Global Sensitivity Analysis of Structural Reliability Using Cliff Delta

Citation: Kala, Z. Global Sensitivity Analysis of Structural Reliability Using Cliff Delta. Mathematics 2024, 12, 2129. https://doi.org/10.3390/ math12132129 Academic Editor: David Greiner Received: 9 June 2024 Revised: 3 July 2024 Accepted: 4 July 2024 Published: 7 July 2024 Copyright: © 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article Global Sensitivity Analysis of Structural Reliability Using Cliff Delta Zdenˇek Kala Institute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic; [email protected] or [email protected].cz Abstract: This paper introduces innovative sensitivity indices based on Cliff’s Delta for the global sensitivity analysis of structural reliability. These indices build on the Sobol’ method, using binary outcomes (success or failure), but avoid the need to calculate failure probability P f and the associated distributional assumptions of resistance Rand load F. Cliff’s Delta, originally used for ordinal data, evaluates the dominance of resistance over load without specific assumptions. The mathematical formulations for computing Cliff’s Delta between Rand Fquantify structural reliability by assessing the random realizations of R>Fusing a double-nested-loop approach. The derived sensitivity indices, based on the squared value of Cliff’s Delta δ2 C , exhibit properties analogous to those in the Sobol’ sensitivity analysis, including first-order, second-order, and higher-order indices. This provides a framework for evaluating the contributions of input variables on structural reliability. The results demonstrate that the Cliff’s Delta method provides a more accurate estimate of P f . In one case study, the Cliff’s Delta approach reduces the standard deviation of P f estimates across various Monte Carlo run counts. This method is particularly significant for FEM applications, where repeated simulations of Ror Fare computationally intensive. The double-nested-loop algorithm of Cliff’s Delta maximizes the extraction of information about structural reliability from these simulations. However, the high computational demand of Cliff’s Delta is a disadvantage. Future research should optimize computational demands, especially for small values of Pf. Keywords: sensitivity analysis; Cliff’s Delta; reliability analysis; importance measure; failure probability; uncertainty MSC: 65C50; 60H99; 82B31 1. Introduction Global sensitivity analysis (GSA) focuses on attributing the uncertainties of model outputs, or related performance indicators, to their inputs, thereby assessing the impact of input uncertainties on outputs or performance indicators [ 1 , 2 ]. Various GSA methods have been developed for this purpose [ 3 , 4 ]. These methods include the screening method [ 5 , 6 ], variance-based methods [ 7 , 8 ], moment-independent methods [ 9 , 10 ], and derivative-based methods [ 11 , 12 ]. Among these methods, variance-based sensitivity indices, also known as Sobol’ indices [ 7 , 8 ], are particularly notable for their mathematical elegance in measuring the individual, interaction, and total contributions of each input to the model output uncertainty, see, e.g., [13,14]. In the limit state method, probabilistic reliability analysis is based on the estimation of the failure probability [ 15 ]. Sobol’ indices have been widely used in structural reliability analysis to pinpoint variables that significantly influence failure probability [ 16 , 17 ]. These sensitivity indices, which focus on failure probability P f , are derived from the variance decomposition of a binary function representing failure and success [ 16 , 17 ]. Fort et al. [ 18 ] expanded on Sobol’ sensitivity indices by introducing a contrast function in place of variance, allowing the indices to be oriented towards variance, probability, and quantile. Mathematics 2024,12, 2129. https://doi.org/10.3390/math12132129 https://www.mdpi.com/journal/mathematics Mathematics 2024,12, 2129 2 of 18 This approach maintains the non-negative property of the indices and ensures that their sum is equal to one, as in the traditional Sobol’ sensitivity analysis. Extending GSA to P f and design quantile represented a significant advancement in civil engineering, as these quantities are crucial in structural reliability assessments [19,20]. The development of GSA methods focused on reliability requires the precise estimation of P f [ 21 ]. However, the complexity of mathematical models and the difficulty of uncertainty propagation using sampling-based methods, such as Monte Carlo (MC) simulations, present significant challenges, due to