<s>
In	O
statistics	O
and	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
isotonic	B-General_Concept
regression	I-General_Concept
or	O
monotonic	B-General_Concept
regression	I-General_Concept
is	O
the	O
technique	O
of	O
fitting	O
a	O
free-form	O
line	O
to	O
a	O
sequence	O
of	O
observations	O
such	O
that	O
the	O
fitted	O
line	O
is	O
non-decreasing	O
(	O
or	O
non-increasing	O
)	O
everywhere	O
,	O
and	O
lies	O
as	O
close	O
to	O
the	O
observations	O
as	O
possible	O
.	O
</s>
<s>
Isotonic	B-General_Concept
regression	I-General_Concept
has	O
applications	O
in	O
statistical	O
inference	O
.	O
</s>
<s>
A	O
benefit	O
of	O
isotonic	B-General_Concept
regression	I-General_Concept
is	O
that	O
it	O
is	O
not	O
constrained	O
by	O
any	O
functional	O
form	O
,	O
such	O
as	O
the	O
linearity	O
imposed	O
by	O
linear	B-General_Concept
regression	I-General_Concept
,	O
as	O
long	O
as	O
the	O
function	O
is	O
monotonic	O
increasing	O
.	O
</s>
<s>
Isotonic	B-General_Concept
regression	I-General_Concept
is	O
used	O
iteratively	O
to	O
fit	O
ideal	O
distances	O
to	O
preserve	O
relative	O
dissimilarity	O
order	O
.	O
</s>
<s>
Isotonic	B-General_Concept
regression	I-General_Concept
is	O
also	O
used	O
in	O
probabilistic	B-General_Concept
classification	I-General_Concept
to	O
calibrate	O
the	O
predicted	O
probabilities	O
of	O
supervised	B-General_Concept
machine	I-General_Concept
learning	I-General_Concept
models	O
.	O
</s>
<s>
Isotonic	B-General_Concept
regression	I-General_Concept
for	O
the	O
simply	O
ordered	O
case	O
with	O
univariate	O
has	O
been	O
applied	O
to	O
estimating	O
continuous	O
dose-response	O
relationships	O
in	O
fields	O
such	O
as	O
anesthesiology	O
and	O
toxicology	O
.	O
</s>
<s>
Narrowly	O
speaking	O
,	O
isotonic	B-General_Concept
regression	I-General_Concept
only	O
provides	O
point	O
estimates	O
at	O
observed	O
values	O
of	O
Estimation	O
of	O
the	O
complete	O
dose-response	O
curve	O
without	O
any	O
additional	O
assumptions	O
is	O
usually	O
done	O
via	O
linear	O
interpolation	O
between	O
the	O
point	O
estimates	O
.	O
</s>
<s>
Software	O
for	O
computing	O
isotone	O
(	O
monotonic	O
)	O
regression	O
has	O
been	O
developed	O
for	O
R	B-Language
,	O
Stata	B-Algorithm
,	O
and	O
Python	B-Language
.	O
</s>
<s>
Isotonic	B-General_Concept
regression	I-General_Concept
seeks	O
a	O
weighted	O
least-squares	B-Algorithm
fit	I-Algorithm
for	O
all	O
,	O
subject	O
to	O
the	O
constraint	O
that	O
whenever	O
.	O
</s>
<s>
This	O
gives	O
the	O
following	O
quadratic	B-Algorithm
program	I-Algorithm
(	O
QP	O
)	O
in	O
the	O
variables	O
:	O
</s>
<s>
Problems	O
of	O
this	O
form	O
may	O
be	O
solved	O
by	O
generic	O
quadratic	B-Algorithm
programming	I-Algorithm
techniques	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
a	O
simple	O
iterative	B-Algorithm
algorithm	I-Algorithm
for	O
solving	O
the	O
quadratic	B-Algorithm
program	I-Algorithm
is	O
the	O
pool	O
adjacent	O
violators	O
algorithm	O
.	O
</s>
<s>
To	O
complete	O
the	O
isotonic	B-General_Concept
regression	I-General_Concept
task	O
,	O
we	O
may	O
then	O
choose	O
any	O
non-decreasing	O
function	O
such	O
that	O
for	O
all	O
i	O
.	O
</s>
<s>
A	O
simple	O
improvement	O
for	O
such	O
applications	O
,	O
named	O
centered	O
isotonic	B-General_Concept
regression	I-General_Concept
(	O
CIR	O
)	O
,	O
was	O
developed	O
by	O
Oron	O
and	O
Flournoy	O
and	O
shown	O
to	O
substantially	O
reduce	O
estimation	O
error	O
for	O
both	O
dose-response	O
and	O
dose-finding	O
applications	O
.	O
</s>
<s>
Both	O
CIR	O
and	O
the	O
standard	O
isotonic	B-General_Concept
regression	I-General_Concept
for	O
the	O
univariate	O
,	O
simply	O
ordered	O
case	O
,	O
are	O
implemented	O
in	O
the	O
R	B-Language
package	O
"	O
cir	O
"	O
.	O
</s>
