Derivative-Variance Hybrid Global Sensitivity Measure with Optimal Sampling Method Selection

Abstract

This paper proposes a derivative-variance hybrid global sensitivity measure with optimal sampling method selection. The proposed sensitivity measure is as computationally efficient as the derivative-based global sensitivity measure, which also serves as the conservative estimation of the corresponding variance-based global sensitivity measure. Moreover, the optimal sampling method for the proposed sensitivity measure is studied. In search of the optimal sampling method, we investigated the performances of six widely used sampling methods, namely Monte Carlo sampling, Latin hypercube sampling, stratified sampling, Latinized stratified sampling, and quasi-Monte Carlo sampling using the Sobol and Halton sequences. In addition, the proposed sensitivity measure is validated through its application to a rural bridge.

1. Introduction

Sensitivity analysis serves as a fundamental instrument for importance ranking and the dimension reduction of input parameters [1], and its applications are spreading across diverse topics in civil engineering, such as strength analysis [2], failure analysis [3], rock mechanics [4], slope stability analysis [5], and structural reliability analysis [6].

Sensitivity analysis examines how the output, or model’s response, is affected by the inputs and therefore calculates their corresponding importance. Compared to qualitative or subjective sensitivity measures [7,8], quantitative sensitivity measures have been demonstrated to be much more trustworthy [9,10,11]. Quantitative sensitivity measures can be roughly classified into local sensitivity measures and global sensitivity measures [12].

The local sensitivity measures, also referred to as the elementary effect, assess how the input factor locally affects the model output by varying the input of interest while keeping the rest of the inputs constant, which is the one factor of a time method [13]. The main drawback of local sensitivity measures is that they are fixed to the corresponding nominal points in the input space, which means they cannot provide information on the inputs’ total effects or detect interactions among inputs. In contrast to local sensitivity measures, global sensitivity measures evaluate the inputs’ overall effects on the model output, which can also account for interactions between inputs. Obviously, global sensitivity measures offer a more comprehensive sensitivity analysis and are thus more widely accepted in practical engineering. Invented in the 1970s, the Fourier amplitude sensitivity test [14] is one of the quickest global sensitivity measures. It is, however, merely a first-order measure and can be insufficient if higher-order terms dominate the model output [9]. Another widely adopted global sensitivity measure is the variance-based global sensitivity measure, which was first proposed by Sobol and further developed by Saltelli and his colleagues [1,11]. The variance-based global sensitivity measure calculates the model output’s variances of different subsets of input variables, which usually requires a large number of model evaluations in order to achieve a reasonable convergence and then becomes computationally unaffordable. To this end, a number of alternative global sensitivity measures have been recently proposed, one of which is the Morris measure [7,15]. The Morris measure relies on averaging the absolute values of local sensitivity measures, which is considerably cheaper in terms of computation than the variance-based global sensitivity measure. Nevertheless, the Morris method uses local sensitivity measures of random points on a fixed grid for averaging, which cannot correctly reflect effects if the model’s characteristic dimensions are smaller than the grid size [16]. To solve this issue, the derivative-based global sensitivity measure was recently proposed by employing local sensitivity measures of randomly distributed points rather than a fixed grid to estimate the integral representing the global effects [17]. The derivative-based global sensitivity measure has been demonstrated to be good at quickly identifying unimportant input factors, while the variance-based global sensitivity measure is more comprehensive but also more computationally expensive. Another advantage of the variance-based global sensitivity measure over the derivative-based global sensitivity measure is that the variance-based global sensitivity measure provides absolute values for the sensitivity measures, while the derivative-based global sensitivity measure only provides relative values for sensitivity measures. In other words, in using a derivative-based global sensitivity measure for the sensitivity analysis, one has to compute the derivative-based global sensitivity measure of each input before determining their sensitivities or importance. Recently, a variety of innovative sensitivity measures have been proposed for various application scopes. A novel global sensitivity measure, called PAWN, was proposed to efficiently compute density-based sensitivity indices [18]. As for structural engineering, a novel vine copula-based approach for analyzing the variance-based global sensitivity measure is proposed [19]. In reliability engineering, a new reliability-oriented global sensitivity measure, which is obtained by replacing the variance with entropy, was proposed [20]. Connections between entropy and variance in reliability engineering have also been recently explored [21].

It should also be pointed out that the computational efficiency of global sensitivity measures heavily depends on different sampling methods [12,22,23]. Although recently developed meta-models appear to be more efficient in terms of sampling [24,25,26], the traditional quasi-Monte Carlo or Monte Carlo methods remain the primary sampling methods for practical sensitivity analysis. It is well known that quasi-Monte Carlo sampling should outperform Monte Carlo sampling in a sensitivity analysis for a sufficiently large sample size [23,27]. A board comparison including other popular sampling methods, such as Latin hypercube sampling, stratified sampling, and Latinized stratified sampling, has not been conducted yet. In addition, the impact of nonlinear dependency among random variables is also important in sensitivity analysis, which can be investigated through the copula theory [28].