the large number of runs required [ 22 ]. Nonlinear finite element models are particularly demanding on CPU time [ 23 ], which further complicates the process of structural reliability estimation, see, e.g., [ 24 , 25 ]. To ensure the most accurate estimate of P f , it is essential to develop methods that provide precise P f estimates, while minimizing the computational costs associated with repeated calls to the computational model [26,27]. The computational burden can be reduced by using metamodels, also known as surrogate models, which approximate the behavior of complex models with simpler ones [ 28 , 29 ]. Methods, such as the polynomial response surface [ 30 , 31 ], the response surface-based multifidelity model [ 32 ], polynomial chaos expansion (PCE) [ 33 , 34 ], the Gaussian process [ 35 , 36 ], Kriging [37,38], and neural networks [39–41] are used to create such metamodels. However, while the use of metamodels significantly reduces the computational burden, there are some critical drawbacks to this approach [ 42 – 44 ]. Metamodel-based approaches lack reliable error measures and may not accurately represent the limit-state functions, leading to potential inaccuracies in reliability analysis [ 45 , 46 ]. Models with discontinuities or sharp changes in behavior (such as buckling) present particular challenges for metamodels, which may struggle to fit such complexities accurately [47]. Consequently, despite the advancements and widespread use of metamodels, the traditional (quasi-) Monte Carlo methods have not been significantly displaced and remain an integral part of reliability and sensitivity analysis methods. Their enduring use is attributed to their flexibility, unbiased estimations, and effective integration with variance reduction techniques [3,4]. The existing research frequently explores the rate of advancements across various Monte Carlo-based reliability applications [ 48 , 49 ], but there is a notable shortage of studies focused on the implications of these advancements for enhancing the efficiency of reliability-oriented global sensitivity analyses (GSAs). Ensuring the accurate estimation of failure probability P f with the available number of simulations is critical because it directly influences reliability assessments and decision-making processes, see, e.g., [ 50 – 53 ]. A more accurate P f estimation is advantageous when using both the original model and the metamodel, depending on the available computational resources. The solution proposed in this article involves adopting an alternative measure of structural reliability based on Cliff’s Delta [ 54 ], which can be calculated using doublenested-loop simulations. This approach enhances the precision of P f estimation, albeit with higher computational demands for calculating Cliff’s Delta, but it more effectively utilizes existing model outputs compared to P f estimation using the Monte Carlo method. Given the substantial cost of obtaining data using FEM models, this approach is justified. By maximizing the utility of existing simulations, this method ensures more accurate reliability analyses. 2. Cliff’s Delta Cliff’s delta, denoted as δC , was initially devised by Norman Cliff, primarily for the handling of ordinal data [ 54 ]. It serves as a metric to assess the frequency with which values from one distribution exceed those in another distribution. A key feature of δC is that it does not necessitate any specific assumptions regarding the distributions’ form or variability. Mathematics 2024,12, 2129 3 of 18 The formula for computing the sample estimate of δCis expressed as follows: δC=∑m i=1∑n j=1yi>yj−yi<yj m·n, where δC∈[−1, 1]. (1) In this equation, the two distributions are characterized by sizes nand m, with respective elements y i and y j . Here, the notation [ · ] refers to the Iverson bracket notation, resulting in 1 if the condition within the brackets holds, and 0 otherwise. Formally, the Iverson bracket notation can be defined as follows: yi>yj=1i f yi>yj; 0otherwise ,yi<yj=1i f yi<yj; 0otherwise . (2) This statistical approach allows for an intuitive comparison of two distributions by quantifying the dominance of one distribution over the other. Building upon the foundational description of δC , this measure is specifically applied to assess the relationship between resistance Rand load Fwithin the framework of structural reliability. In the limit state, a structure is reliable if R ≥ F; otherwise, failure occurs [ 19 ]. Let the difference between Rand Fbe denoted as follows: Z=R−F, (3) where Rand Fare statistically independent random variables. In Monte Carlo simulations, these