A novel approach to uncertainty analysis using methods of hybrid dimension reduction and improved maximum entropy

Abstract

Methods of uncertainty analysis based on statistical moments are more convenient than methods that use a Taylor series expansion because the moments methods require neither an iteration process to locate the most probable point nor the computation of derivatives of the performance function. However, existing moments estimation methods are either computationally expensive (e.g., the full factorial numerical integration method) or produce large errors (e.g., the univariate dimension-reduction method). In this paper, a hybrid dimension-reduction method taking account of interactions among variables is presented for estimating the probability moments of the system performance function. In this method, a contribution-degree analysis with finite changes is implemented to identify the relative importance of the input variables on the output. Then, an approximate performance function is generated with the hybrid dimension-reduction method that is based on the results of contribution-degree analysis. Finally, the statistical moments of the performance function can be calculated from the approximate performance function. Once the probability moments are obtained, an improved maximum entropy method is used to generate the probability density function of the performance function. The uncertainty analysis can be implemented by using the approximation probability density function. Five illustrative numerical examples are presented, and different methods are compared in those examples. The statistical moments estimation results reveal that the proposed moments estimation method can dramatically improve efficiency and also guarantee accuracy. Compared with the other probability density function approximation methods, our improved maximum entropy method, using more statistical moments, is more accurate and robust.

Appendices

Appendix 1

If the number of variables N is odd, considering a situation for all of the random variablex = [{x₁, x₂}, {x₃, x₄}, ⋯, {x_N − 1, x_N}], the{x_i, x_j} means variables x_i and x_j have interaction and the variable x_N has no interaction with other variables. The function y(x) can be approximated as

$$ {\displaystyle \begin{array}{c}\mathrm{y}\left(\mathbf{x}\right)=\hat{\mathrm{y}}\left(\mathbf{x}\right)={y}^1\left({x}_1,{x}_2\right)+{y}^2\left({x}_3,{x}_4\right),+\cdots +{y}^o\left({x}_{N-1},{x}_N\right)\\ {}-\left(\frac{N}{2}-1\right)y\left(\boldsymbol{\upmu} \right)\kern1.1em o=1,\cdots, N/2\end{array}} $$

(64)

Without loss of generality, the entire interaction situations {x_i, x_j} can generate a series of combinations as follows.

$$ \Big\{{\displaystyle \begin{array}{l}{\hat{\mathbf{x}}}_1=\left[\left\{{x}_{1,1},{x}_{1,2}\right\},\left\{{x}_{1,3},{x}_{1,4}\right\},\cdots, \left\{{x}_{1,N-1},{x}_{1,N}\right\}\right]\\ {}{\hat{\mathbf{x}}}_2=\left[\left\{{x}_{2,1},{x}_{2,2}\right\},\left\{{x}_{2,3},{x}_{2,4}\right\},\cdots, \left\{{x}_{2,N-1},{x}_{2,N}\right\}\right]\\ {}\vdots \\ {}{\hat{\mathbf{x}}}_k=\left[\left\{{x}_{k,1},{x}_{k,2}\right\},\left\{{x}_{k,3},{x}_{k,4}\right\},\cdots, \left\{{x}_{k,N-1},{x}_{k,N}\right\}\right]\end{array}} $$

(65)

In (65), each combination ^_i(i = 1, ⋯, k) contains all of the variables of x and each variable without repetition. It means that each combination contains all of the variables of x and all terms are mutually independent. The entire interaction situations {x_i, x_j} can be contained in those combinations without repetition.

According to (64) and (65), a series of approximation functions can be achieved:

$$ \Big\{{\displaystyle \begin{array}{l}{\hat{\mathrm{y}}}_1\left({\hat{\mathbf{x}}}_1\right)=y\left({x}_{1,1},{x}_{1,2}\right)+y\left({x}_{1,3},{x}_{1,4}\right)+\cdots +y\left({x}_{1,N-1},{x}_{1,N}\right)\\ {}-\left(\frac{N}{2}-1\right)y\left(\boldsymbol{\upmu} \right)\\ {}{\hat{\mathrm{y}}}_2\left({\hat{\mathbf{x}}}_2\right)=y\left({x}_{2,1},{x}_{2,2}\right)+y\left({x}_{2,3},{x}_{2,4}\right)+\cdots +y\left({x}_{2,N-1},{x}_{2,N}\right)\\ {}-\left(\frac{N}{2}-1\right)y\left(\boldsymbol{\upmu} \right)\\ {}\vdots \\ {}{\hat{\mathrm{y}}}_k\left({\hat{\mathbf{x}}}_{\mathrm{k}}\right)=y\left({x}_{k,1},{x}_{k,2}\right)+y\left({x}_{k,3},{x}_{k,4}\right)+\cdots +y\left({x}_{k,N-1},{x}_{k,N}\right)\\ {}-\left(\frac{N}{2}-1\right)y\left(\boldsymbol{\upmu} \right)\end{array}} $$

