<s>
The	O
total	B-General_Concept
operating	I-General_Concept
characteristic	I-General_Concept
(	O
TOC	O
)	O
is	O
a	O
statistical	O
method	O
to	O
compare	O
a	O
Boolean	O
variable	O
versus	O
a	O
rank	O
variable	O
.	O
</s>
<s>
Each	O
threshold	O
generates	O
a	O
two-by-two	O
contingency	B-Application
table	I-Application
,	O
which	O
contains	O
four	O
entries	O
:	O
hits	O
,	O
misses	O
,	O
false	B-Error_Name
alarms	I-Error_Name
,	O
and	O
correct	O
rejections	O
.	O
</s>
<s>
The	O
receiver	B-Algorithm
operating	I-Algorithm
characteristic	I-Algorithm
(	O
ROC	O
)	O
also	O
characterizes	O
diagnostic	O
ability	O
,	O
although	O
ROC	O
reveals	O
less	O
information	O
than	O
the	O
TOC	O
.	O
</s>
<s>
For	O
each	O
threshold	O
,	O
ROC	O
reveals	O
two	O
ratios	O
,	O
hits/	O
( hits	O
+	O
misses	O
)	O
and	O
false	O
alarms/	O
( false	O
alarms	O
+	O
correct	O
rejections	O
)	O
,	O
while	O
TOC	O
shows	O
the	O
total	O
information	O
in	O
the	O
contingency	B-Application
table	I-Application
for	O
each	O
threshold	O
.	O
</s>
<s>
the	O
size	O
of	O
every	O
entry	O
in	O
the	O
contingency	B-Application
table	I-Application
for	O
each	O
threshold	O
.	O
</s>
<s>
TOC	O
also	O
provides	O
the	O
popular	O
area	B-Algorithm
under	I-Algorithm
the	I-Algorithm
curve	I-Algorithm
(	O
AUC	O
)	O
of	O
the	O
ROC	O
.	O
</s>
<s>
TOC	O
is	O
applicable	O
to	O
measure	O
diagnostic	O
ability	O
in	O
many	O
fields	O
including	O
but	O
not	O
limited	O
to	O
:	O
land	O
change	O
science	O
,	O
medical	B-Application
imaging	I-Application
,	O
weather	O
forecasting	O
,	O
remote	B-Application
sensing	I-Application
,	O
and	O
materials	O
testing	O
.	O
</s>
<s>
The	O
contingency	B-Application
table	I-Application
that	O
results	O
from	O
the	O
comparison	O
between	O
the	O
Boolean	O
variable	O
and	O
the	O
diagnosis	O
for	O
a	O
single	O
threshold	O
has	O
four	O
central	O
entries	O
.	O
</s>
<s>
The	O
four	O
central	O
entries	O
are	O
hits	O
(	O
H	O
)	O
,	O
misses	O
(	O
M	O
)	O
,	O
false	B-Error_Name
alarms	I-Error_Name
(	O
F	O
)	O
,	O
and	O
correct	O
rejections	O
(	O
C	O
)	O
.	O
</s>
<s>
The	O
terms	O
“	O
true	O
positives	O
”	O
,	O
“	O
false	O
negatives	O
”	O
,	O
“	O
false	O
positives	O
”	O
and	O
“	O
true	O
negatives	O
”	O
are	O
equivalent	O
to	O
hits	O
,	O
misses	O
,	O
false	B-Error_Name
alarms	I-Error_Name
and	O
correct	O
rejections	O
,	O
respectively	O
.	O
</s>
<s>
The	O
entries	O
can	O
be	O
formulated	O
in	O
a	O
two-by-two	O
contingency	B-Application
table	I-Application
or	O
confusion	O
matrix	O
,	O
as	O
follows	O
:	O
</s>
<s>
Four	O
bits	O
of	O
information	O
determine	O
all	O
the	O
entries	O
in	O
the	O
contingency	B-Application
table	I-Application
,	O
including	O
its	O
marginal	O
totals	O
.	O
</s>
<s>
Two	O
bits	O
of	O
information	O
are	O
not	O
