Introduction to Variance-based sensitivity analysis (VBSA)

The aim of sensitivity analysis is to examine which parameter in a model affects output most. In our model, there are 18 parameters (shown below) that have impact on GFP concentrations. The mathematical basis of variance-based sensitivity analysis is to estimate the effect of a parameter using variance. That is, how much portion of variance of GFP could the variance of a particular parameter explain?

Let $X = \{x_i\}|_{i=1}^{n}$ be $n$ parameters in out model, and that these parameters are in interval $[0,1]$ by transformation. It can be assumed that output $y = f(x_1, x_2, \ldots, x_n)$ can be decomposed as summation of several functions:

$$  f(X) = f_0 + \sum_{i=1}^{n} f_i(x_i) + \sum_{i<j}^n f_{i,j}(x_i,x_j) + \ldots  $$

By assumption of independence, one can derive:
$$ \int_0^1 f_{i_1,i_2,\ldots , i_s} (x_{i_1}, x_{i_2}, \ldots , x_{i_s})  dx_{i_k} = 0 \mbox{ for } k = 1 \mbox{ to } s $$
(where $\{i_a\}|_{a=1}^s$ is a sequence. For example, if $s=2, i_1=2, i_2=5$ then the former formula is equivalent to
$\int_0^1 f_{2,5} (x_{2}, x_{5})  dx_{2} = 0 \mbox{ and} \int_0^1 f_{2,5} (x_{2}, x_{5})  dx_{5} = 0 $）

This assumption implies orthogonality, which makes calculations of expected value much easier due to orthogonality:
$$ \begin{cases}
E(y) = f_0
\\
E(y|x_i) = f_0 + f_i
\\
E(y|x_i,x_j) = f_0 + f_i + f_j + f_{i,j} \mbox{ and so on}
\end{cases}$$

Therefore, the variance can be derived as follows:
$$ Var(y) = E(y^2) - (E(y))^2 = \int_{K^n} f^2 dX - f_0^2 = D  $$
Let D be abbreviation of integral defined below
$$ D_{i_1, i_2 \ldots , i_s} = \int_0^1\int_0^1\ldots\int_0^1 f_{i_1, i_2 \ldots , i_s}^2 dx_{i_1} \ldots dx_{i_s}. $$
Then the variance of output, denoted as $D,$ will be:
$$ D = \sum_{i=1}^n D_i + \sum_{i=1}^n\sum_{j=i+1}^n D_{i,j} + \ldots + D_{1,\ldots , n} $$
The index of how much portion of variance of GFP could the variance of a particular parameter explain is:
$$ S_i = \dfrac{D_i}{D} $$

Bibliography

1. Andrea Saltelli, Paola Annoni, Ivano Azzini, Francesca Campolongo, Marco Ratto, and Stefano Tarantola. Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Computer Physics Communications, 181(2):259{270, 2010
2. Karen Chan, Andrea Saltelli, Stefano Tarantola. SENSITIVITY ANALYSIS OF MODEL OUTPUT: VARIANCE-BASED METHODS MAKE THE DIFFERENCE. Proceedings of the 1997 Winter Simulation Conference (http://www.informs-sim.org/wsc97papers/0261.PDF)