<s>
In	O
statistics	O
,	O
propagation	B-General_Concept
of	I-General_Concept
uncertainty	I-General_Concept
(	O
or	O
propagation	B-General_Concept
of	I-General_Concept
error	I-General_Concept
)	O
is	O
the	O
effect	O
of	O
variables	O
 '	O
uncertainties	O
(	O
or	O
errors	O
,	O
more	O
specifically	O
random	O
errors	O
)	O
on	O
the	O
uncertainty	O
of	O
a	O
function	O
based	O
on	O
them	O
.	O
</s>
<s>
It	O
may	O
be	O
defined	O
by	O
the	O
absolute	B-Algorithm
error	I-Algorithm
.	O
</s>
<s>
Uncertainties	O
can	O
also	O
be	O
defined	O
by	O
the	O
relative	B-Algorithm
error	I-Algorithm
,	O
which	O
is	O
usually	O
written	O
as	O
a	O
percentage	O
.	O
</s>
<s>
Most	O
commonly	O
,	O
the	O
uncertainty	O
on	O
a	O
quantity	O
is	O
quantified	O
in	O
terms	O
of	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
,	O
which	O
is	O
the	O
positive	O
square	O
root	O
of	O
the	O
variance	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
68%	O
confidence	O
limits	O
for	O
a	O
one-dimensional	O
variable	O
belonging	O
to	O
a	O
normal	O
distribution	O
are	O
approximately	O
±	O
one	O
standard	B-General_Concept
deviation	I-General_Concept
from	O
the	O
central	O
value	O
,	O
which	O
means	O
that	O
the	O
region	O
will	O
cover	O
the	O
true	O
value	O
in	O
roughly	O
68%	O
of	O
cases	O
.	O
</s>
<s>
In	O
a	O
general	O
context	O
where	O
a	O
nonlinear	O
function	O
modifies	O
the	O
uncertain	O
parameters	O
(	O
correlated	O
or	O
not	O
)	O
,	O
the	O
standard	O
tools	O
to	O
propagate	O
uncertainty	O
,	O
and	O
infer	O
resulting	O
quantity	O
probability	O
distribution/statistics	O
,	O
are	O
sampling	O
techniques	O
from	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
family	O
.	O
</s>
<s>
For	O
very	O
expensive	O
data	O
or	O
complex	O
functions	O
,	O
the	O
calculation	O
of	O
the	O
error	B-General_Concept
propagation	I-General_Concept
may	O
be	O
very	O
expensive	O
so	O
that	O
a	O
surrogate	B-Algorithm
model	I-Algorithm
or	O
a	O
parallel	B-Operating_System
computing	I-Operating_System
strategy	O
may	O
be	O
necessary	O
.	O
</s>
<s>
In	O
some	O
particular	O
cases	O
,	O
the	O
uncertainty	B-General_Concept
propagation	I-General_Concept
calculation	O
can	O
be	O
done	O
through	O
simplistic	O
algebraic	O
procedures	O
.	O
</s>
<s>
This	O
is	O
the	O
most	O
general	O
expression	O
for	O
the	O
propagation	B-General_Concept
of	I-General_Concept
error	I-General_Concept
from	O
one	O
set	O
of	O
variables	O
onto	O
another	O
.	O
</s>
<s>
Since	O
f0	O
is	O
a	O
constant	O
it	O
does	O
not	O
contribute	O
to	O
the	O
error	O
on	O
f	O
.	O
Therefore	O
,	O
the	O
propagation	B-General_Concept
of	I-General_Concept
error	I-General_Concept
follows	O
the	O
linear	O
case	O
,	O
above	O
,	O
but	O
replacing	O
the	O
linear	O
coefficients	O
,	O
Aki	O
and	O
Akj	O
by	O
the	O
partial	O
derivatives	O
,	O
and	O
.	O
</s>
<s>
Neglecting	O
correlations	O
or	O
assuming	O
independent	O
variables	O
yields	O
a	O
common	O
formula	O
among	O
engineers	O
and	O
experimental	O
scientists	O
to	O
calculate	O
error	B-General_Concept
propagation	I-General_Concept
,	O
the	O
variance	O
formula	O
:	O
</s>
<s>
where	O
represents	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
function	O
,	O
represents	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
,	O
represents	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
,	O
and	O
so	O
forth	O
.	O
</s>
<s>
It	O
is	O
important	O
to	O
note	O
that	O
this	O
formula	O
is	O
based	O
on	O
the	O
linear	O
characteristics	O
of	O
the	O
gradient	O
of	O
and	O
therefore	O
it	O
is	O
a	O
good	O
estimation	O
for	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
as	O
long	O
as	O
are	O
small	O
enough	O
.	O
</s>
<s>
where	O
is	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
function	O
,	O
is	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
,	O
is	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
and	O
is	O
the	O
covariance	O
between	O
and	O
.	O
</s>
<s>
For	O
highly	O
non-linear	O
functions	O
,	O
there	O
exist	O
five	O
categories	O
of	O
probabilistic	O
approaches	O
for	O
uncertainty	B-General_Concept
propagation	I-General_Concept
;	O
see	O
Uncertainty	O
quantification	O
for	O
details	O
.	O
</s>
<s>
This	O
table	O
shows	O
the	O
variances	O
and	O
standard	B-General_Concept
deviations	I-General_Concept
of	O
simple	O
functions	O
of	O
the	O
real	O
variables	O
,	O
with	O
standard	B-General_Concept
deviations	I-General_Concept
covariance	O
,	O
and	O
correlation	O
.	O
</s>
<s>
In	O
the	O
columns	O
"	O
Variance	O
"	O
and	O
"	O
Standard	B-General_Concept
Deviation	I-General_Concept
"	O
,	O
A	O
and	O
B	O
should	O
be	O
understood	O
as	O
expectation	O
values	O
(	O
i.e.	O
</s>
<s>
An	O
increasing	O
positive	O
correlation	O
(	O
)	O
will	O
decrease	O
the	O
variance	O
of	O
the	O
difference	O
,	O
converging	O
to	O
zero	O
variance	O
for	O
perfectly	O
correlated	O
variables	O
with	O
the	O
same	B-General_Concept
variance	I-General_Concept
.	O
</s>
<s>
We	O
can	O
calculate	O
the	O
uncertainty	B-General_Concept
propagation	I-General_Concept
for	O
the	O
inverse	O
tangent	O
function	O
as	O
an	O
example	O
of	O
using	O
partial	O
derivatives	O
to	O
propagate	O
error	O
.	O
</s>
<s>
where	O
is	O
the	O
absolute	B-Algorithm
uncertainty	I-Algorithm
on	O
our	O
measurement	O
of	O
.	O
</s>