Sensitivity measures that combine the advantages of both the variance-based global sensitivity measures and derivative-based global sensitivity measures would provide better tools for sensitivity analysis. In light of such an idea, a new global sensitivity measure based on derivative-integral and variance decomposition was recently proposed; however, this measure is too complicated for many engineering applications [29], and a simpler derivate-variance hybrid sensitivity analysis is needed. Therefore, this research presents a simple derivative-variance hybrid sensitivity measure, which we named the derivative-variance hybrid global sensitivity measure. The proposed derivative-variance hybrid global sensitivity measure is as computationally efficient as the derivative-based global sensitivity measure, which can also serve as a conservative approximation of the corresponding variance-based global sensitivity measure. Furthermore, in seeking the optimal sampling method, we investigated the performances of six frequently used sampling methods, namely Monte Carlo samplings, Latin hypercube samplings, stratified samplings, Latinized stratified samplings, and two quasi-Monte Carlo samplings using the Sobol sequence and the Halton sequence.

The rest of this paper is organized as follows: Section 2 reviews the necessary backgrounds of sensitivity measures; Section 3 proposes the derivative-variance hybrid global sensitivity measure; Section 4 determines the optimal sampling method via an investigation of the performances of six widely used sampling methods; and Section 5 validates the proposed sensitivity measure through its application to the sensitivity analysis of a rural bridge.

2. Fundamentals of Sensitivity Measures

Quantitative sensitivity measures are used to calculate the influences of inputs on the model output as certain numbers. In this section, we review three important quantitative sensitivity measures; we consider a model output represented by the function $f(x)=f(x_1,\dots ,x_n)$ defined in the unit hypercube $I^n={[0,1]}^n$, where $f(x)$ is both differentiable and square-integrable.

2.1. Local Sensitivity Measure

For local sensitivity measures, the sensitivity of a variable is measured locally via a nominal point in the space of the inputs, and the value of this variable is represented by the corresponding partial derivative or the approximated partial difference, i.e.,

$${\epsilon }_i(x)=\frac{\partial f(x)}{\partial x_i}\approx \frac{f(x+{\Delta }_i)-f(x)}{{\Delta }_i}$$

(1)

where ε_i is the local sensitivity measure associated with the nominal point x₀ and ${\Delta }_i=[0,\dots ,\Delta ,\dots ,0]$ is a vector of length ∆ along the direction of x_i.

2.2. Derivative-Based Global Sensitivity Measure

In light of the fact that a high value of the output’s derivative indicates a large variation in the model output, the derivative-based global sensitivity measures are achieved globally via stochastic integrals of the randomly distributed local sensitivity measures. There exist various definitions of the derivative-based global sensitivity measures. Here we present a widely used one:

$$v_i={\displaystyle \underset{I^n}{\int }{\left[{\epsilon }_i(x)\right]}^2}\mathrm{d}x\approx \frac{1}{N}{\displaystyle \sum _{k=1}^N{\left[{\epsilon }_i(x_k)\right]}^2}$$

(2)

where x_k is obtained through random sampling, N is the total number of sample points, and v_i is Sobol’s derivative-based global sensitivity measure [15]. It is noted that a derivative-based global sensitivity measure does not reflect the relative importance of x_i. Thus, one has to rank all the inputs according to their derivative-based global sensitivity measures to obtain their relative importance, while the value of a derivative-based global sensitivity measure itself does not provide any useful information on sensitivity.

2.3. Variance-Based Global Sensitivity Measure

Compared to various versions of the derivative-based global sensitivity measures, the definition of variance-based global sensitivity measures is unique, and their core is the analysis of variance decomposition [30]. The analysis of variance decomposition of $f(x)$ has the following form:

$$f(x)=f_0+{\displaystyle \sum _{i=0}^n{f}_i+{\displaystyle \sum _{1\le i<j\le n}{f}_{ij}(x_i,x_j)+\cdots +f_{1\cdots n}(x_i,\dots ,x_n)}}$$

(3)

which should satisfy

$${\displaystyle {\int }_0^1{f}_{j\cdots i}(x_j,\dots ,x_i)\mathrm{d}}{x}_k=0$$

(4)

where 1 ≤ j ≤ k ≤ i ≤ n.

