<s>
Type-2	B-General_Concept
fuzzy	I-General_Concept
sets	I-General_Concept
and	I-General_Concept
systems	I-General_Concept
generalize	O
standard	O
Type-1	O
fuzzy	B-General_Concept
sets	I-General_Concept
and	I-General_Concept
systems	I-General_Concept
so	O
that	O
more	O
uncertainty	O
can	O
be	O
handled	O
.	O
</s>
<s>
From	O
the	O
beginning	O
of	O
fuzzy	O
sets	O
,	O
criticism	O
was	O
made	O
about	O
the	O
fact	O
that	O
the	O
membership	B-General_Concept
function	I-General_Concept
of	O
a	O
type-1	O
fuzzy	O
set	O
has	O
no	O
uncertainty	O
associated	O
with	O
it	O
,	O
something	O
that	O
seems	O
to	O
contradict	O
the	O
word	O
fuzzy	O
,	O
since	O
that	O
word	O
has	O
the	O
connotation	O
of	O
much	O
uncertainty	O
.	O
</s>
<s>
So	O
,	O
what	O
does	O
one	O
do	O
when	O
there	O
is	O
uncertainty	O
about	O
the	O
value	O
of	O
the	O
membership	B-General_Concept
function	I-General_Concept
?	O
</s>
<s>
A	O
type-2	O
fuzzy	O
set	O
lets	O
us	O
incorporate	O
uncertainty	O
about	O
the	O
membership	B-General_Concept
function	I-General_Concept
into	O
fuzzy	O
set	O
theory	O
,	O
and	O
is	O
a	O
way	O
to	O
address	O
the	O
above	O
criticism	O
of	O
type-1	O
fuzzy	O
sets	O
head-on	O
.	O
</s>
<s>
Type1	O
fuzzy	B-General_Concept
systems	I-General_Concept
are	O
working	O
with	O
a	O
fixed	O
membership	B-General_Concept
function	I-General_Concept
,	O
while	O
in	O
type-2	O
fuzzy	B-General_Concept
systems	I-General_Concept
the	O
membership	B-General_Concept
function	I-General_Concept
is	O
fluctuating	O
.	O
</s>
<s>
The	O
membership	B-General_Concept
function	I-General_Concept
of	O
a	O
general	O
type-2	O
fuzzy	O
set	O
,	O
Ã	O
,	O
is	O
three-dimensional	O
(	O
Fig	O
.	O
</s>
<s>
1	O
)	O
,	O
where	O
the	O
third	O
dimension	O
is	O
the	O
value	O
of	O
the	O
membership	B-General_Concept
function	I-General_Concept
at	O
each	O
point	O
on	O
its	O
two-dimensional	O
domain	O
that	O
is	O
called	O
its	O
"	O
footprint	O
of	O
uncertainty	O
"	O
( FOU	O
)	O
.	O
</s>
<s>
The	O
FOU	O
represents	O
the	O
blurring	O
of	O
a	O
type-1	O
membership	B-General_Concept
function	I-General_Concept
,	O
and	O
is	O
completely	O
described	O
by	O
its	O
two	O
bounding	O
functions	O
(	O
Fig	O
.	O
</s>
<s>
2	O
)	O
,	O
a	O
lower	O
membership	B-General_Concept
function	I-General_Concept
(	O
LMF	O
)	O
and	O
an	O
upper	O
membership	B-General_Concept
function	I-General_Concept
(	O
UMF	O
)	O
,	O
both	O
of	O
which	O
are	O
type-1	O
fuzzy	O
sets	O
!	O
</s>
<s>
This	O
changed	O
in	O
the	O
latter	O
part	O
of	O
the	O
1990s	O
as	O
a	O
result	O
of	O
Jerry	O
Mendel	O
and	O
his	O
student	O
's	O
works	O
on	O
type-2	B-General_Concept
fuzzy	I-General_Concept
sets	I-General_Concept
and	I-General_Concept
systems	I-General_Concept
.	O
</s>
<s>
Since	O
then	O
,	O
more	O
and	O
more	O
researchers	O
around	O
the	O
world	O
are	O
writing	O
articles	O
about	O
type-2	B-General_Concept
fuzzy	I-General_Concept
sets	I-General_Concept
and	I-General_Concept
systems	I-General_Concept
.	O
</s>
<s>
Interval	O
type-2	O
fuzzy	O
sets	O
have	O
received	O
the	O
most	O
attention	O
because	O
the	O
mathematics	O
that	O
is	O
needed	O
for	O
such	O
sets	O
—	O
primarily	O