variables are represented as arrays Rand F, corresponding to resistance and load, respectively. Cliff’s delta, δC , is utilized to quantitatively evaluate the extent to which values of the resistance exceed or fall below those of the load, thereby providing fundamental insights into system reliability. Assuming arrays Rand Fare equal in size, with nentries each, the formula for calculating δCis simplified to: δC=∑n i=1∑n j=1Ri>Fj−Ri<Fj n2, where δC∈[−1, 1]. (4) In this expression, R i and F j represent the i-th and j-th entries in the Rand Farrays, respectively, denoting random realizations of resistance and load. The Iverson brackets, [ · ], return 1 when the enclosed condition is true, and 0 otherwise. This formulation facilitates a direct comparison between resistance and load across all sampled scenarios. The estimation of δC between the measurements of resistance and force provides a metric quantifying the frequency with which the values of resistance surpass those of load. Employed as a statistical tool, this measure assesses how frequently resistance can withstand or exceed the applied loads. 3. Sensitivity Measures of Cliff’s Delta Although Cliff’s Delta has been applied in numerous studies, see, e.g., [ 55 – 59 ], its utilization in the global sensitivity analysis of reliability is absent. This chapter demonstrates that the sensitivity measures based on the squared value of Cliff’s Delta, δ2 C , exhibit properties similar to the variance in the Sobol’ sensitivity analysis [ 7 , 8 ], oriented to reliability [16,17]. 3.1. Approximation of Failure Probability with Cliff’s Delta in Sensitivity Analysis In the limit state, a structure is considered reliable if R ≥ F; otherwise, failure occurs [ 15 ]. The probability of failure P f can be defined as the overload probability that F>R(i.e., Z< 0). The failure probability can be expressed as the mean value of the binary reliability function of the Bernoulli distribution, where 1 occurs if Z< 0, and 0 otherwise [21]: Pf=P(Z<0)=E(1Z<0), (5) Mathematics 2024,12, 2129 4 of 18 where 1Z<0=|Z|−Z 2|Z|. (6) The conventional measure of reliability is 1 − P f . The variance of the Bernoulli distribution of the random variable 1Z< 0 can be written as follows: V(1Z<0)=Pf1−Pf. (7) The second moment is a function of P f , which is useful for the formulation of the Sobol’ sensitivity indices, quantifiable through the estimation of the conditional realizations of Pf[21]. If the variance V(1Z< 0) is used in the decomposition within the Sobol’ sensitivity analysis, the first-order sensitivity index of the variance function 1 Z < 0 can be expressed as follows: Si=V(E(1Z<0|Xi) ) V(1Z<0)=V(1Z<0)−E(V(1Z<0|Xi) ) V(1Z<0)=Pf1−Pf−EPf|Xi1−Pf|Xi Pf1−Pf. (8) The concept of sensitivity analysis based on Cliff’s Delta is predicated on the assumption that Cliff’s Delta can be expressed as follows: δC=P(Z>0)−P(Z<0)=1−Pf−Pf=1−2Pf. (9) The failure probability Pfcan be calculated using Cliff’s Delta, as follows: Pf=1−δC 2. (10) Similarly, the second moment can be approximated and written as a function of Cliff’s Delta, as follows: V(1Z<0)=Pf1−Pf=1−δC 21−1−δC 2=1−δ2 C 4. (11) Substituting this into Equation (8), the Sobol’ sensitivity index can be expressed using δC, as follows: Si= 1−δ2 C 4−E1−δ2 C|Xi 4 1−δ2 C 4 =Eδ2 C|Xi−δ2 C 1−δ2 C . (12) This formulation provides an alternative method for calculating the sensitivity index, which carries all the advantages and disadvantages associated with the estimation of Cliff’s Delta compared to failure probability. 3.2. Sensitivity Indices Based on Cliff Delta In the development of sensitivity indices for the evaluation of structural reliability, the use of the squared measure of Cliff’s delta, δ2 C , has been proposed. This chapter delineates the formal definitions of these indices, categorized from first to higher orders. The sensitivity indices are computed as ratios of differences normalized by the constant, 1−C0, where C0is defined as the square of Cliff’s delta, δ2 C. C0=δ2 C, (13) where δ2 C represents the measure calculated when all input random variables; X 1 ,X 2 , . . . , XM, of Rand Fare random and statistically independent. Mathematics 2024,12, 2129 5 of 18 The first-order sensitivity index, S i , is defined to quantify the effect of a single variable Xion the change observed in the squared Cliff’s delta, δ2 C. It is calculated as follows: Si=Eδ2 C|Xi−C0 1−C0 . (14) In Equation (14), having frozen one potential source of variation (X i ), the resulting Eδ2 C|Xi will be higher than the corresponding total or unconditional Cliff’s Delta, where C 0 = δ2 C . For example, if X i were the sole source of