First Order Second Moment (FOSM)

Use Case

FOSM stands for First-Order Second Moment. It is a technique used to express how uncertainty in a model’s inputs affects the uncertainty in its outputs

It is called “First-Order” because it creates a linear (first-order) approximation of the model and “Second-Moment” because it focuses on the second statistical moment (variance) of the model output.

Mathematical Framework

• A random variable in terms of deterministic and stochastic components is given by

  $$X=\bar{X}+\tilde{X}$$

  Here we assume that $E[X]=\bar{X}$ and $E[\tilde{X}]=0$

• Model $\mathcal{M}(X)$ can be approximated using first order Taylor expansion around mean

  $$\mathcal{M}(X)=\mathcal{M}(\bar{X}+\tilde{X})\approx \mathcal{M}(\bar{X})+{\mathcal{M}}^{\prime }(\bar{X})\tilde{X}$$

Note

It’s not a good idea to use FOSM if the model is highly non-linear. Because:

• FOSM is modeled around the mean. 
• It requires only the model’s output and it’s derivatives at a single point (the mean).