Sensitivity Analysis Of Dynamic Model Research Articles

Efficient multivariate sensitivity analysis for dynamic models based on cubature formula

Abstract Variance-based sensitivity analysis (SA) has been frequently applied to dynamic models with multivariate output. The generalized sensitivity indices are defined by combining principal components analysis with analysis of variance to synthesize the influence of each input on the whole dynamic output. In order to efficiently perform global SA on dynamic models, two efficient algorithms based on cubature formula are proposed in this paper to estimate these generalized variance-based sensitivity indices. The new algorithms are double-loop nested cubature formula (DLCF) and single loop cubature formula (SLCF). The DLCF method estimates the sensitivity indices by decreasing the dimensionality of the input variables procedurally, while SLCF method performs SA through extending the dimensionality of the inputs. Both of them can make full use of the advantages of the cubature formula, and provide efficient estimates for the generalized variance-based sensitivity indices. The numerical and engineering examples demonstrate that both the proposed algorithms can avoid the expensive computational cost associated with the sampling methods, and improve the SA of the dynamic models significantly.

Multivariate sensitivity analysis for dynamic models with both random and random process inputs

Dynamic models with both random and random process inputs are frequently used in engineering. However, sensitivity analysis (SA) for such models is still a challenging problem. This paper, therefore, proposes a new multivariate SA technique to aid the safety design of these models. The new method can decompose the SA of dynamic models into a series of SA of their principle components based on singular value decomposition, which will make the SA of dynamic models much more efficient. It is shown that the effect of both random and random process inputs on the uncertainty of dynamic output can be measured from their effects on both the distributions and directions of the principle components, based on which the individual sensitivities are defined. The generalized sensitivities are then proposed to synthesize the information that is spread between the principal components to assess the influence of each input on the entire uncertainty of dynamic output. The properties of the new sensitivities are derived and an efficient estimation algorithm is proposed based on unscented transformation. Numerical results are discussed with application to a hydrokinetic turbine blade model, where the new method is compared with the existing variance-based method.

Multivariate global sensitivity analysis for dynamic models based on wavelet analysis

Dynamic models with time-dependent output are widely used in engineering for risk assessment and decision making. Global sensitivity analysis for these models is very useful for simplifying the model, improving the model performance, etc. The existent covariance decomposition based global sensitivity analysis method combines the variance based sensitivity analysis results of the model output at all the instants, which just utilizes the information of the time-dependent output in time domain. However, many significant features of time-dependent output may not be obtained from the time domain. Thus, performing global sensitivity analysis for dynamic models just with the information in time domain may be incomplete. In this paper, a new kind of sensitivity indices based on wavelet analysis is proposed. The energy distribution of model output over different frequency bands is extracted as a quantitative feature of the time-dependent output, and it contains the information of model output in both time and frequency domains. Then, a vector projection method is utilized to measure the effects of input variables on the energy distribution of model output. An efficient algorithm is also proposed to estimate the new sensitivity indices. The numerical examples show the difference between the new sensitivity indices and the covariance decomposition based sensitivity indices. Finally, the new sensitivity indices are applied to an environmental model to tell the relative importance of the input variables, which can be useful for improving the model performance.

Multivariate global sensitivity analysis for dynamic models based on energy distance

In this paper, a new kind of multivariate global sensitivity index based on energy distance is proposed. The covariance decomposition based index has been widely used for multivariate global sensitivity analysis. However, it just considers the variance of multivariate model output and ignores the correlation between different outputs. The proposed index considers the whole probability distribution of dynamic output based on characteristic function and contains more information of uncertainty than the covariance decomposition based index. The multivariate probability integral transformation based index is an extension of the popularly used moment-independent sensitivity analysis index. Although it considers the whole probability distribution of dynamic output, it is difficult to estimate the joint cumulative distribution function of dynamic output. The proposed sensitivity index can be easily estimated, especially for models with high dimensional outputs. Compared to the classic sensitivity indices, the proposed sensitivity index can be easily used for dynamic systems and obtain reasonable results. An efficient method based on the idea of the given-data method is used to estimate the proposed sensitivity index with only one set of input-output samples. The numerical and engineering examples are employed to compare the proposed index and the covariance decomposition based index. The results show that the input variables may have different effect on the whole probability distribution and variance of dynamic model output since the proposed index and the covariance decomposition based index measure the effects of input variables on the whole distribution and variance of model output separately.

