Statistical and experimental methods in sensitivity analysis

We describe statistical methods for sensitivity and performance analysis of complex computer simulation experiments. Graphical methods, such as trellis plots, are suggested for exploratory analysis of individual or aggregate performance metrics, conditional on different experiment inputs. More formal statistical methods, such as analysis of variance based methods and regression tree analysis, are used to determine variables having substantive influence on the experimental results and to investigate the structure of the underlying relationship between inputs and outputs. The methods are discussed in relation to a supply chain model of the textile manufacturing process having many possible input and output variables of interest and for a computer model used to describe the flow of material in an ecosystem.

Design sensitivity analysis of structures deals with the calculation of the response derivatives with respect to the design variables. These derivatives, called the sensitivity coefficients, are used in the solution of various problems. In design optimization, the sensitivity coefficients are often required to select a search direction. These coefficients are used also in generating approximations for the response of a modified system. In addition, the sensitivities are required for assessing the effects of uncertainties in the structural properties on the system response. Calculation of the sensitivities involves much computational effort, particularly in large structural systems with many design variables. As a result, there has been much interest in efficient procedures for calculating the sensitivity coefficients. Developments in methods for sensitivity analysis are discussed in many studies (e.g. Haug et. al. 1986; Haftka and Adelman, 1989; Haftka and Gurdal, 1993; van Keulen et. al. in press). Methods of sensitivity analysis for discretized systems can be divided into the following classes: a. Finite-difference methods, which are easy to implement but might involve numerous repeated analyses and high computational cost, particularly in problems with many design or response variables. In addition, finite-difference approximations might have accuracy problems. The efficiency can be improved by using fast reanalysis techniques. b. Analytical methods, which provide exact solutions but might not be easy to implement in some problems such as shape optimization. c. "Semi-analytical" methods, which are based on a compromise between finite-difference methods and analytical methods. These methods use finite-difference evaluation of the right-hand-side vector. They are easy to implement but might provide inaccurate results. In general, the following factors are considered in choosing a suitable sensitivity analysis method for a specific application: the accuracy of the calculations, the computational effort involved and the ease-of-implementation. The implementation effort is weighted against the performance of the algorithms as reflected in their computational efficiency and accuracy. The quality of the results and efficiency of the calculations are usually two conflicting factors. That is, higher accuracy is often achieved at the expense of more computational effort. Dynamic sensitivity analysis has been demonstrated by several authors. Using the mode superposition approach and assuming harmonic loading, the response sensitivities were evaluated by direct differentiation of the equations of motion in the generalized coordinates (Kramer and Grierson, 1989). In cases of earthquake loading the ground acceleration is usually given in discrete time steps, thus the loading is not given analytically. In several studies (Kim and Choi, 2000; van Keulen et. al, in press) the unconditionally stable implicit numerical equation was directly derived. It was found that the analysis equations and the sensitivity equations have the same left-hand side expression. Thus, it was possible to use the available factorized coefficient matrix. A numerical procedure was applied for calculation of the sensitivity of the response.

PurposeThe objectives of this paper are the application of sensitivity analysis (SA) methods in atmospheric dispersion modeling to the emission dispersion model (EDM) to study the prediction of atmospheric dispersion of NO2 generated by an industrial fire, whose results are useful for fire safety applications. The EDM is used to predict the level concentration of nitrogen dioxide (NO2) emitted by an industrial fire in a plant located in an industrial region site in Algeria.Design/methodology/approachThe SA was defined for the following input parameters: wind speed, NO2 emission rate and viscosity and diffusivity coefficients by simulating the air quality impacts of fire on an industrial area. Two SA methods are used: a local SA by using a one at a time technique and a global SA, for which correlation analysis was conducted on the EDM using the standardized regression coefficient.FindingsThe study demonstrates that, under ordinary weather conditions and for the fields near to the fire, the NO2 initial concentration has the most influence on the predicted NO2 levels than any other model input. Whereas, for the far field, the initial concentration and the wind speed have the most impact on the NO2 concentration estimation.Originality/valueThe study shows that an effective decision-making process should not be only based on the mean values, but it should, in particular, consider the upper bound plume concentration.

This paper is concerned with the shape sensitivity analysis of a class of boundary control, constrained, optimal control problems for parabolic systems. The notion of Euler and Lagrange derivatives of a boundary optimal control in the direction of a vector field is introduced. The derivatives are obtained in the form of optimal solutions of auxiliary optimal control problems. The method of sensitivity analysis used in this paper is based on related results on the differential stability of metric projections in Hilbert space onto a convex, closed subset, combined with the material derivative method of shape sensitivity analysis. Parabolic initial-boundary value problems with Dirichlet and Neumann boundary conditions are considered in this paper.

