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2.2 Time Series Analysis
2.2.1 Preliminaries
2.2.2 Various Types of Stochastic Processes
2.2.3 Parameters of Univariate and Bivariate Time Series
2.2.4 Estimating Covariance Functions and Spectra
2.2 Time Series Analysis
The central statistical model used in time series analysis is stochastic processes
A stochastic process is an ordered set of random variables,
indexed with an integer t, which usually represents time. A time series is realization of a stochastic process
},{ Z∈tX t
Definition
2.2 Time Series Analysis
Example: White Noise
A simplest stochastic process is a white noise, which is an infinite sequence of zero mean iid normal random variable
5.0
)(
)()(
)(
),(
)0|0( nonzeroany for i.e. memory, no has noise A white
0
00
0
0 0
=
×=
=
>>
∫
∫∫
∫
∫ ∫
∞
∞∞
∞
∞ ∞
+=
dxxf
dyyfdxxf
dxxf
dxdyyxf
XXP
t
ts
t
ts
tts τ
τ
2.2 Time Series Analysis
Properties of a Stochastic Process
A stochastic process is said to be stationary if all stochastic properties are independent of index t. It follows
• Xt has the same distribution function
• for all t and s, the parameters of the joint distribution function of X1 and Xs depend only |t-s|
},{ Z∈tX t
A stochastic process is weakly stationary, if the mean E(Xt) is independent of time and the second moments E(Xt,Xs) are functions of |t-s| only
• The assumption of week stationary is less restrictive than that of stationary and is often sufficient for the methods used in climate research
A stochastic process is weakly cyclo-stationary, if the mean is a function of the time within a deterministic cycle and the central second moments are functions of |t-s| and the phase of the cycle
2.2 Time Series Analysis
Example of Non-stationary
1958-1977 time series of monthly mean atmospheric CO2 concentration measured at the Mauna Loa Observatory in Hawaii. The time series can be considered as a superposition of a stationary process Xt , a linear trend αt and an oscillation with period of 12 months
2.2 Time Series Analysis
Example: Random Walk
Given a white noise Zt, is a random walk. Xt is non-stationary in variance:
∑=
=t
jjt ZX
1
22
11,
2
1
1
)()()(
0)(
σtZEZZEZEXVar
ZEXE
j
t
j
t
kjKj
t
jjt
t
jjt
===⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
=⎟⎟⎠
⎞⎜⎜⎝
⎛=
∑∑∑
∑
===
=
If an ensemble of random walks is considered, the center of gravity will not move, but the scatter increases continuously
2.2 Time Series Analysis
Ergodicity
Ergodicity has to be assumed, since stationary or weak stationary alone is not enough to ensure that the moments of a process can be estimated from a single time series
Definition (loose)
A time series is ergodic, if it varies quickly enough in time that increasing amounts of information about process parameters can be obtained by extending the time series
Example of a Non-Ergodic Process
xt=a, where a is a realization of a random variable A with mean µ
µ==≠== ∑=
)(average ensemble1 average time1
AEaxn
n
tt
2.2 Time Series Analysis
Various Types of Stochastic Processes
We will concentrate on the auto-regressive (AR) processes as the most relevant type of stochastic processes for climate research, and discuss briefly their relation to a more general class of short memory processes, the auto-regressive moving average (ARMA) processes, and the relation to a class of long memory processes, the fractional auto-regressive integrated moving average (ARIMA) processes.
2.2 Time Series Analysis
Definition: AR-Processes
{ } is an AR-process of order p, if there exist real constants αk, k=0,…,p, with and a white noise process { } such that
Z∈tX t ,Z∈tZt ,0≠pα
tkt
p
kkt ZXX ++= −
=∑
10 αα
• AR-processes are important, since given any weakly stationary ergodic process {Xt}, it is possible to find an AR-process {Yt}, that approximates {Xt} arbitrarily closely
• AR-processes are popular, since they represent discretized ordinary differential equations
• The mean and variance are
where ρ (k) is the auto-correlation function
• Yt=Xt-µ is a process with zero mean
∑∑==
−=
−== p
kk
ttp
kk
ot
k
ZVarXVarXE
11)(1
)()( ,1
)(ραα
αµ
2.2 Time Series Analysis
AR(1)-Process: Xt=α1Xt-1+Zt
• AR(1)-Processes (p=1) have only one degree of freedom unable to oscillate
One finds
Thus, the variance of the process is a linear function of the variance of the input white noise Zt and a non-linear function of the parameter α1.
