<s>
In	O
information	O
theory	O
,	O
redundancy	B-General_Concept
measures	O
the	O
fractional	O
difference	O
between	O
the	O
entropy	O
of	O
an	O
ensemble	O
,	O
and	O
its	O
maximum	O
possible	O
value	O
.	O
</s>
<s>
Data	B-General_Concept
compression	I-General_Concept
is	O
a	O
way	O
to	O
reduce	O
or	O
eliminate	O
unwanted	O
redundancy	B-General_Concept
,	O
while	O
forward	B-Error_Name
error	I-Error_Name
correction	I-Error_Name
is	O
a	O
way	O
of	O
adding	O
desired	O
redundancy	B-General_Concept
for	O
purposes	O
of	O
error	B-Error_Name
detection	I-Error_Name
and	I-Error_Name
correction	I-Error_Name
when	O
communicating	O
over	O
a	O
noisy	O
channel	O
of	O
limited	O
capacity	O
.	O
</s>
<s>
In	O
describing	O
the	O
redundancy	B-General_Concept
of	O
raw	O
data	O
,	O
the	O
rate	O
of	O
a	O
source	O
of	O
information	O
is	O
the	O
average	O
entropy	O
per	O
symbol	O
.	O
</s>
<s>
the	O
logarithm	O
of	O
the	O
cardinality	B-Application
of	O
the	O
message	O
space	O
,	O
or	O
alphabet	O
.	O
</s>
<s>
The	O
quantity	O
is	O
called	O
the	O
relative	O
redundancy	B-General_Concept
and	O
gives	O
the	O
maximum	O
possible	O
data	B-General_Concept
compression	I-General_Concept
ratio	I-General_Concept
,	O
when	O
expressed	O
as	O
the	O
percentage	O
by	O
which	O
a	O
file	O
size	O
can	O
be	O
decreased	O
.	O
</s>
<s>
Complementary	O
to	O
the	O
concept	O
of	O
relative	O
redundancy	B-General_Concept
is	O
efficiency	O
,	O
defined	O
as	O
so	O
that	O
.	O
</s>
<s>
A	O
memoryless	O
source	O
with	O
a	O
uniform	O
distribution	O
has	O
zero	O
redundancy	B-General_Concept
(	O
and	O
thus	O
100%	O
efficiency	O
)	O
,	O
and	O
cannot	O
be	O
compressed	O
.	O
</s>
<s>
A	O
measure	O
of	O
redundancy	B-General_Concept
between	O
two	O
variables	O
is	O
the	O
mutual	O
information	O
or	O
a	O
normalized	O
variant	O
.	O
</s>
<s>
A	O
measure	O
of	O
redundancy	B-General_Concept
among	O
many	O
variables	O
is	O
given	O
by	O
the	O
total	O
correlation	O
.	O
</s>
<s>
Redundancy	B-General_Concept
of	O
compressed	B-General_Concept
data	I-General_Concept
refers	O
to	O
the	O
difference	O
between	O
the	O
expected	O
compressed	B-General_Concept
data	I-General_Concept
length	O
of	O
messages	O
(	O
or	O
expected	O
data	O
rate	O
)	O
and	O
the	O
entropy	O
(	O
or	O
entropy	O
rate	O
)	O
.	O
</s>
<s>
(	O
Here	O
we	O
assume	O
the	O
data	O
is	O
ergodic	O
and	O
stationary	B-Algorithm
,	O
e.g.	O
,	O
a	O
memoryless	O
source	O
.	O
)	O
</s>
<s>
Redundancy	B-General_Concept
in	O
an	O
information-theoretic	O
contexts	O
can	O
also	O
refer	O
to	O
the	O
information	O
that	O
is	O
redundant	O
between	O
two	O
mutual	O
informations	O
.	O
</s>
<s>
This	O
formulation	O
of	O
redundancy	B-General_Concept
is	O
complementary	O
to	O
the	O
notion	O
of	O
synergy	O
,	O
which	O
occurs	O
when	O
the	O
joint	O
mutual	O
information	O
is	O
greater	O
than	O
the	O
sum	O
of	O
the	O
marginals	O
,	O
indicating	O
the	O
presence	O
of	O
information	O
that	O
is	O
only	O
disclosed	O
by	O
the	O
joint	O
state	O
and	O
not	O
any	O
simpler	O
collection	O
of	O
sources	O
.	O
</s>