change in the distance between Rand F, fixing it to x∗ iwould result in δ2 C|Xi=x∗ i= 1. The second-order sensitivity index, S ij , extends the analysis to pairs of variables, evaluating the joint effect of X i and X j on the change of δ2 C . This index is expressed as follows: Sij =Eδ2 C|Xi,Xj−C0 1−C0 −Si−Sj. (15) Similarly, the third-order index, S ijk , considers the combined influence of three variables Xi,Xj, and Xk. It is calculated as follows: Sijk =Eδ2 C|Xi,Xj,Xk−C0 1−C0 −Si−Sj−Sk−Sij −Sik −Sjk. (16) Other Cliff’s sensitivity indices, which quantify higher-order interaction effects, are defined analogously. The sensitivity index of the last order can be expressed as follows: S1,2,...,M=Eδ2 C|X1,X2, . . . , XM−C0 1−C0 −∑M−1 p=1∑1≤i1<...<ip≤MSi1,i2,...,ip=1−∑M−1 p=1∑1≤i1<...<ip≤MSi1,i2,...,ip, (17) where E( δ2 C |X 1 ,X 2 , . . . ,X M ) = 1 is ensured due to the nature of Cliff’s Delta, which assumes a value of either 1 or − 1 when all input random variables are fixed. Consequently, each element in the array Radopts a consistent identical value denoted as v 1 , and similarly, each element in the array Fmaintains another consistent identical value, denoted as v 2 . It should be noted that the constants v1and v2are generally different. The sum of all indices is equal to one. This characteristic is guaranteed by the computation of the last-order sensitivity index, as shown in Equation (17), which is derived from the difference between 1 and the sum of all lower-order sensitivity indices. ∑ i Si+∑ i ∑ j>i Sij +∑ i ∑ j>i ∑ k>j Sijk +. . . +S123...M=1. (18) The non-negativity of sensitivity indices is proven by their association with variance—see Equation (11)—and the Sobol’ decomposition of variance [ 7 , 8 ]. The fixing of multiple input variables typically leads to a higher value of δ2 C (with a limit of one in the last-order sensitivity index) compared to the constant C 0 . The properties of sensitivity indices based on Cliff’s delta and their comparison with the classical Sobol’ sensitivity analysis will be further explored in later chapters. 3.3. Sensitivity Indices Based on Failure Probability The impact of input variables on the failure probability P f can be analyzed using sensitivity analysis based on contrast functions [ 18 ]. Unlike the Sobol’ sensitivity analysis, contrast-oriented sensitivity analysis employs a contrast function, whose minimizer is of primary interest [ 60 , 61 ]. Fort [ 18 ] demonstrated that employing the quadratic contrast function, which calculates the mean of the squared deviations from the average, results in the well-known Sobol’ sensitivity indices [ 7 , 8 ]. In reliability-oriented sensitivity analysis, the variance from the Sobol’ sensitivity analysis is considered to be the variance of the binary reliability function V(1Z<0) = Pf(1 −Pf). Mathematics 2024,12, 2129 6 of 18 The first-order sensitivity index, C i , quantifies the main effect of a single variable X i on the variance of the binary reliability function, as follows: Ci=V(E(1Z<0|Xi) ) V(1Z<0)=Pf1−Pf−EPf|Xi1−Pf|Xi Pf1−Pf. (19) In this equation, freezing the source of variation, X i , affects P f . The sensitivity index C i indicates the extent to which one could reduce, on average, the output variance of the binary function 1Z<0, if Xicould be fixed. Hence, it is a measure of the main effect. The second-order sensitivity index, C ij , measures the pair effects of X i and X j on the variance of the binary function 1Z<0. This index can be written as follows: Cij =VE1Z<0|Xi,Xj V(1Z<0)−Si−Sj. (20) Similarly, the third-order index, C ijk , considers the combined influence of three variables Xi,Xj, and Xk, calculated as follows: Cij =VE1Z<0|Xi,Xj,Xk V(1Z<0)−Si−Sj−Sk−Sij −Sik −Sjk. (21) Higher-order contrast sensitivity indices, which quantify higher-order interaction effects, are defined analogously. The sum of all sensitivity indices is equal to one. ∑ i Ci+∑ i ∑ j>i Cij +∑ i ∑ j>i ∑ k>j Cijk +. . . +C123...M=1. (22) The non-negativity of sensitivity indices is guaranteed by their derivation from the Sobol’ sensitivity analysis [6–8], see also [19,21]. 