# Global Sensitivity Analysis Based on Entropy: From Differential Entropy to Alternative Measures

## Abstract

**:**

## 1. Introduction

## 2. Entropy of a Random Variable

_{i}) in Equation (1) are in [0, 1]. This is also an important difference from differential entropy.

_{R}of f(r) is very small, which is illustrated by the example with mean value μ

_{R}= 0, where σ

_{R}is parameter of the graph—see Figure 1.

_{R}decreases with the limit H(R) → −∞ when σ

_{R}→ 0—see Figure 1. The same is true for other classic pdfs. Unlike the discrete case, the entropy of a continuous system does not remain invariant during the transformation of the coordinate systems [65].

_{R}) is linear, similarly, the dependence H(R) vs. ln(σ

_{R}·σ

_{R}) is also linear. It can be noted that the linear dependence of H(R) vs. ln(σ

_{R}) is observed for the Gauss pdf of R but does not occur generally for each pdf. For example, the dependence ln(σ

_{R}) vs. H(R) is linear if the variation coefficient is constant for log-normal pdf of R.

## 3. Entropy-Based Sensitivity Analysis

_{1}, X

_{2}, … X

_{M}}, where statistical independence is assumed between inputs.

#### 3.1. Sensitivity Indices based on Differential Entropy H(R)

_{i}can be written as:

_{i}. H(R|X

_{i}) is the conditional differential entropy, which represents the average loss of random variability on model output R when the input value of X

_{i}is known.

_{i}) must be such that T

_{i}∈[0, 1]. In the limit case, if R|X

_{i}loses all random variability (σ

_{R|Xi}= 0), then the expected influence of X

_{i}on R is 100%, which means T

_{i}= 1. Therefore, H(R|X

_{i}) must be equal to zero and not −∞ as given by Equation (2)—see Figure 1. Equation (2) has the drawback that it can give negative entropy, which allows a sensitivity greater than 100%, T

_{i}> 1, which is not desired in the SA concept. On the other hand, Equation (2) satisfies the second extreme H(R) = H(R|X

_{i}), T

_{i}= 0, where fixing X

_{i}does not influence the pdf of output R. From the point of view of the SA concept, there are problematic cases where the variance of the output decreases to zero.

_{ij}is computed with the fixing of pairs X

_{i}, X

_{j}:

_{i}and X

_{j}. The third-order sensitivity index, E

_{ijk}, is computed analogously:

_{Ti}can be written as:

_{i}and fixed variables (X

_{1}, X

_{2}, …, X

_{i−1}, X

_{i+1}, …, X

_{M}).

#### 3.2. Approximation of Differential Entropy by Functional $\tilde{H}\left(R\right)$ for Sensitivity Indices

_{R}and also differ as little as possible from the differential entropy if f(r) < 1. The function should be increasing and decreasing approximately according to Equation (2) to fit Equation (2) well in the unproblematic areas. One such function, which is useful in modifying Equation (2), is the hyperbolic tangent:

_{R}→ 0. The right side of Figure 2 shows examples of the plots of the natural logarithm functions that are used in Equation (9).

_{R}; otherwise,$\tilde{H}\left(R\right)$ ≈ H(R) approximately according to Equation (2).

_{R}are used. It can be noted that there is not much difference between the values plotted from Equation (9) for t = 4 (red curve) and t = ∞ (black curve)—see Figure 3.

_{R}> 0, so a gradual decrease to zero is more appropriate.

#### 3.3. Approximation of Differential Entropy by Functional $\widehat{H}\left(R\right)$ and Sensitivity Indices

_{1}can be computed from the condition

_{1}, z

_{2}into (11), c

_{1}can be computed as

_{b}(z) for both small and large values of z = f(r)—see the left part of Figure 4. Using the Gauss pdf of f(r), it is shown that large deviations in the approximation of the differential entropy are observed for both large and small values of σ

_{R}—see the right part of Figure 4.

_{R}= 0, each index is in the interval [0, 1] and the sum of all sensitivity indices is equal to one.

## 4. Standard Distribution-Based Sensitivity Analyzes

#### 4.1. Cramér-von Mises Sensitivity Indices

_{R}be the distribution function of R, where R is a model output, and X is a vector of M uncertain model inputs {X

_{1}, X

_{2}, … X

_{M}} with the assumption of statistical independence.

_{i}:

_{i}is determined by measuring the distance between probability Φ

_{Z}(t) and conditional probability ${\mathsf{\Phi}}_{Z}^{i}$(t) when an input is fixed [68].

_{ij}can be written using [68] as

#### 4.2. Borgonovo Moment-Independent Sensitivity Indices

_{R│}

_{Xi}(r) is the conditional pdf of R with fixed parameter X

_{i}[70].