Each term in Equation (3) can be calculated in the following recursive manner:

$$f_0={\displaystyle \underset{I^n}{\int }f\left(x\right)}\mathrm{d}x$$

(5)

$$f_i(x_i)={\displaystyle \underset{I^n}{\int }f\left(x\right){\displaystyle {\prod }_{k\ne i}\mathrm{d}{x}_k}}-f_0$$

(6)

$$f_{ij}(x_i,x_j)={\displaystyle \underset{I^n}{\int }f\left(x\right){\displaystyle {\prod }_{k\ne i,j}\mathrm{d}{x}_k}}-f_0-f_i(x_i)-f_j(x_j)$$

(7)

$$\begin{array}{l}{f}_{i\dots j}(x_i,\dots ,x_j)={\displaystyle \underset{I^n}{\int }f\left(x\right){\displaystyle {\prod }_{k\ne i,j}\mathrm{d}{x}_k}}-f_0-{\displaystyle \sum _{k=i}^j{f}_k(x_k)}-{\displaystyle \sum _{k=i}^j{\displaystyle \sum _{l=i,l\ne k}^j{f}_{kl}(x_k,x_k)}}-\\ -\cdots -{\displaystyle \sum _{k=i}^j{f}_{i\cdots (k-1)(k+1)\cdots j}(x_i,\dots ,x_{k-1},x_{k+1},\dots ,x_j)}\end{array}$$

(8)

Let $x_A=\left\{x_{A_1},\cdots ,x_{A_s}\right\}$ denote an arbitrary subset of $x=\left\{x_1,\cdots ,x_n\right\}$, where 1 ≤ A₁ < A_s ≤ n and 1 ≤ s ≤ n, and let the corresponding complement denote $x_B=\left\{x_{B_1},\cdots ,x_{B_{n-s}}\right\}$, which satisfies $x=\left\{x_A,x_B\right\}$ and $\left\{A_1,\dots ,A_s,B_1,\dots ,B_{n-s}\right\}=\left\{1,2,\dots ,n\right\}$. The variance with respect to x_A is defined as

$$V_{x_A}={\displaystyle \underset{I^n}{\int }{f}_{i\dots j}{(x_i,\dots ,x_j)}^2{x}_i}\cdots x_j.$$

(9)

The total variance is defined by

$$V={\displaystyle \sum _{s=1}^n{\displaystyle \sum _{A_1<\cdots <A_s}^s{V}_{A_1\cdots A_s}}}.$$

(10)

Finally, the variance-based global sensitivity measure with respect to x_A is defined as

$$S_{x_A}=\frac{V_{x_A}}{V}.$$

(11)

The total variance-based global sensitivity measure with respect to x_A is defined as

$$S_{x_A}^T=1-S_{x_B}.$$

(12)

The first-order variance-based global sensitivity measure with respect to x_i is defined as

$$S_i=\frac{V_i}{V}.$$

(13)

where V_i is referred to as the first-order variance.

And the first-order total variance-based global sensitivity measure with respect to x_i is computed as

$$S_i^T=1-\frac{V_{1\cdots (i-1)(i+1)\cdots n}}{V}$$

(14)

Sobol proposed a Monte Carlo sampling-based computation method for the variance-based global sensitivity measure [11], which consists of the following four steps: Firstly, construct two input matrixes U and V, i.e.,

$$U=\left[\begin{array}{ccc}{x}_{11}& \cdots & x_{1n}\\ ⋮& \ddots & ⋮\\ x_{N1}& \cdots & x_{Nn}\end{array}\right]$$

(15)

$$V=\left[\begin{array}{ccc}{x^{\prime }}_{11}& \cdots & {x^{\prime }}_{1n}\\ ⋮& \ddots & ⋮\\ {x^{\prime }}_{N1}& \cdots & {x^{\prime }}_{Nn}\end{array}\right]$$

(16)

Secondly, according to the relationship of x_A and x_B to x, compute U_A and U_B from U, and compute V_A and V_B from V, i.e.,

$$U_A=\left[\begin{array}{ccc}{x}_{1A_1}& \cdots & x_{1A_s}\\ ⋮& \ddots & ⋮\\ x_{N{A}_1}& \cdots & x_{N{A}_s}\end{array}\right]$$

(17)

$$U_B=\left[\begin{array}{ccc}{x}_{1B_1}& \cdots & x_{1B_{n-s}}\\ ⋮& \ddots & ⋮\\ x_{N{B}_1}& \cdots & x_{N{B}_{n-s}}\end{array}\right]$$

(18)

$$V_A=\left[\begin{array}{ccc}{x^{\prime }}_{1A_1}& \cdots & {x^{\prime }}_{1A_s}\\ ⋮& \ddots & ⋮\\ {x^{\prime }}_{N{A}_1}& \cdots & {x^{\prime }}_{N{A}_s}\end{array}\right]$$

