# Modification of Variance-Based Sensitivity Indices for Stochastic Evaluation of Monitoring Measures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Variance-Based Sensitivity Indices

#### 2.1. Variance-Based Sensitivity Indices by Sobol’

_{i}, cf. Equation (1). The uncertainty of each variable V(X

_{i}) propagates through the model and results in uncertain model output V(Y). Variance-based SA determine the contribution of each parameter’s uncertainty to the uncertainty of the output. For complex models, in addition to the direct variance of each parameter V(X

_{i}), covariances V(X

_{i},X

_{j}) and higher order variances V(X

_{i}, … X

_{n}) arise and can be relevant. Then, more sophisticated SA methods serve to identify each parameter’s impact.

_{1}, X

_{2}, …, X

_{q}in a q-dimensional unit hyperspace Ω

_{q}(0 ≤ X

_{i}≤ 1).

_{0}, q linear terms f

_{i}(X

_{i}), q over 2 s-order terms f

_{ij}and higher-order terms. Overall, the model consists of 2

^{q}terms; each square-integrable again.

_{0}to be zero, Sobol’ proved all terms being orthogonal [18]. Thus, the zero-order term f

_{0}corresponds to the model’s expectation E(Y):

_{i}) (see [37]), where d

**x**

_{−i}denotes an integration over all dimensions except i; equivalently, d

**x**

_{−i,j}indicates an integration over all dimensions except i and j.

_{Ωq}f²(

**x**) d

**x**− f

_{0}² can be decomposed in first- and higher-order terms, respectively (Equation (6)). Here, the second-order variance V

_{ij}of two parameters X

_{i}and X

_{j}is the covariance and is a measure of interaction.

_{i}follows as the ratio of the model’s conditional variance V(E(Y|X

_{i})), when all parameters but X

_{i}are fixed, to the variance of the entire model V(Y) with all parameters variable.

_{i}-values quantify the direct impact of an individual variance on the model result Y. Higher-order sensitivity indices (S

_{ij}, … S

_{1,2…q}) can be deduced from conditional variances as well (see [10]). Those are measures of interaction of two (covariance) or more parameters (higher-order variance). Restricting on a single parameter X

_{i}for convenience, its total sensitivity index S

_{Ti}represents the ratio of its direct influence on the model’s variance V

_{i}(conditional variance) and all covariances of X

_{i}(V

_{ij}, V

_{ijk}, …) in relation to the model’s variance. To prevent the conditional variance from being dependent on a fixed value X

_{i}= x

_{i}*, the expectation of the conditional variance V

_{x−i}(X

_{i}= x

_{i}*) over the range of x

_{i}* denoted E

_{Xi}(V

_{x−i}(X

_{i}= x

_{i}*)) = E(V(Y|

**X**

_{−i}) is calculated [10].

_{Ti}for the parameter X

_{i}follows as the ratio of the parameter’s direct variance V

_{i}and all higher-order variances (V

_{ij}, V

_{ijk}…) involving X

_{i}to the variance of the entire model V(Y). Thus, it quantifies the impact of all interactions involving X

_{i}and their direct influences:

_{Ti}= S

_{i}.

#### 2.2. Computation of Variance-Based Sensitivity Indices

**A**and

**B**are generated, each containing n realizations of q variables. For reliable results, the stochastic independence of

**A**and

**B**is essential [41]. It can be found through Pearson’s coefficient of correlation ρ

**≈ 0. Otherwise, even small correlations can be found as spurious correlations in the resulting sensitivity indices [39]. Especially for models with several parameters, the sensitivity indices are likely to be close to zero. Here, spurious correlations can distort the results.**

_{A,B}**C**(with i = 1, … q) are assembled from

_{i}**A**and

**B**. They are generated by interchanging columns. This means that the new matrix

**C**is identical to

_{i}**B**, solely column i (for parameter i) is taken from matrix

**A**instead. Other matrices

**C**are obtained analogously.

_{i}**a**= Y(

**A**),

**b**= Y(

**B**) and

**c**= Y(

_{1}**C**) to

_{1}**c**= Y(

_{q}**C**) are determined. Finally, the sensitivity indices are obtained based on correlation properties between

_{q}**c**and

_{i}**a**or

**c**and

_{i}**b**, respectively. For details see [39].

## 3. Method: Modification of Variance-Based Sensitivity Indices

#### 3.1. Proposed Method Based on SOBOL’ Indices

_{0}(cf. Equation (1)), the best model with utmost reduced variances is denoted Y*. It comprises q parameters X

_{i}*, each having a reduced variance V* (represented by a reduced standard deviation σ* in Figure 1).

