Power spectral density . frequency-side,  , vs. time-side, t

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