sufficient	O
to	O
complete	O
the	O
contingency	B-Application
table	I-Application
.	O
</s>
<s>
The	O
TOC	O
curve	O
with	O
four	O
boxes	O
indicates	O
how	O
a	O
point	O
on	O
the	O
TOC	O
curve	O
reveals	O
the	O
hits	O
,	O
misses	O
,	O
false	B-Error_Name
alarms	I-Error_Name
,	O
and	O
correct	O
rejections	O
.	O
</s>
<s>
The	O
TOC	O
curve	O
is	O
an	O
effective	O
way	O
to	O
show	O
the	O
total	O
information	O
in	O
the	O
contingency	B-Application
table	I-Application
for	O
all	O
thresholds	O
.	O
</s>
<s>
"	O
The	O
total	B-General_Concept
operating	I-General_Concept
characteristic	I-General_Concept
to	O
measure	O
diagnostic	O
ability	O
for	O
multiple	O
thresholds.	O
"	O
</s>
<s>
The	O
following	O
four	O
pieces	O
of	O
information	O
are	O
the	O
central	O
entries	O
in	O
the	O
contingency	B-Application
table	I-Application
for	O
each	O
threshold	O
:	O
</s>
<s>
The	O
number	O
of	O
false	B-Error_Name
alarms	I-Error_Name
at	O
each	O
threshold	O
is	O
the	O
distance	O
between	O
threshold	O
's	O
point	O
and	O
the	O
blue	O
dashed	O
Maximum	O
line	O
that	O
bounds	O
the	O
left	O
side	O
of	O
the	O
TOC	O
space	O
.	O
</s>
<s>
These	O
figures	O
are	O
the	O
TOC	O
and	O
ROC	B-Algorithm
curves	I-Algorithm
using	O
the	O
same	O
data	O
and	O
thresholds	O
.	O
</s>
<s>
Additionally	O
,	O
the	O
TOC	O
curve	O
shows	O
that	O
the	O
number	O
of	O
false	B-Error_Name
alarms	I-Error_Name
is	O
4	O
and	O
the	O
number	O
of	O
correct	O
rejections	O
is	O
16	O
.	O
</s>
<s>
At	O
any	O
given	O
point	O
in	O
the	O
ROC	B-Algorithm
curve	I-Algorithm
,	O
it	O
is	O
possible	O
to	O
glean	O
values	O
for	O
the	O
ratios	O
of	O
false	O
alarms/	O
( false	O
alarms+correct	O
rejections	O
)	O
and	O
hits/	O
( hits+misses	O
)	O
.	O
</s>
<s>
However	O
,	O
these	O
two	O
values	O
are	O
insufficient	O
to	O
construct	O
all	O
entries	O
of	O
the	O
underlying	O
two-by-two	O
contingency	B-Application
table	I-Application
.	O
</s>
<s>
It	O
is	O
common	O
to	O
report	O
the	O
area	B-Algorithm
under	I-Algorithm
the	I-Algorithm
curve	I-Algorithm
(	O
AUC	O
)	O
to	O
summarize	O
a	O
TOC	O
or	O
ROC	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
When	O
measuring	O
diagnostic	O
ability	O
,	O
a	O
commonly	O
reported	O
measure	O
is	O
the	O
area	B-Algorithm
under	I-Algorithm
the	I-Algorithm
curve	I-Algorithm
(	O
AUC	O
)	O
.	O
</s>
<s>
The	O
value	O
of	O
the	O
AUC	O
is	O
consistent	O
for	O
the	O
same	O
data	O
whether	O
you	O
are	O
calculating	O
the	O
area	B-Algorithm
under	I-Algorithm
the	I-Algorithm
curve	I-Algorithm
for	O
a	O
TOC	O
curve	O
or	O
a	O
ROC	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
When	O
using	O
normalized	O
units	O
,	O
the	O
area	B-Algorithm
under	I-Algorithm
the	I-Algorithm
curve	I-Algorithm
(	O
often	O
referred	O
to	O
as	O
simply	O
the	O