_{0}) of the initial model Y

_{0}consists of q first-order variances V

_{i,0}and (q over 2) covariance terms V

_{i,j,0}(cf. Equation (6)). Y* is treated equivalently. Thus, the difference of the two decomposed variances in Equations (15) and (16) captures the extent of potential variance reduction ΔV*.

_{i}* of a single parameter is interpreted as its individual improvement and it reads:

_{i}* out of the q variables is improved, the model can be written as a function of q-1 “original” parameters X

_{j≠i,0}and one “improved” parameter X

_{i}*.

_{j≠i,0}, X

_{i}*)), the variance reduction by an improved knowledge about one parameter’s variance follows:

_{i}* and X

_{j}*, the variance reduction of the entire model follows:

_{j}* except one single variable X

_{i,0}exhibit a reduced variance. This leads to:

_{i}

^{*}are obtained from the ratio of single variance reductions ΔV

_{i}* (Equation (20)) to the total one ΔV* (acc. to Equation (18)). These indices capture the benefit of improving the knowledge about a single parameter X

_{i}but neglecting any interactions with other reduced variances.

_{j≠i}* would require n additional model evaluations. In addition, relevant parameter combinations would have to be estimated in advance. Thus, such indices are omitted in the following evaluations.

#### 3.2. Computational Implementation

**A***and

**B**are determined analogously to the procedure in Section 2.2. Since the indices are no longer determined with correlation coefficients approach, it does not matter, whether they are stochastically independent or not. However, the pre-defined correlation within the matrices (usually the parameters are supposed to be independent) still needs to be maintained. Now,

**A***contains the realizations of reduced variance;

**B**contains the realizations with the highest (initial) scatter. As before, both result vectors

**a***and

**b**are evaluated by employing the model.

**C**are generated. These matrices are assembled mainly from realizations of matrix

_{i}***B**. Only column i comes from matrix

**A***. For each

**c**the model needs to be evaluated again n times.

_{i}*## 4. Application Case: A Model for Fatigue Lifetime Prediction of Pre-Stressed Concrete Bridges

#### 4.1. Reference Structure and Measurements

#### 4.2. Fatigue Lifetime Prediction Model for Pre-Stressed Concrete Bridges

- estimation and prognosis of loads (traffic loads and frequencies, temperature loads);
- calculation of stresses, including the nonlinearity after cracking (typically affected by the structural FE-model, cross-sectional and geometric parameters, material parameters, and stiffness);
- fatigue-related properties of the material resistance (represented by the S–N curve).

_{FL}of a pre-stressed concrete road-bridge, based on the accumulated fatigue damage D [46]. Fatigue failure occurs at damage D = 1 and is calculated by Miner’s rule [47,48]:

_{i}arise from traffic counts and prognosis and relative frequencies of individual vehicle types according to Eurocode 1–2 [45]. Load cycles until failure N

_{i}are obtained from the S–N curve (Figure 4) according to the stress ranges Δσ

_{i}. Stress ranges are determined at different load levels i by combining a structural model and an iterative computation method of stresses. For convenience, the internal forces have been determined on a linear-elastic FE beam model (Figure 5) separated from the computation of stresses on cross-sectional level to reduce the computational costs. An advantage of this approach is that superposition of the internal forces can still be applied for all load cases and evaluations of the complex FE-model can be reduced. Then the stresses are computed, employing an iterative procedure by Krüger and Mertzsch [49] to get accurate stresses in cracked concrete conditions. To also cover non-cracked (linear elastic) conditions, the original approach was slightly modified and now permits the neutral strain fiber to lie outside the cross-section (0 < ξ ≤ 2, with ξ = x/d and x being the height of compression zone and d the effective height), too.

_{CS}(t) of the reduced pre-stress (details in [46]). Furthermore, hardening of concrete (compressive strength f

_{c}(t) and Young’s modulus E

_{c}) is considered by a root function according to Eurocode 2 [52].

- the width of the deck-slab b
_{f}, which represents the variability of the entire geometry; - the effective height d
_{p1}of the pre-stressed cross-section concerning tendon layer no. 1; - a scaling factor for pre-stress losses by creep and shrinkage a
_{CS}; - five (relevant) linear temperature gradients ΔT
_{i}; - a scaling factor w
_{3}for the traffic loads from FLM 4, truck no. 3; - the cross-sectional area of a tendon A
_{p1}; - Young’s moduli of concrete E
_{c}and pre-stressing steel E_{p}; - two parameters to describe the S–N curve: its knee point Δσ(N*) at 10
^{6}load cycles and the slope of the high-cycle fatigue range (k_{2}); for simplification k_{1}is set equal to k_{2}.

