<s>
The	O
analysis	O
of	O
autocorrelation	O
is	O
a	O
mathematical	O
tool	O
for	O
finding	O
repeating	O
patterns	O
,	O
such	O
as	O
the	O
presence	O
of	O
a	O
periodic	O
signal	O
obscured	O
by	O
noise	B-Algorithm
,	O
or	O
identifying	O
the	O
missing	B-Application
fundamental	I-Application
frequency	I-Application
in	O
a	O
signal	O
implied	O
by	O
its	O
harmonic	O
frequencies	O
.	O
</s>
<s>
Unit	O
root	O
processes	O
,	O
trend-stationary	O
processes	O
,	O
autoregressive	B-Algorithm
processes	O
,	O
and	O
moving	O
average	O
processes	O
are	O
specific	O
forms	O
of	O
processes	O
with	O
autocorrelation	O
.	O
</s>
<s>
Then	O
is	O
the	O
value	O
(	O
or	O
realization	O
)	O
produced	O
by	O
a	O
given	O
run	B-General_Concept
of	O
the	O
process	O
at	O
time	O
.	O
</s>
<s>
If	O
is	O
a	O
wide-sense	O
stationary	B-Algorithm
process	I-Algorithm
then	O
the	O
mean	O
and	O
the	O
variance	O
are	O
time-independent	O
,	O
and	O
further	O
the	O
autocovariance	O
function	O
depends	O
only	O
on	O
the	O
lag	O
between	O
and	O
:	O
the	O
autocovariance	O
depends	O
only	O
on	O
the	O
time-distance	O
between	O
the	O
pair	O
of	O
values	O
but	O
not	O
on	O
their	O
position	O
in	O
time	O
.	O
</s>
<s>
The	O
autocorrelation	O
of	O
a	O
continuous-time	O
white	O
noise	B-Algorithm
signal	O
will	O
have	O
a	O
strong	O
peak	O
(	O
represented	O
by	O
a	O
Dirac	O
delta	O
function	O
)	O
at	O
and	O
will	O
be	O
exactly	O
for	O
all	O
other	O
.	O
</s>
<s>
The	O
Wiener	O
–	O
Khinchin	O
theorem	O
relates	O
the	O
autocorrelation	O
function	O
to	O
the	O
power	B-Algorithm
spectral	I-Algorithm
density	I-Algorithm
via	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
:	O
</s>
<s>
For	O
real-valued	O
functions	O
,	O
the	O
symmetric	B-Algorithm
autocorrelation	O
function	O
has	O
a	O
real	O
symmetric	B-Algorithm
transform	O
,	O
so	O
the	O
Wiener	O
–	O
Khinchin	O
theorem	O
can	O
be	O
re-expressed	O
in	O
terms	O
of	O
real	O
cosines	O
only	O
:	O
</s>
<s>
The	O
(	O
potentially	O
time-dependent	O
)	O
auto-correlation	O
matrix	O
(	O
also	O
called	O
second	O
moment	O
)	O
of	O
a	O
(	O
potentially	O
time-dependent	O
)	O
random	B-General_Concept
vector	I-General_Concept
is	O
an	O
matrix	O
containing	O
as	O
elements	O
the	O
autocorrelations	O
of	O
all	O
pairs	O
of	O
elements	O
of	O
the	O
random	B-General_Concept
vector	I-General_Concept
.	O
</s>
<s>
The	O
autocorrelation	B-Algorithm
matrix	I-Algorithm
is	O
used	O
in	O
various	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
algorithms	O
.	O
</s>
<s>
Here	O
denotes	O
Hermitian	B-Algorithm
transposition	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
if	O
is	O
a	O
random	B-General_Concept
vector	I-General_Concept
,	O
then	O
is	O
a	O
matrix	O
whose	O
-th	O
entry	O
is	O
.	O
</s>
<s>
The	O
autocorrelation	B-Algorithm
matrix	I-Algorithm
is	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
for	O
complex	O
random	B-General_Concept
vectors	I-General_Concept
and	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
for	O
real	O
random	B-General_Concept
vectors	I-General_Concept
.	O
</s>
<s>
The	O
autocorrelation	B-Algorithm
matrix	I-Algorithm
is	O
a	O
positive	B-Algorithm
semidefinite	I-Algorithm
matrix	I-Algorithm
,	O
i.e.	O
</s>
<s>
for	O
a	O
real	O
random	B-General_Concept
vector	I-General_Concept
,	O
and	O
respectively	O
in	O
case	O
of	O
a	O
complex	O
random	B-General_Concept
vector	I-General_Concept
.	O
</s>
