Global Sensitivity Analysis

Global Sensitivity Analysis is activated via the keyword variance_based_decomp in the respective method. The two main methods in Dakota to assess global sensitivity analysis via the variance are with sampling or with PCEs. Sampling requires many more samples to assess the sensitivity so it is almost always best done with a surrogate. PCEs can be built in numerous ways either from existing data or by directly choosing the evaluation points.

Test Function

For all of these examples, we will use the so-called Ishigami Function:

f(x,y,z) = sin(x) + a*sin^2(y) + b*z**4*sin(x)

with a = 7 and b = 0.1 on [-pi,pi]^3

The Sobol Index can be solved analytically in terms of a, and b as presented below:

Index	Analytical Form	a = 7 & b = 0.1
Sx	(pi^8*b^2/50 + pi^4*b/5 + 1/2)/(a^2/8 + pi^8*b^2/18 + pi^4*b/5 + 1/2)	0.313905191147811
Sy	a^2/(8*(a^2/8 + pi^8*b^2/18 + pi^4*b/5 + 1/2))	0.442411144790041
Sz	0	0
Tx	-a^2/(8*(a^2/8 + pi^8*b^2/18 + pi^4*b/5 + 1/2)) + 1	0.557588855209959
Ty	-(pi^8*b^2/18 + pi^4*b/5 + 1/2)/(a^2/8 + pi^8*b^2/18 + pi^4*b/5 + 1/2) + 1	0.442411144790041
Tz	-(a^2/8 + pi^8*b^2/50 + pi^4*b/5 + 1/2)/(a^2/8 + pi^8*b^2/18 + pi^4*b/5 + 1/2) + 1	0.243683664062148

Or, in terms of contribution to the variance:

Index	Analytical Form	a = 7 & b = 0.1
x	(pi**8*b**2/50 + pi**4*b/5 + 1/2)/(a**2/8 + pi**8*b**2/18 + pi**4*b/5 + 1/2))	0.313905191147811
y	a**2/(8*(a**2/8 + pi**8*b**2/18 + pi**4*b/5 + 1/2)))	0.442411144790041
z	0	0
xy	0	0
yz	0	0
zx	8*pi**8*b**2/(225*(a**2/8 + pi**8*b**2/18 + pi**4*b/5 + 1/2)))	0.243683664062148
xyz	0	0

Or, as Dakota would format it (if calculated analytically):

                                  Main             Total
                      3.1390519115e-01  5.5758885521e-01 x
                      4.4241114480e-01  4.4241114480e-01 y
                      0.0000000000e-00  2.4368366406e-01 z
                           Interaction
                      0.0000000000e-00 x y
                      2.4377949556e-01 x z
                      0.0000000000e-00 y z
                      0.0000000000e-00 x y z

Run Dakota

See each sub example

Math Equation

Without going into too much detail, the "Main" sensitivity of a direction $d$ is

$$ S_{{d}} = \frac{ \mathbb{V} \left ( \mathbb{E} (f|x_{\{d\}}) \right ) }{ \mathbb{V}(f) } $$

And the "Total" sensitivity is

$$ T_{{d}} = 1- \frac{ \mathbb{V} \left ( \mathbb{E} (f|x_{\{\sim d\}}) \right ) }{ \mathbb{V}(f) } $$

where $x_{{\sim d}}$ is all direction but $\{d\}$

What method will we use?

Each sub example uses a different method. See each example separately

Analysis Driver

Inputs

Outputs

Interpret the results