Second order stationary p.p.

dN(t) = ∫ exp{ i λ t} dZ N (λ ) dt Kolmogorov

N(t) =

Power spectrum

cov{dZ N(λ ), dZ N (μ } = δ (λ - μ) f NN d λ d μ

remember Z complex-valued

Bivariate p.p. process {M,N}

Coherency

R MN (λ) = f MN / √{f MM f NN} cp. r

Coherence

| R MN (λ) | 2 cp. r2

The Empirical FT.

)()0( Note

real ),(}exp{}exp{)(:

}...{0

1 0

1

TNd

tdNtiidEFT

TData

T

N

N T

j

T

N

N

What is the large sample distribution of the EFT?

The complex normal.

2var

)1,0( ...

onentialexp2/2/)(|)1,0(|

2/)()2/1,0()1,0(

||var

:Notes

)2/,(Im ),2/,(Ret independen are V and Uwhere

V i U Y

form theof variatea is ),,( normal,complex The

22

22

1

2

2

2

2

2

2

1

2

21

22

22

2

E

INZZZ

ZZN

iZZINN

YEY

EY

NN

N

j

C

C

C

Theorem. Suppose p.p. N is stationary mixing, then

))(2,0( asymp are

~/2 with 0 integersdistinct ,..., ),2

(),...,2

( ).

)(2,0(

asymp are 0 anddistinct ,..., ),(),...,( ).

0 )),(2,0(

0 )),0(2,(

allyasymptotic is )( ).

11

L11

NN

C

lLLTT

lNN

C

L

T

N

T

N

NN

C

NNN

T

N

TfIN

TrrrT

rd

Tr

diii

TfIN

ddii

TfN

TfTpN

di

Evaluate first and second-order cumulants

Bound higher cumulants

Normal is determined by its moments

Proof. Write

dN(t) = ∫ exp{ i λ t} dZ N (λ ) dt Kolmogorov

then

d NT(λ) = ∫ δ T (λ – α) dZ N (α)

with

δ T (λ )= ∫ 0 T exp{ -i λ t} dt

Consider

)(2)()(

saw We

)()(2~

)()()(

)()()()()}(),(cov{

TTT

NN

T

NN

TT

NN

TTT

N

T

N

d

f

df

ddfdd

Comments.

Already used to study rate estimate

Tapering makes

dfHd NN

TT

N )(|)(|~)(var

Get asymp independence for different frequencies

The frequencies 2r/T are special, e.g. T(2r/T)=0, r 0

Also get asymp independence if consider separate stretches

p-vector version involves p by p spectral density matrix fNN( )

Estimation of the (power) spectrum.

22

2

2

2

2

2

2

2

)(}2/)(var{ and

)(}2/)({ notebut

0)( unlessnt inconsiste appears Estimate

lexponentia ,2/)(~

|))(2,0(|2

1~

|)(|2

1)(

m,periodogra heconsider t ,0For

NNNN

NNNN

NN

NN

NN

C

T

N

T

NN

ff

ffE

f

f

TfNT

dT

I

An estimate whose limit is a random variable

Some moments.

2222

0

22

|)(|)(|)(||)(|

)(}exp{)(

|| var||

r.v.Complex

T

NNN

TT

N

T T

NN

T

N

pdfdE

so

pdtptiEd

EE

The estimate is asymptotically unbiased

Final term drops out if = 2r/T 0

Best to correct for mean, work with

)()( TT

N

T

N pd

Periodogram values are asymptotically independent since dT values are -

independent exponentials

Use to form estimates

sL' severalTry variance.controlCan

/)(}2/)(var{ )(}2/)({

Now

ondistributiin 2/)()(

gives CLT

/)/2()(

Estimate

near /2

0 integersdistinct ,...,Consider .

22

2

2

2

2

2

1

LfLffLfE

Lff

LTrIf

Tr

rrmperiodograSmoothed

NNLNNNNLNN

LNN

T

NN

ll

TT

NN

l

L

Approximate marginal confidence intervals

LVT

L

ff

fT

T

data,split might

FT or taperedmean, weightedmight take

estimate consistentfor might take

ondistributi valueextreme viaband ussimultaneo

levelmean about CIset

logby stabilized variance

Notes.