_{i}, X

_{j}, we obtain the second-order index B

_{ij}, where i < j. Upon fixing triplets X

_{i}, X

_{j}, X

_{k}, we obtain the third-order index B

_{ijk}, where i < j < k, etc. The higher the order of the index, the greater its value, and the index of the last order is equal to one, 0 ≤ B

_{i}≤ B

_{ij}≤ … ≤ B

_{1,2,…,M}≤ 1 [70]. Compared to SA, which has the sum of all indices equal to one, Borgonovo indices are less practical, especially the last index with fixed inputs, which is always equal to one and does not provide any new information. Identification of the influence of X

_{i}using total indices is not possible for Borgonovo indices. The advantage of Borgonovo indices is their evaluation of the whole distribution of the output in a more transparent way than presented by [68]—see Equation (18) vs. Equation (16). The second advantage is that the input variables can be statistically correlated, which is difficult to ensure for other types of indices based on decomposition, such as Sobol indices.

## 5. Variance-Based Sensitivity Analysis

_{i}) is the conditional variance of the model output R.

## 6. The Case Studies

#### 6.1. Computational Model

_{y}, thickness t

_{2}, and width b. Statistical characteristics of f

_{y}, t

_{2}and b are taken into consideration using the results of experimental research [75,76], where steel grade S355 was studied for selected steel products—see Table 1.

_{y}and cross-sectional area t

_{2}·b—see Equation (20).

_{R}of the product R can be obtained from Equation (21):

_{R}is the standard deviation.

_{R}—see Equation (23).

_{R}= 495.216 kN, σ

_{R}= 40.822 kN, a

_{R}= 0.11 are obtained upon substituting the statistical characteristics of the input random variables from Table 1—see Figure 5.

_{R}, σ

_{R}, and a

_{R}[69]. The three-parameter log-normal pdf can be used to estimate the sensitivity index, even if one of the three variables in Equation (20) is fixed as deterministic [32]. If two input variables are fixed, a

_{R}= 0 and product R has a Gauss pdf.

#### 6.2. The Results of the Case Studies

_{R}− 10σ

_{R}, μ

_{R}+ 10σ

_{R}]. If the lower bound of the domain of f(r) is greater than μ

_{R}− 10σ

_{R}, then integration is performed from the lower bound of f(r). Numerical integration is not used for Sobol sensitivity indices, which are computed analytically using Equation (22). Cramér-von Mises sensitivity indices are estimated using the algorithm described in [39,69], with the difference that three input random variables are used in this article. Borgonovo sensitivity indices were estimated according to the procedure in [39]. Further details of the numerical estimates of the sensitivity indices can be found in [39,69,72]. The results of sensitivity analyzes are shown in Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11.

_{1}, X

_{2}, X

_{3}) = 0.

_{1}, X

_{2}, X

_{3}) = 0.

_{1}, X

_{2}, X

_{3}) = 0.

## 7. Discussion

_{1}, X

_{2}, X

_{3})) = 0—see Equation (5). This means that the entropy of the sensitivity index of the last order (deterministic variables) must be calculated according to Equation (1).

_{y}, t

_{2}, b. This sensitivity order was determined using total indices, except for Borgonovo SA, where total indices do not exist. The large value of the sensitivity index of the last order causes the difference between the total indices of certain SA types to be very small—see Figure 6 and Figure 7. In contrast, Sobol SA (Figure 11) and SA based on $\widehat{H}\left(R\right)$—see Figure 8—which clearly identify a strong influence of f

_{y}, provide clear identification of the influential and non-influential inputs.

_{i}is 0.998) and very small higher-order sensitivity indices is given by Sobol SA—see Figure 11. This has also been observed for other tasks [31,47,71]. If the sum of all S

_{i}is equal to one, then the sensitivity order can be determined using only the S

_{i}indices, which are the same as the total indices, and higher order sensitivity indices do not have to be calculated. Easy interpretation of SA results, often carried out only with S

_{i}, is one of the features that makes Sobol SA popular.

## 8. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Approximation of H(R) by $\tilde{H}\left(R\right)$ using natural logarithm and Gauss pdf of f(r).

**Figure 4.**Approximation of H(R) by $\widehat{H}\left(R\right)$ using three types of logarithm and Gauss pdf of f(r).

Characteristic | Index | Symbol | Mean Value μ | Standard Deviation σ |
---|---|---|---|---|

Yield Strength | 1 | f_{y} | 412.68 MPa | 27.941 MPa |

Thickness | 2 | t_{2} | 12 mm | 0.55 mm |

Width | 3 | b | 100 mm | 1 mm |

^{1}All inputs have Gauss pdf and are uncorrelated.

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Kala, Z.
Global Sensitivity Analysis Based on Entropy: From Differential Entropy to Alternative Measures. *Entropy* **2021**, *23*, 778.
https://doi.org/10.3390/e23060778

**AMA Style**

Kala Z.
Global Sensitivity Analysis Based on Entropy: From Differential Entropy to Alternative Measures. *Entropy*. 2021; 23(6):778.
https://doi.org/10.3390/e23060778

**Chicago/Turabian Style**

Kala, Zdeněk.
2021. "Global Sensitivity Analysis Based on Entropy: From Differential Entropy to Alternative Measures" *Entropy* 23, no. 6: 778.
https://doi.org/10.3390/e23060778