)1()( , 2
1
2
11 ασαρ−
== ztXVar
• A non-zero value of xt at time t tends to be damped with an average damping rate of α1 per time step• An AR(1)-process with a negative coefficient will flip around zero. Such a process is considered to be inappropriate for the description of a climate series
2.2 Time Series Analysis
AR(2)-Process: Xt=α1Xt-1+α2Xt-2+Zt
AR(2)-processes (p=2) have only two degrees of freedom and can oscillate with one preferred frequency
The AR(2)-process with α1=0.9 and α2=-0.8 exhibits quasi-periodic behavior with a period of about 6 time steps
The AR(2)-process with α1=0.3 and α2=0.3 has behavior comparable to that of an AR(1)-process with long memory
2.2 Time Series Analysis
Stationarity of AR-Processes
AR-processes can be non-stationary. An AR(1) process with α1=2 and µ=0 is stationary with respect to the mean but non-stationary with respect to the variance, since one has for Xt starting from X0
⎟⎠⎞
⎜⎝⎛ −=+==
+=
∑
∑
=
−
=
−
t
tt
i
ittt
tt
t
ii
ittt
ZVarXVarXVarXEXE
ZXX
411
34)(4)(4)( ),2()(
22
100
10
An AR(p)-process with AR coefficients αk, k=1,…p, is stationary if and only if all roots of the characteristic polynomial
lie outside the circle |y|=1
kp
kk yyp ∑
=
−=1
1)( α
• The characteristic polynomial has p roots, yi, i=1,…,p
• They can be real or appear in complex conjugate pairs
2.2 Time Series Analysis
Condition for a stationary AR(1) process
An AR(1)-process is stationary, if α1<1
111 /1 ,01)( αα =→=−= yyyp
Condition for a stationary AR(2) process
• An AR(2)-process is stationary, if α2+α1<1, α2-α1<1,|α2|<1
• AR(2) coefficients that satisfy these conditions lie in the triangle depicted in the figure
2
2211
2,12
21 24
,01)(α
ααααα
+±−==−−= yyyyp
Region where the characteristic polynomials have a pair of conjugate roots
Region where the characteristic polynomials have two real solutions
α12+4α2=0
2.2 Time Series Analysis
More about the roots of the characteristic polynomial: The roots of a characteristic polynomial describe the ‘typical temporal behavior’ of the corresponding process
Each root yi identifies a set of ‘typical initial conditions that lead to Xt=1 when the noise is disregarded. Since these initial conditions’ are linearly independent, any set of states (Xt-1,…,Xt-p) can be represented as a linear combination of the initial states. In the absence of noise, the future evolution of these will be
∑=
−+ =
p
jiit yX
1
ττ β
Let yi, i=1,…,p be the roots of characteristic polynomial p(y). Given a fixed i, set
substitute these values into the corresponding process
disregard the noise yields
.,1,, pkyX kiikt L==−
tkt
p
kkt ZXX += −
=∑
1α
1=tX
Example I
An AR(1) process has only one solution y1=1/a1,
ττ αβ 11=+tX
Example II
The AR(2) process with (a1,a2)=(0.3,0.3) has solutions y1=1.39 and y2=-2.39
τττ ββ −−
+ += 2211 yyX t
Example III
The AR(2) process with (a1,a2)=(0.9,-0.9) has solutions y1=r exp( iφ), r=1.11, φ=π/3
)exp(1
1*
1112211
φτ
ββββττ
τττττ
iry
yyyyX t
−=
+=+=−−
−−−+
±
2.2 Time Series Analysis
The ‘typical temporal behavior’ of an AR process in the absence of the noise is characterized by damped modes with or without oscillations
• The damping is necessary, since the presence of the noise makes the process non-stationary (random walk)
• The rate of damping is determined by the amplitude of the roots of the corresponding characteristic polynomial
• complex roots lead to oscillations
2.2 Time Series Analysis
Definition: Moving Average (MA) Processes
A process is said to be a moving average process of order q (MA(q)), if
process noise whitea is };{ .3
0such that constants are ,, 2.process theofmean theis 1.
