اندازه گیری اهمیت از متغیرهای طبیعی مرتبط و تجزیه و تحلیل حساسیت آن

In order to explore the contributions by correlated input variables to the variance of the polynomial output in general engineering problems, the correlated and uncorrelated contributions by correlated inputs to the variance of model output are derived analytically by taking the quadratic polynomial output without cross term as an illustration. The analytical sensitivities of the variance contributions with respect to the distribution parameters of input variables are derived, which can explicitly expose the basic factors affecting the variance contributions. Numeric examples are employed and their results demonstrate that the derived analytical expressions are correct, and then they are applied to two engineering examples. The derived analytical expressions can be used directly in recognition of the contributions by input variables and their influencing factors in quadratic or linear polynomial output without cross term. Additionally, the analytical method can be extended to the case of higher order polynomial output, and the results obtained by the proposed method can provide the reference for other new methods.

Sensitivity analysis (SA) aims at quantifying the relative importance of each input model parameter in determining the value of an assigned output variable [1]. SA can be classified into two categories, i.e., the local SA and the global SA [2]. Since 1960s, many researchers have focused on the sensitivity analysis of the partial derivatives of structural responses, characters or indices with respect to input variables. However, those sensitivities are solved at nominal values, and cannot take the variation effects of input variables into account, so those sensitivities are local [2]. Saltelli defines uncertainty sensitivity analysis as the determination of how “uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input” [3], which is the global SA compared with local SA. Global sensitivity analysis (GSA) is also called the importance measure analysis [4]. It can recognize contributions of different input variables to the uncertainty of model output response, and then the priority level of the input variables can be determined in experiments or research. The order determined by the importance of model input variables can help designers define the unknown parameters better to reduce the uncertain scope of response and to get an acceptable uncertain response range [5], [6] and [7]. Thus, the importance measure of input variables provides a feasible way for improving the structure models. The existing importance measures include three categories, i.e., non-parameter techniques (correlation coefficient model) [8] and [9], variance based methods [1], [10], [11] and [12], and moment independent model [2] and [13]. The variance based methods are widely applied because it can directly illustrate the contributions by input variables to the variance of output response. Suppose that the input–output model is described as follows: equation(1) y=f(x)=f(x1,x2,…,xn),y=f(x)=f(x1,x2,…,xn), Turn MathJax on where x i is the i th input and y is the output. As a convention, we use upper-case letters, (i.e., X i, Y ) when referring to the generic aspects of variables and lower-case letters (i.e., x i, y ) represent their observed values. The variance-based methods are based on the decomposition of output variance [10] and [11]: equation(2) View the MathML sourceV(Y)=∑i=1nVi+∑i=1,j>inVij+⋯+V1,2,…,n, Turn MathJax on where V (Y ) is the total variance of output y , V i is the variance contribution by x i to output y , Vi1,…,isVi1,…,is is the variance contributed by the interactions between {xi1,…,xis}{xi1,…,xis}. V i is defined as follows [10] and [11]: equation(3) Vi=var(E(Y|Xi)),Vi=var(E(Y|Xi)), Turn MathJax on where var(radical dot)var() and E(radical dot)E() represent the variance and expected value, respectively. So if output response expression (1) includes no intersection and input variables are independent with each other, Eq. (2) can be simplified as equation(4) View the MathML sourceV(Y)=∑i=1nVi, Turn MathJax on which indicates that in this case the variance of output response is the sum of the variance contributions by every individual input variable. Considering the fact that input variables are correlated in many practical engineering problems, Xu and George [7] pointed out that for models with correlated inputs, the contribution by an individual input variable to uncertainty of the output response should be divided into two parts: the correlated one (by the correlated variations, i.e. the variations of a variable which are correlated with others) and the uncorrelated one (by the uncorrelated variations, i.e. the unique variations of a variable which cannot be explained by any other variables). Xu and George [7] illuminated that equation(5) View the MathML sourceVi=ViU+ViC, Turn MathJax on where View the MathML sourceViC denotes the correlated variance contribution by x i to output y . View the MathML sourceViU is the uncorrelated variance contribution by x i to output y and can be defined as equation(6) View the MathML sourceViU=V(Y)−var[E(Y|X(−i))]=E[var(Y|X(−i))], Turn MathJax on where X (−i) indicates the vector of all inputs except x i and V (−i) represents var[E(Y|X(−i))]var[E(Y|X(−i))] for convenience. Eq. (6) holds with respect to (4) if there is no intersection between inputs in the model output y . From Eq. (6) we can obtain that the result of the variance of output y minus the variance contribution by all the inputs except x i is the uncorrelated contribution by x i. The variance contribution V (−i) naturally contains the contribution by X (−i) correlated with x i, in other words, the contribution by x i correlated with other variables, namely View the MathML sourceViC. Variance based importance measure analysis of correlated input variables can help us comprehend the uncorrelated and correlated contributions by the input variables to output variance. Then we can focus on the important part to minimize the output variance. To improve the important part, we just need to obtain the sensitivities of variance contributions with respect to the distribution parameters, which are the partial derivatives of variance contributions with respect to the distribution parameters. So the importance measure analysis and the local SA are both necessary for minimizing the variance of output response, the former is to recognize the contributions by correlated input variables and the latter is to analyze influencing factors. In engineering analysis, some input–output modelling methods are applied frequently, such as the response surface methods (RSM) [14] and [15], the artificial neural networks (ANN) [15], [16] and [17] etc., by which we can generally get linear or quadratic input–output polynomial without cross term. Taking the quadratic polynomial output without cross term as an illustration, the correlated and uncorrelated contributions by correlated variables to the variance of output response are derived analytically in Section 2. The analytical sensitivities of the variance contributions with respect to the distribution parameters of the input variables are derived in Section 3. In Section 4, numerical examples are used to verify the analytical method, which is then applied to engineering examples. In Section 5 the mechanism of correlated and uncorrelated contributions is primarily discussed. Some conclusions are drawn in Section 6.

Linear or quadratic polynomial output responses without cross term are common in uncertainty analysis. The recognition of the contributions by correlated input variables through variance based importance measure analysis and influencing factors analysis through local sensitivity analysis are both necessary for minimizing the variance of output response. In this work, we have taken the quadratic polynomial output without cross term as an illustration, analytically derived the correlated contribution and the uncorrelated contribution by correlated variables to variance of output response. Based on the analytical variance contribution expressions, analytical sensitivities of the variance contributions with respect to the distribution parameters of the input variables are derived. The results of numeric examples demonstrate that the derived analytical expressions are correct. The derived analytical expressions can be used directly in recognition of the contribution by input variables, sensitivity analysis and influencing factors analysis in quadratic or one-order polynomial output without cross term. It also can be the comparison for other new algorithms. We have then discussed the application of the above work to the LSF of a cantilever beam and the in-plane tensile modulus in the main direction of a symmetrical balanced π/4 laminate. The results we obtained are meaningful and helpful to practical engineering. We have discussed the mechanism of correlated and uncorrelated contribution primarily. It would be difficult to accurately express their causality but a simple understanding may interpret it primarily.