AUC	O
)	O
is	O
equal	O
to	O
the	O
probability	O
that	O
a	O
classifier	O
will	O
rank	O
a	O
randomly	O
chosen	O
positive	O
instance	O
higher	O
than	O
a	O
randomly	O
chosen	O
negative	O
one	O
(	O
assuming	O
'	O
positive	O
 '	O
ranks	O
higher	O
than	O
'	O
negative	O
 '	O
)	O
.	O
</s>
<s>
It	O
can	O
further	O
be	O
shown	O
that	O
the	O
AUC	O
is	O
closely	O
related	O
to	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
,	O
which	O
tests	O
whether	O
positives	O
are	O
ranked	O
higher	O
than	O
negatives	O
.	O
</s>
<s>
It	O
is	O
also	O
equivalent	O
to	O
the	O
Wilcoxon	B-General_Concept
test	I-General_Concept
of	I-General_Concept
ranks	I-General_Concept
.	O
</s>
<s>
Another	O
problem	O
with	O
TOC	O
AUC	O
is	O
that	O
reducing	O
the	O
TOC	O
Curve	O
to	O
a	O
single	O
number	O
ignores	O
the	O
fact	O
that	O
it	O
is	O
about	O
the	O
tradeoffs	O
between	O
the	O
different	O
systems	O
or	O
performance	O
points	O
plotted	O
and	O
not	O
the	O
performance	O
of	O
an	O
individual	O
system	O
,	O
as	O
well	O
as	O
ignoring	O
the	O
possibility	O
of	O
concavity	O
repair	O
,	O
so	O
that	O
related	O
alternative	O
measures	O
such	O
as	O
Informedness	B-General_Concept
or	O
DeltaP	O
are	O
recommended	O
.	O
</s>
<s>
These	O
measures	O
are	O
essentially	O
equivalent	O
to	O
the	O
Gini	O
for	O
a	O
single	O
prediction	O
point	O
with	O
DeltaP	O
 '	O
=	O
informedness	B-General_Concept
=	O
2AUC-1	O
,	O
whilst	O
DeltaP	O
=	O
markedness	O
represents	O
the	O
dual	O
(	O
viz	O
.	O
</s>
<s>
predicting	O
the	O
prediction	O
from	O
the	O
real	O
class	O
)	O
and	O
their	O
geometric	O
mean	O
is	O
the	O
Matthews	B-General_Concept
correlation	I-General_Concept
coefficient	I-General_Concept
.	O
</s>
<s>
Whereas	O
TOC	O
AUC	O
varies	O
between	O
0	O
and	O
1	O
—	O
with	O
an	O
uninformative	O
classifier	O
yielding	O
0.5	O
—	O
the	O
alternative	O
measures	O
known	O
as	O
informedness	B-General_Concept
,	O
Certainty	O
and	O
Gini	O
coefficient	O
(	O
in	O
the	O
single	O
parameterization	O
or	O
single	O
system	O
case	O
)	O
all	O
have	O
the	O
advantage	O
that	O
0	O
represents	O
chance	O
performance	O
whilst	O
1	O
represents	O
perfect	O
performance	O
,	O
and	O
−1	O
represents	O
the	O
"	O
perverse	O
"	O
case	O
of	O
full	O
informedness	B-General_Concept
always	O
giving	O
the	O
wrong	O
response	O
.	O
</s>
<s>
Bringing	O
chance	O
performance	O
to	O
0	O
allows	O
these	O
alternative	O
scales	O
to	O
be	O
interpreted	O
as	O
Kappa	B-General_Concept
statistics	I-General_Concept
.	O
</s>
<s>
Informedness	B-General_Concept
has	O
been	O
shown	O
to	O
have	O
desirable	O
characteristics	O
for	O
machine	O
learning	O
versus	O
other	O
common	O
definitions	O
of	O
Kappa	O
such	O
as	O
Cohen	B-General_Concept
kappa	I-General_Concept
and	O
Fleiss	O
kappa	O
.	O
</s>
