# A Novel Global Sensitivity Measure Based on Probability Weighted Moments

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review of Dominating Global Sensitivity Measure Systems

**X**), where Y is the model output of interest and $\mathit{X}=\left({X}_{1},\text{}{X}_{2},\dots ,{X}_{n}\right)$ is the n-dimensional vector of the model input.

#### 2.1. Variance-Based Global Sensitivity Measures

#### 2.2. Moment-Independent Global Sensitivity Measures

## 3. Probability Weighted Moments

## 4. Global Sensitivity Measure Based on Probability Weighted Moments

_{i}to the difference is bigger, the main measure is bigger as well. Then, we can establish the following properties of the novel global sensitivity measure:

**Property**

**1.**

**Property**

**2.**

_{i}has no effect on the output.

**Proof.**

_{i}, the conditional distribution $Y\left|{X}_{i}\right.$ is equivalent to the distribution of $Y$. It is obvious that ${E}_{{X}_{i}}\left[{\beta}_{{\mathit{X}}_{~i}}^{k}\left(Y\left|{X}_{i}\right.\right)\right]$ is also equivalent to the corresponding ${\beta}_{Y}^{k}\left(Y\right)$. So, Equation (11) is equal to 0 and ${\eta}_{i}^{k}=0$. □

**Property**

**3.**

_{i}affects the output.

**Proof.**

## 5. Numerical Estimation for Global Sensitivity Measures

#### 5.1. Double-Loop-Repeated-Set Numerical Estimators

_{1}samples of the variable vector $\mathit{x}=\left({x}_{1},\text{}{x}_{2},\dots ,{x}_{n}\right)$ by the joint PDF ${f}_{\mathit{X}}\left(x\right)$ and calculate the corresponding responses $Y$ as well as its kth-order PWM. These samples can be generated by Monte Carlo sampling, Latin hypercube sampling, quasi-Monte Carlo simulation with sampling based on Sobol’ sequences [29,30], or other sampling techniques. Here, quasi-Monte Carlo simulation is recommended for higher and faster convergence.

_{2}samples of the input variable X

_{i}by its PDF ${f}_{{\mathit{X}}_{i}}\left({x}_{i}\right)$ and denote these samples as $\left({x}_{{i}_{1}},{x}_{{i}_{2}},\dots ,{x}_{{i}_{{N}_{2}}}\right)$.

_{i}should be fixed at ${x}_{{i}_{m}}\left(m=1,\hspace{1em}2,\hspace{1em}\dots {N}_{2}\right)$ individually and generate N

_{3}samples of the remaining variables according to the joint PDF $\prod _{j\ne i}^{n}{f}_{{\mathit{X}}_{j}}\left({x}_{j}\right)$ (all variables are independent in this paper). So, the conditional PWM of responses ${\beta}_{{\mathit{X}}_{~i}}^{k}\left(Y\left|{X}_{i}={x}_{{i}_{m}}\right.\right)$ can be obtained, and this step needs to be repeated N

_{2}times.

_{1}+ n × (N

_{2}+ N

_{2}× N

_{3}). We denote it as double-loop-repeated-set QMC (DLRS QMC) because this method needs repeated sampling of inputs and outputs in each inner loop. It is obvious that the computational cost would be too heavy.

#### 5.2. Double-Loop-Single-Set Numerical Estimators

_{1}samples of whole variables by the joint PDF ${f}_{\mathit{X}}\left(x\right)$ and assign half of these samples to a sample matrix

**A**:

**B**is generated by the remaining samples, which is:

**A**to the ith column of

**B**:

_{1}, contributing to a large reduction in sampling time compared to DLRS QMC. N

_{1}, N

_{2}, and N

_{3}are in the same order of magnitude, which is not more than 4000 usually. Hence, to reduce the calculation cost, the following examples are all solved by the DLSS QMC method.

## 6. Examples and Discussion

#### 6.1. Numerical Example 1: Linear Function with Normal Distribution Variables

#### 6.2. Numerical Example 2: Linear Function with Exponential Distribution Variables

_{i}(i = 1, 2, 3, 4) all obey the exponential distribution, i.e., ${X}_{i}~Exp\left(\lambda =1\right)$. From the results shown in Table 3, variables have the same influence on the output according to not only the Sobol’ method but also the moment-independent method. However, measures based on PWMs are somewhat different, as presented in Table 4 and Figure 2. The variables that have absolutely the same coefficient have the same importance when ignoring the calculation and sampling error, meaning the measures can reflect the influence of positive and negative coefficients with the increase in order k.

