Time series power spectral density.

frequency-side, , vs. time-side, t

X(t), t= 0, ±1, ±2,… Suppose stationary

cXX(u) = cov{X(t+u),X(t)} u = 0, ±1, ±2, … lag

f XX()= (1/2π) Σ exp {-iu} cXX(u) - < ≤

period 2π non-negative

/2 : frequency (cycles/unit time)

cXX(u) = ∫ -pi pi exp{iuλ} f XX() d

The Empirical FT.

Data X(0), …, X(T-1)

dXT()= Σ exp {-iu} X(u)

dXT(0)= Σ X(u)

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 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

TrrrT

rd

T

rdiii

TfIN

ddii

TfN

TfTcN

di

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

f

df

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

ff

ffE

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

cdfdE

so

ctitXEd

EE

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

Tr

rrmperiodograSmoothed

XXLNNXXLXX

LXX

T

XX

l l

TT

XX

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

11

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

Consider

T

2

T

T

T

lll

lll

lT

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

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

ff

ff

dfdZdZ

dZ

dZit

tY

tX

Crossperiodogram.

T

YY

T

YX

T

XY

T

XXT

T

Y

T

X

T

XY

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 Y via X

)()(}exp{)(

}exp{)()2()(

)()(/)()( :coherency

)(]|)(|1[ :MSE

phase :)(arggain |:)(|

functionfer trans,)()()(

|)(-)(|min

)(by )( predicting

1

2

1

2

X

YYXXYX

YY

XXYX

XYA

XY

dZAtitY

diuAua

fffR

fR

AA

ffA

AdZdZE

dZdZ

Plug in estimates.

LRE

R

LL

RRLLFR

R

fffR

fR

AA

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

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...

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")