<s>
In	O
mathematics	O
,	O
unimodality	B-Algorithm
means	O
possessing	O
a	O
unique	O
mode	O
.	O
</s>
<s>
More	O
generally	O
,	O
unimodality	B-Algorithm
means	O
there	O
is	O
only	O
a	O
single	O
highest	O
value	O
,	O
somehow	O
defined	O
,	O
of	O
some	O
mathematical	O
object	O
.	O
</s>
<s>
In	O
statistics	O
,	O
a	O
unimodal	B-Algorithm
probability	O
distribution	O
or	O
unimodal	B-Algorithm
distribution	O
is	O
a	O
probability	O
distribution	O
which	O
has	O
a	O
single	O
peak	O
.	O
</s>
<s>
If	O
there	O
is	O
a	O
single	O
mode	O
,	O
the	O
distribution	O
function	O
is	O
called	O
"	O
unimodal	B-Algorithm
"	O
.	O
</s>
<s>
Figure	O
1	O
illustrates	O
normal	O
distributions	O
,	O
which	O
are	O
unimodal	B-Algorithm
.	O
</s>
<s>
Other	O
examples	O
of	O
unimodal	B-Algorithm
distributions	O
include	O
Cauchy	O
distribution	O
,	O
Student	O
's	O
t-distribution	O
,	O
chi-squared	O
distribution	O
and	O
exponential	O
distribution	O
.	O
</s>
<s>
Among	O
discrete	O
distributions	O
,	O
the	O
binomial	O
distribution	O
and	O
Poisson	O
distribution	O
can	O
be	O
seen	O
as	O
unimodal	B-Algorithm
,	O
though	O
for	O
some	O
parameters	O
they	O
can	O
have	O
two	O
adjacent	O
values	O
with	O
the	O
same	O
probability	O
.	O
</s>
<s>
Other	O
definitions	O
of	O
unimodality	B-Algorithm
in	O
distribution	O
functions	O
also	O
exist	O
.	O
</s>
<s>
In	O
continuous	O
distributions	O
,	O
unimodality	B-Algorithm
can	O
be	O
defined	O
through	O
the	O
behavior	O
of	O
the	O
cumulative	O
distribution	O
function	O
(	O
cdf	O
)	O
.	O
</s>
<s>
If	O
the	O
cdf	O
is	O
convex	O
for	O
x	O
<	O
m	O
and	O
concave	O
for	O
x>m	O
,	O
then	O
the	O
distribution	O
is	O
unimodal	B-Algorithm
,	O
m	O
being	O
the	O
mode	O
.	O
</s>
<s>
Note	O
that	O
under	O
this	O
definition	O
the	O
uniform	O
distribution	O
is	O
unimodal	B-Algorithm
,	O
as	O
well	O
as	O
any	O
other	O
distribution	O
in	O
which	O
the	O
maximum	O
distribution	O
is	O
achieved	O
for	O
a	O
range	O
of	O
values	O
,	O
e.g.	O
</s>
<s>
Criteria	O
for	O
unimodality	B-Algorithm
can	O
also	O
be	O
defined	O
through	O
the	O
characteristic	O
function	O
of	O
the	O
distribution	O
or	O
through	O
its	O
Laplace	B-Algorithm
–	I-Algorithm
Stieltjes	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Another	O
way	O
to	O
define	O
a	O
unimodal	B-Algorithm
discrete	O
distribution	O
is	O
by	O
the	O
occurrence	O
of	O
sign	O
changes	O
in	O
the	O
sequence	O
of	O
differences	O
of	O
the	O
probabilities	O
.	O
</s>
<s>
A	O
discrete	O
distribution	O
with	O
a	O
probability	O
mass	O
function	O
,	O
,	O
is	O
called	O
unimodal	B-Algorithm
if	O
the	O
sequence	O
has	O
exactly	O
one	O
sign	O
change	O
(	O
when	O
zeroes	O
do	O
n't	O
count	O
)	O
.	O
</s>
<s>
One	O
reason	O
for	O
the	O
importance	O
of	O
distribution	O
unimodality	B-Algorithm
is	O
that	O
it	O
