High-Dimensional Models

The statistical approach to chemical kinetics was developed by Li et al. (2001, 2002), and high-dimensional model representations (HDMR) were proposed as efficient tools to provide a fully global statistical analysis of a model. The work of Feng et al. (2004) was focused on how the network properties are affected by random rate constant changes. The rate constants were transformed to a logarithmic scale to ensure an even distribution over the large space. 

The high dimensional models and representations that most chemical/biological engineering problems require cannot be solved analytically even when they are linear. 

Characterizing the overall uncertainties associated with the PBPK model estimates is also an important component of the PBPK model evaluation and application. This includes characterizing the uncertainties in model outputs resulting from the uncertainty in the PBPK model parameters. Traditionally, Monte Carlo has been employed for performing uncertainty analysis of PBPK models (39, 40). Some of the recent techniques that have been applied for the uncertainty analysis of PBPK models include the stochastic response surface method (SRSM) (38, 41) and the high-dimensional model reduction (HDMR) technique (42). 

Within the present contribution we resorted to the so-called cluster expansion (CE)[10], a variant of the n-mode representational, 22], also called highdimensional model representation or eut-high dimensional model representation (cut-HDMR)[23-26], Within the CE the potential is approximated by n-particle interaction terms. These terms are called clusters. Again the Qa refer to the composite coordinates as detailed above. The PES is approximated by the expansion... 

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Li GY, Hu JS, Wang SW, Georgopoulos PG, Schoendoif J, Rabitz H (2(X)6) Random sampling-high dimensional model representation (RS-HDMR) and orthogonality of its different order component functions. J Phys Chem A 110 2474—2485... 

Manzhos S, Carrington T (2006) A random-sampling high dimensional model representation neural network for building potential eneigy surfaces. J Chem Phys 125 084109... 

Chen, S., Brune, W.H., Oluwole, O.O., Kolb, C.E., Bacon, F., Li, G.Y., Rabitz, H. Global sensitivity analysis of the regional atmospheric chemical mechanism an application of random sampling-high dimensional model representation to urban oxidation chemistry. Environ. Sci. Technol. 46, 11162-11170 (2012)... 

Fell, B., Kucherenko, S., Shah, N., IEEE Comparison of Monte Carlo and Quasi Monte Carlo sampling methods in high dimensional model representation. SIMUL 2009 First International Conference on Advances in System Simulation. IEEE, New York (2009)... 

Li, G., Wang, S.-W., Rabitz, H., Wang, S., Jaffe, P. Global uncertainty assessments by high dimensional model representations (HDMR). Chem. Eng. Sci. 57, 4445-4460 (2002b)... 

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Ziehn, T., Tomlin, A.S. Global sensitivity analysis of a 3D street canyon model - part I the development of high dimensional model representations. Atm. Environ. 42, 1857-1873 (2008a)... 

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Rao BN, Chowdhury R (2009) Enhanced high dimensional model representation for reUability analysis. Int J Numer Methods Eng 77(5) 719-750 Richard RM (1993) PRCONN manual. RMR Design Group, Tucson... 

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