Hierarchical multivariate regression-based sensitivity analysis reveals complex parameter interaction patterns in dynamic models

Dynamic models of biological systems often possess complex and multivariate mappings between input parameters and output state variables, posing challenges for comprehensive sensitivity analysis across the biologically relevant parameter space. In particular, more efficient and robust ways to obtain a solid understanding of how the sensitivity to each parameter depends on the values of the other parameters are sorely needed.We report a new methodology for global sensitivity analysis based on Hierarchical Cluster-based Partial Least Squares Regression (HC-PLSR)-based approximations (metamodelling) of the input–output mappings of dynamic models, which we expect to be generic, efficient and robust, even for systems with highly nonlinear input–output relationships. The two-step HC-PLSR metamodelling automatically separates the observations (here corresponding to different combinations of input parameter values) into groups based on the dynamic model behaviour, then analyses each group separately with Partial Least Squares Regression (PLSR). This produces one global regression model comprising all observations, as well as regional regression models within each group, where the regression coefficients can be used as sensitivity measures. Thereby a more accurate description of complex interactions between inputs to the dynamic model can be revealed through analysis of how a certain level of one input parameter affects the model sensitivity to other inputs. We illustrate the usefulness of the HC-PLSR approach on a dynamic model of a mouse heart muscle cell, and demonstrate how it reveals interaction patterns of probable biological significance not easily identifiable by a global regression-based sensitivity analysis alone.Applied for sensitivity analysis of a complex, high-dimensional dynamic model of the mouse heart muscle cell, several interactions between input parameters were identified by the two-step HC-PLSR analysis that could not be detected in the single-step global analysis. Hence, our approach has the potential to reveal new biological insight through the identification of complex parameter interaction patterns. The HC-PLSR metamodel complexity can be adjusted according to the nonlinear complexity of the input–output mapping of the analysed dynamic model through adjustment of the number of regional regression models included. This facilitates sensitivity analysis of dynamic models of varying complexities.

Sensitivity study of dynamic systems using polynomial chaos

Global sensitivity has mainly been analyzed in static models, though most physical systems can be described by differential equations. Very few approaches have been proposed for the sensitivity of dynamic models and the only ones are local. Nevertheless, it would be of great interest to consider the entire uncertainty range of parameters since they can vary within large intervals depending on their meaning. Other advantage of global analysis is that the sensitivity indices of a given parameter are evaluated while all the other parameters can be varied. In this way, the relative variability of each parameter is taken into account, revealing any possible interactions. This paper presents the global sensitivity analysis for dynamic models with an original approach based on the polynomial chaos (PC) expansion of the output. The evaluation of the PC expansion of the output is less expensive compared to direct simulations. Moreover, at each time instant, the coefficients of the PC decomposition convey the parameter sensitivity and then a sensitivity function can be obtained. The PC coefficients are determined using non-intrusive methods. The proposed approach is illustrated with some well-known dynamic systems.

Sulphate Reduction Processes in Biological Permeable Reactive Barriers: Column Experimentation and Modeling

In this work a mathematical model for simulating biological sulphate reduction processes was developed and validated by experimental data in fixed bed columns as lab-scale representations of permeable reactive barriers (PRBs). The model takes in consideration transport and adsorption of sulphate and heavy metals, sulphate bioreduction, bioprecipitation and chemical precipitation of metals. Firstly, the effect of sorption capacity of solid phase of column filling versus the effect of active biological mechanisms on solute removal was isolated denoting the significant contribution of sorption versus bioprecipitation of heavy metals. Subsequently, the mathematical model was validated using experimental data from laboratory column experiments and a good agreement between experimental data and simulation results was obtained. Sensitivity analysis of dynamic model showed that, after steady state was reached, parameter related to biological sulphate reduction affected the model output in the most significant way.

Discrete first- and second-order adjoints and automatic differentiation for the sensitivity analysis of dynamic models

We describe the use of first- and second-order tangent-linear and adjoint models of the residual of linear-implicit autonomous differential algebraic systems in the context of an extrapolated Euler scheme. The derivative code compiler dcc is applied to a C-implementation of the residual to get first derivative code. Second-(and higher-)order derivative models are obtained by reapplication of dcc to its own output. The resulting solver serves as a first proof of concept of a new platform for source-level manipulation of mathematical models that is currently under development at RWTH Aachen University.

Sensitivity Analysis of Dynamic Models to Uncertainties in Inputs Data With Time-Varying Variances

This article deals with the differential computation of sensitivity functions and confidence intervals for model output, when model inputs are subject to systematic or stochastic uncertainties with time-varying variances. The nonlinear, time-varying systems dealt with correspond to the class of nonlinear systems with time-invariant dynamics and boundary conditions involving algebraic-only equations. It is shown that the first-order kernel of a Volterra series expansion of the time-invariant model, allied with a derivation of the algebraic equations, can be used to derive approached differential formulas. These are applied to the case study of a real-size building thermal dynamic model developed with the Clim2000 software; the results are compared with Monte Carlo sampling and show very good agreement.

Interval analysis as a tool for sensitivity analysis of a hemopoietic model

In this article a method for sensitivity analysis of dynamic models based on interval analysis is introduced and illustrated for a simple model of a cell renewal system. This system is an important subsystem in the hemopoietic system, which is one of the most important cell renewal systems in the human body. The basic tool used is interval analysis, which is briefly introduced in this article. Simulation results showing the successful application of the method conclude the paper.