NDLI: General foundations of high‐dimensional model representations

Content Provider	Springer Nature Link
Author	Rabitz, Herschel Aliş, Ömer F.
Copyright Year	1999
Abstract	A family of multivariate representations is introduced to capture the input–output relationships of high‐dimensional 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 well‐defined physical systems, only relatively low‐order correlations of the input variables are expected to have an impact upon the output. The high‐dimensional model representations (HDMR) utilize this property to present an exact hierarchical representation of the physical system. At each new level of HDMR, higher‐order correlated effects of the input variables are introduced. Tests on several systems indicate that the few lowest‐order terms are often sufficient to represent the model in equivalent form to good accuracy. The input variables may be either finite‐dimensional (i.e., a vector of parameters chosen from the Euclidean space $$\mathcal{R}^n$$ ) or may be infinite‐dimensional as in the function space $${\text{C}}^n \left[ {0,1} \right]$$ . 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: ANOVA‐HDMR is the same as the analysis of variance (ANOVA) decomposition used in statistics. Another cut‐HDMR 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 non‐regressive, non‐parametric learning networks. Selected applications of the HDMR concept are presented along with a discussion of its general utility.
Starting Page	197
Ending Page	233
Page Count	37
File Format	PDF
ISSN	02599791
Journal	Journal of Mathematical Chemistry
Volume Number	25
Issue Number	2-3
e-ISSN	15728897
Language	English
Publisher	Kluwer Academic Publishers-Plenum Publishers
Publisher Date	1999-01-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Physical Chemistry Theoretical and Computational Chemistry Math. Applications in Chemistry
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Chemistry

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4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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