First-order second-moment method

Consider the objective function ${\displaystyle g(x)}$, where the input vector ${\displaystyle x}$ is a realization of the random vector ${\displaystyle X}$ with probability density function ${\displaystyle f_{X}(x)}$. As ${\displaystyle X}$ is randomly distributed, also ${\displaystyle g}$ is randomly distributed. Following the FOSM method, the mean value of ${\displaystyle g}$ is approximated by

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

The variance of ${\displaystyle g}$ is approximated by

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

where ${\displaystyle n}$ is the length/dimension of ${\displaystyle x}$ and ${\displaystyle {\frac {\partial g(\mu )}{\partial x_{i}}}}$ is the partial derivative of ${\displaystyle g}$ at the mean vector ${\displaystyle \mu }$ with respect to the i-th entry of ${\displaystyle x}$. More accurate, second-order second-moment approximations are also available [2]

The objective function is approximated by a Taylor series at the mean 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 mean value of ${\displaystyle g}$ is given by the integral

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

Inserting the first-order Taylor series yields

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

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

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

According to the computational formula for the variance, this can be written as

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

Inserting the Taylor series yields

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

The following abbreviations are introduced.

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

In the following, the entries of the random vector ${\displaystyle X}$ are assumed to be independent. Considering also the second-order terms of the Taylor expansion, the approximation of the mean value is given by

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

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 ${\displaystyle g}$ can be determined from the third central moment ${\displaystyle \mu _{g,3}}$. When considering only linear terms of the Taylor series, but higher-order moments, the third central moment is approximated by

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

For the second-order approximations of the third central moment as well as for the derivation of all higher-order approximations see Appendix D of Ref.[3] Taking into account the quadratic terms of the Taylor series and the third moments of the input variables is referred to as second-order third-moment method.[4] However, the full second-order approach of the variance (given above) also includes fourth-order moments of input parameters,[5] the full second-order approach of the skewness 6th-order moments,[3][6] and the full second-order approach of the kurtosis up to 8th-order moments.[6]

There are several examples in the literature where the FOSM method is employed to estimate the stochastic distribution of the buckling load of axially compressed structures (see e.g. Ref.[7][8][9][10]). For structures which are very sensitive to deviations from the ideal structure (like cylindrical shells) it has been proposed to use the FOSM method as a design approach. Often the applicability is checked by comparison with a Monte Carlo simulation. Two comprehensive application examples of the full second-order method specifically oriented towards the fatigue crack growth in a metal railway axle are discussed and checked by comparison with a Monte Carlo simulation in Ref.[5][6]

In engineering practice, the objective function often is not given as analytic expression, but for instance as a result of a finite-element simulation. Then the derivatives of the objective function need to be estimated by the central differences method. The number of evaluations of the objective function equals ${\displaystyle 2n+1}$. Depending on the number of random variables this still can mean a significantly smaller number of evaluations than performing a Monte Carlo simulation. However, when using the FOSM method as a design procedure, a lower bound shall be estimated, which is actually not given by the FOSM approach. Therefore, a type of distribution needs to be assumed for the distribution of the objective function, taking into account the approximated mean value and standard deviation.