4. The Case Study In the case study, the new sensitivity indices are compared with the results of the Sobol’ sensitivity analysis using a simple example. In Equation (3), the resistance can be considered as follows: R=X1·X2+X2·X3+K, (23) where Kis constant. The load is considered as follows: F=X4·X5. (24) All input random variables, X 1 ,X 2 , . . . ,X 5 , follow a Gaussian probability density function with a mean value of zero and a standard deviation of one. The input random variables are statistically independent. Table 1presents the estimated values of C 0 = δ2 C , where δC is obtained using n= 12,000 runs of the Latin Hypercube Sampling (LHS) method [ 62 , 63 ]. The conditional values of δC are estimated using double-nested-loop computation of the LHS method. When a single input variable, X i , is fixed, it is sampled using n= 12,000 runs, and for each realization, δC is calculated again using n= 12,000 runs of the LHS method. This numerical procedure is analogous to that of Sobol’ [ 7 , 8 ], but the sensitivity measure is not based on variance but on Cliff’s Delta. When computing S i , the computational complexity of E( δ2 C |X i ) is n2= 144,000,000. In Table 1, the data in the last column represent a set of values that are highly consistent, with low variance. Most values range between 0.924 and 0.940, with a few exceptions approaching 1.000. In Table 1, the value 1 is always present in the last row, ensuring that the sum of all indices is equal to one. Mathematics 2024,12, 2129 7 of 18 Table 1. Average values of δ2 Cfor sequentially fixed input variables. Conditional Mean of δ2 CK= 0 K= 1 K= 2 K= 3 K= 4 C0=δ2 C0 0.3150 0.6504 0.8323 0.9221 E(δ2 C|X1)0 0.3223 0.6556 0.8355 0.9240 E(δ2 C|X2)0 0.3508 0.6745 0.8430 0.9266 E(δ2 C|X3)0 0.3223 0.6556 0.8355 0.9240 E(δ2 C|X4)0 0.3284 0.6576 0.8371 0.9239 E(δ2 C|X5)0 0.3284 0.6576 0.8371 0.9239 E(δ2 C|X1,X2)0.1803 0.4633 0.7270 0.8664 0.9367 E(δ2 C|X1,X3)0 0.3509 0.6746 0.8432 0.9267 E(δ2 C|X1,X4)0 0.3362 0.6628 0.8413 0.9265 E(δ2 C|X1,X5)0 0.3362 0.6628 0.8413 0.9265 E(δ2 C|X2,X3)0.1803 0.4633 0.7270 0.8664 0.9367 E(δ2 C|X2,X4)0 0.3704 0.6825 0.8478 0.9286 E(δ2 C|X2,X5)0 0.3704 0.6825 0.8478 0.9286 E(δ2 C|X3,X4)0 0.3370 0.6628 0.8392 0.9252 E(δ2 C|X3,X5)0 0.3370 0.6628 0.8392 0.9252 E(δ2 C|X4,X5)0.2581 0.4820 0.7238 0.8616 0.9326 E(δ2 C|X1,X2,X3)0.4145 0.6274 0.8169 0.9133 0.9604 E(δ2 C|X1,X2,X4)0.1918 0.4852 0.7357 0.8713 0.9385 E(δ2 C|X1,X2,X5)0.1918 0.4852 0.7357 0.8713 0.9385 E(δ2 C|X1,X3,X4)0 0.3702 0.6827 0.8481 0.9287 E(δ2 C|X1,X3,X5)0 0.3702 0.6827 0.8481 0.9287 E(δ2 C|X1,X4,X5)0.2644 0.4923 0.7303 0.8664 0.9356 E(δ2 C|X2,X3,X4)0.1925 0.4844 0.7363 0.8722 0.9395 E(δ2 C|X2,X3,X5)0.1925 0.4844 0.7363 0.8722 0.9395 E(δ2 C|X2,X4,X5)0.3151 0.5425 0.7561 0.8760 0.9393 E(δ2 C|X3,X4,X5)0.2649 0.4928 0.7298 0.8644 0.9343 E(δ2 C|X1,X2,X3,X4)0.4630 0.6677 0.8346 0.9231 0.9648 E(δ2 C|X1,X2,X3,X5)0.4630 0.6677 0.8346 0.9231 0.9648 E(δ2 C|X1,X2,X4,X5)0.5361 0.6802 0.8236 0.9083 0.9539 E(δ2 C|X1,X3,X4,X5)0.3123 0.5449 0.7569 0.8768 0.9397 E(δ2 C|X2,X3,X4,X5)0.5361 0.6802 0.8236 0.9083 0.9539 E(δ2 C|X1,X2,X3,X4,X5)1 1 1 1 1 Table 1presents the average values of δ2 C for various fixed input variables, computed using the LHS method with n= 12,000 runs. The conditional values of δ2 Care estimated using a double-nested-loop computation. The column with K= 0 exhibits values close to zero, indicating the absence of dominance of Rover Fin the observations of δ2 C . Conversely, values far from zero suggest a dominance of Rover Fin the observation of δ2 C. In general, the strong influence of an input variable or variables occurs when the mean value of the fixed realizations of δ2 C is significantly different from C 0 —see Equations (14)–(16). It can be noted that the accuracy of the estimation of the sensitivity indices is lowest for K= 4, where the conditioned realizations of δ2 C in the last column of Table 1are very consistent, with low variance, and are minimally different from C0. The influences of the input variables and their groups, as expressed using sensitivity indices, are displayed in Figures 1–3. The color legend in Figure 1is applied to all subsequent pie charts in this article. In Figure 1, the absence of red in the pie chart demonstrates that all first-order sensitivity indices (represented by the red square in the legend) are zero. The second-order sensitivity indices S 12 = 0.18 and S 23 = 0.18 have the same value, which is due to the nature of Equation 10 and the same characteristics of the input random variables. The dominant influence is the interaction effect of variables X 4 and X 5 , as indicated by the value S45 = 0.26. Mathematics 2024,12, 2129 8 of 18 Mathematics 2024, 12, x FOR PEER REVIEW 8 of 18 Figure 1. Cliff’s Delta sensitivity indices for K = 0. Figure 2. Cliff’s Delta sensitivity indices for K = 1 and K = 2. Figure 3. Cliff’s Delta sensitivity indices for K = 3 and K = 4. In Figure 1, the absence of red in the pie chart demonstrates that all first-order sensitivity indices (represented by the red square in the legend) are zero. The second-order sensitivity indices S12 = 0.18 and S23 = 0.18 have the same value, which is