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

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

Examples.

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

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()

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

Spectral representation approach.

b.v. of ,)()()}(),(cov{

)(}exp{/)(

)(}exp{/)(

NMMNNM

N

M

FddFdZdZ

dZitdttdN

dZitdttdM

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

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)

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.

Regression analysis/system identification.

dZN() = A() dZM() + error()

A() = exp{-iu}a(u)du

Empirical examples.

sea hare

Mississippi river flow

Partial coherency. Trivariate process {M,N,O}

]}||1][||1{[/][ 22

| ONMOONMOMNOMN ffffff

“Removes” the linear time invariant effects of O from M and N