where
q1
1
Z∈
≠
++= ∑=
−
tZ
ZZX
t
q
X
q
lltltXt
βββµ
βµ
L
},{ Z∈tX t
An MA process is stationary with mean µx and variance
⎟⎟⎠
⎞⎜⎜⎝
⎛+= ∑
=
q
lltt ZVarXVar
1
21)()( β
2.2 Time Series Analysis
Definition: Auto-regressive Moving Average (ARMA) Processes
A process is said to be an ARMA process of order (p,q), if},{ Z∈tX t
process noise whitea is };{ .3
0,0such that constants are ,, and ,, 2.process theofmean theis 1.
where
qp11
11
Z∈
≠≠
+=−− ∑∑=
−=
−
tZ
ZZXX
t
qp
X
q
lltlt
p
iitiXt
βαββααµ
βαµ
LL
There is a substantial overlap between the class of MA, AR, and ARMA models
•Any weakly stationary ergodic process can be approximated arbitrarily closely by any of the three types of models
• The ARMA models can approximate the behavior of a given weakly stationary ergodic process to a specific level of accuracy with fewer parameters than a pure AR or MA model does.
2.2 Time Series Analysis
Backward shift operator B
• B acts on the time index of the stochastic process. It is defined by
and satisfies
• AR, MA and ARMA can all formally be written in terms of B. Specifically define the AR operator
and the MA operator
AR, MA and ARMA processes are then formally stochastic processessatisfying
∑=
−=p
i
iiBB
11)( αφ
jq
jj BB ∑
=
+=1
1)( βθ
[ ] 1−= tt XXB 1)1( −−=− ttt XXXB
(ARMA) )()((MA) )( (AR) )(
tt
tt
tt
ZBXBZBX
ZXB
θφθ
φ
===
Why B?
• provide the tool needed to explore the connections between AR and MA models
• introduce other classes of models
2.2 Time Series Analysis
Definition: Auto-regressive-integrated Moving Average (ARIMA) Processes
•A process is said to be an ARIMA process of order (p,q,d), if the dth difference of Xt, (1-B)dXt, satisfies the ARMA operator of order (p,q), I.e.
•If –1/2<d<1/2, Xt is called a fractional ARIMA process
• An ARIMA process with a positive integer d is generally not stationary, whereas a fractional ARIMA process can be stationary
• Fractional ARIMA processes are also known as long-memory processes
},{ Z∈tX t
ttd ZBXBB )()1)(( θφ =−
2.2 Time Series Analysis
Parameters of Time Series: The Auto-covariance Function
Let Xt be a real or complex-valued stationary process with mean µ. Then
is the auto-covariance function of Xt, and the normalized function
is the auto-correlation function of Xt. The argument τ is the lag. An auto-correlation function has the properties
),()))((()( *ττ µµτγ ++ =−−= tttt XXCovXXE
)0()()(
γτγτρ =
1|)(| ),()( ≤−= τρτρτρ
The auto-covariance function and the auto-correlation function have the same shape, but differ in their units. The former is in units of Xt
2, the latter is dimensionless
Example:
Auto-covariance function of a white noise
⎩⎨⎧ =
=otherwise 0,
0 ,1)(
ττρ
2.2 Time Series Analysis
The Yule-Walker equations for an AR(p) process
If we multiply a zero mean AR(p) process Xt by Xt-τ for τ=1,…,p,
and take expectations, we obtain a system of equations
that are known as the Yule-Walker equations. The equation relates the auto-covariances
at lag τ=1,…,p to the process parameters
and the auto-covariances γ(τ) at lags τ=0,…,p-1 through the pXp matrix
∑=
−−−− +=p
itttititt XZXXXX
1τττ α
ppp γα rr=Σ
Tp p))(,),2(),1(( γγγγ Lr
=
Tpp ),,,( 21 αααα L
r=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
−−
=Σ
)0()2()1(
)2()0()1()1()1()0(
γγγ
γγγγγγ
L
MOMM
L
L
pp
pp
p
2.2 Time Series Analysis
The Yule-Walker equations can be used to build an AR model
If covariances γ(τ),τ=0,…,p are known (e.g. have been estimated from data), the parameters of the AR(p) process can be determined by solving the Yule-Walker equations for α1,…,αp.