(66)

According (66) and (2), the function y(x)can be approximated as

$$ y\left(\mathbf{x}\right)\cong \sum \limits_{i=1}^k{\hat{y}}_i\left({\hat{\mathbf{x}}}_i\right)-\left(N-2\right){\hat{y}}_{UDR}\left(\mathbf{x}\right) $$

(67)

Appendix 2

Because x = [{x₁, x₂}, {x₃, x₄}, ⋯, {x_N − 2, x_N − 1}, {x_N}], each term is independent and each term of the right side in (3)is mutually independent.

The first central moment refers to the expected value

$$ {\displaystyle \begin{array}{l}{\mu}_y=E\left[y(x)\right]=E\left[{y}^1+{y}^2+\cdots +{y}^o+{y}^{\prime }-\frac{N-1}{2}{y}^{\mu}\right]\\ {}=\sum \limits_{i=1}^o{\mu}_{y^i}+{\mu}_{y^{\prime }}-\frac{N-1}{2}{y}^{\mu}\end{array}} $$

(68)

where yⁱ and y^μ are functions yⁱ() and y(μ), respectively.

The second order central moment is

$$ {\displaystyle \begin{array}{l}{\sigma}_y^2=E\left[{\left(y(x)-{\mu}_y\right)}^2\right]=E\left[{\left(\left({y}^1+{y}^2+\cdots +{y}^o+y\prime -\frac{N-1}{2}{y}^{\mu}\right)-{\mu}_y\right)}^2\right]\\ {}=E\left[{\left(\left({y}^1+{y}^2+\cdots +{y}^o+y\prime -\frac{N-1}{2}{y}^{\mu}\right)-\left(\sum \limits_{i=1}^o{\mu}_{y^i}+{\mu}_{y^{\prime }}-\frac{N-1}{2}{y}^{\mu}\right)\right)}^2\right]\\ {}=E\left[{\left(\sum \limits_{i=1}^o\left({y}^i-{\mu}_{y^i}\right)+\left(y\prime -{\mu}_{y^{\prime }}\right)\right)}^2\right]\\ {}=E\left[\sum \limits_{i=1}^o{\left({y}^i-{\mu}_{y^i}\right)}^2+{\left(y\prime -{\mu}_{y^{\prime }}\right)}^2+2\left(y^{\prime }-{\mu}_{y^{\prime }}\right)\sum \limits_{i=1}^o\left({y}^i-{\mu}_{y^i}\right)+2\sum \limits_{1\le i<j\le o}\left({y}^i-{\mu}_{y^i}\right)\left({y}^j-{\mu}_{y^j}\right)\right]\\ {}=\sum \limits_{i=1}^oE\left[{\left({y}^i-{\mu}_{y^i}\right)}^2\right]+E\left[\left[{\left(y\prime -{\mu}_{y^{\prime }}\right)}^2\right]+2E\left[\left(y^{\prime }-{\mu}_{y^{\prime }}\right)\right]\sum \limits_{i=1}^oE\right[\left(\left({y}^i-{\mu}_{y^i}\right)\right]+2\sum \limits_{1\le i<j\le o}E\left[\left({y}^i-{\mu}_{y^i}\right)\right]E\left[\left({y}^j-{\mu}_{y^j}\right)\right]=\sum \limits_{i=1}^o{\sigma}_{y^i}^2+{\sigma}_{y^{\prime}}^2\end{array}} $$

(69)

