of motion from a data series, Complex Systems 1 (1987) 417–452.
J.D. Farmer and J.J. Sidorowich, Predicting chaotic time series, Phys. Rev. Lett. 59 (1987) 845–848.
J.Y. Campbell, A.W. Lo and A.C. Mackinlay, The Econometrics of Financial Markets (Princeton University Press, Princeton, NJ, 1997) chapter 12.
P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors, Phys. D 9 (1983) 189–208.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Alış, Ö.F., Rabitz, H. Efficient Implementation of High Dimensional Model Representations. Journal of Mathematical Chemistry 29, 127–142 (2001). https://doi.org/10.1023/A:1010979129659
Issue Date:
DOI: https://doi.org/10.1023/A:1010979129659