Interval	B-Algorithm
arithmetic	I-Algorithm
—	O
is	O
much	O
simpler	O
than	O
the	O
mathematics	O
that	O
is	O
needed	O
for	O
general	O
type-2	O
fuzzy	O
sets	O
.	O
</s>
<s>
Type-2	O
fuzzy	O
sets	O
are	O
finding	O
very	O
wide	O
applicability	O
in	O
rule-based	O
fuzzy	O
logic	O
systems	B-General_Concept
(	O
FLSs	O
)	O
because	O
they	O
let	O
uncertainties	O
be	O
modeled	O
by	O
them	O
whereas	O
such	O
uncertainties	O
cannot	O
be	O
modeled	O
by	O
type-1	O
fuzzy	O
sets	O
.	O
</s>
<s>
Membership	B-General_Concept
functions	I-General_Concept
are	O
used	O
to	O
describe	O
these	O
fuzzy	O
sets	O
,	O
and	O
in	O
a	O
type-1	O
FLS	O
they	O
are	O
all	O
type-1	O
fuzzy	O
sets	O
,	O
whereas	O
in	O
an	O
interval	O
type-2	O
FLS	O
at	O
least	O
one	O
membership	B-General_Concept
function	I-General_Concept
is	O
an	O
interval	O
type-2	O
fuzzy	O
set	O
.	O
</s>
<s>
Membership	B-General_Concept
function	I-General_Concept
parameters	O
—	O
because	O
when	O
those	O
parameters	O
are	O
optimized	O
using	O
uncertain	O
(	O
noisy	O
)	O
training	O
data	O
,	O
the	O
parameters	O
become	O
uncertain	O
.	O
</s>
<s>
In	O
a	O
type-1	O
FLS	O
,	O
output	O
processing	O
,	O
called	O
"	O
defuzzification	B-General_Concept
"	O
,	O
maps	O
a	O
type-1	O
fuzzy	O
set	O
into	O
a	O
number	O
.	O
</s>
<s>
There	O
are	O
many	O
ways	O
for	O
doing	O
this	O
,	O
e.g.	O
,	O
compute	O
the	O
union	O
of	O
the	O
fired-rule	O
output	O
fuzzy	O
sets	O
(	O
the	O
result	O
is	O
another	O
type-1	O
fuzzy	O
set	O
)	O
and	O
then	O
compute	O
the	O
center	O
of	O
gravity	O
of	O
the	O
membership	B-General_Concept
function	I-General_Concept
for	O
that	O
set	O
;	O
compute	O
a	O
weighted	O
average	O
of	O
the	O
centers	O
of	O
gravity	O
of	O
each	O
of	O
the	O
fired	O
rule	O
consequent	O
membership	B-General_Concept
functions	I-General_Concept
;	O
etc	O
.	O
</s>
<s>
There	O
are	O
as	O
many	O
type-reduction	O
methods	O
as	O
there	O
are	O
type-1	O
defuzzification	B-General_Concept
methods	O
.	O
</s>
<s>
The	O
second	O
step	O
of	O
Output	O
Processing	O
,	O
which	O
occurs	O
after	O
type-reduction	O
,	O
is	O
still	O
called	O
"	O
defuzzification	B-General_Concept
"	O
.	O
</s>
<s>
Freeware	O
MATLAB	O
implementations	O
,	O
which	O
cover	O
general	O
and	O
interval	O
type-2	B-General_Concept
fuzzy	I-General_Concept
sets	I-General_Concept
and	I-General_Concept
systems	I-General_Concept
,	O
as	O
well	O
as	O
type-1	O
fuzzy	B-General_Concept
systems	I-General_Concept
,	O
are	O
available	O
at	O
:	O
.	O
</s>
<s>
Software	O
supporting	O
discrete	O
interval	O
type-2	O
fuzzy	O
logic	O
systems	B-General_Concept
is	O
available	O
at	O
:	O
</s>
<s>
Java	O
libraries	O
including	O
source	O
code	O
for	O
type-1	O
,	O
interval	O
-	O
and	O
general	O
type-2	O
fuzzy	B-General_Concept
systems	I-General_Concept
are	O
available	O
at	O
:	O
.	O
</s>
<s>
Python	O
library	O
for	O
interval	O
type	O
2	O
fuzzy	B-General_Concept
sets	I-General_Concept
and	I-General_Concept
systems	I-General_Concept
is	O
available	O
at	O
:	O
</s>
<s>
An	O
open	O
source	O
Matlab/Simulink	O
Toolbox	O
for	O
Interval	O
Type-2	O
Fuzzy	O
Logic	O
Systems	B-General_Concept
is	O
available	O
at	O
:	O
</s>