The third order central moment is

$$ {\displaystyle \begin{array}{l}{M}_y^3=E\left[{\left(y(x)-{\mu}_y\right)}^3\right]\\ {}=E\left[{\left(\sum \limits_{i=1}^o\left({y}^i-{\mu}_{y^i}\right)+\left(y\prime -{\mu}_{y^{\prime }}\right)\right)}^3\right]\\ {}=E\left[{\left(\sum \limits_{i=1}^o\left({y}^i-{\mu}_{y^i}\right)\right)}^3+{\left(y\prime -{\mu}_{y^{\prime }}\right)}^3+3{\left(y\prime -{\mu}_{y^{\prime }}\right)}^2\sum \limits_{i=1}^o\left({y}^i-{\mu}_{y^i}\right)+3\left(y^{\prime }-{\mu}_{y^{\prime }}\right){\left(\sum \limits_{i=1}^o\left({y}^i-{\mu}_{y^i}\right)\right)}^2\right]\\ {}=\sum \limits_{i=1}^oE\left[{\left({y}^i-{\mu}_{y^i}\right)}^3\right]+E\left[{\left(y\prime -{\mu}_{y^{\prime }}\right)}^3\right]+3E\left[{\left(y\prime -{\mu}_{y^{\prime }}\right)}^2\right]\sum \limits_{i=1}^oE\left[\left({y}^i-{\mu}_{y^i}\right)\right]+3E\left[\left(y^{\prime }-{\mu}_{y^{\prime }}\right)\right]E\left[{\left(\sum \limits_{i=1}^o\left({y}^i-{\mu}_{y^i}\right)\right)}^2\right]\\ {}=\sum \limits_{i=1}^o{M}_{y^i}^3+{M}_{y^{\prime}}^3\end{array}} $$

(70)

The fourth-order central moment is

$$ {\displaystyle \begin{array}{l}{M}_y^4=E\Big[\left[{\left(y(x)-{\mu}_y\right)}^4\right]\\ {}=E\left[{\left(\sum \limits_{i=1}^o\left({y}^i-{\mu}_{y^i}\right)+\left(y\prime -{\mu}_{y^{\prime }}\right)\right)}^4\right]\\ {}=E\left[{\left(\sum \limits_{i=1}^o\left({y}^i-{\mu}_{y^i}\right)\right)}^4+{\left(y\prime -{\mu}_{y^{\prime }}\right)}^4+4{\left(y\prime -{\mu}_{y^{\prime }}\right)}^3\sum \limits_{i=1}^o\left({y}^i-{\mu}_{y^i}\right)+4\left(y^{\prime }-{\mu}_{y^{\prime }}\right){\left(\sum \limits_{i=1}^o\left({y}^i-{\mu}_{y^i}\right)\right)}^3+6{\left(y\prime -{\mu}_{y^{\prime }}\right)}^2{\left(\sum \limits_{i=1}^o\left({y}^i-{\mu}_{y^i}\right)\right)}^2\right]\\ {}=\sum \limits_{i=1}^oE\left[{\left({y}^i-{\mu}_{y^i}\right)}^4\right]+E\Big[\left[{\left(y\prime -{\mu}_{y^{\prime }}\right)}^4\right]+4E\left[{\left(y\prime -{\mu}_{y^{\prime }}\right)}^3\right]\sum \limits_{i=1}^oE\left[\left({y}^i-{\mu}_{y^i}\right)\right]\\ {}+4E\left[\left(y^{\prime }-{\mu}_{y^{\prime }}\right)\right]E\Big[\left[{\left(\sum \limits_{i=1}^o\left({y}^i-{\mu}_{y^i}\right)\right)}^3\right]+6E\left[{\left(y\prime -{\mu}_{y^{\prime }}\right)}^2\right]E\left[{\left(\sum \limits_{i=1}^o\left({y}^i-{\mu}_{y^i}\right)\right)}^2\right]\\ {}+6\sum \limits_{1\le i<j\le o}E\left[{\left({y}^i-{\mu}_{y^i}\right)}^2\right]E\left[{\left({y}^j-{\mu}_{y^j}\right)}^2\right]\\ {}=\sum \limits_{i=1}^o{M}_{y^i}^4+{M}_{y^{\prime}}^4+6{\sigma}_{y^{\prime}}^2\sum \limits_{i=1}^o{\sigma}_{y^i}^2+6\sum \limits_{1\le i<j\le o}{\sigma}_{y^i}^2{\sigma}_{y^j}^2\end{array}} $$

(71)