Global sensitivity analysis for calculating the contribution of genetic parameters to the variance of crop model prediction

Dynamic models are often used to predict the effects of farmers’ practices on crop yield, crop quality, and environment. These models usually include many parameters that must be estimated from experimental data before practical use. Parameter estimation is a difficult problem especially when some of the parameters vary across genotypes. These genetic parameters may be estimated from plant breeding experiments but this is very costly and requires a lot of experimental work. Moreover, some of the genetic parameters may account for only a very small part of the output variance and, so, do not deserve an accurate determination. This paper shows how methods of global sensitivity analysis can be used to evaluate the contributions of the genetic parameters to the variance of model prediction. Two methods are applied to a complex crop model for estimating the sensitivity indices associated to 13 genetic parameters. The results show that only five genetic parameters have a significant effect on crop yield and grain quality.     

A mechanism motion error sensitivity analysis method based on principal component analysis and artificial neural network

Present sensitivity analysis of motion error usually focuses on the trajectory deviation of the mechanism, which inevitably introduces an intractable time dependent problem. For efficiently and accurately measuring the motion error of the planar mechanism with dimension and clearance uncertainties by global sensitivity analysis (GSA), a novel method is proposed in this work. By applying the principal component analysis (PCA), the motion error is transformed into new vector output and cleverly avoids the time dependent problem. To ensure the accuracy of PCA in the case of small samples, the Bootstrap method is introduced. Based on the PCA results, the artificial neural network (ANN) surrogate model is established between the input variables and the vector output. Then the classical variance-based GSA method is applied to obtain the variable importance ranking for different PCs, and the synthesized GSA indices are introduced. Four representative examples are studied to demonstrate the versatility and effectiveness of the proposed method.     

Reliability importance analysis of Markovian systems at steady state using perturbation analysis

Sensitivity analysis has been primarily defined for static systems, i.e. systems described by combinatorial reliability models (fault or event trees). Several structural and probabilistic measures have been proposed to assess the components importance. For dynamic systems including inter-component and functional dependencies (cold spare, shared load, shared resources, etc.), and described by Markov models or, more generally, by discrete events dynamic systems models, the problem of sensitivity analysis remains widely open. In this paper, the perturbation method is used to estimate an importance factor, called multi-directional sensitivity measure, in the framework of Markovian systems. Some numerical examples are introduced to show why this method offers a promising tool for steady-state sensitivity analysis of Markov processes in reliability studies.     

A general first-order global sensitivity analysis method

Fourier amplitude sensitivity test (FAST) is one of the most popular global sensitivity analysis techniques. The main mechanism of FAST is to assign each parameter with a characteristic frequency through a search function. Then, for a specific parameter, the variance contribution can be singled out of the model output by the characteristic frequency. Although FAST has been widely applied, there are two limitations: (1) the aliasing effect among parameters by using integer characteristic frequencies and (2) the suitability for only models with independent parameters. In this paper, we synthesize the improvement to overcome the aliasing effect limitation [Tarantola S, Gatelli D, Mara TA. Random balance designs for the estimation of first order global sensitivity indices. Reliab Eng Syst Safety 2006; 91(6):717–27] and the improvement to overcome the independence limitation [Xu C, Gertner G. Extending a global sensitivity analysis technique to models with correlated parameters. Comput Stat Data Anal 2007, accepted for publication]. In this way, FAST can be a general first-order global sensitivity analysis method for linear/nonlinear models with as many correlated/uncorrelated parameters as the user specifies. We apply the general FAST to four test cases with correlated parameters. The results show that the sensitivity indices derived by the general FAST are in good agreement with the sensitivity indices derived by the correlation ratio method, which is a non-parametric method for models with correlated parameters.     

A methodology for global-sensitivity analysis of time-dependent outputs in systems biology modelling

One of the main challenges in the development of mathematical and computational models of biological systems is the precise estimation of parameter values. Understanding the effects of uncertainties in parameter values on model behaviour is crucial to the successful use of these models. Global sensitivity analysis (SA) can be used to quantify the variability in model predictions resulting from the uncertainty in multiple parameters and to shed light on the biological mechanisms driving system behaviour. We present a new methodology for global SA in systems biology which is computationally efficient and can be used to identify the key parameters and their interactions which drive the dynamic behaviour of a complex biological model. The approach combines functional principal component analysis with established global SA techniques. The methodology is applied to a model of the insulin signalling pathway, defects of which are a major cause of type 2 diabetes and a number of key features of the system are identified.     