due to the nature of Equation 10 and the same characteristics of the input random variables. The dominant influence is the interaction effect of variables X4 and X5, as indicated by the value S45 = 0.26. Increasing the value of the constant K distances the random realizations of resistance R from load action F and enhances the value of Cliff’s Delta. The interaction effects indicated by sensitivity indices S12, S23, and S45 decrease with increasing K, while the proportion of third-, fourth-, and fifth-order sensitivity indices increases—see Figures 2 and 3. In the sensitivity analysis based on Cliff’s Delta, the impact of input variables on the observed changes in 𝛿  is quantified using sensitivity indices of the first order and higher orders. To derive meaningful conclusions and to categorize input variables into influential, less influential, and non-influential groups, it is essential to assign the effects of each input variable without the complexity of interpreting numerous sensitivity indices. To achieve this, the concept of the total effect index is employed. This index captures the comprehensive contribution of a factor, Xi, to the changes observed in 𝛿 . Specifically, it encompasses both the first-order effects and all higher-order effects resulting from interactions. The total effect index provides a robust measure of the impact that each input variable has on the 𝛿 , accounting for all potential interactions. For instance, in a five-factor model, the total effect of the factor X1 is calculated by adding all terms in Equation (18) where the factor X1 is included. Figure 1. Cliff’s Delta sensitivity indices for K= 0. Mathematics 2024, 12, x FOR PEER REVIEW 8 of 18 Figure 1. Cliff’s Delta sensitivity indices for K = 0. Figure 2. Cliff’s Delta sensitivity indices for K = 1 and K = 2. Figure 3. Cliff’s Delta sensitivity indices for K = 3 and K = 4. In Figure 1, the absence of red in the pie chart demonstrates that all first-order sensitivity indices (represented by the red square in the legend) are zero. The second-order sensitivity indices S12 = 0.18 and S23 = 0.18 have the same value, which is due to the nature of Equation 10 and the same characteristics of the input random variables. The dominant influence is the interaction effect of variables X4 and X5, as indicated by the value S45 = 0.26. Increasing the value of the constant K distances the random realizations of resistance R from load action F and enhances the value of Cliff’s Delta. The interaction effects indicated by sensitivity indices S12, S23, and S45 decrease with increasing K, while the proportion of third-, fourth-, and fifth-order sensitivity indices increases—see Figures 2 and 3. In the sensitivity analysis based on Cliff’s Delta, the impact of input variables on the observed changes in 𝛿  is quantified using sensitivity indices of the first order and higher orders. To derive meaningful conclusions and to categorize input variables into influential, less influential, and non-influential groups, it is essential to assign the effects of each input variable without the complexity of interpreting numerous sensitivity indices. To achieve this, the concept of the total effect index is employed. This index captures the comprehensive contribution of a factor, Xi, to the changes observed in 𝛿 . Specifically, it encompasses both the first-order effects and all higher-order effects resulting from interactions. The total effect index provides a robust measure of the impact that each input variable has on the 𝛿 , accounting for all potential interactions. For instance, in a five-factor model, the total effect of the factor X1 is calculated by adding all terms in Equation (18) where the factor X1 is included. Figure 2. Cliff’s Delta sensitivity indices for K= 1 and K= 2. Mathematics 2024, 12, x FOR PEER REVIEW 8 of 18 Figure 1. Cliff’s Delta sensitivity indices for K = 0. Figure 2. Cliff’s Delta sensitivity indices for K = 1 and K = 2. Figure 3. Cliff’s Delta sensitivity indices for K = 3 and K = 4. In Figure 1, the absence of red in the pie chart demonstrates that all first-order sensitivity indices (represented by the red square in the legend) are zero. The second-order sensitivity indices S12 = 0.18 and S23 = 0.18 have the same value, which is due to the nature of Equation 10 and the same characteristics of the input random variables. The dominant influence is the interaction effect of variables X4 and X5, as indicated by the value S45 = 0.26. Increasing the value of the constant K distances the random realizations of resistance R from load action F and enhances the value of Cliff’s Delta. The interaction