A data-driven polynomial chaos method considering correlated random variables

Abstract

Variable correlation commonly exists in practical engineering applications. However, most of the existing polynomial chaos (PC) approaches for uncertainty propagation (UP) assume that the input random variables are independent. To address variable correlation, an intrusive PC method has been developed for dynamic system, which however is not applicable to problems with black-box-type functions. Therefore, based on the existing data-driven PC method, a new non-intrusive data-driven polynomial chaos approach that can directly consider variable correlation for UP of black-box computationally expensive problems is developed in this paper. With the proposed method, the multivariate orthogonal polynomial basis corresponding to the correlated input random variables is conveniently constructed by solving the moment-matching equations based on the correlation statistical moments to consider the variable correlation. A comprehensive comparative study on several numerical examples of UP and design optimization under uncertainty with correlated input random variables is conducted to verify the effectiveness and advantage of the proposed method. The results show that the proposed method is more accurate than the existing data-driven PC method with Nataf transformation when the variable distribution is known, and it can produce accurate results with unknown variable distribution, demonstrating its effectiveness.

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Abbreviations

b i :

The ith coefficient of PC model

d :

Dimension of random inputs

x :

Random input vector

y :

Stochastic response value

H :

Order of PC model

P (k) :

The kth orthogonal polynomials for correlated variables

\( \overline{P} \) :

The orthogonal polynomials for independent variables

Q + 1:

Number of PC coefficients

μ :

Mean value

μ a, b :

Correlation statistical moment

ρ :

Correlation coefficient

σ :

Standard deviation value

Ωc :

Original correlated random variable space

Γ(x):

Joint cumulative distribution function

DD-PC:

The data-driven polynomial chaos method

gPC:

The generalized polynomial chaos method

GS-PC:

The Gram-Schmidt polynomial chaos method

ME-PC:

The multi-element generalized polynomial chaos method

PC:

Polynomial chaos

UP:

Uncertainty propagation

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Funding

Grant support was received from Science Challenge Project (No. TZ2018001) and Hongjian Innovation Foundation (No.BQ203-HYJJ-Q2018002).

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Correspondence to Fenfen Xiong.

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The authors declare that they have no conflict of interest.

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Replication of results

The results shown in the manuscript can be re-produced. Considering the size limit of the uploaded supplementary material, the codes for one of the mathematical example for UP (Function 1 and Function 2 in Sect. 3.1) is uploaded as supplementary material. For the rest of the examples, it is very easy to implement by changing the response functions and sample points based on the codes provided to obtain the results shown in the manuscript

Responsible Editor: Byeng D Youn

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Appendix

Appendix

The correlated statistical moments and the polynomial coefficients for Function 1 (Case 1, ρ = 0.8) in Sect. 3.1 are provided as below.

For this example, the dimension of correlated random variable is 2 and the order of model is set as H = 3. Therefore, there are ten two-dimensional orthogonal polynomial bases with polynomial order on more than 3. The correlated statistical moments that are in the form of matrix in (17) are shown as below:

$$ {P}^{(0)}\left({x}_1,{x}_2\right)=\left[1\right]\left(k=0\right) $$
(21)
$$ {P}^{(1)}\left({x}_1,{x}_2\right)=\left[\begin{array}{cc}{\int}_{x_1,{x}_2\in {\Omega}_c}1\boldsymbol{d}\Gamma \left({x}_1,{x}_2\right)& {\int}_{x_1,{x}_2\in {\Omega}_c}{x}_1\boldsymbol{d}\Gamma \left({x}_1,{x}_2\right)\\ {}0& 1\end{array}\right]\left(k=1\right) $$
(22)
$$ {P}^{(2)}\left({x}_1,{x}_2\right)=\left[\begin{array}{ccc}{\int}_{x_1,{x}_2\in {\varOmega}_c}1\mathrm{d}\varGamma \left({x}_1,{x}_2\right)& {\int}_{x_1,{x}_2\in {\varOmega}_c}{x}_1\mathrm{d}\varGamma \left({x}_1,{x}_2\right)& {\int}_{x_1,{x}_2\in {\varOmega}_c}{x}_2\mathrm{d}\varGamma \left({x}_1,{x}_2\right)\\ {}{\int}_{x_1,{x}_2\in {\varOmega}_c}{x}_1\mathrm{d}\varGamma \left({x}_1,{x}_2\right)& {\int}_{x_1,{x}_2\in {\varOmega}_c}{x_1}^2\mathrm{d}\varGamma \left({x}_1,{x}_2\right)& {\int}_{x_1,{x}_2\in {\varOmega}_c}{x}_1{x}_2\mathrm{d}\varGamma \left({x}_1,{x}_2\right)\\ {}0& 0& 1\end{array}\right]\left(k=2\right) $$
(23)
$$ {P}^{(9)}\left({x}_1,{x}_2\right)=\left[\begin{array}{cccc}{\int}_{x_1,{x}_2\in {\varOmega}_c}1\mathrm{d}\varGamma \left({x}_1,{x}_2\right)& {\int}_{x_1,{x}_2\in {\varOmega}_c}{x}_1\mathrm{d}\varGamma \left({x}_1,{x}_2\right)& \cdots & {\int}_{x_1,{x}_2\in {\varOmega}_c}{x_2}^3\mathrm{d}\varGamma \left({x}_1,{x}_2\right)\\ {}{\int}_{x_1,{x}_2\in {\varOmega}_c}{x}_1\mathrm{d}\varGamma \left({x}_1,{x}_2\right)& {\int}_{x_1,{x}_2\in {\varOmega}_c}{x_1}^2\mathrm{d}\varGamma \left({x}_1,{x}_2\right)& \cdots & {\int}_{x_1,{x}_2\in {\varOmega}_c}{x}_1{x_2}^3\mathrm{d}\varGamma \left({x}_1,{x}_2\right)\\ {}\vdots & \vdots & \vdots & \vdots \\ {}{\int}_{x_1,{x}_2\in {\varOmega}_c}{x}_1{x_2}^2\mathrm{d}\varGamma \left({x}_1,{x}_2\right)& {\int}_{x_1,{x}_2\in {\varOmega}_c}{x_1}^2{x_2}^2\mathrm{d}\varGamma \left({x}_1,{x}_2\right)& \cdots & {\int}_{x_1,{x}_2\in {\varOmega}_c}{x}_1{x_2}^5\mathrm{d}\varGamma \left({x}_1,{x}_2\right)\\ {}0& 0& \cdots & 1\end{array}\right]\left(k=9\right) $$
(24)

