Elicitation of multiattribute value functions through high dimensional model representations: Monotonicity and interactions

Hybrid high dimensional model representation (HHDMR) on the partitioned data

Based on the high dimensional model representation (HDMR) and Monte Carlo simulation (MCS), this paper presents the improved method used to evaluate the failure probability of the system with multi-failure models. The HDMR is a general set of quantitative model assessment and analysis tools for capturing the high-dimensional relationships between sets of input and output model variables. Once the limit state function is defined by using the HDMR, the failure probability can be obtained by using the MCS without increasing computational efforts. The series and parallel system are considered in this paper, a numerical example is presented to demonstrate the efficiency and the accuracy of the proposed method. It is shown that the efficiency of the HDMR are both high in terms of series system and parallel system, the accuracy can be acceptable with respect to series system, and the accuracy can not be acceptable with respect to parallel system.

This work presents an adapted Random Sampling - High Dimensional Model Representation (RS-HDMR) algorithm for synergistically addressing three key problems in network biology: (1) identifying the structure of biological networks from multivariate data, (2) predicting network response under previously unsampled conditions, and (3) inferring experimental perturbations based on the observed network state. RS-HDMR is a multivariate regression method that decomposes network interactions into a hierarchy of non-linear component functions. Sensitivity analysis based on these functions provides a clear physical and statistical interpretation of the underlying network structure. The advantages of RS-HDMR include efficient extraction of nonlinear and cooperative network relationships without resorting to discretization, prediction of network behavior without mechanistic modeling, robustness to data noise, and favorable scalability of the sampling requirement with respect to network size. As a proof-of-principle study, RS-HDMR was applied to experimental data measuring the single-cell response of a protein-protein signaling network to various experimental perturbations. A comparison to network structure identified in the literature and through other inference methods, including Bayesian and mutual-information based algorithms, suggests that RS-HDMR can successfully reveal a network structure with a low false positive rate while still capturing non-linear and cooperative interactions. RS-HDMR identified several higher-order network interactions that correspond to known feedback regulations among multiple network species and that were unidentified by other network inference methods. Furthermore, RS-HDMR has a better ability to predict network response under unsampled conditions in this application than the best statistical inference algorithm presented in the recent DREAM3 signaling-prediction competition. RS-HDMR can discern and predict differences in network state that arise from sources ranging from intrinsic cell-cell variability to altered experimental conditions, such as when drug perturbations are introduced. This ability ultimately allows RS-HDMR to accurately classify the experimental conditions of a given sample based on its observed network state.

Any engineering structure under the action of various internal and external factors like changes in the material properties, inadequate design, faulty construction, deterioration due to malfunctioning are susceptible to damages. In the past, many methods have attempted to identify damage by solving an inverse problem, which inevitably needs an analytical model. However, often the construction of these analytical model requires considerable effort in building a mathematical framework with acceptable level of accuracy and reliability which makes these approaches less attractive. To circumvent this complexity, this work presents a computationally efficient approach in structural damage identification using high dimensional model representation.

Uncertainty propagation (UP) of the frequency response is essential for the robust design of viscoelastic damping structures. One challenge in solving this problem is enormous computation cost from the UP analysis. This paper aims to develop a novel computational effective method based on the combination of the adaptive sparse grid collocation (ASGC) method and the high dimensional model representation (HDMR) technique to evaluate the variability of the frequency response of viscoelastic damping structures. First, a well-validated layer-wise finite element method is employed to model the viscoelastic damping structures. The direct frequency response (DFR) method is utilized to calculate the response. Then, a modified adaptive strategy using expectation increments as the sampling indicator is proposed for cost-effective sparse grid construction. Lastly, high dimensional model representation (HDMR) technique is introduced to address the difficulty in moderate and high random dimensional situations. Two numerical examples are provided to assess the performances of the proposed method. Variations in constitutive parameters of viscoelastic material and thicknesses of the viscoelastic layer are considered. Numerical results show that, compared to the original high-dimensional ASGC method and Monte Carlo Simulation (MCS), the proposed method accurately predicts the variability of frequency responses and significantly improves the computational efficiency.