(19)

$$V_B=\left[\begin{array}{ccc}{x^{\prime }}_{1B_1}& \cdots & {x^{\prime }}_{1B_{n-s}}\\ ⋮& \ddots & ⋮\\ {x^{\prime }}_{N{B}_1}& \cdots & {x^{\prime }}_{N{B}_{n-s}}\end{array}\right]$$

(20)

Thirdly, composite the following two matrices: W_A = [U_A,V_B] and W_B = [V_A,U_B]. Lastly, compute the variance-based global sensitivity measure using Equations (9)–(14), and apply the following stochastic integrals:

$$V_{x_A}={\displaystyle \underset{I^n}{\int }f(x)}f(x_A,{x^{\prime }}_B)\mathrm{d}x\mathrm{d}{x^{\prime }}_B-{f_0}^2\approx \frac{1}{N}{\displaystyle \sum _{i=1}^{N}f(U_A)}f(W_A)-{f_0}^2$$

(21)

$$V_{x_B}={\displaystyle \underset{I^n}{\int }f(x)}f({x^{\prime }}_A,x_B)\mathrm{d}x\mathrm{d}{x^{\prime }}_A-{f_0}^2\approx \frac{1}{N}{\displaystyle \sum _{i=1}^{N}f(U_A)}f(W_B)-{f_0}^2$$

(22)

$$V={{\displaystyle \underset{I^n}{\int }f(x)}}^2\mathrm{d}x-{f_0}^2\approx \frac{1}{N}{\displaystyle \sum _{i=1}^{N}f{(U_A)}^2}-{f_0}^2$$

(23)

$$f_0={\displaystyle \underset{I^n}{\int }f(x)}\mathrm{d}x\approx \frac{1}{N}{\displaystyle \sum _{i=1}^{N}f(U_A)}$$

(24)

where x’ is another input space that is independent from x.

If the aim is to compute the total of the variance-based global sensitivity measure, the computational cost is nN in terms of model evaluations, which linearly increases with both the sample size and the dimensions of the inputs. What hampers many engineering applications of the variance-based global sensitivity measure is the fact that the sample size N required is usually too large to afford, which is significantly greater than that of the derivative-based global sensitivity measures [31]. In addition, the uniformity and stochasticity of the samples can dramatically influence the convergence rate [32].

3. The Derivative-Variance Hybrid Global Sensitivity Measure

In this section, the derivative-variance hybrid global sensitivity measure is proposed, which is as computationally efficient as the derivative-based global sensitivity measure while providing as much meaningful absolute value for sensitivity analysis as the variance-based global sensitivity measure. Section 3.1 proposes the sensitivity measure; Section 3.2 presents a method for computing the proposed sensitivity measure; Section 3.3 compares the derivative-variance hybrid global sensitivity measure with the derivative-based global sensitivity measure and the variance-based global sensitivity measure using three benchmark examples; and Section 3.4 discusses the advantages and limitations of the proposed sensitivity measure.

3.1. Presentation of the Sensitivity Measure

The derivative-based global sensitivity measure and the variance-based global sensitivity measure can be linked using [17]:

$${\pi }^{2}V\cdot S_i^T\le v_i$$

(25)

where $S_i^T$ is the variance-based global sensitivity measure computed via Equation (14), V is the total variance computed via Equation (23), and v_i is the derivative-based global sensitivity measure computed via Equation (2).

On the basis of Equation (25), the following sensitivity measure, referred to as the derivative-variance hybrid global sensitivity measure, is proposed:

$${\mathsf{\Upsilon }}_i=\frac{{\nu }_i}{{\pi }^{2}V}\ge S_i^T.$$

(26)

Evidently, the derivative-variance hybrid global sensitivity measure ϒ_i is obtained by dividing the derivative-based global sensitivity measure v_i by π²V, so they should share a similar computational cost that is much lower than that of the variance-based global sensitivity measure. Derivative-variance hybrid global sensitivity measures can be treated as reduced derivative-based global sensitivity measures in which the reduction allows the value of the derivative-variance hybrid global sensitivity measure to serve as a conservative approximation of the corresponding variance-based global sensitivity measure.

3.2. Computational Method

As shown in Figure 1, the derivative-variance hybrid global sensitivity measure can be computed using the following steps: (1) sample the input matrix x; (2) compute the total variance V using Equations (23) and (24); (3) enumerate each input x_i with Steps 4 and 5; (4) compute the derivative-based global sensitivity measure v_i using Equation (2); and (5) compute the derivative-variance hybrid global sensitivity measure ϒ_i using Equation (26).

3.3. Comparison with Other Global Sensitivity Measures

In this subsection, the performances of the derivative-variance hybrid global sensitivity measure, the derivative-based global sensitivity measure, and the variance-based global sensitivity measure using three benchmark examples are compared and analyzed, and the corresponding results are obtained through Monte Carlo simulations, in which the sample size is 10⁸.

Example 1. 