)}2/(/log)(log)(log

)2/1(/log)(Pr{log2

2

More on choice of L

biased constant,not is If

radians /2

(.) of width affects L of Choice

)()(W

)2/)/(()2/)(sin2

11

/2/2 },/)({)}({

Consider

T

2

T

T

T

lll

lll

l

T

ff

TL

W

df

dfTTL

TrTrLIEfE

Approximation to bias

0)( symmetricFor W

...2/)()(")()(')()(

...]2/)(")(')()[(

)()(

)()(

)()( Suppose

22

22

11

dW

dWfBdWfBdWf

dfBfBfW

BfW

dfW

BWBW

TTT

TT

T

T

TT

T

Indirect estimate

duuquwuip T

NN

TT

N )()(}exp{21

2

Could approximate p.p. by 0-1 time series and use t.s. estimate

choice of cells?

Estimation of finite dimensional .

approximate likelihood (assuming IT values independent exponentials)

)}/2(/)/2(exp{)/2()(

in ),;( spectrum

1 TrfTrITrfL

f

r

T

Bivariate case.

)( )(

)( )(

matrixdensity spectral

)()()}(),(cov{

)(

)(/]1}[exp{

)(

)(

NNNM

MNMM

MNNM

N

M

ff

ff

dfdZdZ

dZ

dZitit

tN

tM

Crossperiodogram.

T

NN

T

NM

T

MN

T

MMT

T

N

T

M

T

MN

II

II

ddT

I

)( formmatrix

)()(2

1)(

I

Smoothed periodogram.

LlTrLTr ll

l

TT

NN ,...,1 ,/2 ,/)/2()( If

Complex Wishart

ΣW

XXWΣ

0XX

nE

nW

IN

n T

jj

C

r

rn

squared-chi diagonals

~),(

),(~,...,

1

1

Predicting N via M

)()(1}exp{

)(

}exp{)()2()(

)()(/)()( :coherency

)(]|)(|1[ :MSE

phase :)(arggain |:)(|

functionfer trans,)()()(

|)(-)(|min

)(by )( predicting

1

2

1

2

M

NNMMNM

NN

MMNM

MNA

MN

dZAiit

tN

diuAua

fffR

fR

AA

ffA

AdZdZE

dZdZ

Plug in estimates.

LRE

R

LL

RRLLFR

R

fffR

fR

AA

ffA

T

TL

T

NNMMNM

T

T

NN

T

TT

T

MM

T

NM

T

/1||

)-(1-1

point %100approx 0|| If

)1()1()(

)||||;1;,()||1(

|| ofDensity

)()(/)()(

)(]|)(|1[ :MSE

)(arg |)(|

)()()(

2

1)-1/(L

2

22

122

2

2

1

Large sample distributions.

var log|AT| [|R|-2 -1]/L

var argAT [|R|-2 -1]/L

Networks.

partial spectra - trivariate (M,N,O)

Remove linear time invariant effect of M from N and O and examine coherency of residuals,

)(

)()()()()(

M of effects

invariant lineartime removed having O and N of spectrum-cross partial

),()()()(

),()()()(

|

1

|

1

|

1

|

T

MNO

MNMMNMNOMNO

MMOMMOMO

MMNMMNMN

R

fffff

ffBdZBdZdZ

ffAdZAdZdZ

Point process system.

An operation carrying a point process, M, into another, N

N = S[M]

S = {input,mapping, output}

Pr{dN(t)=1|M} = { + a(t-u)dM(u)}dt

System identification: determining the charcteristics of a system from inputs and corresponding outputs

{M(t),N(t);0 t<T}

Like regression vs. bivariate

Realizable case.

a(u) = 0 u<0

A() is of special form

e.g. N(t) = M(t-)

arg A( ) = -

Advantages of frequency domain approach.

techniques formany stationary processes look the same

approximate i.i.d sample values

assessing models (character of departure)

time varying variant

...