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

The Fourier transform. regularity conditions

Functions, A(), - < <

|A()|d finite

FT. a(t) = exp{it)A()d - < <

Inverse A() =(2)-1 exp{-it} a(t) dt

unique

C()= A() + B()

c(t) = c(t) + b(t)

2 1

Convolution (filtering).

d(t) = b(t-s) c( s)ds

D() = B()C()

Discrete FT.

a(t) = T-1 exp{i2ts/T} A(2s/T) s, t = 0,1,...,T-1

A(2s/T) = exp {-i2st/T) a(t)

FFTs exist

Dirac delta.

g() () d = g(0)

exp {it}() d = 1

inverse

() = (2)-1 exp {-it}dt

Heavyside function H() = signum ()

() = dH()/d

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

dZiittN

})(){(},cov{ increments orthogonal

)()()}(),(cov{order of spectrumcumulant

...),...,()...()}(),...,({)()}({

)()(dZ valued,-complex random, :

111...11

N

YX

NNNN

KKNNKKNN

NN

NN

YXEYX

ddfdZdZK

ddfdZdZcumdpdZE

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()

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

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

ffffffA

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

Time series variants.

details later

continuous time case

Mixing.

cov{Y(t+u),Y(t)} = cYY(u)

small for large |u|

|cYY(u)|du <

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

/2 : frequency (cycles/unit time)

fYY() = (2)-1 exp{-iu}cov{Y(t+u),Y(t)}

= (2)-1 exp{-iu}cYY(u)du -<<

Non-negative, symmetric

Approach unifies analyses of processes of widely varying types

Things in the frequency domain look the same

Spectral representation.

Y(t) = exp{it}dZY() - < t <

ZY() random, complex-valued conj{ZY()} = ZY(-)

E{dZY()} = ()cNd

cov{dZY(),dZY()}=(-)f NN()dd

cum{dZY(1),...,dZY(K)} = ...

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()

Bivariate time series case.

(X(t),Y(t)) - < t <

Cross-covariance function. general case

cov{X(s),Y(t)}

= cXY(s,t)

Spectral representation approach.

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

)(}exp{/)(

)(}exp{/)(

NMXYYX

Y

X

FddFdZdZ

dZitdttdY

dZitdttdX

FXY(.): cross-spectral measure

Frequency domain approach. Coherency, coherence

Cross-spectrum.

f XY()= (2)-1 exp{-iu)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

Regression analysis/system identification.

dZY() = A() dZX() + error()

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

Partial coherency. Trivariate process {X,Y,O}

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

| OYXOOYXOXYOXYffffff

“Removes” the linear time invariant effects of O from X and Y