The analysis of variance decomposition is provided below [33]:

$$f(x)={\displaystyle \sum }_{i=1}^3{c}_i\left(x_i-\frac{1}{2}\right)+c_{12}\left(x_1-\frac{1}{2}\right){\left(x_2-\frac{1}{2}\right)}^5$$

(27)

where 0 < x_i < 1, c₁ = 10, c₂ = 1, c₃ = 0.1, and c₁₂ = 10. In this example, x₁ is the most sensitive input, x₃ is trivial, the importance of x₂ is in-between, and the sensitivity of a highly non-linear term is controlled by c₁₂. The corresponding results are shown in

Table 1. Results of the derivative-variance hybrid global sensitivity measure, the derivative-based global sensitivity measure, and the variance-based global sensitivity measure from Example 1.

	Variance-Based Global Sensitivity Measure	Derivative-Based Global Sensitivity Measure	Derivative-Variance Hybrid Global Sensitivity Measure
x1	0.9901	100.0	1.204
x2	0.0100	1.091	0.0131
x3	9.657 × 10−5	0.010	1.204 × 10−4

, from which it is seen that: (1) the derivative-based global sensitivity measures cannot directly indicate sensitivities of the corresponding inputs, while the variance-based global sensitivity measure and the derivative-variance hybrid global sensitivity measure can; (2) the derivative-variance hybrid global sensitivity measure can serve as a conservative estimation of the corresponding variance-based global sensitivity measure since the value of the derivative-variance hybrid global sensitivity measure is slightly larger than that of the corresponding variance-based global sensitivity measure; the value of derivative-variance hybrid global sensitivity measure can be greater than 1.

Example 2. 

The g-function is provided below [33]:

$$f(x)={\displaystyle \prod }_{i=1}^n\frac{\left|4x_i-2\right|+a_i}{1+a_i}$$

(28)

where 0 < x_i < 1, n = 3, a₁ = 1, a₂ = 10, and a₃ = 100. In this example, a larger value of a_i corresponds to a smaller importance of x_i. The corresponding results are shown in

Table 2. Results of the derivative-variance hybrid global sensitivity measure, the derivative-based global sensitivity measure, and the variance-based global sensitivity measure from Example 2.

	Variance-Based Global Sensitivity Measure	Derivative-Based Global Sensitivity Measure	Derivative-Variance Hybrid Global Sensitivity Measure
x1	0.9679	4.011	4.7063
x2	0.0352	0.1433	0.1681
x3	4.125 × 10−4	0.0017	0.0020

. It is seen from Table 2 that the derivative-variance hybrid global sensitivity measure can still effectively identify insensitive input x₃; however, the ratio between the derivative-variance hybrid global sensitivity measure and the corresponding variance-based global sensitivity measure can be significantly greater than 1.

Example 3. 

The Ishigami function is provided below [34]:

$$f(x)=\sin \left(2\pi x_1-\pi \right)+a\sin {\left(2\pi x_2-\pi \right)}^2+b{\left(2\pi x_3-\pi \right)}^4\sin \left(2\pi x_1-\pi \right)$$

(29)

where 0 < x_i < 1, i = 1, 2, or 3, and a and b in the following examples were set to 7 and 0.1, respectively. The corresponding results are shown in

Table 3. Results of the derivative-variance hybrid global sensitivity measure, the derivative-based global sensitivity measure, and the variance-based global sensitivity measure from Example 3.

	Variance-Based Global Sensitivity Measure	Derivative-Based Global Sensitivity Measure	Derivative-Variance Hybrid Global Sensitivity Measure
x1	0.5574	304.7	2.230
x2	0.4421	967.2	7.078
x3	0.2433	433.7	3.174

, from which it is seen that the derivative-variance hybrid global sensitivity measure may not be efficient with regard to the ranking of inputs of significant sensitivities.

3.4. Advantages and Limitations

Since the derivative-variance hybrid global sensitivity measure is the conservative approximation of the corresponding variance-based global sensitivity measure, the values of the variance-based global sensitivity measures can directly reflect the inputs’ sensitivities, which provides a solid argument for identifying insensitive inputs. For instance, if it is found that ϒ_i is negligibly small, then x_i can be safely reduced without further analyzing the rest of the inputs. For the identification of insensitive inputs using the derivative-based global sensitivity measures, one has to exhaust each input and rank their derivative-based global sensitivity measures in ascending order. On the other hand, the computational cost of the derivative-variance hybrid global sensitivity measure is similar to that of the derivative-based global sensitivity measure, which is considerably lower than that of the variance-based global sensitivity measure. At this point, we list the advantages of the proposed derivative-variance hybrid global sensitivity measure as follows: (1) inputs with a small derivative-variance hybrid global sensitivity measure can be labeled as unimportant without investigating the rest of the inputs; (2) the computational cost of the derivative-variance hybrid global sensitivity measure is comparable to that of the derivative-based global sensitivity measure, which only requires the additional computation of the total variance V; (3) since the derivative-variance hybrid global sensitivity measure is the upper bound for the corresponding variance-based global sensitivity measure, the derivative-based global sensitivity measures can serve as a conservative approximation of the variance-based global sensitivity measure. Thus, the derivative-variance hybrid global sensitivity measure is recommended to replace derivative-based global sensitivity measure or the variance-based global sensitivity measure for sensitivity analysis.