#### 6.3. Numerical Example 3: Ishigami Function

_{i}(i = 1, 2, 3) are uniformly distributed on interval $\left[-\pi ,\pi \right]$. Table 5 lists the Sobol’ indices calculated by the analytical method and delta indices. Measures based on PWMs calculated by the DLRS QMC are listed in Table 6, and the corresponding figure follows. By analyzing the results, the importance rankings obtained by the main Sobol’ indices, moment-independent indices, and new measures are the same, whereas discrepancy exists with the Sobol’ total indices. This phenomenon can be explained by comparing Equation (1) with Equation (11). The definitions of the Sobol’ main index and the presented measure have a similar form, thus leading to a certain relationship between them.

#### 6.4. Engineering Example: A Roof Truss Structure

_{C}of the top node C can be derived as:

_{C}and E

_{C}represent the sectional area and elastic modulus of concrete bars, respectively; A

_{S}and E

_{S}are the sectional area and elastic modulus of steel bars, respectively, which are also denoted in Figure 3b. Considering the serviceability criterion that the vertical displacement ∆

_{C}should not exceed an admissible maximal deflection of 3 cm, the performance function can be constructed as $Y=\text{}0.03-{\Delta}_{C}$. Then, we assume that six random inputs q, l, A

_{C}, E

_{C}, A

_{S}, and E

_{S}are independently and normally distributed. Their distribution parameters are given in Table 7.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

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Same Value of All Variables | Different Values of All Variables | Parameters | |
---|---|---|---|

Case 1 | ${\mu}_{i}$,${\sigma}_{i}$,$\left|{a}_{i}\right|$ | / | $\mu =\left[5,\text{}5,\text{}5,\text{}5\right]$, $\sigma =\left[1,\text{}1,\text{}1,\text{}1\right]$, $\mathit{a}=\left[1,\text{}-1,\text{}1,\text{}-1\right]$ |

Case 2 | ${\mu}_{i}$,${\sigma}_{i}$, | $\left|{a}_{i}\right|$ | $\mu =\left[5,\text{}5,\text{}5,\text{}5\right]$, $\sigma =\left[1,\text{}1,\text{}1,\text{}1\right]$, $\mathit{a}=\left[1,\text{}2,\text{}3,\text{}4\right]$ |

Case 3 | ${a}_{i}$,${\sigma}_{i}$,$\left|{\mu}_{i}\right|$ | / | $\mu =\left[5,-5,\text{}5,\text{}-5\right]$, $\sigma =\left[1,\text{}1,\text{}1,\text{}1\right]$, $\mathit{a}=\left[1,\text{}1,\text{}1,\text{}1\right]$ |

Case 4 | ${a}_{i}$,${\sigma}_{i}$ | $\left|{\mu}_{i}\right|$ | $\mu =\left[3,\text{}5,\text{}7,\text{}9\right]$, $\sigma =\left[1,\text{}1,\text{}1,\text{}1\right]$, $\mathit{a}=\left[1,\text{}1,\text{}1,\text{}1\right]$ |

Case 5 | ${a}_{i}$,${\mu}_{i}$ | ${\sigma}_{i}$ | $\mu =\left[5,\text{}5,\text{}5,\text{}5\right]$, $\sigma =\left[0.5,\text{}1,\text{}1.5,\text{}2\right]$, $\mathit{a}=\left[1,\text{}1,\text{}1,\text{}1\right]$ |

Measures | S_{i} (S_{Ti}) | ${\mathit{\delta}}_{\mathit{i}}$ | ${\mathit{\eta}}_{\mathit{i}}^{\mathit{k}}$ |
---|---|---|---|

Case 1 | [0.2500, 0.2500, 0.2500, 0.2500] | [0.1808, 0.1808, 0.1808, 0.1808] | [0.2500, 0.2500, 0.2500, 0.2500] |

Case 2 | [0.0333, 0.1333, 0.3000, 0.5334] | [0.0567, 0.1231, 0.2041, 0.3182] | [0.0021, 0.0361, 0.2019, 0.7599] |

Case 3 | [0.2500, 0.2500, 0.2500, 0.2500] | [0.1808, 0.1808, 0.1808, 0.1808] | [0.2500, 0.2500, 0.2500, 0.2500] |

Case 4 | [0.2500, 0.2500, 0.2500, 0.2500] | [0.1808, 0.1808, 0.1808, 0.1808] | [0.2500, 0.2500, 0.2500, 0.2500] |

Case 5 | [0.0333, 0.1333, 0.3000, 0.5334] | [0.0567, 0.1231, 0.2041, 0.3182] | [0.0021, 0.0361, 0.2019, 0.7599] |

