1. Power spectral density. frequency-side, , vs. time-side, t
/2 : frequency (cycles/unit time)
Non-negative
Unifies analyses of processes of widely varying types

2. Examples.

4. Spectral representation. stationary increments - Kolmogorov

5. 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

6. 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

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

8. Empirical examples.
sea hare

10. Muscle spindle

11. Spectral representation approach.
Filtering.
dO(t)/dt =  a(t-v)dM(v) =  a(t-j )
=  exp{it}dZM()

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