<s>
Variance-based	B-General_Concept
sensitivity	I-General_Concept
analysis	I-General_Concept
(	O
often	O
referred	O
to	O
as	O
the	O
Sobol’	O
method	O
or	O
Sobol’	O
indices	O
,	O
after	O
Ilya	O
M	O
.	O
Sobol’	O
)	O
is	O
a	O
form	O
of	O
global	O
sensitivity	O
analysis	O
.	O
</s>
<s>
From	O
a	O
black	B-Device
box	I-Device
perspective	O
,	O
any	O
model	O
may	O
be	O
viewed	O
as	O
a	O
function	O
Y	O
=	O
f(X )	O
,	O
where	O
X	O
is	O
a	O
vector	O
of	O
d	O
uncertain	O
model	O
inputs	O
{	O
X1	O
,	O
X2	O
,	O
...	O
Xd}	O
,	O
and	O
Y	O
is	O
a	O
chosen	O
univariate	O
model	O
output	O
(	O
note	O
that	O
this	O
approach	O
examines	O
scalar	O
model	O
outputs	O
,	O
but	O
multiple	O
outputs	O
can	O
be	O
analysed	O
by	O
multiple	O
independent	O
sensitivity	O
analyses	O
)	O
.	O
</s>
<s>
all	O
the	O
terms	O
in	O
the	O
functional	O
decomposition	O
are	O
orthogonal	B-Application
.	O
</s>
<s>
From	O
which	O
it	O
can	O
be	O
seen	O
that	O
fi	O
is	O
the	O
effect	O
of	O
varying	O
Xi	O
alone	O
(	O
known	O
as	O
the	O
main	B-General_Concept
effect	I-General_Concept
of	O
Xi	O
)	O
,	O
and	O
fij	O
is	O
the	O
effect	O
of	O
varying	O
Xi	O
and	O
Xj	O
simultaneously	O
,	O
additional	O
to	O
the	O
effect	O
of	O
their	O
individual	O
variations	O
.	O
</s>
<s>
Now	O
,	O
further	O
assuming	O
that	O
the	O
f(X )	O
is	O
square-integrable	B-Algorithm
,	O
the	O
functional	O
decomposition	O
may	O
be	O
squared	O
and	O
integrated	O
to	O
give	O
,	O
</s>
<s>
A	O
direct	O
variance-based	O
measure	O
of	O
sensitivity	O
Si	O
,	O
called	O
the	O
"	O
first-order	O
sensitivity	O
index	O
"	O
,	O
or	O
"	O
main	B-General_Concept
effect	I-General_Concept
index	O
"	O
is	O
stated	O
as	O
follows	O
,	O
</s>
<s>
This	O
is	O
the	O
contribution	O
to	O
the	O
output	O
variance	O
of	O
the	O
main	B-General_Concept
effect	I-General_Concept
of	O
Xi	O
,	O
therefore	O
it	O
measures	O
the	O
effect	O
of	O
varying	O
Xi	O
alone	O
,	O
but	O
averaged	O
over	O
variations	O
in	O
other	O
input	O
parameters	O
.	O
</s>
<s>
However	O
,	O
in	O
the	O
vast	O
majority	O
of	O
cases	O
they	O
are	O
estimated	O
–	O
this	O
is	O
usually	O
done	O
by	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
Monte	O
Carlo	O
approach	O
involves	O
generating	O
a	O
sequence	O
of	O
randomly	O
distributed	O
points	O
inside	O
the	O
unit	O
hypercube	O
(	O
strictly	O
speaking	O
these	O
will	O
be	O
pseudorandom	B-Algorithm
)	O
.	O
</s>
<s>
This	O
is	O
then	O
known	O
as	O
the	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
To	O
calculate	O
the	O
indices	O
using	O
the	O
(	O
quasi	O
)	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
,	O
the	O
following	O
steps	O
are	O
used	O
:	O
</s>