allows	O
for	O
several	O
important	O
results	O
.	O
</s>
<s>
Several	O
inequalities	O
are	O
given	O
below	O
which	O
are	O
only	O
valid	O
for	O
unimodal	B-Algorithm
distributions	O
.	O
</s>
<s>
Thus	O
,	O
it	O
is	O
important	O
to	O
assess	O
whether	O
or	O
not	O
a	O
given	O
data	O
set	O
comes	O
from	O
a	O
unimodal	B-Algorithm
distribution	O
.	O
</s>
<s>
Several	O
tests	O
for	O
unimodality	B-Algorithm
are	O
given	O
in	O
the	O
article	O
on	O
multimodal	O
distribution	O
.	O
</s>
<s>
This	O
inequality	O
depends	O
on	O
unimodality	B-Algorithm
.	O
</s>
<s>
The	O
Vysochanskiï	O
–	O
Petunin	O
inequality	O
refines	O
this	O
to	O
even	O
nearer	O
values	O
,	O
provided	O
that	O
the	O
distribution	O
function	O
is	O
continuous	O
and	O
unimodal	B-Algorithm
.	O
</s>
<s>
where	O
the	O
median	O
is	O
ν	O
,	O
the	O
mean	O
is	O
μ	O
and	O
ω	O
is	O
the	O
root	B-General_Concept
mean	I-General_Concept
square	I-General_Concept
deviation	I-General_Concept
from	O
the	O
mode	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
for	O
a	O
unimodal	B-Algorithm
distribution	O
that	O
the	O
median	O
ν	O
and	O
the	O
mean	O
μ	O
lie	O
within	O
(	O
3/5	O
)	O
1/2	O
≈	O
0.7746	O
standard	B-General_Concept
deviations	I-General_Concept
of	O
each	O
other	O
.	O
</s>
<s>
Moreover	O
,	O
when	O
,	O
the	O
bound	O
is	O
equal	O
to	O
,	O
which	O
is	O
the	O
maximum	O
distance	O
between	O
the	O
median	O
and	O
the	O
mean	O
of	O
a	O
unimodal	B-Algorithm
distribution	O
.	O
</s>
<s>
A	O
similar	O
relation	O
holds	O
between	O
the	O
median	O
and	O
the	O
mode	O
θ	O
:	O
they	O
lie	O
within	O
31/2	O
≈	O
1.732	O
standard	B-General_Concept
deviations	I-General_Concept
of	O
each	O
other	O
:	O
</s>
<s>
Rohatgi	O
and	O
Szekely	O
claimed	O
that	O
the	O
skewness	B-General_Concept
and	O
kurtosis	B-Error_Name
of	O
a	O
unimodal	B-Algorithm
distribution	O
are	O
related	O
by	O
the	O
inequality	O
:	O
</s>
<s>
where	O
κ	O
is	O
the	O
kurtosis	B-Error_Name
and	O
γ	O
is	O
the	O
skewness	B-General_Concept
.	O
</s>
<s>
Klaassen	O
,	O
Mokveld	O
,	O
and	O
van	O
Es	O
showed	O
that	O
this	O
only	O
applies	O
in	O
certain	O
settings	O
,	O
such	O
as	O
the	O
set	O
of	O
unimodal	B-Algorithm
distributions	O
where	O
the	O
mode	O
and	O
mean	O
coincide	O
.	O
</s>
<s>
They	O
derived	O
a	O
weaker	O
inequality	O
which	O
applies	O
to	O
all	O
unimodal	B-Algorithm
distributions	O
:	O
</s>
<s>
The	O
definition	O
of	O
"	O
unimodal	B-Algorithm
"	O
was	O
extended	O
to	O
functions	O
of	O
real	O
numbers	O
as	O
well	O
.	O
</s>
<s>
A	O
common	O
definition	O
is	O
as	O
follows	O
:	O
a	O
function	O
f(x )	O
is	O
a	O
unimodal	B-Algorithm
function	O
if	O
for	O
some	O
value	O
m	O
,	O
it	O
is	O
monotonically	O
increasing	O
for	O
x≤m	O
and	O
monotonically	O
decreasing	O
for	O
x≥m	O
.	O
</s>
<s>
Proving	O
unimodality	B-Algorithm
