<s>
In	O
machine	O
learning	O
and	O
data	B-Application
mining	I-Application
,	O
quantification	B-General_Concept
(	O
variously	O
called	O
learning	O
to	O
quantify	O
,	O
or	O
supervised	O
prevalence	O
estimation	O
,	O
or	O
class	O
prior	O
estimation	O
)	O
is	O
the	O
task	O
of	O
using	O
supervised	B-General_Concept
learning	I-General_Concept
in	O
order	O
to	O
train	O
models	O
(	O
quantifiers	O
)	O
that	O
estimate	O
the	O
relative	O
frequencies	O
(	O
also	O
known	O
as	O
prevalence	O
values	O
)	O
of	O
the	O
classes	O
of	O
interest	O
in	O
a	O
sample	O
of	O
unlabelled	B-General_Concept
data	I-General_Concept
items	I-General_Concept
.	O
</s>
<s>
For	O
instance	O
,	O
in	O
a	O
sample	O
of	O
100,000	O
unlabelled	O
tweets	B-Application
known	O
to	O
express	O
opinions	O
about	O
a	O
certain	O
political	O
candidate	O
,	O
a	O
quantifier	O
may	O
be	O
used	O
to	O
estimate	O
the	O
percentage	O
of	O
these	O
100,000	O
tweets	B-Application
which	O
belong	O
to	O
class	O
`Positive	O
 '	O
(	O
i.e.	O
,	O
which	O
manifest	O
a	O
positive	O
stance	O
towards	O
this	O
candidate	O
)	O
,	O
and	O
to	O
do	O
the	O
same	O
for	O
classes	O
`Neutral	O
 '	O
and	O
`Negative	O
 '	O
.	O
</s>
<s>
Quantification	B-General_Concept
may	O
also	O
be	O
viewed	O
as	O
the	O
task	O
of	O
training	O
predictors	O
that	O
estimate	O
a	O
(	O
discrete	O
)	O
probability	O
distribution	O
,	O
i.e.	O
,	O
that	O
generate	O
a	O
predicted	O
distribution	O
that	O
approximates	O
the	O
unknown	O
true	O
distribution	O
of	O
the	O
items	O
across	O
the	O
classes	O
of	O
interest	O
.	O
</s>
<s>
Quantification	B-General_Concept
is	O
different	O
from	O
classification	B-General_Concept
,	O
since	O
the	O
goal	O
of	O
classification	B-General_Concept
is	O
to	O
predict	O
the	O
class	O
labels	O
of	O
individual	O
data	O
items	O
,	O
while	O
the	O
goal	O
of	O
quantification	B-General_Concept
it	O
to	O
predict	O
the	O
class	O
prevalence	O
values	O
of	O
sets	O
of	O
data	O
items	O
.	O
</s>
<s>
Quantification	B-General_Concept
is	O
also	O
different	O
from	O
regression	O
,	O
since	O
in	O
regression	O
the	O
training	O
data	O
items	O
have	O
real-valued	O
labels	O
,	O
while	O
in	O
quantification	B-General_Concept
the	O
training	O
data	O
items	O
have	O
class	O
labels	O
.	O
</s>
<s>
that	O
performing	O
quantification	B-General_Concept
by	O
classifying	O
all	O
unlabelled	O
instances	O
and	O
then	O
counting	O
the	O
instances	O
that	O
have	O
been	O
attributed	O
to	O
each	O
class	O
(	O
the	O
'	O
classify	O
and	O
count	O
 '	O
method	O
)	O
usually	O
leads	O
to	O
suboptimal	O
quantification	B-General_Concept
accuracy	O
.	O
</s>
<s>
In	O
our	O
case	O
,	O
the	O
problem	O
to	O
be	O
solved	O
directly	O
is	O
quantification	B-General_Concept
,	O
while	O
the	O
more	O
general	O
intermediate	O
problem	O
is	O
classification	B-General_Concept
.	O
</s>
<s>
As	O
a	O
result	O
of	O
the	O
suboptimality	O
of	O
the	O
'	O
classify	O
and	O
count	O
 '	O