This work presents the random sampling - high dimensional model representation (RS-HDMR) algorithm for identifying complex bionetwork structures from multivariate data. RS-HDMR describes network interactions through a hierarchy of input-output (IO) functions of increasing dimensionality. Sensitivity analysis based on the calculated RS-HDMR component functions provides a statistically interpretable measure of network interaction strength, and can be used to efficiently infer network structure. Advantages of RS-HDMR include the ability to capture nonlinear and cooperative realtionships among network components, the ability to handle both continuous and discrete relationships, the ability to be used as a high-dimensional IO model for quantitative property prediction, and favorable scalability with respect to the number of variables. To demonstrate, RS-HDMR was applied to experimental data measuring the single-cell response of a protein-protein signaling network to various perturbations. The resultant analysis identified the network structure comparable to that reported in the literature and to the results from a previous Bayesian network (BN) analysis. The IO model also revealed several nonlinear feedback and cooperative mechanisms that were unidentified through BN analysis.

Affine transformational HDMR and linearised rational least squares approximation

The effects of parametric uncertainties in simulations of a reactive plume using a Lagrangian stochastic model

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.

A comparative assessment of efficient uncertainty analysis techniques for environmental fate and transport models: application to the FACT model

High Dimensional Model Representation (HDMR) method does not have the random data partitioning capability because of its nature. This paper focuses on how we can bring the random data partitioning feature to the HDMR method by virtue of the fluctuation free approximation method for the multivariate interpolation problems. The main aim of this work is to construct the universal fluctuation free matrix and to determine the best eigenvalues and the corresponding eigenvectors needed in our new method for partitioning the given multivariate random data through the HDMR method.

This work focuses on the development of a multivariate function approximating method by using quartic Transformational High Dimensional Model Representation (THDMR). The method uses the target function’s image under a quartic transformation for High Dimensional Model Representation(HDMR) instead of the function’s itself.

This paper presents a new and alternative computational tool for predicting failure probability of structural/mechanical systems subject to random loads, material properties, and geometry based on high‐dimensional model representation (HDMR) generated from low‐order function components. HDMR is a general set of quantitative model assessment and analysis tools for capturing the high‐dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher‐order variable correlations are weak, allowing the physical model to be captured by the lower‐order terms and facilitating lower‐dimensional approximation of the original high‐dimensional implicit limit state/performance function. When first‐order HDMR approximation of the original high‐dimensional implicit limit state/performance function is not adequate to provide the desired accuracy to the predicted failure probability, this paper presents an enhanced HDMR (eHDMR) method to represent the higher‐order terms of HDMR expansion by expressions similar to the lower‐order ones with monomial multipliers. The accuracy of the HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input–output samples without directly invoking the determination of second‐ and higher‐order terms. The mathematical foundation of eHDMR is presented along with its applicability to approximate the original high‐dimensional implicit limit state/performance function for subsequent reliability analysis, given that conventional methods for reliability analysis are computationally demanding when applied in conjunction with complex finite element models. This study aims to assess how accurately and efficiently the eHDMR approximation technique can capture complex model output uncertainty. The limit state/performance function surrogate is constructed using moving least‐squares interpolation formula by component functions of eHDMR expansion. Once the approximate form of implicit response function is defined, the failure probability can be obtained by statistical simulation. Results of five numerical examples involving elementary mathematical functions and structural/solid‐mechanics problems indicate that the failure probability obtained using the eHDMR approximation method for implicit limit state/performance function, provides significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations. Copyright © 2008 John Wiley & Sons, Ltd.

A novel method for multispectral image pansharpening based on high dimensional model representation

This paper presents an efficient uncertain analysis method for estimating the bounds on the reliability of structural systems in the presence of mixed uncertain variables. The proposed method involves high dimensional model representation (HDMR) for the limit state function approximation, transformation technique to obtain the contribution of the fuzzy variables to the convolution integral and fast Fourier transform for solving the convolution integral. In the proposed method, efforts are required in evaluating conditional responses at a selected input determined by sample points, as compared to full scale simulation methods. Therefore, the proposed technique estimates the failure probability accurately with significantly less computational effort compared to the direct Monte Carlo simulation. The methodology developed is applicable for structural reliability estimation involving any number of fuzzy variables and random variables with any kind of distribution. The accuracy and efficiency of the proposed method is demonstrated through numerical examples.