<s>
All	O
eigenvalues	O
of	O
the	O
autocorrelation	B-Algorithm
matrix	I-Algorithm
are	O
real	O
and	O
non-negative	O
.	O
</s>
<s>
The	O
auto-covariance	O
matrix	O
is	O
related	O
to	O
the	O
autocorrelation	B-Algorithm
matrix	I-Algorithm
as	O
follows:Respectively	O
for	O
complex	O
random	B-General_Concept
vectors	I-General_Concept
:	O
</s>
<s>
For	O
processes	O
that	O
are	O
not	O
stationary	B-Algorithm
,	O
these	O
will	O
also	O
be	O
functions	O
of	O
,	O
or	O
.	O
</s>
<s>
For	O
processes	O
that	O
are	O
also	O
ergodic	B-Algorithm
,	O
the	O
expectation	O
can	O
be	O
replaced	O
by	O
the	O
limit	O
of	O
a	O
time	O
average	O
.	O
</s>
<s>
These	O
definitions	O
have	O
the	O
advantage	O
that	O
they	O
give	O
sensible	O
well-defined	O
single-parameter	O
results	O
for	O
periodic	O
functions	O
,	O
even	O
when	O
those	O
functions	O
are	O
not	O
the	O
output	O
of	O
stationary	B-Algorithm
ergodic	B-Algorithm
processes	O
.	O
</s>
<s>
(	O
See	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
for	O
a	O
related	O
process	O
.	O
)	O
</s>
<s>
These	O
properties	O
hold	O
for	O
wide-sense	O
stationary	B-Algorithm
processes	O
.	O
</s>
<s>
By	O
using	O
the	O
symbol	O
to	O
represent	O
convolution	B-Language
and	O
is	O
a	O
function	O
which	O
manipulates	O
the	O
function	O
and	O
is	O
defined	O
as	O
,	O
the	O
definition	O
for	O
may	O
be	O
written	O
as	O
:	O
</s>
<s>
For	O
data	O
expressed	O
as	O
a	O
discrete	O
sequence	O
,	O
it	O
is	O
frequently	O
necessary	O
to	O
compute	O
the	O
autocorrelation	O
with	O
high	O
computational	B-General_Concept
efficiency	I-General_Concept
.	O
</s>
<s>
then	O
we	O
get	O
a	O
circular	O
autocorrelation	O
(	O
similar	O
to	O
circular	B-Algorithm
convolution	I-Algorithm
)	O
where	O
the	O
left	O
and	O
right	O
tails	O
of	O
the	O
previous	O
autocorrelation	O
sequence	O
will	O
overlap	O
and	O
give	O
which	O
has	O
the	O
same	O
period	O
as	O
the	O
signal	O
sequence	O
The	O
procedure	O
can	O
be	O
regarded	O
as	O
an	O
application	O
of	O
the	O
convolution	B-Language
property	O
of	O
Z-transform	B-Algorithm
of	O
a	O
discrete	O
signal	O
.	O
</s>
<s>
While	O
the	O
brute	O
force	O
algorithm	O
is	O
order	O
,	O
several	O
efficient	B-General_Concept
algorithms	I-General_Concept
exist	O
which	O
can	O
compute	O
the	O
autocorrelation	O
in	O
order	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Wiener	O
–	O
Khinchin	O
theorem	O
allows	O
computing	O
the	O
autocorrelation	O
from	O
the	O
raw	O
data	O
with	O
two	O
fast	O
Fourier	B-Algorithm
transforms	I-Algorithm
(	O
FFT	O
)	O
:	O
</s>
<s>
where	O
IFFT	O
denotes	O
the	O
inverse	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
In	O
regression	O
analysis	O
using	O
time	O
series	O
data	O
,	O
autocorrelation	O
in	O
a	O
variable	O
of	O
interest	O
is	O
typically	O
modeled	O
either	O
with	O
an	O
autoregressive	B-Algorithm
model	I-Algorithm
(	O
AR	O
)	O
,	O
a	O
moving	O
average	O
model	O
(	O
MA	O
)	O
,	O
their	O
combination	O
as	O
an	O
autoregressive-moving-average	O
model	O
(	O
ARMA	O
)	O
,	O
or	O
an	O
extension	O
of	O
the	O
latter	O
called	O
an	O
autoregressive	B-Algorithm
integrated	O
moving	O
average	O
model	O
(	O
ARIMA	O
)	O
.	O
</s>
<s>
With	O
multiple	O
interrelated	O
data	O
series	O
,	O
vector	O
autoregression	B-Algorithm
(	O
VAR	O
)	O
or	O
its	O
extensions	O
are	O
used	O
.	O
</s>
<s>
In	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
(	O
OLS	O
)	O