Principal components analysis in sensitivity studies of dynamic systems

Local sensitivity analysis is a modeling tool for determining the effects of single parameter variations on the output of a first order differential system. To determine the effects of multi-parameter variations, the local sensitivity matrix can be used in a first order Taylor series to approximately model the variance of the system output. For dynamic systems, a principal component analysis based on this time varying approximation reveals the evolution of the directions and magnitudes of greatest variation of the system output derived from input variability in the parameters. Such an analysis acts as a means of modeling the robustness of dynamic differential systems.     

Sensitivity analysis in presence of model uncertainty and correlated inputs

The first motivation of this work is to take into account model uncertainty in sensitivity analysis (SA). We present with some examples, a methodology to treat uncertainty due to a mutation of the studied model. Development of this methodology has highlighted an important problem, frequently encountered in SA: how to interpret sensitivity indices when random inputs are non-independent? This paper suggests a strategy for the problem of SA of models with non-independent random inputs. We propose a new application of the multidimensional generalization of classical sensitivity indices, resulting from group sensitivities (sensitivity of the output of the model to a group of inputs), and describe an estimation method based on Monte-Carlo simulations. Practical and theoretical applications illustrate the interest of this method.     

Random balance designs for the estimation of first order global sensitivity indices

We present two methods for the estimation of main effects in global sensitivity analysis. The methods adopt Satterthwaite's application of random balance designs in regression problems, and extend it to sensitivity analysis of model output for non-linear, non-additive models. Finite as well as infinite ranges for model input factors are allowed. The methods are easier to implement than any other method available for global sensitivity analysis, and reduce significantly the computational cost of the analysis. We test their performance on different test cases, including an international benchmark on safety assessment for nuclear waste disposal originally carried out by OECD/NEA.     

A global sensitivity index based on Fréchet derivative and its efficient numerical analysis

Sensitivity analysis plays an important role in reliability evaluation, structural optimization and structural design, etc. The local sensitivity, i.e., the partial derivative of the quantity of interest in terms of parameters or basic variables, is inadequate when the basic variables are random in nature. Therefore, global sensitivity such as the Sobol’ indices based on the decomposition of variance and the moment-independent importance measure, among others, have been extensively studied. However, these indices are usually computationally expensive, and the information provided by them has some limitations for decision making. Specifically, all these indices are positive, and therefore they cannot reveal whether the effects of a basic variable on the quantity of interest are positive or adverse. In the present paper, a novel global sensitivity index is proposed when randomness is involved in structural parameters. Specifically, a functional perspective is firstly advocated, where the probability density function (PDF) of the output quantity of interest is regarded as the output of an operator on the PDF of the source basic random variables. The Fréchet derivative is then naturally taken as a measure for the global sensitivity. In some sense such functional perspective provides a unified perspective on the concepts of global sensitivity and local sensitivity. In the case the change of the PDF of a basic random variable is due to the change of parameters of the PDF of the basic random variable, the computation of the Fréchet-derivative-based global sensitivity index can be implemented with high efficiency by incorporating the probability density evolution method (PDEM) and change of probability measure (COM). The numerical algorithms are elaborated. Several examples are illustrated, demonstrating the effectiveness of the proposed method.     

Gaussian Surrogates for Computer Models With Time-Varying Inputs and Outputs

Computer models of dynamic systems produce outputs that are functions of time; models that solve systems of differential equations often have this character. In many cases, time series output can be usefully reduced via principal components to simplify analysis. Time-indexed inputs, such as the functions that describe time-varying boundary conditions, are also common with such models. However, inputs that are functions of time often do not have one or a few “characteristic shapes” that are more common with output functions, and so, principal component representation has less potential for reducing the dimension of input functions. In this article, Gaussian process surrogates are described for models with inputs and outputs that are both functions of time. The focus is on construction of an appropriate covariance structure for such surrogates, some experimental design issues, and an application to a model of marrow cell dynamics.     

Polynomial chaos expansion for sensitivity analysis

In this paper, the computation of Sobol's sensitivity indices from the polynomial chaos expansion of a model output involving uncertain inputs is investigated. It is shown that when the model output is smooth with regards to the inputs, a spectral convergence of the computed sensitivity indices is achieved. However, even for smooth outputs the method is limited to a moderate number of inputs, say 10-20, as it becomes computationally too demanding to reach the convergence domain. Alternative methods (such as sampling strategies) are then more attractive. The method is also challenged when the output is non-smooth even when the number of inputs is limited.     

