<s>
A	O
sigmoid	B-Algorithm
function	I-Algorithm
is	O
a	O
mathematical	O
function	O
having	O
a	O
characteristic	O
"	O
S	O
"	O
-shaped	O
curve	O
or	O
sigmoid	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
A	O
common	O
example	O
of	O
a	O
sigmoid	B-Algorithm
function	I-Algorithm
is	O
the	O
logistic	O
function	O
shown	O
in	O
the	O
first	O
figure	O
and	O
defined	O
by	O
the	O
formula	O
:	O
</s>
<s>
Other	O
standard	O
sigmoid	B-Algorithm
functions	I-Algorithm
are	O
given	O
in	O
the	O
Examples	O
section	O
.	O
</s>
<s>
In	O
some	O
fields	O
,	O
most	O
notably	O
in	O
the	O
context	O
of	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
,	O
the	O
term	O
"	O
sigmoid	B-Algorithm
function	I-Algorithm
"	O
is	O
used	O
as	O
an	O
alias	O
for	O
the	O
logistic	O
function	O
.	O
</s>
<s>
Special	O
cases	O
of	O
the	O
sigmoid	B-Algorithm
function	I-Algorithm
include	O
the	O
Gompertz	O
curve	O
(	O
used	O
in	O
modeling	O
systems	O
that	O
saturate	O
at	O
large	O
values	O
of	O
x	O
)	O
and	O
the	O
ogee	O
curve	O
(	O
used	O
in	O
the	O
spillway	O
of	O
some	O
dams	O
)	O
.	O
</s>
<s>
Sigmoid	B-Algorithm
functions	I-Algorithm
have	O
domain	O
of	O
all	O
real	O
numbers	O
,	O
with	O
return	O
(	O
response	O
)	O
value	O
commonly	O
monotonically	O
increasing	O
but	O
could	O
be	O
decreasing	O
.	O
</s>
<s>
Sigmoid	B-Algorithm
functions	I-Algorithm
most	O
often	O
show	O
a	O
return	O
value	O
(	O
y	O
axis	O
)	O
in	O
the	O
range	O
0	O
to	O
1	O
.	O
</s>
<s>
A	O
wide	O
variety	O
of	O
sigmoid	B-Algorithm
functions	I-Algorithm
including	O
the	O
logistic	O
and	O
hyperbolic	O
tangent	O
functions	O
have	O
been	O
used	O
as	O
the	O
activation	B-Algorithm
function	I-Algorithm
of	O
artificial	B-Algorithm
neurons	I-Algorithm
.	O
</s>
<s>
Sigmoid	B-Algorithm
curves	I-Algorithm
are	O
also	O
common	O
in	O
statistics	O
as	O
cumulative	O
distribution	O
functions	O
(	O
which	O
go	O
from	O
0	O
to	O
1	O
)	O
,	O
such	O
as	O
the	O
integrals	O
of	O
the	O
logistic	O
density	O
,	O
the	O
normal	O
density	O
,	O
and	O
Student	O
's	O
t	O
probability	O
density	O
functions	O
.	O
</s>
<s>
The	O
logistic	O
sigmoid	B-Algorithm
function	I-Algorithm
is	O
invertible	O
,	O
and	O
its	O
inverse	O
is	O
the	O
logit	O
function	O
.	O
</s>
<s>
A	O
sigmoid	B-Algorithm
function	I-Algorithm
is	O
a	O
bounded	O
,	O
differentiable	O
,	O
real	O
function	O
that	O
is	O
defined	O
for	O
all	O
real	O
input	O
values	O
and	O
has	O
a	O
non-negative	O
derivative	B-Algorithm
at	O
each	O
point	O
and	O
exactly	O
one	O
inflection	O
point	O
.	O
</s>
<s>
In	O
general	O
,	O
a	O
sigmoid	B-Algorithm
function	I-Algorithm
is	O
monotonic	O
,	O
and	O
has	O
a	O
first	B-Algorithm
derivative	I-Algorithm
which	O
is	O
bell	B-Algorithm
shaped	I-Algorithm
.	O
</s>
<s>
Conversely	O
,	O
the	O
integral	O
of	O
any	O
continuous	O
,	O
non-negative	O
,	O
bell-shaped	B-Algorithm
