First-order second-moment method

In probability theory , the first-order second-moment method ( FOSM for short ), also known as the mean value first-order second-moment method ( MVFOSM for short ), is an approximation method for determining the stochastic moments of a function with randomly distributed input variables. The English name is derived from the derivation, in a Taylor series of the first order ( first-order ) and the first two moments ( second moment ) of the input variables are used.

Approximation

The objective function is given , the vector being a realization of the random vector with the probability density function . Since it is random, it is also random. The FOSM method approximates the expected value of the objective function ${\displaystyle g(x)}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle f_{X}(x)}$ ${\displaystyle X}$ ${\displaystyle g}$

${\displaystyle \mu _{g}\approx g(\mu )}$

The variance of loud FOSM method approximately ${\displaystyle g}$

${\displaystyle \sigma _{g}^{2}\approx \sum _{i=1}^{n}\sum _{j=1}^{n}{\frac {\partial g(\mu )}{\partial x_{i}}}{\frac {\partial g(\mu )}{\partial x_{j}}}\operatorname {cov} (X_{i},X_{j})}$

where is the length / dimension of and the partial derivative of the mean value vector after the i -th entry of . ${\displaystyle n}$ ${\displaystyle x}$ ${\displaystyle {\frac {\partial g(\mu )}{\partial x_{i}}}}$ ${\displaystyle x}$

Derivation

The objective function is approximated by a Taylor series on the mean value vector. ${\displaystyle \mu }$

${\displaystyle g(x)=g(\mu )+\sum _{i=1}^{n}{\frac {\partial g(\mu )}{\partial x_{i}}}(x_{i}-\mu _{i})+{\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}{\frac {\partial ^{2}g(\mu )}{\partial x_{i}\,\partial x_{j}}}(x_{i}-\mu _{i})(x_{j}-\mu _{j})+\cdots }$

The expected value of is given by the following integral. ${\displaystyle g}$

${\displaystyle \mu _{g}=E[g(x)]=\int _{-\infty }^{\infty }g(x)f_{X}(x)\,dx}$

If you insert the Taylor series, you get

${\displaystyle {\begin{aligned}\mu _{g}&\approx \int _{-\infty }^{\infty }\left[g(\mu )+\sum _{i=1}^{n}{\frac {\partial g(\mu )}{\partial x_{i}}}\right]f_{X}(x)\,dx\\&=\int _{-\infty }^{\infty }g(\mu )f_{X}(x)\,dx+\int _{-\infty }^{\infty }\sum _{i=1}^{n}{\frac {\partial g(\mu )}{\partial x_{i}}}(x_{i}-\mu _{i})f_{X}(x)\,dx\\&=g(\mu )\underbrace {\int _{-\infty }^{\infty }f_{X}(x)\,dx} _{1}+\sum _{i=1}^{n}{\frac {\partial g(\mu )}{\partial x_{i}}}\underbrace {\int _{-\infty }^{\infty }(x_{i}-\mu _{i})f_{X}(x)\,dx} _{0}\\&=g(\mu ).\end{aligned}}}$

The variance of is given by the following integral. ${\displaystyle g}$

${\displaystyle \sigma _{g}^{2}=E([g(x)-\mu _{g}]^{2})=\int _{-\infty }^{\infty }[g(x)-\mu _{g}]^{2}f_{X}(x)\,dx.}$

With the displacement theorem one obtains

${\displaystyle \sigma _{g}^{2}=E([g(x)-\mu _{g}]^{2})=E(g(x)^{2})-\mu _{g}^{2}=\int _{-\infty }^{\infty }g(x)^{2}f_{X}(x)\,dx-\mu _{g}^{2}}$