effects indicated by sensitivity indices S12, S23, and S45 decrease with increasing K, while the proportion of third-, fourth-, and fifth-order sensitivity indices increases—see Figures 2 and 3. In the sensitivity analysis based on Cliff’s Delta, the impact of input variables on the observed changes in 𝛿  is quantified using sensitivity indices of the first order and higher orders. To derive meaningful conclusions and to categorize input variables into influential, less influential, and non-influential groups, it is essential to assign the effects of each input variable without the complexity of interpreting numerous sensitivity indices. To achieve this, the concept of the total effect index is employed. This index captures the comprehensive contribution of a factor, Xi, to the changes observed in 𝛿 . Specifically, it encompasses both the first-order effects and all higher-order effects resulting from interactions. The total effect index provides a robust measure of the impact that each input variable has on the 𝛿 , accounting for all potential interactions. For instance, in a five-factor model, the total effect of the factor X1 is calculated by adding all terms in Equation (18) where the factor X1 is included. Figure 3. Cliff’s Delta sensitivity indices for K= 3 and K= 4. Increasing the value of the constant Kdistances the random realizations of resistance R from load action Fand enhances the value of Cliff’s Delta. The interaction effects indicated by sensitivity indices S 12 ,S 23 , and S 45 decrease with increasing K, while the proportion of third-, fourth-, and fifth-order sensitivity indices increases—see Figures 2and 3. In the sensitivity analysis based on Cliff’s Delta, the impact of input variables on the observed changes in δ2 C is quantified using sensitivity indices of the first order and higher orders. To derive meaningful conclusions and to categorize input variables into influential, less influential, and non-influential groups, it is essential to assign the effects of each input variable without the complexity of interpreting numerous sensitivity indices. To achieve this, the concept of the total effect index is employed. This index captures the comprehensive contribution of a factor, X i , to the changes observed in δ2 C . Specifically, it encompasses both the first-order effects and all higher-order effects resulting from interactions. The total effect index provides a robust measure of the impact that each input variable has on the δ2 C, accounting for all potential interactions. For instance, in a five-factor model, the total effect of the factor X 1 is calculated by adding all terms in Equation (18) where the factor X1is included. ST1=S1+S12 +S13 +S14 +S15 +S123 +S124 +S125 +S134 +S135 +S145 +S1234 +S1235 +S1245 +S1345 +S12345. (25) This sum accounts for X 1 direct influence on δ2 C , as well as its synergistic effects with other factors. The total effect measure provides an educated answer to the following Mathematics 2024,12, 2129 9 of 18 question: which factor can be fixed anywhere over its range of variability without affecting the δ2 C ? The total effect index reflects both the main and the interaction influences of X 1 on the outcome, providing a comprehensive view of its relative importance in the system’s reliability, measured by the distance from Fto R. The total effects for the case study are displayed in Figure 4. Mathematics 2024, 12, x FOR PEER REVIEW 9 of 18 1234513451245123512341451351341251241231514131211 SSSSSSSSSSSSSSSSS T+++++++++++++++= . (25) This sum accounts for X1 direct influence on 𝛿 , as well as its synergistic effects with other factors. The total effect measure provides an educated answer to the following question: which factor can be fixed anywhere over its range of variability without affecting the 𝛿 ? The total effect index reflects both the main and the interaction influences of X1 on the outcome, providing a comprehensive view of its relative importance in the system’s reliability, measured by the distance from F to R. The total effects for the case study are displayed in Figure 4. Figure 4. Cliff’s Delta total sensitivity indices for all K. The sensitivity analysis results show the total sensitivity indices for each input variable (X1, X2, X3, X4, and X5) across five distinct constant values (K = 0, 1, 2, 3, and 4), using Cliff’s delta as the sensitivity measure. As K increases from 0 to 4, a general trend of increasing total sensitivity indices is observed for X1 and X2, as depicted in Figure 4. For X3, the total sensitivity indices exhibit a slightly convex pattern. An increasing trend for X3 is observed