The Yule-Walker equations can be used to determine auto-covariance functions
If is known, the Yule-Walker equations can be solved for γ(1),…,γ(p), given the variance of the process, γ(0). The full auto-covariance function can be derived by recursively extending the Yule-Walker equations. This is done by multiplying the original AR process by Xt-τ for τ p to obtain
pαr
≥
∑=
−=p
kk k
1)()( τγατγ
2.2 Time Series Analysis
Uniqueness of the AR(p) Approximation to an Arbitrary StationaryProcess
The following theorem is useful when fitting an AR(p) process to an observed time series
Let Xt be a stationary process with auto-correlation function ρ. For each p>0 there is a unique AR(p) process with auto-correlation function ρp such that
The parameters of the approximating process of order p are recursively related to those of the approximating process of order p-1 by
where
starting from α1,1=ρ(1).
pp ≤= ||for )()( ττρτρ
Tpppp ),,( ,1, ααα L
r=
1,,1 )(),1(,),1(, −=−= −−− pkkppppkpkp Lαααα
∑
∑−
=−−
−
=−
−−
−−= 1
1)(),1(
1
1),1(
,
)(1
)()(
p
kkpp
p
kkp
pp
kp
kpp
ρα
ραρα
2.2 Time Series Analysis
Auto-covariance and auto-correlation functions of some AR(p) processes
p=1:
The Yule-Walker equation is
τατραρ
γγα
11
1
)( ,(1) )1()0(
==⇒
=
p=2:
The Yule-Walker equation is
2
222
21
2
1
2
1
1)2(
-1(1)
)2()1(
)0()1()1()0(
ααααρ
ααρ
γγ
αα
γγγγ
−+−
=
=⇒
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
Auto-correlation functions of two AR(1) processes with α1=0.3 (hatched) and 0.9 (solid)
Auto-correlation functions of two AR(2) processes with (α1 α2)=(0.3,0.3) (hatched) and (0.9,-0.8) (solid)
•The auto-covariance function is a sum of auto-covariance functions of AR(1) and AR(2) processes
•The weak stationary assumption ensures that |yk|>1 for all k. Thus, each real root contributes a component to the auto-correlation function that decays exponentially and each pair of complex conjugate roots contributes an exponentially damped oscillation
2.2 Time Series Analysis
The General Form of the Auto-correlation Function of an AR(p) Process
•The auto-correlation function of a weakly stationary AR(p) process can beexpressed as
where yk,k=1,…,p, are the roots of the characteristic polynomial and are either real or appear in complex conjugate pairs. ak can be derived from the process parameters . When yk is real (complex), the corresponding ak is complex.
τ
ττ
τ
ψτφ
τρ
⎟⎟⎠
⎞⎜⎜⎝
⎛∝
++=
=
∑∑
∑=
−
||1)cos(
)(
i
1
||
kk
kk
kk
i
i
p
kkk
yra
ya
ya
pαr
Definition I (H. von Storch & F. Zwiers)
Let Xt be an ergodic weakly stationary stochastic process with auto-covariance function γ(τ),τ=…,-1,0,1,…. The the spectrum (or power spectrum) Γ of Xt is the Fourier transform F of the auto-covariance function γ, i.e.
for all .
