#
On Taylor Series Expansion for Statistical Moments of Functions of Correlated Random Variables^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Taylor Series Expansion

#### 2.1. Linear Terms of Taylor Series Expansion

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

#### 2.2. Higher-Order Taylor Series Expansion

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

## 3. Numerical Computation

#### 3.1. Methodology of ECoV by TSE

- (1)
- linear TSE with a simple differencing scheme using Equation (3)—${n}_{sim}=n+1$,
- (2)
- linear TSE with an advanced differencing scheme using Equation (9)—${n}_{sim}=2n+1$,
- (3)
- TSE truncated to quadratic terms with a differencing scheme using Equation (9) for the first-order derivatives, Equation (10) for the second-order partial derivatives, and Equation (11) for the mixed derivatives—number of calculation is ${n}_{sim}=2n+\left(\genfrac{}{}{0pt}{}{n}{2}\right)+1$ in total.

#### 3.2. Reference Solution

#### 3.3. Example 1: Simple Linear Model

#### 3.4. Example 2: Linear Model with Interactions

#### 3.5. Example 3: Approximation of Industrial Example

#### 3.6. Example 4: Non-Linear Function

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Proposed methodology composed of the three levels of Taylor series expansion (TSE) approximation using asymmetric differencing schemes adapted for civil engineering. Iso-lines of bivariate standard Gaussian probability distribution are represented by dotted circles.

**Figure 2.**Realizations (red dots) generated by Latin Hypercube Sampling (LHS) and iso-lines of joint probability density of input random vector in uncorrelated (

**left**) and correlated (

**right**) space.

**Figure 3.**Estimation of coefficient of variance (CoV) (

**top**) and variance (

**bottom**) of the first example by the presented methods.

**Figure 4.**Estimation of CoV (

**top**) and variance (

**bottom**) of the second example by the presented methods.

**Figure 5.**Estimation of CoV (

**top**) and variance (

**bottom**) of the third example by the presented methods.

**Figure 6.**Estimation of CoV (

**top**) and variance (

**bottom**) of the fourth example by the presented methods.

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**MDPI and ACS Style**

Novák, L.; Novák, D.
On Taylor Series Expansion for Statistical Moments of Functions of Correlated Random Variables. *Symmetry* **2020**, *12*, 1379.
https://doi.org/10.3390/sym12081379

**AMA Style**

Novák L, Novák D.
On Taylor Series Expansion for Statistical Moments of Functions of Correlated Random Variables. *Symmetry*. 2020; 12(8):1379.
https://doi.org/10.3390/sym12081379

**Chicago/Turabian Style**

Novák, Lukáš, and Drahomír Novák.
2020. "On Taylor Series Expansion for Statistical Moments of Functions of Correlated Random Variables" *Symmetry* 12, no. 8: 1379.
https://doi.org/10.3390/sym12081379