The occurrence of missing data due to protocol deviations is inevitable in clinical trials. When missing data exist, analyses rely on assumptions about the behaviour of the individuals after dropping out. As a result, sensitivity analysis, which is now advocated by regulatory bodies, should be performed to explore the robustness of the inference to those assumptions. These assumptions should be relevant to the estimand of the study and be practically accessible by all parties. The aim of this document is twofold: to assess the statistical validity of a new method for sensitivity analysis, and apply this method to a published Alzheimer’s study. At the beginning of the thesis, a description of the Alzheimer’s study and issues with missing data encountered therein, take place. This study was mainly set up to investigate the effect rosiglitazone, as an adjunct therapy in Alzheimer’s patients. Two different doses of the drug were compared to placebo. The study suffered from a moderate degree of missing data in each treatment arm. The thesis proceeds with a critique on the per-protocol and intention-to-treat estimands, and revisits their meaning when missing data occur. Two new estimands are introduced, which are particularly amenable to studies with missing data. They are termed de-jure and de-facto. Following that, the main methods for dealing with missing data are introduced, with a particular emphasis on multiple imputation, and how it can easily incorporate missing not at random (MNAR) analyses. A thorough presentation of the new methodology is given. This is built around a set of assumptions, that reflect possible distributional behaviours of the subjects after protocol deviation. The assumptions are Randomised-treatment Missing at Random (MAR), Jump to Reference, Last Mean Carried Forward, Copy Increments in Reference, and Copy Reference. Estimation and inference is achieved via multiple imputation, and it is shown how the predictive distribution of the imputation model can be constructed from parameters borrowed from an MAR model and manipulated in a pattern-mixture model approach, to obtain the five assumptions for the 4 unobserved component of groups of individuals. A number of simulations whose aim is to explore the statistical properties of the new method, are carried out. The simulated datasets, which are based on the parameters from the Alzheimer’s study, focus on the estimator bias, the size and power of the methods, the bias of the variance estimator, and coverage. The results obtained from the simulations show the method has sensible properties; no bias for the estimator was detectable and the sizes and power of the methods agreed closely with their theoretical equivalents. The main result however, pertains to Rubin’s variance estimator, which proves to appropriately reflect the loss of information from missing data. It is therefore argued, it is the right estimator to use in this setting. The results from the application of the proposed method on the Alzheimer’s dataset are presented in tables. Inferences from the sensitivity analysis assumptions were consistent with those from the original MAR analyses. The comparison between high dose rosiglitazone and placebo did not show any evidence in favour of the treatment under any sensitivity assumption. Some evidence of treatment difference existed when the low dose treatment was compared to placebo. This finding though, should be interpreted with caution, as the differences were obtained from analyses not subjected to the rigorous inferential process that was used in the Harringtons study. It was further argued, this finding might have been due to chance, and it was not replicated in a different study.

Efficient sensitivity analysis method for chaotic dynamical systems

Sensitivity analysis of CHF parameters under flow instability by using a neural network method

Sensitivity analysis for decision-making using the MORE method—A Pareto approach

Mathematical modelling is one of the most powerful methods for the study and understanding of the Earth’s climate system and its components. Modern climate models used in variety of applications are derived from a set of multi-dimensional non-linear differential equations in partial derivatives, which describe dynamical, physical and chemical processes and cycles in the climate system. Climate models are mostly deterministic with a large-phase space dimension containing a vast number of parameters that have various meanings. Most of them are not well-known a priori and, hence, are not well defined. Parameter errors and their time and space variabilities generate parametric uncertainty. Some model parameters describe external forcing that can strongly influence the climate model behaviour. It is, therefore, very important to estimate the influence of variations in parameters on the model behaviour and results of simulations. Questions like these can be answered by performing sensitivity analysis. This chapter considers various methods of sensitivity analysis that can be used: first, to estimate the influence of model parameter variations on its behaviour; second, to identify parameters of climate models and third, to study the model response to external forcing.

A numerical procedure is presented for implementing the Green’s function method of sensitivity analysis in chemical kinetics. The procedure is applied to three sets of chemical reactions: the Chapman mechanism for ozone kinetics, a mechanism for methane combustion and a model for formaldehyde oxidation in the presence of carbon monoxide. Whenever possible, comparisons with alternative methods of sensitivity analysis are made. It is shown that carefully analyzed sensitivity profiles can be used in conjunction with experiments and/or models to obtain useful information about chemical kinetic behavior. By using methods from multivariable calculus an entire family of sensitivity coefficients may be derived from the elementary sensitivities obtained by solving differential equations. Each elementary or derived sensitivity coefficient has a unique physical interpretation in terms of an experiment or modeling calculation. A simple nonlinear interpolation formula is suggested for easily estimating higher-order sensitivity information. Finally the overall computational efficacy of the Green’s function method of sensitivity analysis is assessed.