_{i,0}, CV

_{i,0}), the reduced variances (as improved variation coefficient CV* with aleatory uncertainty only) are summarized in Table 1. The ratios of improved to original variances V*/V

_{0}= σ*²/σ

_{0}² are a measure to assess the improvement of a single parameter. For values close to zero the improvement is significant, for V

_{i}*/V

_{i,0}= 1 there is no reduction of the variance (by measurements). The given values are assumptions based on measurement data from the reference structure and information from the literature.

_{i}and S

_{Ti}according to Section 2.1 (Figure 7, left) and reduced variances (as improved standard deviations σ* and as ratios of improved to original variances V*/V

_{0}) are given as well in Table 1. Before the sensitivity indices are determined, the result is logarithmically transformed (y

_{i}= log(x

_{i})) to be more robust. Consequently, the data appears Gaussian distributed.

#### 4.3. Stochastic Lifetime Prediction, Sensitivity Analysis and Evaluation of Modified Sensitivity Indices

**b**(no improved variances) and the “best” model

**a***(all variances improved) are listed in the lower rows, too. Additionally, two columns of the prognosis result’s fractiles D

_{0,90}and D

_{0,99}have been computed as characteristic values. For the fatigue damage, the upper bound of the distribution is of interest and focused. Additionally, in comparison to the improvement from the initial model (

**b**) to the best model (

**a***) the individual gains are quantified by relative specific improvements in the 6th and 7th column. The modified variance-based sensitivity indices are determined according to Equation (29) and are given in Table 2. They are also shown in Figure 7, right.

_{i}and S

_{Ti}) and modified sensitivity indices (S

_{i}*) are opposed in Figure 7. For the presented fatigue lifetime model, at least six different variables show a significant impact (S

_{i}and S

_{Ti}) on the variance of the model. As a result of non-uniformly improved variances the results of the modified sensitivity indices (S

_{i}*) are individually shifted in comparison to the original sensitivity indices S

_{i}and S

_{Ti}in Table 2. Obviously, only relevant parameters characterized by a high sensitivity index S

_{Ti}and a seriously reduced variance (σ

_{i,0}>> σ

_{i}*) lead to a significant variance reduction of the model. In contrast, a parameter without any variance reduction (σ

_{i,0}= σ

_{i}*), like the scaling factor of traffic loads w

_{3}, does not change the model’s variance.

_{CS}. Since the variance of a

_{CS}can be reduced at most (cf. Table 1) it also delivers the highest modified sensitivity index S

_{i}* (Figure 7, right)—as expected. The knee point of the S–N curve Δσ(N*) is a moderately important variable (S

_{i}≈ S

_{Ti}≈ 0.11), but its variance can hardly be reduced by measurements. Thus, it exhibits a low modified sensitivity index S

_{i}* = 0.03.

_{0}with the highest variance—given by the computational result in b and marked by a dashed black line. This distribution function represents the initial prediction of the fatigue lifetime, having little knowledge about the parameters. Its total scatter comprises both, aleatory (random) and epistemic (lack of knowledge) parts. The “best” prediction model Y* using the computational result in a* yields another pdf marked as a solid black line in Figure 8. It visualizes the response when all variances are reduced to the greatest extent. Then the scatter is caused by aleatory parts uniquely. Compared to Y

_{0}the variance of Y* is significantly lower while the mean is shifted, too.

_{CS}and the linear temperature gradient ΔT(−7 K) are the most important. They shift both distributions (CDF and PDF) significantly towards the best model’s one. However, an irrelevant parameter would yield curves similar to the initial one.

#### 4.4. Convergence

^{4}. For each n, the modified sensitivity indices were determined ten times with different sampling sets to assess their variation around the mean x

_{m}by means of the 5% and 95% fractiles (x

_{m}± 1.645∙σ, assuming a Gaussian distribution) as a measure of variance. The results of the modified sensitivity indices for some of the 16 parameters are drawn in Figure 9. The selection comprises three types of parameters:

- The scaling factor of the pre-stress loss a
_{CS}in Figure 9 top left is a relevant parameter (S_{Ti}-value is high) and can be reduced significantly (V_{i}*/V_{i,0}<< 1). Therefore, its modified sensitivity index S_{i}* is expected to be high. - The knee point of the S–N curve Δσ(N*) in Figure 9 top right is a relevant parameter (high S
_{Ti}-value) without significant variance reduction (V_{i}*/V_{i,0}≈ 1); thus, S_{i}* is expected to be low. - Third, the effective depth of the pre-stressing steel (d
_{p1}) on the lower left of Figure 9 has a low (original) total sensitivity index S_{Ti}and even in case of a significant reduction of its variance, the modified sensitivity index can be expected to be low.

