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

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

Approximation of Mean

$$\begin{array}{rl}\mathbb{E}[\mathcal{M}(\bar{X})+{\mathcal{M}}^{\prime }(\bar{X})\tilde{X}]& =\mathbb{E}[\mathcal{M}(\bar{X})]+\mathbb{E}[{\mathcal{M}}^{\prime }(\bar{X})\tilde{X}]\\ & =\mathcal{M}(\bar{X})+{\mathcal{M}}^{\prime }(\bar{X})\text{cancelto}0\mathbb{E}[\tilde{X}]\end{array}$$ $$\boxed{\mathbb{E}[\mathcal{M}(X)]\approx \mathcal{M}(\bar{X})}$$

This shows that evaluating the model at the mean input is an approximation of the mean output.

Approximation of Variance

For the variance, we proceed similarly by computing the variance of the Taylor expansion:

$$\begin{array}{rl}V[\mathcal{M}(X)]& \approx V\left[\mathcal{M}(\bar{X})+{\mathcal{M}}^{\prime }(\bar{X})\tilde{X}\right]\\ & =V\left[{\mathcal{M}}^{\prime }(\bar{X})\tilde{X}\right]\\ & ={\left({\mathcal{M}}^{\prime }(\bar{X})\right)}^{2}V[\tilde{X}]\end{array}$$ $$\boxed{V[\mathcal{M}(X)]\approx {\left({\mathcal{M}}^{\prime }(\bar{X})\right)}^{2}V[\tilde{X}]}$$

Where $\mathbb{E}[\widetilde{X^2}]=\mathbb{V}[\tilde{X}]$

This result shows that the output variance is approximately the product of the squared sensitivity (derivative) at the mean and the input variance, providing a simple way to propagate uncertainty through models.