Measures | S_{i/}S_{Ti} | ${\mathit{\delta}}_{\mathit{i}}$ |
---|---|---|

X_{1} | 0.2500 (1) | 0.1930 (1) |

X_{2} | 0.2500 (1) | 0.1930 (1) |

X_{3} | 0.2500 (1) | 0.1930 (1) |

X_{4} | 0.2500 (1) | 0.1930 (1) |

Measures | ${\mathit{\eta}}_{\mathit{i}}^{\mathit{k}}$ | |||
---|---|---|---|---|

k | 1 | 2 | 3 | 4 |

X_{1} | 0.2459 (4) | 0.3092 (2) | 0.3469 (2) | 0.3708 (2) |

X_{2} | 0.2559 (1) | 0.1931 (3) | 0.1556 (3) | 0.1319 (3) |

X_{3} | 0.2476 (3) | 0.3103 (1) | 0.3477 (1) | 0.3715 (1) |

X_{4} | 0.2505 (2) | 0.1874 (4) | 0.1497 (4) | 0.1258 (4) |

Sample size | 3000 |

Measures | S_{i} | S_{Ti} | ${\mathit{\delta}}_{\mathit{i}}$ |
---|---|---|---|

X_{1} | 0.3139 (2) | 0.5576 (1) | 0.2259 (2) |

X_{2} | 0.4424 (1) | 0.4424 (2) | 0.4086(1) |

X_{3} | 0 (3) | 0.2437 (3) | 0.1798 (3) |

Measures | ${\mathit{\eta}}_{\mathit{i}}^{\mathit{k}}$ | |||
---|---|---|---|---|

k | 1 | 2 | 3 | 4 |

X_{1} | 0.2109 (2) | 0.2126 (2) | 0.2410 (2) | 0.2771 (2) |

X_{2} | 0.7758 (1) | 0.7731 (1) | 0.7348 (1) | 0.6837 (1) |

X_{3} | 0.013l (3) | 0.0141 (3) | 0.0242 (3) | 0.0382 (3) |

Sample size | 3000 |

Variable X | Mean | Coefficient of Variance |
---|---|---|

q (N·m^{−1}) | 20,000 | 0.07 |

l (m) | 12 | 0.01 |

A_{s} (m^{2}) | 9.82 × 10^{−4} | 0.06 |

A_{c} (m^{2}) | 0.04 | 0.12 |

E_{s} (MPa) | 2 × 10^{11} | 0.06 |

E_{c} (MPa) | 3 × 10^{10} | 0.06 |

Measures | S_{i} | S_{Ti} | ${\mathit{\delta}}_{\mathit{i}}$ |
---|---|---|---|

q | 0.4581 (1) | 0.4608 (1) | 0.2831 (1) |

l | 0.0374 (5) | 0.0378 (5) | 0.0613 (5) |

A_{s} | 0.1710 (2) | 0.1725 (2) | 0.1427 (3) |

A_{c} | 0.1287 (4) | 0.1298 (4) | 0.1185 (4) |

E_{s} | 0.1709 (3) | 0.1724 (3) | 0.1428 (2) |

E_{c} | 0.0300 (6) | 0.0306 (6) | 0.0541 (6) |

Sample size | 3 × 10^{5} |

Measure | ${\mathit{\eta}}_{\mathit{i}}^{\mathit{k}}$ | |||
---|---|---|---|---|

k | 1 | 2 | 3 | 4 |

q | 0.7657 (1) | 0.7780 (1) | 0.7859 (1) | 0.7913 (1) |

L | 0.0039 (5) | 0.0037 (5) | 0.0035 (5) | 0.0036 (5) |

A_{s} | 0.0891 (3) | 0.0847 (3) | 0.0817 (3) | 0.0795 (3) |

A_{c} | 0.0491 (4) | 0.0454 (4) | 0.0433 (4) | 0.0418 (4) |

E_{s} | 0.0898 (2) | 0.0858 (2) | 0.0831 (2) | 0.0812 (2) |

E_{c} | 0.0024 (6) | 0.0025 (6) | 0.0025 (6) | 0.0026 (6) |

Sample size | 4000 |

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**MDPI and ACS Style**

Song, S.; Wang, L.
A Novel Global Sensitivity Measure Based on Probability Weighted Moments. *Symmetry* **2021**, *13*, 90.
https://doi.org/10.3390/sym13010090

**AMA Style**

Song S, Wang L.
A Novel Global Sensitivity Measure Based on Probability Weighted Moments. *Symmetry*. 2021; 13(1):90.
https://doi.org/10.3390/sym13010090

**Chicago/Turabian Style**

Song, Shufang, and Lu Wang.
2021. "A Novel Global Sensitivity Measure Based on Probability Weighted Moments" *Symmetry* 13, no. 1: 90.
https://doi.org/10.3390/sym13010090