On the other hand, since the computation of the derivative-variance hybrid global sensitivity measure requires the value of the corresponding derivative-based global sensitivity measure, it shares the same limitations as the derivative-based global sensitivity measure, such as: (1) it requires the output function to be continuous so that the corresponding derivatives are available; (2) its applications on highly nonlinear output functions may result in considerable errors; and (3) it cannot detect the interactive effect between different inputs.

4. Selection of the Optimal Sampling Method

In this section, six frequently used sampling methods are evaluated in terms of their performances of sensitivity analysis using the derivative-variance hybrid global sensitivity measure, namely: Monte Carlo sampling [23]; Latin hypercube sampling [12]; stratified sampling [22]; Latinized stratified sampling [22]; and two quasi-Monte Carlo samplings using the Sobol sequence and Halton sequence [23]. Figure 2 presents one realization of random points in domain I² that are generated from the above six sampling methods. It is noted that, when the simple size N equals kⁿ and k is a positive integer, stratified sampling and Latinized stratified sampling can guarantee that there is exactly one point contained in each stratum (a hypercube with a length of 1/k defined by a corresponding grid).

The performances of the derivative-variance hybrid global sensitivity measure of the aforementioned six sampling methods were investigated using the aforementioned three examples described by Equations (27)–(29). Since all these sampling methods produce unbiased estimations for random integrals, only the stochastic errors were studied, and we chose the standard deviation of the relative error as the index for the stochastic errors. At this point, the relative error of variable χ is defined as: $r({\chi }_i)=\left|\left({\chi }_i-{\chi }^0\right)/{\chi }^0\right|$, where ${\chi }_i$ is one realization of χ and ${\chi }^0$ is the corresponding analytical solution. Figure 3 shows the standard deviations (STD) of the relative errors for the derivative-variance hybrid global sensitivity measure ϒ_i as functions of sample size N, in which N equals 10³, 25³, and 50³. In the calculation, 10⁷ sample points were divided into the corresponding number of sample sets according to a different sample size N, and each sample set yields one group of relative errors. As can be seen from Figure 3, Monte Carlo sampling and Latin hypercube sampling always have the largest stochastic errors among these six sampling methods, suggesting a ban of these two sampling methods on sensitivity analysis. Figure 3 presents the stochastic errors for derivative-variance hybrid global sensitivity measure evaluations. It is noted that, due to the derivative of x₃ of Example 1 being a constant, the errors’ STDs obtained using different sampling methods are equal to zero. In Figure 3, it can be seen that the Latinized stratified sampling and the two quasi-Monte Carlo samplings produce the lowest stochastic error, and their relative performances may vary for different cases. In this instance, the two quasi-Monte Carlo samplings outperform the Latinized stratified sampling in the evaluations of ϒ₁ and ϒ₃, while the Latinized stratified sampling provides the lowest stochastic error for the evaluations of ϒ₂. Thus, Latinized stratified sampling and quasi-Monte Carlo sampling are recommended.

An explanation for the above phenomenon can involve the sample’s stochasticity and uniformity. On the one hand, a random integral requires the sample to have good uniformity to cover sudden changes in small areas. On the other hand, reasonable stochasticity is also necessary to avoid periodic related issues. Nevertheless, the dilemma is that high uniformity and high stochasticity cannot be achieved simultaneously since the uniformity decreases as the stochasticity increases. Monte Carlo sampling and Latin hypercube sampling have poor uniformity, resulting in undesirable performances. In contrast, quasi-Monte Carlo samplings and Latinized stratified sampling have the right balance between uniformity and stochasticity, which contributes to their satisfactory accuracy and convergence rate in sensitivity analysis.

5. Numerical Application and Validation

In order to demonstrate its validity and practicability, the proposed sensitivity measure is applied to the sensitivity analysis of a rural bridge. The bridge, located in the Xinzhou district (Wuhan city, Hubei Province, China), is a solid plate bridge with a span-combination of 4 × 6 m, whose width equals 10 m (shown in Figure 4). Each span consists of 10 solid plates of 0.99 m wide.

