The Empirical FT .

The Empirical FT.

)()0( Note

1 0 real }exp{)()(:

1)}-TX(1),...X( {X(0),

tXTXd

N TtitXTXdEFT

Data

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 UwhereV i U Y

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

22

221

2

22

22

21

2

21

22

22

2

E

INZZZ

ZZNiZZINN

YEYEY

NN

N

j

C

C

C

Theorem. Suppose X 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

XXC

lLLTT

lXXC

LTX

TX

XXC

XXX

TX

TfIN

TrrrTrd

Trdiii

TfIN

ddii

TfN

TfTcNdi

Evaluate first and second-order cumulants

Bound higher cumulants

Normal is determined by its moments

Proof. Write

)()()( X

TT

XdZd

Consider

)(2)()( have We

)()(2~ )()()(

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

TTT

XX

T

XX

TT

XX

TTT

X

T

X

d

fdf

ddfdd

Comments.

Already used to study mean estimate

Tapering, h(t)X(t). makes

dfHdXX

TT

X)(|)(|~)(var 2

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 fXX( )

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

XXXX

XXXX

XX

XX

XX

C

T

X

T

XX

ffffE

f

f

TfNT

dT

I

An estimate whose limit is a random variable

Some moments.

2222

0

22

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

)(}exp{)()( || var||

T

NXX

TT

X

T T

X

T

X

cdfdEso

ctitXEdEE

The estimate is asymptotically unbiased

Final term drops out if = 2r/T 0

Best to correct for mean, work with

)()( TT

X

T

Xcd

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

TrrrmperiodograSmoothed

XXLNNXXLXX

LXX

T

XX

l l

TT

XX

l

L

Approximate marginal confidence intervals

LVT

L

fff

T

T

data,split might FT or taperedmean, weightedmight take

estimate consistentfor might takeondistributi valueextreme viaband ussimultaneo

levelmean about CIset logby stabilized variance

Notes.

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

2

2

More on choice of L

biased constant,not is If

radians /2 (.) of width affects L of Choice

)()(W

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

11

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

Consider

T

2

T

T

T

lll

lll

lT

ff

TLW

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

duucuwui T

XX

T )()(}exp{21

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{

)()(

}exp{)()(

YYYX

XYXX

XYYX

Y

X

ffff

dfdZdZ

dZdZ

ittYtX

Crossperiodogram.

T

YY

T

YX

T

XY

T

XXT

T

Y

T

X

T

XY

IIII

ddT

I

)( formmatrix

)()(2

1)(

I

Smoothed periodogram.

LlTrLTr ll

lTT

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

Complex Wishart

ΣW

XXWΣ

0XX

nE

nW

IN

n Tjj

Cr

rn

squared-chi diagonals

~),(

),(~,...,

1

1

Predicting Y via X

)()(}exp{)(

}exp{)()2()()()(/)()( :coherency

)(]|)(|1[ :MSEphase :)(arggain |:)(|

functionfer trans,)()()(

|)(-)(|min )(by )( predicting

1

2

1

2

X

YYXXYX

YY

XXYX

XYA

XY

dZAtitY

diuAuafffR

fRAA

ffA

AdZdZEdZdZ

Plug in estimates.

LRE

R

LLRRLLFR

R

fffR

fRAA

ffA

T

TL

T

T

YY

T

XX

T

YX

T

T

YY

T

TT

T

XX

T

YX

T

/1||)-(1-1

point %100approx 0|| If

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

|| ofDensity

)()(/)()(

)(]|)(|1[ :MSE)(arg |)(|

)()()(

2

1)-1/(L

2

22

12

2

2

2

1

Large sample distributions.

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

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

Berlin andVienna monthlytemperatures

RecifeSOI

Furnace data

RXZ|Y =

(R XZ – R XZ R ZY )/[(1- |R XZ|2 )(1- |RZY | 2 )]

Partial coherence/coherency. Mississipi dams

Cleveland, RB, Cleveland, WS, McRae, JE & Terpenning, I (1990), ‘STL: a seasonal-trend decomposition procedurebased on loess’, Journal of Official Statistics

Y(t) = S(t) + T(t) + E(t)

Seasonal, trend. error

London water usage

Dynamic spectrum, spectrogram: IT (t,). London water

Earthquake? Explosion?

Lucilia cuprina

nobs = length(EXP6) # number of observations wsize = 256 # window size overlap = 128 # overlap ovr = wsize-overlap nseg = floor(nobs/ovr)-1; # number of segments krnl = kernel("daniell", c(1,1)) # kernel ex.spec = matrix(0, wsize/2, nseg)for (k in 1:nseg){ a = ovr*(k-1)+1 b = wsize+ovr*(k-1) ex.spec[,k] = mvspec(EXP6[a:b], krnl, taper=.5, plot=FALSE)$spec } x = seq(0, 10, len = nrow(ex.spec)/2) y = seq(0, ovr*nseg, len = ncol(ex.spec)) z = ex.spec[1:(nrow(ex.spec)/2),] # below is text version filled.contour(x,y,log(z),ylab="time",xlab="frequency (Hz)",nlevels=12 ,col=gray(11:0/11),main="Explosion") dev.new() # a nicer version with color filled.contour(x, y, log(z), ylab="time", xlab="frequency(Hz)", main="Explosion") dev.new() # below not shown in text persp(x,y,z,zlab="Power",xlab="frequency(Hz)",ylab="time",ticktype="detailed",theta=25,d=2,main="Explosion")

Advantages of frequency domain approach.

techniques for many stationary processes look the same

approximate i.i.d sample values

assessing models (character of departure)

time varying variant...