Inserting the Taylor series yields

${\displaystyle {\begin{aligned}\sigma _{g}^{2}&\approx \int _{-\infty }^{\infty }\left[g(\mu )+\sum _{i=1}^{n}{\frac {\partial g(\mu )}{\partial x_{i}}}(x_{i}-\mu _{i})\right]^{2}f_{X}(x)\,dx-\mu _{g}^{2}\\&=\int _{-\infty }^{\infty }\left\{g(\mu )^{2}+2g(\mu )\sum _{i=1}^{n}{\frac {\partial g(\mu )}{\partial x_{i}}}(x_{i}-\mu _{i})+\left[\sum _{i=1}^{n}{\frac {\partial g(\mu )}{\partial x_{i}}}(x_{i}-\mu _{i})\right]^{2}\right\}f_{X}(x)\,dx-\mu _{g}^{2}\\&=\int _{-\infty }^{\infty }g(\mu )^{2}f_{X}(x)\,dx+\int _{-\infty }^{\infty }2\,g(\mu )\sum _{i=1}^{n}{\frac {\partial g(\mu )}{\partial x_{i}}}(x_{i}-\mu _{i})f_{X}(x)\,dx\\&{}\quad {}+\int _{-\infty }^{\infty }\left[\sum _{i=1}^{n}{\frac {\partial g(\mu )}{\partial x_{i}}}(x_{i}-\mu _{i})\right]^{2}f_{X}(x)\,dx-\mu _{g}^{2}\\&=g(\mu )^{2}\underbrace {\int _{-\infty }^{\infty }f_{X}(x)\,dx} _{1}+2g(\mu )\sum _{i=1}^{n}{\frac {\partial g(\mu )}{\partial x_{i}}}\underbrace {\int _{-\infty }^{\infty }(x_{i}-\mu _{i})f_{X}(x)\,dx} _{0}\\&{}\quad {}+\int _{-\infty }^{\infty }\left[\sum _{i=1}^{n}\sum _{j=1}^{n}{\frac {\partial g(\mu )}{\partial x_{i}}}{\frac {\partial g(\mu )}{\partial x_{j}}}(x_{i}-\mu _{i})(x_{j}-\mu _{j})\right]f_{X}(x)\,dx-\mu _{g}^{2}\\&=\underbrace {g(\mu )^{2}} _{\approx \,\mu _{g}^{2}}+\sum _{i=1}^{n}\sum _{j=1}^{n}{\frac {\partial g(\mu )}{\partial x_{i}}}{\frac {\partial g(\mu )}{\partial x_{j}}}\underbrace {\int _{-\infty }^{\infty }(x_{i}-\mu _{i})(x_{j}-\mu _{j})f(x)\,dx} _{\operatorname {cov} (X_{i},X_{j})}-\mu _{g}^{2}\\&\approx \sum _{i=1}^{n}\sum _{j=1}^{n}{\frac {\partial g(\mu )}{\partial x_{i}}}{\frac {\partial g(\mu )}{\partial x_{j}}}\operatorname {cov} (X_{i},X_{j}).\end{aligned}}}$

Higher order approximation

The following abbreviations are introduced.

${\displaystyle g_{\mu }=g(\mu ),\quad g_{,i}={\frac {\partial g(\mu )}{\partial x_{i}}},\quad g_{,ij}={\frac {\partial g^{2}(\mu )}{\partial x_{i}\,\partial x_{j}}},\quad \mu _{i,j}=E[(x_{i}-\mu _{i})^{j}]}$

In the following it is assumed that the entries are independent of. If the terms of the second order are also taken into account in the Taylor series, then the approximation for the expected value is ${\displaystyle X}$

${\displaystyle \mu _{g}\approx g_{\mu }+{\frac {1}{2}}\sum _{i=1}^{n}g_{,ii}\;\mu _{i,2}}$