The performance function [35] of the weakest plate is chosen as the model output, which can be expressed as

$$M=f_{sd}{A}_s\left(h-a_s-\frac{f_{sd}{A}_s}{2f_{cd}b}\right)-bh{\rho }_1{l}^2/8-b{h}^{\prime }{\rho }_2{l}^2/8-F\alpha l/4$$

(30)

in which the specific parameters of the bridge are obtained from in situ investigations and reasonable error analysis (shown in

Table 4. List of the inputs.

Input Parameter	Median	Discrepancy	Coefficient of Variance
Tensile strength of reinforcements fsd (kPa)	388,280	26,600	0.0685
Cross-sectional area of reinforcements As (m2)	0.002815	0.000099	0.035
Thickness of reinforcement protection layer as (m)	0.05	0.003	0.06
Axial compressive strength of concrete fcd (kPa)	31,200	4680	0.15
Length of the solid plate l (m)	5.6	0.015	0.0027
Width of the solid plate b (m)	0.99	0.01	0.01
Thickness of the solid plate h (m)	0.32	0.01	0.031
Density of the solid plate ρ1 (kN/m3)	25	2	0.08
Thickness of the deck pavement h′ (m)	0.1	0.01	0.1
Density of the deck pavement ρ2 (kN/m3)	26	2.08	0.08
Vehicle load F (kN)	300	30	0.1
Transverse distribution coefficient α	0.26	0.026	0.1

).

Firstly, the derivative-variance hybrid global sensitivity measure is applied to the sensitivity analysis of the rural bridge. In the analysis, the sampling method used was quasi-Monte Carlo sampling with Sobol sequence, and the convergence threshold in terms of absolute error was set to 0.01. The values of the derivative-based global sensitivity measures obtained from 10⁸ samples are considered the target for error computation. As shown in

Table 5. Sensitivity analysis of the bridge using the derivative-variance hybrid global sensitivity measure.

	fsd	As	as	fcd	l	b	h	ρ1	h′	ρ2	F	α
N = 1000	0.4851	0.1321	0.0174	0.0114	4.51 × 10−4	8.7 × 10−5	0.1609	0.0094	0.0016	0.0011	0.1929	0.1929
N = 108	0.4894	0.1332	0.0176	0.0115	4.60 × 10−4	8.21 × 10−5	0.1624	0.0095	0.0017	0.0011	0.1947	0.1947
Absolute error	0.0043	0.0011	0.0002	0.0001	0	0	0.0015	0.0001	0.0001	0	0.0018	0.0018
Relative error	0.88%	0.88%	0.88%	0.89%	0.89%	0.89%	0.88%	0.88%	0.88%	0.88%	0.89%	0.89%

, it was found that 1000 samples can already guarantee the above criterion of convergence, and such a good convergence using merely 1000 samples indicates the convergence rate of derivative-variance hybrid global sensitivity measure using quasi-Monte Carlo sampling is sufficiently high. In addition, it is seen from Table 5 that the six insensitive inputs, namely l, b, $h^{\prime }$, ${\rho }_1$, ${\rho }_2$, and $f_{cd}$, are found (highlighted in gray), and their derivative-variance hybrid global sensitivity measures are less than the corresponding sensitivity threshold of 1.5%. At this juncture, it is noted that the relative errors for the different inputs’ derivative-variance hybrid global sensitivity measures are surprisingly close to each other, and the reason for this remains to be further studied.

On the other hand, in order to compare the derivative-variance hybrid global sensitivity measure and the variance-based global sensitivity measure in terms of computational efficiency, the variance-based global sensitivity measure was also imposed on the sensitivity analysis of the above example, and the identical convergence threshold of 0.01 was adopted. The target variance-based global sensitivity measure values for error computation were obtained from 10⁸ samples. It is found that, if the sample size equals 240,000, the above criterion for convergence was just met, and the corresponding result is shown in

Table 6. Sensitivity analysis of the bridge using the variance-based global sensitivity measure.

	fsd	As	as	fcd	l	b	h	ρ1	h′	ρ2	F	α
N = 240,000	0.3942	0.1079	0.016	0.0085	6.04 × 10−4	2.24 × 10−5	0.1285	0.0084	0.0008	0.001	0.1649	0.156
N = 108	0.4021	0.1102	0.0147	0.0093	3.62 × 10−4	9.18 × 10−5	0.1323	0.0079	0.0014	0.0009	0.1602	0.1606
Absolute error	0.0079	0.0023	0.0013	0.0008	2.42 × 10−4	6.94 × 10−5	0.0038	0.0005	0.0006	0.0001	0.0047	0.0046
Relative error	1.96%	2.09%	2.09%	8.60%	66.90%	75.60%	2.87%	6.33%	42.80%	11.10%	2.93%	2.86%

. By comparing Table 5 and Table 6, it is evident that the computational efficiency of the derivative-variance hybrid global sensitivity measure is much higher than that of the variance-based global sensitivity measure, and the required sample sizes for the derivative-variance hybrid global sensitivity measure and the variance-based global sensitivity measure to meet the convergence threshold of 0.01 are 1000 and 240000, respectively.