^{4}simulations. At least 5000 simulation runs are recommended in this case. For smaller sample sizes, all modified sensitivity indices possess large variance. For less than 1000 simulations, the results should not be used at all. Then, some results with too-small sample sizes take on values even outside the reasonable range 0 ≤ S

_{i}* ≤ 1. As it could be expected, this is more likely for S

_{i}*-values close to zero. On the lower right, Figure 9 illustrates the convergence by means of the standard deviation. All three parameters converge similarly. Differences are seen purely caused by chance.

## 5. Conclusions

_{i}*) are those impairing a model’s variance significantly (characterized by high-variance-based indices S

_{i}and S

_{Ti}) and simultaneously having a great potential for variance reduction by monitoring. In comparison to others, they possess the highest modified sensitivity indices.

_{i}*. In view of an ever-increasing stock of aged infrastructure buildings worldwide, the modified indices might help to save money and resources, avoiding unnecessary measurements in the future.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Probability density functions of a variable with original and reduced variance due to increased knowledge.

**Figure 2.**Reference structure Pariser Straße in Düsseldorf, Germany—side and top views along with a general cross-section.

**Figure 5.**Finite-element model of the reference structure; beam elements (cross-sections for visualization) and boundary conditions by springs and supports.

**Figure 6.**Damage evolution from a deterministic model with time-dependent stress ranges and three different approaches of the load frequency.

**Figure 7.**Original and total (

**left**) versus modified sensitivity indices (

**right**) for the reference bridge.

**Figure 8.**Probability and cumulative density functions of the original (Y

_{0}) and the best model (Y*) along with two cases when only one parameter’s scatter is reduced by monitoring.

**Figure 9.**Convergence of the modified sensitivity index for three parameters of the lifetime prediction model.

**Table 1.**Stochastic variables with initial (original) distribution characteristics (μ

_{i,0}, σ

_{i,0}) and reduced variances (σ

_{i}*), as well as Sobol’s sensitivity indices.

Variable i | Distribution | Orig. Distribution | Improvement | Sensitivity Indices | |||||
---|---|---|---|---|---|---|---|---|---|

μ_{i,0} | CV_{i,0} | CV_{i}* | V_{i}*/V_{i,0} | S_{i} | S_{Ti} | ||||

Pre-strain | ε_{p}^{(0)} | [‰] | N | 2.175 | 0.046 | 0.014 | 0.09 | 0.05 | 0.12 |

Young’s modulus of steel | E_{p} | [N/mm^{2}] | N | 205,000 | 0.030 | 0.024 | 0.63 | ~0 | 0.06 |

Scaling factor for creep and shrinkage | a_{CS} | [-] | N | 1 | 0.100 | 0.050 | 0.25 | 0.18 | 0.26 |

Young’s modulus of concrete | E_{c} | [N/mm^{2}] | LN | 33,000 | 0.091 | 0.045 | 0.25 | 0.02 | 0.04 |

S–N curve: | LN | 120 | 0.008 | 0.063 | 0.88 | 0.11 | 0.11 | ||

knee point | Δσ (N*) | [N/mm^{2}] | |||||||

slope | k_{2} | [-] | LN | 7 | 0.071 | 0.043 | 0.36 | 0.004 | 0.01 |

Width of the deck-slab | b_{f} | [m] | N | 4.95 | 0.101 | 0.001 | <0.01 | 0.01 | 0.05 |

Area of a tendon | A_{p1} | [cm^{2}] | N | 26.55 | 0.016 | 0.007 | 0.21 | 0.01 | 0.04 |

Effective height for tendon layer 1 | d_{p1} | [m] | N | 1.31 | 0.008 | 0.002 | 0.04 | 0.002 | 0.02 |

Gradient of load cycles per year | dn/dt | N | 15,000 | 0.333 | 0.317 | 0.9 | 0.02 | 0.02 | |

Scaling factor for FLM4-type 3 | w_{3} | [-] | N | 1 | 0.100 | 0.100 | 1 | ~0 | 0.05 |

Temperature gradients (scaled): | 1 | 0.008 | 0.141 | 0.5 | ~0 | 0.02 | |||

ΔT(−4 K) | N | ||||||||

ΔT(−5 K) | N | 1 | 0.200 | 0.141 | 0.5 | 0.01 | 0.04 | ||

ΔT(−6 K) | N | 1 | 0.200 | 0.141 | 0.5 | 0.06 | 0.18 | ||

ΔT(−7 K) | N | 1 | 0.200 | 0.141 | 0.5 | 0.18 | 0.29 | ||

ΔT(−8 K) | N | 1 | 0.200 | 0.141 | 0.5 | 0.14 | 0.24 |

**Table 2.**Improved distribution parameters, specific improvements of the target variable and modified sensitivity indices.