function	I-Algorithm
(	O
with	O
one	O
local	O
maximum	O
and	O
no	O
local	O
minimum	O
,	O
unless	O
degenerate	O
)	O
will	O
be	O
sigmoidal	O
.	O
</s>
<s>
A	O
sigmoid	B-Algorithm
function	I-Algorithm
is	O
constrained	O
by	O
a	O
pair	O
of	O
horizontal	O
asymptotes	O
as	O
.	O
</s>
<s>
A	O
sigmoid	B-Algorithm
function	I-Algorithm
is	O
convex	O
for	O
values	O
less	O
than	O
a	O
particular	O
point	O
,	O
and	O
it	O
is	O
concave	O
for	O
values	O
greater	O
than	O
that	O
point	O
:	O
in	O
many	O
of	O
the	O
examples	O
here	O
,	O
that	O
point	O
is	O
0	O
.	O
</s>
<s>
Up	O
to	O
shifts	O
and	O
scaling	O
,	O
many	O
sigmoids	B-Algorithm
are	O
special	O
cases	O
of	O
where	O
is	O
the	O
inverse	O
of	O
the	O
negative	O
Box	O
–	O
Cox	O
transformation	O
,	O
and	O
and	O
are	O
shape	O
parameters	O
.	O
</s>
<s>
Many	O
natural	O
processes	O
,	O
such	O
as	O
those	O
of	O
complex	O
system	O
learning	B-General_Concept
curves	I-General_Concept
,	O
exhibit	O
a	O
progression	O
from	O
small	O
beginnings	O
that	O
accelerates	O
and	O
approaches	O
a	O
climax	O
over	O
time	O
.	O
</s>
<s>
When	O
a	O
specific	O
mathematical	O
model	O
is	O
lacking	O
,	O
a	O
sigmoid	B-Algorithm
function	I-Algorithm
is	O
often	O
used	O
.	O
</s>
<s>
The	O
van	O
Genuchten	O
–	O
Gupta	O
model	O
is	O
based	O
on	O
an	O
inverted	O
S-curve	B-Algorithm
and	O
applied	O
to	O
the	O
response	O
of	O
crop	O
yield	O
to	O
soil	O
salinity	O
.	O
</s>
<s>
Examples	O
of	O
the	O
application	O
of	O
the	O
logistic	O
S-curve	B-Algorithm
to	O
the	O
response	O
of	O
crop	O
yield	O
(	O
wheat	O
)	O
to	O
both	O
the	O
soil	O
salinity	O
and	O
depth	O
to	O
water	O
table	O
in	O
the	O
soil	O
are	O
shown	O
in	O
modeling	O
crop	O
response	O
in	O
agriculture	O
.	O
</s>
<s>
In	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
,	O
sometimes	O
non-smooth	O
functions	O
are	O
used	O
instead	O
for	O
efficiency	O
;	O
these	O
are	O
known	O
as	O
hard	B-Algorithm
sigmoids	I-Algorithm
.	O
</s>
<s>
In	O
audio	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
,	O
sigmoid	B-Algorithm
functions	I-Algorithm
are	O
used	O
as	O
waveshaper	O
transfer	O
functions	O
to	O
emulate	O
the	O
sound	O
of	O
analog	O
circuitry	O
clipping	O
.	O
</s>
<s>
In	O
biochemistry	O
and	O
pharmacology	O
,	O
the	O
Hill	O
and	O
Hill	O
–	O
Langmuir	O
equations	O
are	O
sigmoid	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
In	O
computer	O
graphics	O
and	O
real-time	O
rendering	O
,	O
some	O
of	O
the	O
sigmoid	B-Algorithm
functions	I-Algorithm
are	O
used	O
to	O
blend	O
colors	O
or	O
geometry	O
between	O
two	O
values	O
,	O
smoothly	O
and	O
without	O
visible	O
seams	O
or	O
discontinuities	O
.	O
</s>
<s>
Titration	O
curves	O
between	O
strong	O
acids	O
and	O
strong	O
bases	O
have	O
a	O
sigmoid	B-Algorithm
shape	I-Algorithm
due	O
to	O
the	O
logarithmic	O
nature	O
of	O
the	O
pH	O
scale	O
.	O
</s>
