<s>
In	O
the	O
analysis	O
of	O
data	O
,	O
a	O
correlogram	B-Application
is	O
a	O
chart	O
of	O
correlation	O
statistics	O
.	O
</s>
<s>
If	O
cross-correlation	O
is	O
plotted	O
,	O
the	O
result	O
is	O
called	O
a	O
cross-correlogram	O
.	O
</s>
<s>
The	O
correlogram	B-Application
is	O
a	O
commonly	O
used	O
tool	O
for	O
checking	O
randomness	O
in	O
a	O
data	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
In	O
addition	O
,	O
correlograms	B-Application
are	O
used	O
in	O
the	O
model	O
identification	O
stage	O
for	O
Box	O
–	O
Jenkins	O
autoregressive	O
moving	O
average	O
time	O
series	O
models	O
.	O
</s>
<s>
The	O
correlogram	B-Application
is	O
an	O
excellent	O
way	O
of	O
checking	O
for	O
such	O
randomness	O
.	O
</s>
<s>
In	O
multivariate	O
analysis	O
,	O
correlation	O
matrices	O
shown	O
as	O
color-mapped	B-Algorithm
images	O
may	O
also	O
be	O
called	O
"	O
correlograms	B-Application
"	O
or	O
"	O
corrgrams	O
"	O
.	O
</s>
<s>
The	O
correlogram	B-Application
can	O
help	O
provide	O
answers	O
to	O
the	O
following	O
questions	O
:	O
</s>
<s>
Many	O
commonly	O
used	O
statistical	O
formulae	O
depend	O
on	O
the	O
randomness	O
assumption	O
,	O
the	O
most	O
common	O
formula	O
being	O
the	O
formula	O
for	O
determining	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
sample	O
mean	O
:	O
</s>
<s>
where	O
s	O
is	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
data	O
.	O
</s>
<s>
In	O
the	O
same	O
graph	O
one	O
can	O
draw	O
upper	O
and	O
lower	O
bounds	O
for	O
autocorrelation	O
with	O
significance	B-General_Concept
level	I-General_Concept
:	O
</s>
<s>
If	O
the	O
autocorrelation	O
is	O
higher	O
(	O
lower	O
)	O
than	O
this	O
upper	O
(	O
lower	O
)	O
bound	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
there	O
is	O
no	O
autocorrelation	O
at	O
and	O
beyond	O
a	O
given	O
lag	O
is	O
rejected	O
at	O
a	O
significance	B-General_Concept
level	I-General_Concept
of	O
.	O
</s>
<s>
This	O
test	O
is	O
an	O
approximate	O
one	O
and	O
assumes	O
that	O
the	O
time-series	O
is	O
Gaussian	B-Application
.	O
</s>
<s>
In	O
the	O
above	O
,	O
z1−	O
α/2	O
is	O
the	O
quantile	O
of	O
the	O
normal	O
distribution	O
;	O
SE	O
is	O
the	O
standard	B-General_Concept
error	I-General_Concept
,	O
which	O
can	O
be	O
computed	O
by	O
Bartlett	O
's	O
formula	O
for	O
MA(ℓ )	O
processes	O
:	O
</s>
<s>
In	O
the	O
example	O
plotted	O
,	O
we	O
can	O
reject	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
there	O
is	O
no	O
autocorrelation	O
between	O
time-points	O
which	O
are	O
separated	O
by	O
lags	O
up	O
to	O
4	O
.	O
</s>
<s>
For	O
most	O
longer	O
periods	O
one	O
cannot	O
reject	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
no	O
autocorrelation	O
.	O
</s>
<s>
If	O
the	O
correlogram	B-Application
is	O
being	O
used	O
to	O
test	O
for	O
randomness	O
(	O
i.e.	O
,	O
there	O
is	O
no	O
time	O
dependence	O
in	O
the	O
data	O
)	O
,	O
the	O
following	O
formula	O
is	O
recommended	O
:	O
</s>
<s>
where	O
N	O
is	O
the	O
sample	O
size	O
,	O
z	O
is	O
the	O
quantile	O
function	O
of	O
the	O
standard	O
normal	O
distribution	O
and	O
α	O
is	O
the	O
significance	B-General_Concept
level	I-General_Concept
.	O
</s>
<s>
Correlograms	B-Application
are	O
also	O
used	O
in	O
the	O
model	O
identification	O
stage	O
for	O
fitting	O
ARIMA	O
models	O
.	O
</s>
<s>
Correlograms	B-Application
are	O
available	O
in	O
most	O
general	O
purpose	O
statistical	O
libraries	O
.	O
</s>
<s>
Correlograms	B-Application
:	O
</s>
