In the chemical sciences, many laboratory experiments, environmental and industrial processes, as well as modeling exercises, are characterized by large numbers of input variables. A general objective in such cases is an exploration of the high-dimensional input variable space as thoroughly as possible for its impact on observable system behavior, often with either optimization in mind or simply for achieving a better understanding of the phenomena involved. An important concern when undertaking these explorations is the number of experiments or modeling excursions necessary to effectively learn the system input f output behavior, which is typically a nonlinear relationship. Although simple logic suggests that the number of runs could grow exponentially with the number of input variables, broadscale evidence indicates that the required effort often scales far more comfortably. This paper considers an emerging family of high dimensional model representation concepts and techniques capable of dealing with such input f output problems in a practical fashion. A summary of the state of the subject is presented, along with several illustrations from various areas in the chemical sciences.
A general set of quantitative model assessment and analysis tools, termed high-dimensional model representations (HDMR), has been introduced recently for improving the efficiency of deducing high-dimensional input−output system behavior. HDMR is a particular family of representations where each term in the representation reflects the independent and cooperative contributions of the inputs upon the output. When data are randomly sampled, a RS (random sampling)-HDMR can be constructed. To reduce the sampling effort, different analytical basis functions, such as orthonormal polynomials, cubic B splines, and polynomials may be employed to approximate the RS-HDMR component functions. Only one set of random input−output samples is necessary to determine all the RS-HDMR component functions, and a few hundred samples may give a satisfactory approximation, regardless of the dimension of the input variable space. It is shown in an example that judicious use of orthonormal polynomials can provide a sampling saving of ∼103 in representing a system compared to employing a direct sampling technique. This paper discusses these practical approaches: their formulas and accuracy along with an illustration from atmospheric modeling.
The objective of a global sensitivity analysis is to rank the importance of the system inputs considering their uncertainty and the influence they have upon the uncertainty of the system output, typically over a large region of input space. This paper introduces a new unified framework of global sensitivity analysis for systems whose input probability distributions are independent and/or correlated. The new treatment is based on covariance decomposition of the unconditional variance of the output. The treatment can be applied to mathematical models, as well as to measured laboratory and field data. When the input probability distribution is correlated, three sensitivity indices give a full description, respectively, of the total, structural (reflecting the system structure) and correlative (reflecting the correlated input probability distribution) contributions for an input or a subset of inputs. The magnitudes of all three indices need to be considered in order to quantitatively determine the relative importance of the inputs acting either independently or collectively. For independent inputs, these indices reduce to a single index consistent with previous variance-based methods. The estimation of the sensitivity indices is based on a meta-modeling approach, specifically on the random sampling-high dimensional model representation (RS-HDMR). This approach is especially useful for the treatment of laboratory and field data where the input sampling is often uncontrolled.
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