View source for First-order second-moment method

{{redirect|FOSM|open geographic database project|Free Open Street Map}}
In probability theory, the '''first-order second-moment (FOSM) method''', also referenced as '''mean value first-order second-moment (MVFOSM) method''', is a probabilistic method to determine the stochastic moments of a function with random input variables. The name is based on the derivation, which uses a ''first-order'' [[Taylor series]] and the first and ''second moments'' of the input variables.<ref>A. Haldar and S. Mahadevan, Probability, Reliability, and Statistical Methods in Engineering Design. John Wiley & Sons New York/Chichester, UK, 2000.</ref>

== Approximation ==
Consider the objective function <math>g(x)</math>, where the input vector <math>x</math> is a realization of the random vector <math>X</math> with [[probability density function]] <math>f_X(x)</math>. Because <math>X</math> is randomly distributed, <math>g</math> is also randomly distributed. 
Following the FOSM method, the [[mean value]] of <math>g</math> is approximated by

:<math> \mu_g \approx g(\mu) </math>

The [[variance]] of <math>g</math> is approximated by

:<math> \sigma^2_g \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}\left(X_i, X_j\right) </math>

where <math>n</math> is the length/dimension of <math>x</math> and <math display="inline"> \frac{\partial g( \mu)}{\partial x_i} </math> is the partial derivative of <math>g</math> at the mean vector <math>\mu</math> with respect to the ''i''-th entry of <math>x</math>. More accurate, second-order second-moment approximations are also available <ref>{{cite journal|last1=Crespo|first1=L. G.|last2=Kenny|first2=S. P.|title=A first and second order moment approach to probabilistic control synthesis|journal={AIAA} Guidance Navigation and Control conference|date=2005|hdl=2060/20050232742|hdl-access=free}}</ref>

== Derivation ==
The objective function is approximated by a [[Taylor series]] at the mean vector <math>\mu</math>.

:<math> 
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
</math>

The mean value of <math>g</math> is given by the integral
:<math>\mu_g = E[ g(x)] = \int_{-\infty}^\infty g(x) f_X(x)\, dx</math>

Inserting the first-order Taylor series yields
:<math>\begin{align}

\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{align}</math>

The variance of <math>g</math> is given by the integral
:<math>
\sigma_g^2  = E\left([g(x) - \mu_g]^2\right) = \int_{-\infty}^\infty [g(x) - \mu_g]^2 f_X(x) \, dx.
</math>

According to the computational formula for the variance, this can be written as
:<math>
\sigma _g^2
= E\left([g(x) - \mu_g]^2\right) = E\left(g(x)^2\right) - \mu_g^2
= \int_{-\infty}^\infty g(x)^2 f_X(x) \, dx - \mu_g^2
</math>

Inserting the Taylor series yields
:<math>\begin{align}
\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 + 2 g_\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 + 2 g_\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 (x) \, dx }_{\operatorname{cov}\left(X_i, X_j\right)} - \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}\left(X_i, X_j\right).
\end{align}</math>

== Higher-order approaches ==
The following abbreviations are introduced.

:<math>\begin{align}
       g_\mu &= g(\mu), &
      g_{,i} &= \frac{\partial g(\mu)}{\partial x_i}, &
     g_{,ij} &= \frac{\partial^2 g (\mu)}{\partial x_i \, \partial x_j }, &
  \mu _{i,j} &= E\left[ (x_i  - \mu _i)^j \right]
\end{align}</math>

In the following, the entries of the random vector <math>X</math> are assumed to be independent. 
Considering also the second-order terms of the Taylor expansion, the approximation of the mean value is given by
:<math>
\mu _g \approx g_\mu  + \frac{1}{2}\sum_{i = 1}^n g_{,ii} \;\mu _{i,2}
</math>

The second-order approximation of the variance is given by
:<math>\begin{align}
  \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} \\ 
          &+ \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{align}</math>

The [[skewness]] of <math>g</math> can be determined from the third [[central moment]] <math>\mu_{g,3} </math>. When considering only linear terms of the Taylor series, but higher-order moments, the third central moment is approximated by 
:<math>
\mu _{g,3} \approx \sum_{i = 1}^n g_{,i}^3 \;\mu _{i,3}
</math>

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.<ref name="BK">B. Kriegesmann, "Probabilistic Design of Thin-Walled Fiber Composite Structures", Mitteilungen des Instituts für Statik und Dynamik der Leibniz Universität Hannover 15/2012, {{ISSN|1862-4650}}, Gottfried Wilhelm Leibniz Universität Hannover, Hannover, Germany, 2012, [http://edok01.tib.uni-hannover.de/edoks/e01dh12/722293151.pdf PDF; 10,2MB].</ref> 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.<ref>Y. J. Hong, J. Xing, and J. B. Wang, "A Second-Order Third-Moment Method for Calculating the Reliability of Fatigue", Int. J. Press. Vessels Pip., 76 (8), pp 567–570, 1999.</ref> However, the full second-order approach of the variance (given above) also includes fourth-order moments of input parameters,<ref name="MC">Mallor C, Calvo S, Núñez JL, Rodríguez-Barrachina R, Landaberea A. "Full second-order approach for expected value and variance prediction of probabilistic fatigue crack growth life." International Journal of Fatigue 2020;133:105454. https://doi.org/10.1016/j.ijfatigue.2019.105454.</ref> the full second-order approach of the skewness 6th-order moments,<ref name="BK" /><ref name="MC2">Mallor C, Calvo S, Núñez JL, Rodríguez-Barrachina R, Landaberea A. "Uncertainty propagation using the full second-order approach for probabilistic fatigue crack growth life." International Journal of Numerical Methods for Calculation and Design in Engineering (RIMNI) 2020:11. https://doi.org/10.23967/j.rimni.2020.07.004.</ref> and the full second-order approach of the kurtosis up to 8th-order moments.<ref name="MC2" />

== Practical application ==
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.<ref>I. Elishakoff, S. van Manen, P. G. 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.</ref><ref>I. Elishakoff, "Uncertain Buckling: Its Past, Present and Future", Int. J. Solids Struct., 37 (46–47), pp 6869–6889, Nov. 2000.</ref><ref>J. Arbocz and M. W. Hilburger, "Toward a Probabilistic Preliminary Design Criterion for Buckling Critical Composite Shells", AIAA J., 43 (8), pp 1823–1827, 2005.</ref><ref>B. Kriegesmann, R. Rolfes, C. Hühne, and A. Kling, "Fast Probabilistic Design Procedure for Axially Compressed Composite Cylinders", Compos. Struct., 93, pp 3140–3149, 2011.</ref>). 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.<ref name="MC" /><ref name="MC2" />

In engineering practice, the objective function often is not given as analytic expression, but for instance as a result of a [[Finite element method|finite-element]] simulation. Then the derivatives of the objective function need to be estimated by the [[Finite difference|central differences]] method.  The number of evaluations of the objective function equals <math> 2 n + 1</math>. 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.

== References ==
{{Reflist}}

[[Category:Probabilistic models]]
[[Category:Randomized algorithms]]