∑∞
−∞=
−==Γτ
τωπτγωγω ie 2)()}({)( F
]2/1,2/1[−∈ω
2.2 Time Series Analysis
Parameter of Time Series: Spectrum
The Fourier transform F{} is a mapping from a set of discrete summable series to the set of real functions defined on the interval [-1/2,1/2]. If s is a summable discrete series, its Fourier transform F{s} is a function that takes, for all , the value
The Fourier transform mapping is invertible,
]2/1,2/1[−∈ω
∑∞
−∞=
−=j
ijjess ωπω 2)}({F
ωω ωπ dess ijj ∫
−
=2/1
2/1
2)}({F
2.2 Time Series Analysis
Properties of Γ(ω)
• Γ(ω) of a real-valued process is symmetric:
• Γ(ω) is continuous and differentiable for
•
• γ(τ) can be obtained using the inverse Fourier transform:
• Γ(ω) describes the distribution of variance across time scales
• Γ(ω) is a linear function of the auto-covariance function. That is, if γ(τ)=a1γ1(τ)+a2γ2(τ), then
]2/1 ,2/1[−∈ω
0|)(0 =
Γ=ωω
ωd
d
)()( ωω −Γ=Γ
ωωτγ πωτ de i∫−
Γ=2/1
2/1
2)()(
∫Γ==172
0
)(2)0()( ωγtXVar
)()()( 2211 ωωω Γ+Γ=Γ aa
2.2 Time Series Analysis
Parameter of Time Series: Spectrum
Definition II (Koopmans, Brockwell & Davis)
• A stochastic process is represented as the inverse Fourier transform of a random complex valued function (or measure) Zω that is defined in the frequency domain
(i.e. a stochastic process is a sum of random oscillations)
• The spectrum is defined as the expectation of the squared modulus of the random spectral measure Zω
• The auto-covariance function is shown to be the inverse Fourier transform of the spectrum
ωπωτω deZX i
t ∫= 2
2.2 Time Series Analysis
Consider a periodic weakly stationary process
where ωj=1/Tj, i=-n,…,n, Tj Z, Zj is complex random number statisfying Zj=Z-j*.
tin
njjt
jeZX πω2∑−=
=
∈
The first and second moments of the process are
Thus, Xt is weakly stationary, if
• Zj, j=1,…,n have zero mean
• Zj is uncorrelated with Zk with j k
tin
nj
i
kjkj
n
nj
ij
tt
n
nj
tijtX
kjjj
j
eeZZEeZE
XXE
eZEXE
)(22*22
2
)()|(|
)()(
)()(
ωωπτπωτπω
τ
πω
τγ
µ
−
−= ≠−=
+
−=
∑∑∑
∑
+=
=
==
≠
Under the condition of weak stationary, one has
∞=
+==
∑
∑∑
=∞→
=−=
τ
τ
τπω
τγ
τπωτγ
1
1
2222
)(lim
)2cos()|(|)|(|)|(|)(
j
n
jjjo
in
njj ZEZEeZE j
The spectrum is a line spectrum. It is discrete rather than continuous!
A long memory process
2.2 Time Series Analysis
Definition I is more suitable for short memory process
γ(τ) decays sufficiently fast with increasing, so that
variance attributes to a continuous range of frequency, rather than a few discrete frequencies, since otherwise we would have a process with a periodic covariance function and hence infinite memory.
