<s>
In	O
mathematics	O
and	O
statistics	O
,	O
a	O
stationary	B-Algorithm
process	I-Algorithm
(	O
or	O
a	O
strict/strictly	O
stationary	B-Algorithm
process	I-Algorithm
or	O
strong/strongly	O
stationary	B-Algorithm
process	I-Algorithm
)	O
is	O
a	O
stochastic	O
process	O
whose	O
unconditional	O
joint	O
probability	O
distribution	O
does	O
not	O
change	O
when	O
shifted	O
in	O
time	O
.	O
</s>
<s>
If	O
you	O
draw	O
a	O
line	O
through	O
the	O
middle	O
of	O
a	O
stationary	B-Algorithm
process	I-Algorithm
then	O
it	O
should	O
be	O
flat	O
;	O
it	O
may	O
have	O
'	O
seasonal	O
 '	O
cycles	O
,	O
but	O
overall	O
it	O
does	O
not	O
trend	O
up	O
nor	O
down	O
.	O
</s>
<s>
Since	O
stationarity	B-Algorithm
is	O
an	O
assumption	O
underlying	O
many	O
statistical	O
procedures	O
used	O
in	O
time	O
series	O
analysis	O
,	O
non-stationary	B-Algorithm
data	O
are	O
often	O
transformed	O
to	O
become	O
stationary	O
.	O
</s>
<s>
The	O
most	O
common	O
cause	O
of	O
violation	O
of	O
stationarity	B-Algorithm
is	O
a	O
trend	O
in	O
the	O
mean	O
,	O
which	O
can	O
be	O
due	O
either	O
to	O
the	O
presence	O
of	O
a	O
unit	O
root	O
or	O
of	O
a	O
deterministic	O
trend	O
.	O
</s>
<s>
A	O
trend	O
stationary	B-Algorithm
process	I-Algorithm
is	O
not	O
strictly	B-Algorithm
stationary	I-Algorithm
,	O
but	O
can	O
easily	O
be	O
transformed	O
into	O
a	O
stationary	B-Algorithm
process	I-Algorithm
by	O
removing	O
the	O
underlying	O
trend	O
,	O
which	O
is	O
solely	O
a	O
function	O
of	O
time	O
.	O
</s>
<s>
An	O
important	O
type	O
of	O
non-stationary	B-Algorithm
process	O
that	O
does	O
not	O
include	O
a	O
trend-like	O
behavior	O
is	O
a	O
cyclostationary	O
process	O
,	O
which	O
is	O
a	O
stochastic	O
process	O
that	O
varies	O
cyclically	O
with	O
time	O
.	O
</s>
<s>
For	O
many	O
applications	O
strict-sense	O
stationarity	B-Algorithm
is	O
too	O
restrictive	O
.	O
</s>
<s>
Other	O
forms	O
of	O
stationarity	B-Algorithm
such	O
as	O
wide-sense	O
stationarity	B-Algorithm
or	O
N-th-order	O
stationarity	B-Algorithm
are	O
then	O
employed	O
.	O
</s>
<s>
The	O
definitions	O
for	O
different	O
kinds	O
of	O
stationarity	B-Algorithm
are	O
not	O
consistent	O
among	O
different	O
authors	O
(	O
see	O
Other	O
terminology	O
)	O
.	O
</s>
<s>
White	O
noise	O
is	O
the	O
simplest	O
example	O
of	O
a	O
stationary	B-Algorithm
process	I-Algorithm
.	O
</s>
<s>
An	O
example	O
of	O
a	O
discrete-time	O
stationary	B-Algorithm
process	I-Algorithm
where	O
the	O
sample	O
space	O
is	O
also	O
discrete	O
(	O
so	O
that	O
the	O
random	O
variable	O
may	O
take	O
one	O
of	O
N	O
possible	O
values	O
)	O
is	O
a	O
Bernoulli	O
scheme	O
.	O
</s>
<s>
Other	O
examples	O
of	O
a	O
discrete-time	O
stationary	B-Algorithm
process	I-Algorithm
with	O
continuous	O
sample	O
space	O
include	O
some	O
autoregressive	B-Algorithm
and	O
moving	O
average	O
processes	O
which	O
are	O
both	O
subsets	O
of	O
the	O
autoregressive	B-Algorithm
moving	O
average	O
model	O
.	O
</s>
<s>
Models	O
with	O
a	O
non-trivial	O
autoregressive	B-Algorithm
component	O
may	O
be	O
either	O
stationary	O
or	O
non-stationary	B-Algorithm
,	O
depending	O
on	O
the	O
parameter	O
values	O
,	O
and	O
important	O
non-stationary	B-Algorithm
special	O
cases	O
are	O
where	O
unit	O
roots	O
exist	O
in	O
the	O
model	O
.	O
</s>
<s>
The	O
time	O
average	O
of	O
does	O
not	O
converge	O
since	O
the	O
process	O
is	O
not	O
ergodic	B-Algorithm
.	O
</s>
<s>
Then	O
is	O
strictly	B-Algorithm
stationary	I-Algorithm
since	O
(	O
modulo	O
)	O
follows	O
the	O
same	O
uniform	O
distribution	O
as	O
for	O
any	O
.	O
</s>
<s>
Keep	O
in	O
mind	O
that	O
a	O
white	O
noise	O
is	O
not	O
necessarily	O
strictly	B-Algorithm
stationary	I-Algorithm
.	O
</s>
<s>
So	O
is	O
a	O
white	O