Variable i | Distribution Characteristics When i Is Improved | Relative Fractile Change | Mod. Sensitivity Index | ||||||
---|---|---|---|---|---|---|---|---|---|

λ | ζ | D_{0,90} | D_{0,99} | D_{0,90} | D_{0,99} | S_{i}* | |||

Pre-strain | ε_{p}^{(0)} | [‰] | −1.94 | 3.189 | 8.535 | 238.8 | +11% | +16% | 0.09 |

Young’s modulus of steel | E_{p} | [N/mm^{2}] | −1.93 | 3.254 | 9.350 | 280.2 | +1% | +1% | 0.001 |

Area of the tendon | A_{p1} | [cm^{2}] | −1.94 | 3.243 | 9.156 | 271.3 | +3% | +4% | 0.02 |

Effective depth of the tendon | d_{p1} | [m] | −1.93 | 3.243 | 9.283 | 275.0 | +% | +3% | 0.02 |

Scaling factor creep and shrinkage | a_{CS} | [-] | −1.95 | 3.026 | 6.851 | 161.8 | +31 % | +45% | 0.29 |

S–N curve: | |||||||||

knee point | Δσ(N*) | [N/mm^{2}] | −1.92 | 3.236 | 9.291 | 273.2 | +2% | +4% | 0.03 |

slope | k_{2} | [-] | −1.93 | 3.249 | 9.324 | 278.0 | +1% | +2% | 0.01 |

Young’s modulus of concrete | E_{c} | [N/mm^{2}] | −1.97 | 3.245 | 8.893 | 263.9 | +7% | +7% | 0.01 |

Width of the deck-slab | b_{f} | [m] | −1.96 | 3.235 | 8.857 | 260.1 | +7% | +9% | 0.03 |

Gradient of load cycles in time | dn/dt | −1.93 | 3.257 | 9.406 | 282.7 | +0% | +0% | ~0 | |

Scaling factor FLM4-type 3 | w_{3} | [-] | −1.92 | 3.262 | 9.556 | 288.7 | −% | −2% | ~0 |

Temperature gradient: | |||||||||

ΔT(−4 K) | −1.96 | 3.284 | 9.469 | 292.8 | −0% | −4% | ~0 | ||

ΔT(−5 K) | −1.98 | 3.260 | 9.004 | 271.4 | +5% | +4% | ~0 | ||

ΔT(−6 K) | −2.08 | 3.201 | 7.519 | 213.2 | +23% | +26% | 0.07 | ||

ΔT(−7 K) | −2.25 | 3.094 | 5.570 | 141.1 | +46% | +53% | 0.20 | ||

ΔT(−8 K)
| −2.19 | 3.143 | 6.268 | 167.1 | +38% | +43% | 0.14 | ||

“best” model a* with V* | −2.92 | 2.349 | 1.099 | 12.8 | +100% | +100% | - | ||

initial model b with V_{0} | −1.93 | 3.255 | 9.433 | 283.0 | ±0 | ±0 | - |

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**MDPI and ACS Style**

Sanio, D.; Ahrens, M.A.; Mark, P.
Modification of Variance-Based Sensitivity Indices for Stochastic Evaluation of Monitoring Measures. *Infrastructures* **2021**, *6*, 149.
https://doi.org/10.3390/infrastructures6110149

**AMA Style**

Sanio D, Ahrens MA, Mark P.
Modification of Variance-Based Sensitivity Indices for Stochastic Evaluation of Monitoring Measures. *Infrastructures*. 2021; 6(11):149.
https://doi.org/10.3390/infrastructures6110149

**Chicago/Turabian Style**

Sanio, David, Mark Alexander Ahrens, and Peter Mark.
2021. "Modification of Variance-Based Sensitivity Indices for Stochastic Evaluation of Monitoring Measures" *Infrastructures* 6, no. 11: 149.
https://doi.org/10.3390/infrastructures6110149