Notes on Stationarity- Sandra Chapman (MPAGS: Time series analysis)
Stationarity and stochastic processes
Stationarity
Implies that different realisations/samples are equivalent, useful since it follows that:
⇒ many realisations can be drawn in sequence from a single time series
- for stochastic processes this equivalence is statistical
- for deterministic processes this equivalence is repetition under time shift (periodicity)
⇒ implies time independence of power spectrum
a)
Strong/strict stationarity
for a sequence xk at two times k1 , k2 define
Fk1 , k 2 ( a1 , a2 ) = P [ xk1 ! a1 , xk 2 ! a2 ]
P is just the joint CDF – expresses the correlation structure of xk
Strong stationarity is then:
Fk1 , k2 ...k N ( a1 , a2 ...aN ) = Fk1 +! , k2 +! ...( a1...aN )
- completely stationary under time shift !
- not a practical definition!
b)
Weak/2nd order stationarity
all the joint moments of xk up to order 2 are same as that of xk +!
ie:
xk , xk2
are independent of ! .
Alternatively, a stationarity test is provided by the covariance:
1 N
cov ( x, y ) = " ( xk ! x )( yk ! y ) = ( xk ! x )( yk ! y ) .
N l
Now consider
cov ( xk1 , xk2 ) where k2 is just k1 shifted by !
= cov ( x0 , xk2 !k1 ) = cov ( x0 , x! )
This is only a function of ! , not position in sequence.
This is usually normalized to the autocovariance :
1
cov ( x0 , x! )
cov ( x0 , x0 )
Notes on Stationarity- Sandra Chapman (MPAGS: Time series analysis)
Spectral representation of stochastic processes (outline)
Issues are (i) defining an integral over a stochastic process and (ii) convergence of the integral for
random walks (sums of stochastic steps) which grow without bound (outline not formal proof!).
Consider a finite interval of a discrete stochastic process xk .
recall DFT
1 N "1
xk =
- Not stochastic
# S e2!imk / N
N!t m=0 m
Write instead for our stochastic process
1
xk =
N!t
N (1
)S
m
e
"
%
mk
$ 2!i +i"m '
$#
'&
N
.
m=0
This is a 'random phase' model for the stochastic process xk . The S m are real constants and the !m is
a stochastic, iid random variable.
1 N "1
Dm e2!imk / N where Dm = S m ei!m
Write this as xk =
#
N!t m=0
It follows that the statistical properties of the Dm are that of the !m , ie:
!m = 0
Dm = 0 and Dm2 = S m2
if
This follows since Dm Dm* = S m2 for each m, and the !m are uncorrelated.
We can think of the Dm as a stationary stochastic process. We can build a random walk (which is
not stationary) from stationary stochastic steps ie:
N !1
z ( f ) = " Dm
f m < f # f m+1
m=0
which defines stochastic increment dz ( f ) = z ( f + df ) ! z ( f ) ,
then dz ( f m ) = Dm , this is the "power" in f m , so in an appropriate continuous limit is equivalent to
z( f )df . The continuous limit is via the definition of a Riemann-Stieltjes integral.
Then the spectral representation theorem is:
1
2
1
!
2
x ( t ) = " e2!ift dz ( f
)
formally equivalent to xk =
1 N !1
$ D e2!ifmtk
N#t m=0 m
where dz is stochastic – this is a Riemann-Stieltjes integral.
All the usual results of Fourier theory follow (since we have an expansion in an orthogonal set). In
particular, we are interested in scaling – random walks (coloured noises) with power law power
spectra.
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