noise	O
,	O
however	O
it	O
is	O
not	O
strictly	B-Algorithm
stationary	I-Algorithm
.	O
</s>
<s>
N-th-order	O
stationarity	B-Algorithm
is	O
a	O
weaker	O
form	O
of	O
stationarity	B-Algorithm
where	O
this	O
is	O
only	O
requested	O
for	O
all	O
up	O
to	O
a	O
certain	O
order	O
.	O
</s>
<s>
A	O
weaker	O
form	O
of	O
stationarity	B-Algorithm
commonly	O
employed	O
in	O
signal	O
processing	O
is	O
known	O
as	O
weak-sense	O
stationarity	B-Algorithm
,	O
wide-sense	O
stationarity	B-Algorithm
(	O
WSS	O
)	O
,	O
or	O
covariance	O
stationarity	B-Algorithm
.	O
</s>
<s>
Any	O
strictly	B-Algorithm
stationary	I-Algorithm
process	O
which	O
has	O
a	O
finite	O
mean	O
and	O
a	O
covariance	O
is	O
also	O
WSS	O
.	O
</s>
<s>
The	O
main	O
advantage	O
of	O
wide-sense	O
stationarity	B-Algorithm
is	O
that	O
it	O
places	O
the	O
time-series	O
in	O
the	O
context	O
of	O
Hilbert	O
spaces	O
.	O
</s>
<s>
The	O
same	O
result	O
holds	O
for	O
a	O
discrete-time	O
stationary	B-Algorithm
process	I-Algorithm
,	O
with	O
the	O
spectral	O
measure	O
now	O
defined	O
on	O
the	O
unit	O
circle	O
.	O
</s>
<s>
When	O
processing	O
WSS	O
random	O
signals	O
with	O
linear	O
,	O
time-invariant	B-Algorithm
(	O
LTI	O
)	O
filters	O
,	O
it	O
is	O
helpful	O
to	O
think	O
of	O
the	O
correlation	O
function	O
as	O
a	O
linear	B-Architecture
operator	I-Architecture
.	O
</s>
<s>
Since	O
it	O
is	O
a	O
circulant	O
operator	O
(	O
depends	O
only	O
on	O
the	O
difference	O
between	O
the	O
two	O
arguments	O
)	O
,	O
its	O
eigenfunctions	B-Algorithm
are	O
the	O
Fourier	O
complex	O
exponentials	O
.	O
</s>
<s>
Additionally	O
,	O
since	O
the	O
eigenfunctions	B-Algorithm
of	O
LTI	O
operators	O
are	O
also	O
complex	O
exponentials	O
,	O
LTI	O
processing	O
of	O
WSS	O
random	O
signals	O
is	O
highly	O
tractable	O
—	O
all	O
computations	O
can	O
be	O
performed	O
in	O
the	O
frequency	O
domain	O
.	O
</s>
<s>
The	O
concept	O
of	O
stationarity	B-Algorithm
may	O
be	O
extended	O
to	O
two	O
stochastic	O
processes	O
.	O
</s>
<s>
The	O
terminology	O
used	O
for	O
types	O
of	O
stationarity	B-Algorithm
other	O
than	O
strict	B-Algorithm
stationarity	I-Algorithm
can	O
be	O
rather	O
mixed	O
.	O
</s>
<s>
Priestley	O
uses	O
stationary	O
up	O
to	O
order	O
m	O
if	O
conditions	O
similar	O
to	O
those	O
given	O
here	O
for	O
wide	O
sense	O
stationarity	B-Algorithm
apply	O
relating	O
to	O
moments	O
up	O
to	O
order	O
m	O
.	O
Thus	O
wide	O
sense	O
stationarity	B-Algorithm
would	O
be	O
equivalent	O
to	O
"	O
stationary	O
to	O
order	O
2	O
"	O
,	O
which	O
is	O
different	O
from	O
the	O
definition	O
of	O
second-order	O
stationarity	B-Algorithm
given	O
here	O
.	O
</s>
<s>
Honarkhah	O
and	O
Caers	O
also	O
use	O
the	O
assumption	O
of	O
stationarity	B-Algorithm
in	O
the	O
context	O
of	O
multiple-point	O
geostatistics	O
,	O
where	O
higher	O
n-point	O
statistics	O
are	O
assumed	O
to	O
be	O
stationary	O
in	O
the	O
spatial	O
domain	O
.	O
</s>
<s>
Tahmasebi	O
and	O
Sahimi	O
have	O
presented	O
an	O
adaptive	O
Shannon-based	O
methodology	O
that	O
can	O
be	O
used	O
for	O
modeling	O
of	O
any	O
non-stationary	B-Algorithm
systems	O
.	O
</s>
<s>
One	O
of	O
the	O
ways	O
for	O
identifying	O
non-stationary	B-Algorithm
times	O
series	O
is	O
the	O
ACF	O
plot	O
.	O
</s>
<s>
Another	O
approach	O
to	O
identifying	O
non-stationarity	O
is	O
to	O
look	O
at	O
the	O
Laplace	O
transform	O
of	O
a	O
series	O
,	O
which	O
will	O
identify	O
both	O
exponential	O
trends	O
and	O
sinusoidal	O
seasonality	O
(	O
complex	O
exponential	O
trends	O
)	O
.	O
</s>
<s>
Related	O
techniques	O
from	O
signal	O
analysis	O
such	O
as	O
the	O
wavelet	B-Algorithm
transform	I-Algorithm
and	O
Fourier	B-Algorithm
transform	I-Algorithm
may	O
also	O
be	O
helpful	O
.	O
</s>
