Mixing . Stationary case unless otherwise indicated cov{dN(t+u),dN(t)} small for large |u|
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Transcript of Mixing . Stationary case unless otherwise indicated cov{dN(t+u),dN(t)} small for large |u|
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Mixing. Stationary case unless otherwise indicated
cov{dN(t+u),dN(t)} small for large |u|
|pNN(u) - pNpN| small for large |u|
hNN(u) = pNN(u)/pN ~ pN for large |u|
qNN(u) = pNN(u) - pNpN u 0
|qNN(u)|du <
cov{dN(t+u),dN(t)}= [(u)pN + qNN(u)]dtdu
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Power spectral density. frequency-side, , vs. time-side, t
/2 : frequency (cycles/unit time)
fNN() = (2)-1 exp{-iu}cov{dN(t+u),dN(t)}/dt
= (2)-1 exp{-iu}[(u)pN+qNN(u)]du
= (2)-1pN + (2)-1 exp{-iu}qNN(u)]du
Non-negative, symmetric
Approach unifies analyses of processes of widely varying types
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Examples.
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Spectral representation. stationary increments - Kolmogorov
)(}exp{/)(
)(1}exp{
)(
N
N
dZitdttdN
dZiit
tN
})(){(},cov{
increments orthogonal
)()()}(),(cov{
order of spectrumcumulant
...),...,()...()}(),...,({
)()}({
)()(dZ valued,-complex random, :
111...11
N
YX
NNNN
KKNNKKNN
NN
NN
YXEYX
ddfdZdZ
K
ddfdZdZcum
dpdZE
dZZ
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Filtering.
dN(t)/dt = a(t-v)dM(v) = a(t-j )
= exp{it}A()dZM()
with
a(t) = (2)-1 exp{it}A()d
dZN() = A() dZM()
fNN() = |A()|2 fMM()
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Bivariate point process case.
Two types of points (j ,k)
Crossintensity. a rate
Prob{dN(t)=1|dM(s)=1}
=(pMN(t,s)/pM(s))dt
Cross-covariance density.
cov{dM(s),dN(t)}
= qMN(s,t)dsdt no () often
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Spectral representation approach.
b.v. of ,)()()}(),(cov{
)(}exp{/)(
)(}exp{/)(
NMMNNM
N
M
FddFdZdZ
dZitdttdN
dZitdttdM
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Frequency domain approach. Coherency, coherence
Cross-spectrum.
duuquif MNMN )(}exp{21
)(
Coherency.
R MN() = f MN()/{f MM() f NN()}
complex-valued, 0 if denominator 0
Coherence
|R MN()|2 = |f MN()| 2 /{f MM() f NN()|
|R MN()|2 1, c.p. multiple R2
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where
A() = exp{-iu}a(u)du
fOO () is a minimum at A() = fNM()fMM()-1
Minimum: (1 - |RMN()|2 )fNN()
0 |R MN()|2 1
AAfAfAfff MMNMMNNNOO
Proof. Filtering. M = {j }
a(t-v)dM(v) = a(t-j )
Consider
dO(t) = dN(t) - a(t-v)dM(v)dt, (stationary increments)
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Proof.
0
Take
0
sderivative second andfirst Consider
1
1
MNMMNMNN
MMNM
OO
MMNMMNNNOO
ffff
ffA
f
AAfAfAfff
Coherence, measure of the linear time invariant association of the components of a stationary bivariate process.
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Regression analysis/system identification.
dZN() = A() dZM() + error()
A() = exp{-iu}a(u)du
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Empirical examples.
sea hare
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Mississippi river flow
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Partial coherency. Trivariate process {M,N,O}
]}||1][||1{[/][ 22
| ONMOONMOMNOMN ffffff
“Removes” the linear time invariant effects of O from M and N
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