,	O
the	O
adequacy	O
of	O
a	O
model	O
specification	O
can	O
be	O
checked	O
in	O
part	O
by	O
establishing	O
whether	O
there	O
is	O
autocorrelation	O
of	O
the	O
regression	O
residuals	O
.	O
</s>
<s>
Autocorrelation	O
of	O
the	O
errors	O
violates	O
the	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
assumption	O
that	O
the	O
error	O
terms	O
are	O
uncorrelated	O
,	O
meaning	O
that	O
the	O
Gauss	O
Markov	O
theorem	O
does	O
not	O
apply	O
,	O
and	O
that	O
OLS	O
estimators	O
are	O
no	O
longer	O
the	O
Best	O
Linear	O
Unbiased	O
Estimators	O
(	O
BLUE	O
)	O
.	O
</s>
<s>
While	O
it	O
does	O
not	O
bias	O
the	O
OLS	O
coefficient	O
estimates	O
,	O
the	O
standard	B-General_Concept
errors	I-General_Concept
tend	O
to	O
be	O
underestimated	O
(	O
and	O
the	O
t-scores	O
overestimated	O
)	O
when	O
the	O
autocorrelations	O
of	O
the	O
errors	O
at	O
low	O
lags	O
are	O
positive	O
.	O
</s>
<s>
The	O
traditional	O
test	O
for	O
the	O
presence	O
of	O
first-order	O
autocorrelation	O
is	O
the	O
Durbin	B-General_Concept
–	I-General_Concept
Watson	I-General_Concept
statistic	I-General_Concept
or	O
,	O
if	O
the	O
explanatory	O
variables	O
include	O
a	O
lagged	O
dependent	O
variable	O
,	O
Durbin	O
's	O
h	O
statistic	O
.	O
</s>
<s>
The	O
Durbin-Watson	B-General_Concept
can	O
be	O
linearly	O
mapped	O
however	O
to	O
the	O
Pearson	O
correlation	O
between	O
values	O
and	O
their	O
lags	O
.	O
</s>
<s>
Another	O
application	O
of	O
autocorrelation	O
is	O
the	O
measurement	O
of	O
optical	O
spectra	O
and	O
the	O
measurement	O
of	O
very-short-duration	O
light	O
pulses	O
produced	O
by	O
lasers	O
,	O
both	O
using	O
optical	O
autocorrelators	B-Algorithm
.	O
</s>
<s>
The	O
small-angle	O
X-ray	O
scattering	O
intensity	O
of	O
a	O
nanostructured	O
system	O
is	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
spatial	O
autocorrelation	O
function	O
of	O
the	O
electron	O
density	O
.	O
</s>
<s>
In	O
signal	O
processing	O
,	O
autocorrelation	O
can	O
give	O
information	O
about	O
repeating	O
events	O
like	O
musical	O
beats	O
(	O
for	O
example	O
,	O
to	O
determine	O
tempo	B-Application
)	O
or	O
pulsar	B-Application
frequencies	O
,	O
though	O
it	O
cannot	O
tell	O
the	O
position	O
in	O
time	O
of	O
the	O
beat	O
.	O
</s>
<s>
The	O
SEQUEST	B-Application
algorithm	O
for	O
analyzing	O
mass	O
spectra	O
makes	O
use	O
of	O
autocorrelation	O
in	O
conjunction	O
with	O
cross-correlation	O
to	O
score	O
the	O
similarity	O
of	O
an	O
observed	O
spectrum	O
to	O
an	O
idealized	O
spectrum	O
representing	O
a	O
peptide	O
.	O
</s>
<s>
In	O
astrophysics	O
,	O
autocorrelation	O
is	O
used	O
to	O
study	O
and	O
characterize	O
the	O
spatial	O
distribution	O
of	O
galaxies	B-Application
in	O
the	O
universe	O
and	O
in	O
multi-wavelength	O
observations	O
of	O
low	O
mass	O
X-ray	O
binaries	O
.	O
</s>
<s>
In	O
analysis	O
of	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
data	O
,	O
autocorrelation	O
must	O
be	O
taken	O
into	O
account	O
for	O
correct	O
error	O
determination	O
.	O
</s>
<s>
In	O
medical	B-Application
ultrasound	I-Application
imaging	O
,	O
autocorrelation	O
is	O
used	O
to	O
visualize	O
blood	O
flow	O
.	O
</s>
<s>
If	O
a	O
time	O
series	O
is	O
stationary	B-Algorithm
,	O
then	O
statistical	O
dependence	O
between	O
the	O
pair	O
would	O
imply	O
that	O
there	O
is	O
statistical	O
dependence	O
between	O
all	O
pairs	O
of	O
values	O
at	O
the	O
same	O
lag	O
.	O
</s>