∑=
∞→∞<<
τ
ττγ
1)(lim
j
Definition II is more suitable for long memory processes or processes with distinct oscillations
γ(τ) decays not sufficiently fast with increasing τ, so that γ(τ) is not summable
Variance attributes to a few discrete frequencies
∑=
∞→∞=
τ
ττγ
1)(lim
j
2.2 Time Series Analysis
The Spectrum of AR(p) and MA(q) Processes
The spectrum of an AR(p) process with process parameters {α1,…,αp} and noise variance Var(Zt)=σ2
Z is
221
2
|1|)(
ωπασω
ikp
k k
Z
e−=∑−
=Γ
The spectrum of an MA(q) process with process parameters {β1,…,βq} and noise variance Var(Zt)=σ2
Z is
ωπβσω ilq
l lZ e 21
2 1|)( −=∑+=Γ
2.2 Time Series Analysis
The Spectrum of a White Noise Process:2)( Zσω =Γ
The Spectrum of an AR(1) Process:
No extremes in the interior of [0,1/2], since
When α1>0, the spectrum speak is located at frequency ω=0. Such processes are referred to as red noise processes
⎪⎪⎩
⎪⎪⎨
⎧
−>>
−<<
−=
<<+−
≈
−+=
−=Γ −
1
212
21
21
212
21
2
21
21
21
21
2
221
2
)1()(2for )2(
)1()(2for )1(
1)(2for )2()1(
)2cos(21
|1|)(
ααπω
πωασ
ααπω
ασ
πωπωαα
σ
πωαασ
ασω ωπ
Z
Z
Z
Z
iZ
e
0)2sin()(2)( 21 ≠Γ−=Γ πωωαω
ωdd
Power spectra of AR(1) processes with α1=0.3(L) and α1=0.9(R)
2.2 Time Series Analysis
Plotting Formats
Power spectra of AR(1) processes with α1=0.3(L) and α1=0.9(R)
The same spectra plotted in log-log format, that
• Emphasizes low-frequency variations
• emphasizes certain power law
• but is not variance conserving
2.2 Time Series Analysis
Spectrum of an AR(2) Process
Power spectrum of an AR(2) process with parameters (α1,α2) is
where
The spectrum has an extreme for ω (0,1/2) when
It is a maximum when α2<0 and a minimum when α2>0
∈
)(21)( 2
221
2
ωαασω
gZ
−++=Γ
)4cos()2cos()1()( 221 πωαπωααω +−=g
||4|)1(| 221 ααα <−
Power spectra of AR(2) processes with (α1,α2) =(0.3,0.3) (L) and (α1,α2) =(0.9,-0.8) (R). The former has a minimum at ω~0.28, while the latter has a maximum at ω~0.17
2.2 Time Series Analysis
The general form of the spectra of AR processes
Since the auto-covariance function is a sum of auto-covariance functions of AR(1) and AR(2) processes, and since the Fourier transform is linear, the spectrum of an AR(p) process is the Fourier transform of the sum of auto-covariance functions of AR(1) and AR(2) processes and hence the sum of auto-spectra of AR(1) and AR(2) processes
Interpretation of spectra of AR processes
The exponential decay of auto-covariance functions of AR(p) processes implies that the spectra of AR(p) processes are continuous. A peak in the spectra cannot reflect the presence of an oscillatory component, even though it may indicate the presence of damped eigen-oscillations with eigen-frequencies close to that of the peak
2.2 Time Series Analysis
Parameters of Time Series: The Cross-covariance Function
Let (Xt,Yt) represent a pair of stochastic processes that are jointly weakly stationary. Then the cross-covariance function γxy is given by
Where µx is the mean of Xt and µy is the mean of Yt, and the cross-correlation function ρxy is given by
Where σx and σy are the standard deviations of processes Xt and Yt, respectively
)))((()( *YtXtxy YXE µµτγ τ −−= +
YXxyxy σστγτρ /)()( =
To ensure that the cross-correlation function exists and is absolutely summable, one needs to assure
•The mean µx and µy are independent of time
•
•
|)(|)))((( |),(|)))(((
|)(|)))((( |),(|)))(((
stXYEstYXE
stYYEstXXE
yxYsYtxyXsXt
yyYsYtxxXsXt
−=−−−=−−
−=−−−=−−
γµµγµµ
γµµγµµ
yyxyxxabab ,,for |)(| =∞<∑∞=
−∞=
τ
τ
τγ
Properties
)()()(.3
)0()0(|)(|.2
)()( .1
*,
*
ταγτγαβτγ
γγτγ
τγτγ
βα xzxyzyx
yyxxyx
xyyx
+=
≤
−=
+
2.2 Time Series Analysis
Examples:
Cross-correlation between Xt and Yt=αXt)()( ταγτγ xxxy = symmetric!