Layerwise modeling of progressive damage in fiber-reinforced composite laminates

This paper investigates the effects of discrete layer transverse shear strain and discrete layer transverse normal strain on the predicted progressive damage response and global failure of fiber-reinforced composite laminates. These effects are isolated using a hierarchical, displacement-based 2-D finite element model that includes the first-order shear deformation model (FSD), type-I layerwise models (LW1) and type-II layerwise models (LW2) as special cases. Both the LW1 layerwise model and the more familiar FSD model use a reduced constitutive matrix that is based on the assumption of zero transverse normal stress; however, the LW1 model includes discrete layer transverse shear effects via in-plane displacement components that are C ⁰ continuous with respect to the thickness coordinate. The LW2 layerwise model utilizes a full 3-D constitutive matrix and includes both discrete layer transverse shear effects and discrete layer transverse normal effects by expanding all three displacement components as C ⁰ continuous functions of the thickness coordinate. The hierarchical finite element model incorporates a 3-D continuum damage mechanics (CDM) model that predicts local orthotropic damage evolution and local stiffness reduction at the geometric scale represented by the homogenized composite material ply. In modeling laminates that exhibit either widespread or localized transverse shear deformation, the results obtained in this study clearly show that the inclusion of discrete layer kinematics significantly increases the rate of local damage accumulation and significantly reduces the predicted global failure load compared to solutions obtained from first-order shear deformable models. The source of this effect can be traced to the improved resolution of local interlaminar shear stress concentrations, which results in faster local damage evolution and earlier cascading of localized failures into widespread global failure.     

Global sensitivity analysis of computer models with functional inputs

Global sensitivity analysis is used to quantify the influence of uncertain model inputs on the response variability of a numerical model. The common quantitative methods are appropriate with computer codes having scalar model inputs. This paper aims at illustrating different variance-based sensitivity analysis techniques, based on the so-called Sobol's indices, when some model inputs are functional, such as stochastic processes or random spatial fields. In this work, we focus on large cpu time computer codes which need a preliminary metamodeling step before performing the sensitivity analysis. We propose the use of the joint modeling approach, i.e., modeling simultaneously the mean and the dispersion of the code outputs using two interlinked generalized linear models (GLMs) or generalized additive models (GAMs). The “mean model” allows to estimate the sensitivity indices of each scalar model inputs, while the “dispersion model” allows to derive the total sensitivity index of the functional model inputs. The proposed approach is compared to some classical sensitivity analysis methodologies on an analytical function. Lastly, the new methodology is applied to an industrial computer code that simulates the nuclear fuel irradiation.     

Second-level global sensitivity analysis of numerical simulators with application to an accident scenario in a sodium-cooled fast reactor

Numerical simulators are widely used to model physical phenomena and global sensitivity analysis (GSA) aims at studying the global impact of the input uncertainties on the simulator output. To perform GSA, statistical tools based on inputs/output dependence measures are commonly used. We focus here on the Hilbert–Schmidt independence criterion (HSIC). Sometimes, the probability distributions modeling the uncertainty of inputs may be themselves uncertain and it is important to quantify their impact on GSA results. We call it here the second-level global sensitivity analysis (GSA2). However, GSA2, when performed with a Monte Carlo double-loop, requires a large number of model evaluations, which is intractable with CPU time expensive simulators. To cope with this limitation, we propose a new statistical methodology based on a Monte Carlo single-loop with a limited calculation budget. First, we build a unique sample of inputs and simulator outputs, from a well-chosen probability distribution of inputs. From this sample, we perform GSA for various assumed probability distributions of inputs by using weighted HSIC measures estimators. Statistical properties of these weighted estimators are demonstrated. Subsequently, we define 2nd-level HSIC-based measures between the distributions of inputs and GSA results, which constitute GSA2 indices. The efficiency of our GSA2 methodology is illustrated on an analytical example, thereby comparing several technical options. Finally, an application to a test case simulating a severe accidental scenario on nuclear reactor is provided.     

Modelling and simulation of manufacturing systems with first-order hybrid Petri nets

First-order hybrid Petri nets are models that consist of continuous places holding fluid, discrete places containing a non-negative integer number of tokens, and transitions, either discrete or continuous. In the first part of the paper, we provide a framework to describe the overall hybrid net behaviour that combines both time-driven and event-driven dynamics. The resulting model is a linear discrete-time, time-varying state variable model that can be directly used by an efficient simulation tool. In the second part of the paper, we focus on manufacturing systems. Manufacturing systems are discrete-event dynamic systems whose number of reachable states is typically very large, hence approximating fluid models have often been used in this context. We describe the net models of the elementary components of a flexible manufacturing system (machines and buffers) and we show in a final example how these modules can be put together in a bottom-up fashion.     