Lastly, in order to demonstrate that the derivative-variance hybrid global sensitivity measures can serve as good approximations for the corresponding variance-based global sensitivity measures, the derivative-variance hybrid global sensitivity measures, the derivative-based global sensitivity measures, and the variance-based global sensitivity measures for the above example are computed and compared, and the sample sizes for computing the variance-based global sensitivity measures, the derivative-based global sensitivity measures, and the derivative-variance hybrid global sensitivity measures are 240,000, 1000, and 1000, respectively.

Table 7. Comparison between the derivative-based global sensitivity measures, the derivative-based global sensitivity measures, and the variance-based global sensitivity measures.

	fsd	As	as	fcd	l	b	h	ρ1	h′	ρ2	F	α
Derivative-based global sensitivity measure	3.3 × 108	8.9 × 107	1.2 × 107	7.7 × 107	3.1 × 105	5.9 × 104	9.2 × 107	6.4 × 106	1.1 × 106	7.4 × 105	1.3 × 108	1.3 × 108
Variance-based global sensitivity measure	0.3942	0.1079	0.016	0.0084	6.04 × 10−4	2.2 × 10−5	0.1285	0.0084	0.0008	0.001	0.1649	0.156
Derivative-variance hybrid global sensitivity measure	0.4851	0.1321	0.0174	0.0114	4.51 × 10−4	8.7 × 10−5	0.1609	0.0094	0.0016	0.0011	0.1929	0.1929
Absolute error	0.0909	0.0242	0.0014	0.003	1.53 × 10−4	6.5 × 10−5	0.0324	0.001	0.0008	0.0001	0.028	0.0369
Relative error	23.10%	18.30%	8.75%	35.70%	25.30%	295%	25.21%	11.90%	100%	10.00%	16.90%	23.60%

shows the comparison among the derivative-based global sensitivity measures, the variance-based global sensitivity measures, and derivative-variance hybrid global sensitivity measures. From Table 7, it is seen that: (1) the absolute values of the derivative-based global sensitivity measures are meaningless, so one must compute all the inputs’ derivative-based global sensitivity measures and rank them before determining their sensitivities, while the values of the derivative-variance hybrid global sensitivity measures and the variance-based global sensitivity measures directly reflect the sensitivity of the corresponding inputs; (2) for sensitive inputs, the derivative-variance hybrid global sensitivity measure is consistently greater than the corresponding variance-based global sensitivity measure; and (3) for sensitive inputs, the relative errors between the derivative-based global sensitivity measures and the corresponding variance-based global sensitivity measures are all less than 26%. Therefore, it is demonstrated that derivative-variance hybrid global sensitivity measures are good and conservative approximations for the variance-based global sensitivity measures.

6. Conclusions

This research proposes a new sensitivity measure for sensitivity analysis, which is referred to as the derivative-variance hybrid global sensitivity measure. Optimal sampling methods for computing the derivative-variance hybrid global sensitivity measure are investigated by evaluating the corresponding performances of six frequently used sampling methods. To demonstrate its validity and practicality, the proposed sensitivity measure is applied to the sensitivity analysis of a rural bridge, which is further compared to the derivative-based global sensitivity measure and the variance-based global sensitivity measure. We found that the derivative-variance hybrid global sensitivity measure can directly measure the sensitivities of the corresponding inputs of the variance-based global sensitivity measure, while the derivative-based global sensitivity measures cannot directly reflect the inputs’ sensitivities, and the computational efficiency of the derivative-variance hybrid global sensitivity measure is much higher than that of the variance-based global sensitivity measure. Thus, the derivative-variance hybrid global sensitivity measure has promising sensitivity analysis applications in a variety of engineering fields, such as strength analysis, failure analysis, rock mechanics, slope stability analysis, and structural reliability analysis. In addition, the following conclusions are drawn:

• The derivative-variance hybrid global sensitivity measure and the derivative-based global sensitivity measure share similar computational costs, but the values of the derivative-variance hybrid global sensitivity measures directly imply the sensitivities of the corresponding inputs, while the values of the derivative-based global sensitivity measures are meaningless.
• The derivative-variance hybrid global sensitivity measure can serve as a good and conservative approximation for the corresponding variance-based global sensitivity measure, while the computational cost of the derivative-variance hybrid global sensitivity measure is much lower than that of the variance-based global sensitivity measure.
• Latinized stratified sampling and quasi-Monte Carlo sampling are recommended for computing the derivative-variance hybrid global sensitivity measure.