from K = 1 to K = 4. This suggests that these variables become more influential with higher values of K, as measured using Cliff’s delta. Conversely, the sensitivity indices for X4 and X5 decrease with increasing K. The variable X2 consistently exhibits the highest total sensitivity index across all tested values of K, indicating its dominant influence on Cliff’s delta. This effect is attributable to X2’s involvement in both additive terms of the resistance function, as shown in Equation (23). The variables X1 and X3 also demonstrate an increasing influence, although they remain slightly less dominant than X2, but are notably more influential than X4 and X5 as K increases. The influence of X4 and X5, which are involved in the load force equation, decreases especially as K surpasses 1, highlighting their reduced significance in affecting Cliff’s Delta. The sensitivity analysis outcomes reflect a decreasing probability P(R < F), which diminishes as K increases. The Cliff Delta-based sensitivity analysis exhibits characteristics of a reliability-oriented sensitivity analysis [19], describing the change in the influence of each variable on Cliff’s Delta due to K. The influence on the results of the sensitivity indices according to the value of the deterministic quantity K is the main difference compared to the Sobol’ sensitivity analysis. The results of the classical Sobol’ sensitivity analysis can be obtained analytically. Non-zero values of the Sobol’ sensitivity indices were obtained only for second-order sensitivity indices 𝑆  = 1/3, 𝑆  = 1/3, and 𝑆  = 1/3; other Sobol’ indices are zero. The total effect Sobol’ sensitivity indices are 𝑆  = 1/3, 𝑆  = 2/3, 𝑆  = 1/3, 𝑆  = 1/3, and 𝑆  = 1/3. The dominant influence of the input variable X2 confirms the most important conclusions of the newly introduced sensitivity analysis based on Cliff’s Delta; however, Figure 4. Cliff’s Delta total sensitivity indices for all K. The sensitivity analysis results show the total sensitivity indices for each input variable (X 1 ,X 2 ,X 3 ,X 4 , and X 5 ) across five distinct constant values (K= 0, 1, 2, 3, and 4), using Cliff’s delta as the sensitivity measure. As Kincreases from 0 to 4, a general trend of increasing total sensitivity indices is observed for X 1 and X 2 , as depicted in Figure 4. For X 3 , the total sensitivity indices exhibit a slightly convex pattern. An increasing trend for X 3 is observed from K= 1 to K= 4. This suggests that these variables become more influential with higher values of K, as measured using Cliff’s delta. Conversely, the sensitivity indices for X 4 and X 5 decrease with increasing K. The variable X 2 consistently exhibits the highest total sensitivity index across all tested values of K, indicating its dominant influence on Cliff’s delta. This effect is attributable to X 2 ’s involvement in both additive terms of the resistance function, as shown in Equation (23). The variables X 1 and X 3 also demonstrate an increasing influence, although they remain slightly less dominant than X 2 , but are notably more influential than X 4 and X 5 as Kincreases. The influence of X 4 and X 5 , which are involved in the load force equation, decreases especially as Ksurpasses 1, highlighting their reduced significance in affecting Cliff’s Delta. The sensitivity analysis outcomes reflect a decreasing probability P(R<F), which diminishes as Kincreases. The Cliff Delta-based sensitivity analysis exhibits characteristics of a reliability-oriented sensitivity analysis [ 19 ], describing the change in the influence of each variable on Cliff’s Delta due to K. The influence on the results of the sensitivity indices according to the value of the deterministic quantity Kis the main difference compared to the Sobol’ sensitivity analysis. The results of the classical Sobol’ sensitivity analysis can be obtained analytically. Nonzero values of the Sobol’ sensitivity indices were obtained only for second-order sensitivity indices SSob 12 = 1/3, SSob 23 = 1/3, and SSob 45 = 1/3; other Sobol’ indices are zero. The total effect Sobol’ sensitivity indices are SSob T1 = 1/3, SSob T2 = 2/3, SSob T3 = 1/3, SSob T4 = 1/3, and SSob T5 = 1/3. The dominant influence of the input variable X 2 confirms the most important conclusions of the newly introduced sensitivity analysis based on Cliff’s Delta; however, there are differences. The results of the Sobol’ sensitivity analysis are independent of the value of the constant K, because the Sobol’ indices are based on the decomposition of variance, which the deterministic variable Kdoes not affect. 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