1. FT makes the New Yorker, October 4, 2010 page 71

2. 2   1 The Fourier transform.  regularity conditions
Functions, A(), - <  < 
 |A()|d finite
FT a(t) =  exp{it)A()d - <  < 
Inverse A() =(2)-1  exp{-it} a(t) dt
unique
C()=  A() +  B()
c(t) =  c(t) +  b(t)
2   1

3. Convolution (filtering).
d(t) =  b(t-s) c( s)ds
D() = B()C()
Discrete FT.
a(t) = T-1 exp{i2ts/T} A(2s/T) s, t = 0,1,...,T-1
A(2s/T) =  exp {-i2st/T) a(t)
FFTs exist

4. Dirac delta.
 g() () d = g(0)
 exp {it}() d  = 1
inverse
() = (2)-1  exp {-it}dt
Heavyside function H() = signum ()
() = dH()/d

5. 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

6. 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

7. Examples.

9. Spectral representation. stationary increments - Kolmogorov

10. 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()

11. Association. Measuring? Due to chance?
Are two processes associated? Eg. t.s. and p.p.
How strongly?
Can one predict one from the other?
Some characteristics of dependence:
E(XY)  E(X) E(Y)
E(Y|X) = g(X)
X = g (), Y = h(),  r.v.
f (x,y)  f (x) f(y)
corr(X,Y)  0

12. 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

13. Spectral representation approach.

14. 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

15. 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

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

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

18. Empirical examples.
sea hare

19. Mississippi river flow

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

25. Time series variants.
details later
continuous time case
Mixing.
cov{Y(t+u),Y(t)} = cYY(u)
small for large |u|
 |cYY(u)|du < 

26. Power spectral density. frequency-side, , vs. time-side, t
/2 : frequency (cycles/unit time)
fYY() = (2)-1 exp{-iu}cov{Y(t+u),Y(t)}
= (2)-1 exp{-iu}cYY(u)du <<
Non-negative, symmetric
Approach unifies analyses of processes of widely varying types
Things in the frequency domain look the same

27. Spectral representation.
Y(t) = exp{it}dZY() - < t < 
ZY() random, complex-valued conj{ZY()} = ZY(-)
E{dZY()} = ()cNd
cov{dZY(),dZY()}=(-)f NN()dd
cum{dZY(1),...,dZY(K)} = ...

28. Filtering.
Yt) =  a(t-v)X(v)dv
=  exp{it}A()dZX()
with
a(t) = (2)-1  exp{it}A()d
dZY() = A() dZX()
fYY() = |A()|2 fXX()

29. Bivariate time series case.
(X(t),Y(t)) - < t < 
Cross-covariance function. general case
cov{X(s),Y(t)}
= cXY(s,t)

30. Spectral representation approach.
FXY(.): cross-spectral measure

31. Frequency domain approach. Coherency, coherence
Cross-spectrum.
f XY()= (2)-1  exp{-iu)c XY(u)du -< <
complex-valued
Coherency.
R XY() = f XY()/{f XX() f YY()}
0 if denominator 0
Coherence.
|RXY()|2 = |f XY()| 2 /{fXX() fYY()|
|RXY()|2  1, c.p. multiple R2

32. Regression analysis/system identification.
dZY() = A() dZX() + error()
A() =  exp{-iu}a(u)du

33. Partial coherency. Trivariate process {X,Y,O}
“Removes” the linear time invariant effects of O from X and Y