Data-driven modal surrogate model for frequency response uncertainty propagation

A method is developed for propagation of model parameter uncertainties into frequency response functions based on a modal representation of the equations of motion. Individual local surrogate models of the eigenfrequencies and residue matrix elements for each mode are trained to build a global surrogate model. The computational cost of the global surrogate model is reduced in three steps. First, modes outside the range of interest, necessary to describe the in-band frequency response, are approximated with few residual modes. Secondly, the dimension of the residue matrices for each mode is reduced using principal component analysis. Lastly, multiple surrogate model structures are employed in a mixture. Cheap second-order multivariate polynomial models and more expensive Gaussian process models with different kernels are used to model the modal data. Leave-one-out cross-validation is used for model selection of the local surrogate models. The approximations introduced allow the method to be used for modally dense models at a small computational cost, without sacrificing the global surrogate model’s ability to capture mode veering and crossing phenomena. The method is compared to a Monte Carlo based approach and verified on one industrial-sized component and on one assembly of two car components.     

Calculations of Sobol indices for the Gaussian process metamodel

Global sensitivity analysis of complex numerical models can be performed by calculating variance-based importance measures of the input variables, such as the Sobol indices. However, these techniques, requiring a large number of model evaluations, are often unacceptable for time expensive computer codes. A well-known and widely used decision consists in replacing the computer code by a metamodel, predicting the model responses with a negligible computation time and rending straightforward the estimation of Sobol indices. In this paper, we discuss about the Gaussian process model which gives analytical expressions of Sobol indices. Two approaches are studied to compute the Sobol indices: the first based on the predictor of the Gaussian process model and the second based on the global stochastic process model. Comparisons between the two estimates, made on analytical examples, show the superiority of the second approach in terms of convergence and robustness. Moreover, the second approach allows to integrate the modeling error of the Gaussian process model by directly giving some confidence intervals on the Sobol indices. These techniques are finally applied to a real case of hydrogeological modeling.     

Global sensitivity analysis for fuzzy inputs based on the decomposition of fuzzy output entropy

To analyse the component of fuzzy output entropy, a decomposition method of fuzzy output entropy is first presented. After the decomposition of fuzzy output entropy, the total fuzzy output entropy can be expressed as the sum of the component fuzzy entropy contributed by fuzzy inputs. Based on the decomposition of fuzzy output entropy, a new global sensitivity analysis model is established for measuring the effects of uncertainties of fuzzy inputs on the output. The global sensitivity analysis model can not only tell the importance of fuzzy inputs but also simultaneously reflect the structural composition of the response function to a certain degree. Several examples illustrate the validity of the proposed global sensitivity analysis, which is a significant reference in engineering design and optimization of structural systems.     

Sensitivity analysis of infectious disease models: methods,advances and their application

Sensitivity analysis (SA) can aid in identifying influential model parameters and optimizing model structure, yet infectious disease modelling has yet to adopt advanced SA techniques that are capable of providing considerable insights over traditional methods. We investigate five global SA methods—scatter plots, the Morris and Sobol’ methods, Latin hypercube sampling-partial rank correlation coefficient and the sensitivity heat map method—and detail their relative merits and pitfalls when applied to a microparasite (cholera) and macroparasite (schistosomaisis) transmission model. The methods investigated yielded similar results with respect to identifying influential parameters, but offered specific insights that vary by method. The classical methods differed in their ability to provide information on the quantitative relationship between parameters and model output, particularly over time. The heat map approach provides information about the group sensitivity of all model state variables, and the parameter sensitivity spectrum obtained using this method reveals the sensitivity of all state variables to each parameter over the course of the simulation period, especially valuable for expressing the dynamic sensitivity of a microparasite epidemic model to its parameters. A summary comparison is presented to aid infectious disease modellers in selecting appropriate methods, with the goal of improving model performance and design.     

Sensitivity analysis of a complex, proposed geologic waste disposal system using the Fourier Amplitude Sensitivity Test method

The Fourier Amplitude Sensitivity Test (FAST) method has been used to perform a sensitivity analysis of a computer model developed for conducting total system performance assessment of the proposed high-level nuclear waste repository at Yucca Mountain, Nevada, USA. The computer model has a large number of random input parameters with assigned probability density functions, which may or may not be uniform, for representing data uncertainty. The FAST method, which was previously applied to models with parameters represented by the uniform probability distribution function only, has been modified to be applied to models with nonuniform probability distribution functions. Using an example problem with a small input parameter set, several aspects of the FAST method, such as the effects of integer frequency sets and random phase shifts in the functional transformations, and the number of discrete sampling points (equivalent to the number of model executions) on the ranking of the input parameters have been investigated. Because the number of input parameters of the computer model under investigation is too large to be handled by the FAST method, less important input parameters were first screened out using the Morris method. The FAST method was then used to rank the remaining parameters. The validity of the parameter ranking by the FAST method was verified using the conditional complementary cumulative distribution function (CCDF) of the output. The CCDF results revealed that the introduction of random phase shifts into the functional transformations, proposed by previous investigators to disrupt the repetitiveness of search curves, does not necessarily improve the sensitivity analysis results because it destroys the orthogonality of the trigonometric functions, which is required for Fourier analysis.