The Empirical FT continued.

)()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

)()()( N

TT

N dZd

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.

many stationary processes look the same

approximate i.i.d sample values

assessing models (character of departure)

...

Muscle spindle example

Published by AAAS

R. Fuentes et al., Science 323, 1578 -1582 (2009)

Fig. 2. DCS restores locomotion and desynchronizes corticostriatal activity