method	O
,	O
quantification	B-General_Concept
has	O
evolved	O
as	O
a	O
task	O
in	O
its	O
own	O
right	O
,	O
different	O
(	O
in	O
goals	O
,	O
methods	O
,	O
techniques	O
,	O
and	O
evaluation	O
measures	O
)	O
from	O
classification	B-General_Concept
.	O
</s>
<s>
The	O
main	O
variants	O
of	O
quantification	B-General_Concept
,	O
according	O
to	O
the	O
characteristics	O
of	O
the	O
set	O
of	O
classes	O
used	O
,	O
are	O
:	O
</s>
<s>
Binary	B-General_Concept
quantification	I-General_Concept
,	O
corresponding	O
to	O
the	O
case	O
in	O
which	O
there	O
are	O
only	O
classes	O
and	O
each	O
data	O
item	O
belongs	O
to	O
exactly	O
one	O
of	O
them	O
;	O
</s>
<s>
Single-label	O
multiclass	B-General_Concept
quantification	I-General_Concept
,	O
corresponding	O
to	O
the	O
case	O
with	O
classes	O
and	O
each	O
data	O
item	O
belongs	O
to	O
exactly	O
one	O
of	O
them	O
;	O
</s>
<s>
Ordinal	B-General_Concept
quantification	I-General_Concept
,	O
corresponding	O
to	O
the	O
single-label	O
multiclass	O
case	O
in	O
which	O
a	O
total	O
order	O
is	O
defined	O
on	O
the	O
set	O
of	O
classes	O
.	O
</s>
<s>
Most	O
known	O
quantification	B-General_Concept
methods	O
address	O
the	O
binary	O
case	O
or	O
the	O
single-label	O
multiclass	O
case	O
,	O
and	O
only	O
few	O
of	O
them	O
address	O
the	O
ordinal	O
case	O
.	O
</s>
<s>
Methods	O
that	O
can	O
deal	O
with	O
both	O
the	O
binary	O
case	O
and	O
the	O
single-label	O
multiclass	O
case	O
include	O
probabilistic	O
classify	O
and	O
count	O
(	O
PCC	O
)	O
,	O
adjusted	O
classify	O
and	O
count	O
(	O
ACC	O
)	O
,	O
probabilistic	O
adjusted	O
classify	O
and	O
count	O
(	O
PACC	O
)	O
,	O
and	O
the	O
Saerens-Latinne-Decaestecker	O
EM-based	O
method	O
(	O
SLD	O
)	O
.	O
</s>
<s>
Methods	O
for	O
the	O
ordinal	O
case	O
include	O
Ordinal	B-General_Concept
Quantification	I-General_Concept
Tree	O
(	O
OQT	O
)	O
,	O
and	O
ordinal	O
version	O
of	O
the	O
above-mentioned	O
ACC	O
,	O
PACC	O
,	O
and	O
SLD	O
methods	O
.	O
</s>
<s>
Several	O
evaluation	O
measures	O
can	O
be	O
used	O
for	O
evaluating	O
the	O
error	O
of	O
a	O
quantification	B-General_Concept
method	O
.	O
</s>
<s>
Since	O
quantification	B-General_Concept
consists	O
of	O
generating	O
a	O
predicted	O
probability	O
distribution	O
that	O
estimates	O
a	O
true	O
probability	O
distribution	O
,	O
these	O
evaluation	O
measures	O
are	O
ones	O
that	O
compare	O
two	O
probability	O
distributions	O
.	O
</s>
<s>
Most	O
evaluation	O
measures	O
for	O
quantification	B-General_Concept
belong	O
to	O
the	O
class	O
of	O
divergences	O
.	O
</s>
<s>
Quantification	B-General_Concept
is	O
of	O
special	O
interest	O
in	O
fields	O
such	O
as	O
the	O
social	O
sciences	O
,	O
</s>
<s>
and	O
enforcing	O
classifier	B-General_Concept
fairness	O
,	O
performing	O
word	B-General_Concept
sense	I-General_Concept
disambiguation	I-General_Concept
,	O
</s>
<s>
and	O
improving	O
the	O
accuracy	O
of	O
classifiers	B-General_Concept
.	O
</s>