is	O
often	O
hard	O
.	O
</s>
<s>
A	O
general	O
method	O
based	O
on	O
derivatives	B-Algorithm
exists	O
,	O
but	O
it	O
does	O
not	O
succeed	O
for	O
every	O
function	O
despite	O
its	O
simplicity	O
.	O
</s>
<s>
Examples	O
of	O
unimodal	B-Algorithm
functions	O
include	O
quadratic	O
polynomial	O
functions	O
with	O
a	O
negative	O
quadratic	O
coefficient	O
,	O
tent	B-Algorithm
map	I-Algorithm
functions	O
,	O
and	O
more	O
.	O
</s>
<s>
A	O
function	O
f(x )	O
is	O
a	O
weakly	O
unimodal	B-Algorithm
function	O
if	O
there	O
exists	O
a	O
value	O
m	O
for	O
which	O
it	O
is	O
weakly	O
monotonically	O
increasing	O
for	O
x≤m	O
and	O
weakly	O
monotonically	O
decreasing	O
for	O
x≥m	O
.	O
</s>
<s>
An	O
example	O
of	O
a	O
weakly	O
unimodal	B-Algorithm
function	O
which	O
is	O
not	O
strongly	O
unimodal	B-Algorithm
is	O
every	O
other	O
row	O
in	O
Pascal	O
's	O
triangle	O
.	O
</s>
<s>
Depending	O
on	O
context	O
,	O
unimodal	B-Algorithm
function	O
may	O
also	O
refer	O
to	O
a	O
function	O
that	O
has	O
only	O
one	O
local	O
minimum	O
,	O
rather	O
than	O
maximum	O
.	O
</s>
<s>
For	O
example	O
,	O
local	B-Algorithm
unimodal	I-Algorithm
sampling	I-Algorithm
,	O
a	O
method	O
for	O
doing	O
numerical	O
optimization	O
,	O
is	O
often	O
demonstrated	O
with	O
such	O
a	O
function	O
.	O
</s>
<s>
It	O
can	O
be	O
said	O
that	O
a	O
unimodal	B-Algorithm
function	O
under	O
this	O
extension	O
is	O
a	O
function	O
with	O
a	O
single	O
local	O
extremum	O
.	O
</s>
<s>
One	O
important	O
property	O
of	O
unimodal	B-Algorithm
functions	O
is	O
that	O
the	O
extremum	O
can	O
be	O
found	O
using	O
search	B-Application
algorithms	I-Application
such	O
as	O
golden	B-Algorithm
section	I-Algorithm
search	I-Algorithm
,	O
ternary	B-Algorithm
search	I-Algorithm
or	O
successive	B-Algorithm
parabolic	I-Algorithm
interpolation	I-Algorithm
.	O
</s>
<s>
A	O
function	O
f(x )	O
is	O
"	O
S-unimodal	O
"	O
(	O
often	O
referred	O
to	O
as	O
"	O
S-unimodal	O
map	O
"	O
)	O
if	O
its	O
Schwarzian	O
derivative	B-Algorithm
is	O
negative	O
for	O
all	O
,	O
where	O
is	O
the	O
critical	O
point	O
.	O
</s>
<s>
In	O
computational	O
geometry	O
if	O
a	O
function	O
is	O
unimodal	B-Algorithm
it	O
permits	O
the	O
design	O
of	O
efficient	O
algorithms	O
for	O
finding	O
the	O
extrema	O
of	O
the	O
function	O
.	O
</s>
<s>
A	O
more	O
general	O
definition	O
,	O
applicable	O
to	O
a	O
function	O
f(X )	O
of	O
a	O
vector	O
variable	O
X	O
is	O
that	O
f	O
is	O
unimodal	B-Algorithm
if	O
there	O
is	O
a	O
one-to-one	O
differentiable	O
mapping	O
X	O
=	O
G(Z )	O
such	O
that	O
f(G(Z )	O
)	O
is	O
convex	O
.	O
</s>
<s>
Quasiconvex	O
functions	O
and	O
quasiconcave	O
functions	O
extend	O
the	O
concept	O
of	O
unimodality	B-Algorithm
to	O
functions	O
whose	O
arguments	O
belong	O
to	O
higher-dimensional	O
Euclidean	O
spaces	O
.	O
</s>