The second order approximation of the variance is given by

${\displaystyle {\begin{aligned}\sigma _{g}^{2}&\approx g_{\mu }^{2}+\sum _{i=1}^{n}g_{,i}^{2}\,\mu _{i,2}+{\frac {1}{4}}\sum _{i=1}^{n}g_{,ii}^{2}\,\mu _{i,4}+g_{\mu }\sum _{i=1}^{n}g_{,ii}\,\mu _{i,2}+\sum _{i=1}^{n}g_{,i}\,g_{,ii}\,\mu _{i,3}\\&{}\quad {}+{\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=i+1}^{n}g_{,ii}\,g_{,jj}\,\mu _{i,2}\,\mu _{j,2}+\sum _{i=1}^{n}\sum _{j=i+1}^{n}g_{,ij}^{2}\,\mu _{i,2}\,\mu _{j,2}-\mu _{g}^{2}\end{aligned}}}$

The skewness of can be determined from the third central moment . If only linear terms of the Taylor series are taken into account, but higher moments of the input variables, then the third central moment results approximately to ${\displaystyle g}$ ${\displaystyle \mu _{g,3}}$

${\displaystyle \mu _{g,3}\approx \sum _{i=1}^{n}g_{,i}^{3}\;\mu _{i,3}}$

For the second order approximation of the third central moment and for the derivation of all higher order approximations, reference is made to Appendix D of Ref. If the quadratic terms of the Taylor series and the third-order moments of the random vector are taken into account, this is also known as the second-order third-moment method. However, the full second order approximation of the variance also includes fourth order moments, and the full second order approximation of the skewness includes sixth order moments.

Practical use

Various examples can be found in the literature in which the FOSM method is used to determine the stochastic distribution of the buckling load of axially loaded structures (see e.g. Ref.). For structures that are very sensitive to deviations from the ideal structure (such as circular cylindrical shells), it was proposed to use the FOSM method as a design method. The applicability is often checked by comparing it with Monte Carlo simulations . In engineering applications, the objective function is often not an analytical function, but is, for example, the result of a finite element simulation. In this case, the derivatives can be approximated using central differences. The objective function must therefore be evaluated. Depending on the number of random variables, this can be a significantly lower number of evaluations than is necessary for a Monte Carlo simulation. A lower assessment limit must be determined as part of a assessment procedure, but this does not result directly from the FOSM method. Therefore, a distribution type has to be selected for the objective function, taking into account the determined expected value, the variance and the skewness. ${\displaystyle 2n+1}$

literature

1. ^ A. Haldar and S. Mahadevan, Probability, Reliability, and Statistical Methods in Engineering Design. John Wiley & Sons New York / Chichester, UK, 2000.
2. ↑ ^{a b} B. Kriegesmann, "Probabilistic Design of Thin-Walled Fiber Composite Structures", reports from the Institute for Statics and Dynamics of Leibniz University Hannover 15/2012, ISSN  1862-4650 , Gottfried Wilhelm Leibniz University Hannover, Hannover, Germany, 2012 , PDF; 10.2MB .
3. ^ YJ Hong, J. Xing, and JB Wang, "A Second-Order Third-Moment Method for Calculating the Reliability of Fatigue," Int. J. Press. Vessels Pip., 76 (8), pp 567-570, 1999.
4. ↑ I. Elishakoff, S. van Manen, PG Vermeulen, and J. Arbocz, "First-Order Second Moment Analysis of the Buckling of Shells with Random Imperfections", AIAA J., 25 (8), pp 1113-1117, 1987.
5. ↑ I. Elishakoff, "Uncertain Buckling: Its Past, Present and Future", Int. J. Solids Struct., 37 (46-47), pp 6869-6889, Nov. 2000.
6. ↑ J. Arbocz and MW Hilburger, "Toward a Probabilistic Preliminary Design Criterion for Buckling Critical Composite Shells", AIAA J., 43 (8), pp 1823-1827., 2005
7. ^ B. Kriegesmann, R. Rolfes, C. Huehne, and A. Kling, "Fast Probabilistic Design Procedure for Axially Compressed Composite Cylinders", Compos. Struct., 93, pp 3140-3149, 2011.