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Mixing . Stationary case unless otherwise indicated cov{dN(t+u),dN(t)} small for large |u|

Mixing . Stationary case unless otherwise indicated cov{dN(t+u),dN(t)} small for large |u| |p NN (u) - p N p N | small for large |u| h NN (u) = p NN (u)/p N ~ p N for large |u| q NN (u) = p NN (u) - p N p N u  0  |q NN (u)|du < 

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Mixing . Stationary case unless otherwise indicated cov{dN(t+u),dN(t)} small for large |u|

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  1. 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

  2. Power spectral density. frequency-side, , vs. time-side, t /2 : frequency (cycles/unit time) fNN() = (2)-1 exp{-iu}cov{dN(t+u),dN(t)}/dt = (2)-1 exp{-iu}[(u)pN+qNN(u)]du = (2)-1pN + (2)-1  exp{-iu}qNN(u)]du Non-negative, symmetric Approach unifies analyses of processes of widely varying types

  3. Examples.

  4. Spectral representation. stationary increments - Kolmogorov

  5. 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()

  6. 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

  7. Spectral representation approach.

  8. Frequency domain approach. Coherency, coherence Cross-spectrum. 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

  9. 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) where A() =  exp{-iu}a(u)du fOO () is a minimum at A() = fNM()fMM()-1 Minimum: (1 - |RMN()|2 )fNN() 0  |R MN()|2 1

  10. Proof. Coherence, measure of the linear time invariant association of the components of a stationary bivariate process.

  11. Regression analysis/system identification. dZN() = A() dZM() + error() A() =  exp{-iu}a(u)du

  12. Empirical examples. sea hare

  13. Mississippi river flow

  14. Partial coherency. Trivariate process {M,N,O} “Removes” the linear time invariant effects of O from M and N

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