A review of sensitivity analysis methods in building energy analysis

The article analyzes the essence of entrepreneurial risk and methods of its assessment. During the economic, political and social crisis, economic, production or commercial activities of Ukrainian entrepreneurs are impossible without risks. Therefore, the issue of reliable assessment of entrepreneurial risk is relevant and vital for entrepreneurs, which can be solved only by the right choice of appropriate methods. The purpose of the article is to analyze the qualitative and quantitative methods of assessing entrepreneurial risk, to identify their advantages and disadvantages, and to choose the most optimal method. It is noted that the entrepreneurial risk can be interpreted as the probability of income loss in doing business. It is determined that the methods of entrepreneurial risk assessment can be devided into two groups, i.e. qualitative and quantitative. There were considered the following main qualitative metods of entrepreneurial risk assessment, including brainstorming, method of expert estimations, method of associations, method of analogy, and Delphi method. It is concluded that qualitative methods can be used only as an aid in combination with quantitative methods to stimulate creative activity, to generate new ideas and proposals, to use the experience and opinions of experts. It is noted that the problem of quantitative risk assessment is one of the main in business, especially for choosing investment options. It is determined that it is possible to assess the risk in absolute and relative terms by using of quantitative methods. There exist differencies between such terms as costs, losses, and damages. There were considered the following main quantitative metods of entrepreneurial risk assessment, including calculation and analytical method, statistical method, methods of game theory, decision tree method, sensitivity analysis (vulnerability), and simulation methods. It is concluded that the most universal methods for entrepreneurial risk assessment are simulation methods, as they have advantages over other methods, such as the method of analogies, sensitivity analysis (vulnerability), and decision tree method. Also they include some of them, such as the statistical method.

The observational research based on big data in healthcare has attracted increasing attention, with the control and evaluation of residual confounding being the critical issue that needs to be solved urgently. This review summarized the methods for statistical adjustment and sensitivity analysis of residual confounding in the association analysis with a multicenter database. Based on individual-level data, the residual confounding can be adjusted in each subcenter using methods such as regression discontinuity design, while the pooled estimate can be obtained as a weighted average. Based on the center-level results, the Bayesian Meta-analysis method can adjust the pooled estimates. The sensitivity analysis of residual confounding can also be carried out using center-level data to calculate the E-value, p^(q), T^(r, q) and G^r,q. The abovementioned methods should be selected reasonably according to the requirements for practical applications, advantages, and disadvantages. For example, the use of subcenter individual data for residual confounding adjustment usually needs strict study design and frequent coordination; the Bayesian Meta-analysis is based on some strong assumptions; the interpretation of the results in the sensitivity analysis, such as E-value requires professional judgment to assess the risk of residual confounding. Therefore, the methods for controlling and evaluating residual confounding in association analysis based on multicenter databases still need further development and improvement.

Variational methods (VM) sensitivity analysis, which is the continuous alternative to the discrete sensitivity analysis, is employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The determination of the sensitivity derivatives of the performance index or functional entails the coupled solutions of the state and costate equations. As the stable and converged numerical solution of the costate equations with their boundary conditions are a priori unknown, numerical stability analysis is performed on both the state and costate equations. Thereafter, based on the amplification factors obtained by solving the generalized eigenvalue equations, the stability behavior of the costate equations is discussed and compared with the state (Euler) equations. The stability analysis of the costate equations suggests that the converged and stable solution of the costate equations is possible only if the computational domain of the costate equations is transformed to take into account the reverse flow nature of the costate equations. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.

This chapter will review the statistical methods used in RNA sequencing data analysis, including bulk RNA sequencing and single-cell RNA sequencing. RNA sequencing data analysis has been widely used in biomedical and biological research to identify genes associated with certain conditions or diseases. Many statistical methods have been proposed to analyze bulk and single-cell RNA sequencing data. Several studies have compared the performance of different statistical methods for RNA sequencing data analysis through simulation studies and real data evaluations. This chapter will summarize the statistical methods and the evaluation results for comparing different statistical analysis methods used for RNA sequencing data analysis. It will cover the statistical models, model assumptions, and challenges encountered in the RNA sequencing data analysis. It is hoped that this chapter will help researchers learn more about the statistical perspective of the RNA sequencing data analysis and enable them to choose appropriate statistical analysis methods for their own RNA sequencing data analysis.