Based on the probabilistic distribution information of the correlated input random variables, one can obtain these correlated statistical moments that will be employed in the orthogonal polynomial basis construction. As the correlated statistical moments for k = 0, 1,..., 8 are all part of those for k = 9, only those for k = 9 is given as below:

$$ \left[\begin{array}{cccccccccc}1& 1.0037& 2.0047& 1.0470& 2.0432& 4.0581& 1.1307& 2.1618& 4.1988& 8.2947\\ {}1.0037& 1.0470& 2.0432& 1.1307& 2.1618& 4.1988& 1.2600& 2.3652& 4.5040& 8.7086\\ {}2.0047& 2.0432& 4.0581& 2.1618& 4.1988& 8.2947& 2.3652& 4.5040& 8.7086& 17.1153\\ {}1.0470& 1.1307& 2.1618& 1.2600& 2.3652& 4.5040& 1.4445& 2.6671& 4.9901& 9.4668\\ {}2.0432& 2.1618& 4.1988& 2.3652& 4.5040& 8.7086& 2.6671& 4.9901& 9.4668& 18.2262\\ {}4.0581& 4.1988& 8.2947& 4.5040& 8.7086& 17.1153& 4.9901& 9.4668& 18.2262& 35.6458\\ {}1.1307& 1.2600& 2.3652& 1.4445& 2.6671& 4.9901& 1.6991& 3.0911& 5.6923& 10.6171\\ {}2.1618& 2.3652& 4.5040& 2.6671& 4.9901& 9.4668& 3.0911& 5.6923& 10.6171& 20.0705\\ {}4.1988& 4.5040& 8.7086& 4.9901& 9.4668& 18.2262& 5.6923& 10.6171& 20.0705& 38.4842\end{array}\right]\left(k=9\right) $$
(25)

Correspondingly, the polynomial coefficients of the ten two-dimensional orthogonal polynomial bases are listed as below:

$$ {\displaystyle \begin{array}{l}\left[{p}_0^{(0)}\right]=\left[1\right]\\ {}\left[{p}_0^{(1)},{p}_1^{(1)}\right]=\left[1,-1.0037\right]\\ {}\begin{array}{l}\left[{p}_0^{(2)},{p}_1^{(2)},{p}_2^{(2)}\right]=\left[1,-0.7874,-1.2143\right]\\ {}\left[{p}_0^{(3)},{p}_1^{(3)},\dots {p}_3^{(3)}\right]=\left[\mathrm{1,0.0000},-\mathrm{2.0149,0.9733}\right]\\ {}\begin{array}{l}\left[{p}_0^{(4)},{p}_1^{(4)},\dots {p}_4^{(4)}\right]=\left[1,-0.7601,-1.0339,-\mathrm{0.4550,1.2818}\right]\\ {}\left[{p}_0^{(5)},{p}_1^{(5)},\dots {p}_5^{(5)}\right]=\left[1,-\mathrm{1.5269,0.5869},-\mathrm{2.4795,1.8735,1.5374}\right]\\ {}\begin{array}{l}\left[{p}_0^{(6)},{p}_1^{(6)},\dots {p}_6^{(6)}\right]=\left[\mathrm{1,0.0124},-0.0202,-2.9868,-\mathrm{0.0242,2.8980},-0.8728\right]\\ {}\left[{p}_0^{(7)},{p}_1^{(7)},\dots {p}_7^{(7)}\right]=\left[1,-\mathrm{0.7414,0.0434},-\mathrm{2.1284,0.2956,0.9175,1.9156},-1.2221\right]\\ {}\begin{array}{l}\left[{p}_0^{(8)},{p}_1^{(8)},\dots {p}_8^{(8)}\right]=\left[1,-\mathrm{1.5061,0.5835},-1.0560,-\mathrm{0.8766,1.1998,2.6053},-0.3874,-1.6164\right]\\ {}\left[{p}_0^{(9)},{p}_1^{(9)},\dots {p}_9^{(9)}\right]=\left[1,-\mathrm{2.1648,1.5190},-0.3394,-\mathrm{3.8632,5.6408},-\mathrm{2.0132,4.9097},-3.6227,-2.0516\right]\end{array}\end{array}\end{array}\end{array}\end{array}} $$
(26)

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Lin, Q., Xiong, F., Wang, F. et al. A data-driven polynomial chaos method considering correlated random variables. Struct Multidisc Optim (2020). https://doi.org/10.1007/s00158-020-02602-7

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Keywords

  • Uncertainty propagation
  • Polynomial chaos
  • Data-driven
  • Variable correlation