Cross-correlation between Xt and Yt=αXt+Zt with Zt being independent white noise
)()(0 if ),(0 if ,)0(
)( 2
22
ταγτγττγατσγα
τγ
xxxy
xx
zxxyy
=⎩⎨⎧
≠=+
=
Cross-correlation between Xt and Yt=Xt+1-Xt
⎟⎠⎞
⎜⎝⎛ ≈
−−=
)()(
)1()()(
τγτ
τγ
τγτγτγ
xxxy
xxxxxy
dd
2.2 Time Series Analysis
Examples:
Cross-correlation between an AR(1) process and its driving noise
0for 0)()(0for )( 2
1
<=−=≥=
ττγτγτσατγ τ
ZXXZ
zXZ
Highly non-symmetric!
Estimated cross-correlation functions between two monthly indices of SST and SLP over the North Pacific : one from data (thin line) and the other from a stochastic climate model (heavy line)
z leadsx leads
2.2 Time Series Analysis
The Effect of Feedbacks on Cross-correlation Functions
The continuous version of an AR(1) process is a first-order differential equation
Where the ‘forcing’ Zt acts on Xt without feedback.
ttt ZX
tX
+−=∂
∂ λ
A system with feedback can be written as
• λa=0: no feedback, ρxz=0 when X leads
• λa>0: negative feedback, ρxz is anti-symmetric
• λa<0: positive feedback, ρxz is positive everywhere with a maximum near lag zero
zttat
xttt
t
NXZ
NZXt
X
+−=
++−=∂
∂
λ
λ
2.2 Time Series Analysis
The Effect of Feedbacks on Cross-correlation Functions
Predicted correlation between Z (monthly mean turbulent heat flux) and X (monthly mean sea surface temperature) for different feedbacks (Frankignoul 1985)
Estimated correlation between Z (monthly mean turbulent heat flux) and X (monthly mean sea surface temperature), averaged over different latitudinal bands in the Atlantic ocean
2.2 Time Series Analysis
Parameters of Time Series: The Cross-spectrum
Definition:
Let Xt and Yt be two weakly stationary stochastic processes with covariance functions γxx and γyy, and a cross-covariance function γxy . Then the cross-spectrum Γxy is defined as the Fourier transform of γxy:
The cross-spectrum is generally a complex-values function, since the cross-covariance function is neither strictly symmetric nor anti symmetric.
[-1/2,1/2] , )()}({)( 2 ∈==Γ −∞=
−∞=∑ ωτγωγω τωπτ
τ
ixyxyxy eF
2.2 Time Series Analysis
The cross-spectrum can be represented in different ways
1. The cross-spectrum can be decomposed into its real and imaginary parts as
2. The cross-spectrum can be written in polar coordinates as
3. The (squared) coherence spectrum as dimensionless amplitude spectrum
spectrum quadrature the:)( spectrum,-co the:)(
)()()(
ωω
ωωω
xyxy
xyxyxy i
ΨΛ
Ψ+Λ=Γ
0)( when 0)( if 2/0)( if 2/
)(
0)(hen w0)( if 0)( if 0
)(
0)(,0)( when ))(/)((tan)(
))()(()(
spectrum phase the:)( spectrum, amplitude the:)(
)()(
xy
xy
xy
xy
1
2/122
)(
=Λ⎩⎨⎧
<Ψ−>Ψ
=Φ
=Ψ⎩⎨⎧
<Λ±>Λ
=Φ
≠Λ≠ΨΛΨ=Φ
Ψ+Λ=
Φ
=Γ
−
Φ
ωωπωπ
ω
ωωπω
ω
ωωωωω
ωωω
ωω
ωω ω
xyxy
xyxy
xyxyxyxyxy
xyxyxy
xyxy
ixyxy
A
A
eA xy
)()()(
)(2
ωωω
ωκyyxx
xyxy
AΓΓ
=
For jointly weakly stationary processes Xt, Yt and Zt
2.2 Time Series Analysis
Properties of the Cross-spectrum
1)(0 .3
)()( .2
)()()( .12/1
2/1
2
*,
≤≤
Γ=
Γ+Γ=Γ
∫−
+
ωκ
ωωτγ
ωαωαβω
πτω
βα
xy
ixyxy
xzxyzyx
de
2.2 Time Series Analysis
Properties of the Cross-spectrum of Real Weakly Stationary Processes
1. The co-spectrum is the Fourier transform of the symmetric part, γxys(τ), and the
quadrature spectrum is the Fourier transform of the anti-symmetric part of the cross-covariance spectrum, γxy
a(τ)
2. 2
3. The amplitude spectrum is positive and symmetric, and the phase spectrum is anti-symmetric
4. The coherence spectrum is symmetric
5. It is sufficient to consider spectra for positive ω
)()( ),()( ωωωω −Ψ−=Ψ−Λ=Λ xy
( ) ( ))()(21)( ,)()(
21)(
with
)2sin()(2)( ),2cos()(2)0()(11
τγτγτγτγτγτγ
πτωτγωπτωτγγωττ
−−=−+=
−=Ψ+=Λ ∑∑∞
=
∞
=
xyxyaxyxyxy
sxy
axyxy
sxyxyxy
2.2 Time Series Analysis
Examples:
Cross-spectrum between Xt and Yt=αXt
1)(
0)(
)()(
0)(
)()(
)()(
)()(2
=
=Φ
Γ=Α
=Ψ
Γ=Λ
Γ=Γ
Γ=Γ
ωκ
ω
ωαω
ω
ωαω
ωαω
ωαω
xy
xy
xxxy
xy
xxxy
xxyy
xxxy
Cross-spectrum between Xt and Yt=αXt+Zt being an independent white noise
1)(
)()(
)()(
22
2
22
<Γ+
Γ=
+Γ=Γ
ωασωαωκ
σωαω
xxz
xxxy
zxxyy
Cross-spectrum between Xt and Yt=Xt+1-Xt
0for ,1)(
0for ,21))(cot(tan
)2cos(1)2sin(tan)(
)()()())2cos(1(2)(
)()2sin()(
)())2cos(1()(
)())2cos(1(2)(
)()1()(
1
1
22
2
≠=
≥⎟⎠⎞
⎜⎝⎛ −==
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=Φ
ΓΓ=Γ−=Α
Γ=Ψ
Γ−=Λ
Γ−=Γ
Γ−=Γ
−
−
−
ωωκ
ωωππω
πωπωω
ωωωπωω
ωπωω
ωπωω
ωπωω
ωω ωπ
xy
xy
yyxxxxxy
xxxy
xxxy
xxyy
xxi
xy e
2.2 Time Series Analysis
Properties of the Cross-spectrum of Real Weakly Stationary Processes
1. The co-spectrum is the Fourier transform of the symmetric part, γxys(τ), and the
quadrature spectrum is the Fourier transform of the anti-symmetric part of the cross-covariance spectrum, γxy
a(τ)
2.
3. When the cross-covariance function is symmetric, the quadrature and phase spectra are zero for all ω. When the cross-covariance function is anti-symmetric, the co-spectrum vanishes and the phase spectrum is
4. The amplitude spectrum is positive and symmetric, and the phase spectrum is anti-symmetric
5. The coherence spectrum is symmetric
6. It is sufficient to consider spectra for positive ω
)()( ),()( ωωωω −Ψ−=Ψ−Λ=Λ xy
( ) ( ))()(21)( ,)()(
21)(
with
)2sin()(2)( ),2cos()(2)0()(11
τγτγτγτγτγτγ
πτωτγωπτωτγγωττ
−−=−+=
−=Ψ+=Λ ∑∑∞
=
∞
=
xyxyaxyxyxy
sxy
axyxy
sxyxyxy
))(sin(2
)( ωπω xyxy Ψ−=Φ
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