Estimation of the spectral density function. The spectral density function, f( ) The spectral...

Estimation of the spectral density function

cos( )h h f d

0

1 10 cos( )

2 h

f h h

The spectral density function, f() The spectral density function, f(x), is a symmetric function defined on the interval [-,] satisfying

and

The spectral density function, f(x), can be calculated from the autocovariance function and vice versa.

1cos( )

2 h

h h

2. cos sinixe x i x

Some complex number results:

Use

1. where 1z x iy i

2 3

12! 3!

u u ue u

2 4

cos 12! 4!

u uu

3 5

sin3! 5!

u uu u

3. (polar representation)iz x iy Re

2 2where and tanx

R x yy

4. cos2

ix ixe ex

5. sin2

ix ixe ex

i

Expectations of Linear and Quadratic forms of a weakly

stationary Time Series

Expectations, Variances and Covariances of Linear forms

Theorem Let {xt:t T} be a weakly stationary time series.Let

Then

and

where

and Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.

T

ttt xcL

1

T

ttcLE

1

1

11 1

)()(T

Tr Sssrs

T

s

T

tts

r

ccrstccLVar

dfCdfecT

t

tit

22

1

)(

T

t

tit ecC

1

)(

Proof 

T

tt

T

ttt

T

ttt cxEcxcELE

111

T

t

T

ssstt xcxcELVar

1 1

T

t

T

sst stcc

1 1

T

t

T

sstst xxEcc

1 1

1

1

T

Tt

T

Sssrs

r

ccr

Also since

Q.E.D.

dfedfhh hi )()()cos(

T

t

T

s

stist dfeccLVar

1 1

dfececT

t

sit

T

t

tit

11

dfC

2

dfeecc sitiT

t

T

sst

1 1

dfecT

t

tit

2

1

Theorem Let {xt:t T} be a weakly stationary time series.

Let

and

T

tttc xcL

1

T

tttb xbL

1

Expectations, Variances and Covariances of Linear forms

Summary

Theorem Let {xt:t T} be a weakly stationary time series.Let

Then

and

where

and Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.

T

ttt xcL

1

T

ttcLE

1

1

11 1

)()(T

Tr Sssrs

T

s

T

tts

r

ccrstccLVar

2( )C f d

T

t

tit ecC

1

)(

Theorem Let {xt:t T} be a weakly stationary time series.

Let and

T

tttc xcL

1

T

tttb xbL

1

Then

T

s

T

ttsbc stbcLLCov

1 1

)(,

1

1

)(T

Tr Sssrs

r

bcr

dfBC )()(

T

t

tit ecC

1

)(

T

t

tit ebB

1

)( where and

Then

T

s

T

ttsbc stbcLLCov

1 1

)(,

dfebecT

s

sis

T

t

tit

11

T

t

tit ecC

1

)(

T

t

tit ebB

1

)(

1

1

)(T

Tr Sssrs

r

bcr

dfBC )()(

where and

Also Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.

Expectations, Variances and Covariances of Quadratic forms

Theorem Let {xt:t T} be a weakly stationary time series.

Let

Then

T

s

T

ttsst xxaQ

1 1

T

s

T

t

T

Tr Strssst

r

arstaQE1 1

1

1,

dfAdfeaT

s

T

t

stist ),(

1 1

and

T

s

T

t

T

s

T

ttsst ttssaaQVar

1 1 1' 1''' ''2

1

1

1

1,2

T

Tr

T

Tp Ss Stptrsst

r r

aapr

1

1

1

1,, ,,

T

Tr

T

Tp Ss Stptrsrsst pstrstaa

r r

stssst ',',

ddffeaT

s

T

t

stist

2

1 1

2

T

s

T

t

T

s

T

ttsst stssstaa

1 1 1' 1''' ',',

ddffA2

),(2

T

s

T

t

T

s

T

ttsst stssstaa

1 1 1' 1''' ',',

T

s

T

t

stisteaA

1 1

, where

and Sr = {1,2, ..., T-r}, if r ≥ 0,

Sr = {1- r, 2 - r, ..., T} if r ≤ 0,

(h,r,s) = the fourth order cumulant

= E[(xt - )(xt+h - )(xt+r - )(xt+s - )]

- [(h)(r-s)+(r)(h-s)+(s)(h-r)]

Note (h,r,s) = 0 if {xt:t T}is Normal.

Theorem Let {xt:t T} be a weakly stationary time series.

Let

Then

T

s

T

ttsst xxaQ

1 1

T

s

T

ttsst xxbP

1 1

T

s

T

t

T

s

T

ttsst ttssbaPQCov

1 1 1' 1''' ''2,

stssst ',',

1

1

1

1,2

T

Tr

T

Tp Ss Stptrsst

r r

bapr

1

1

1

1,, ,,

T

Tr

T

Tp Ss Stptrsrsst pstrstba

r r

ddffebeaT

s

T

t

stits

T

s

T

t

stist

1' 1'

''''

1 1

2

T

s

T

t

T

s

T

ttsst stssstba

1 1 1' 1''' ',',

ddffBA ),(),(2

T

s

T

t

T

s

T

ttsst stssstba

1 1 1' 1''' ',',

and

where

T

s

T

t

stist eaA

1 1

,

T

s

T

t

stist eaB

1 1

,

ExamplesThe sample mean

1

11

T

sss

T

ss xcx

Tx

TsT

cs ,.,2,1for 1

where

and

Thus

1

11

T

s

T

ss T

cxE

T

s

T

t

T

s

T

tts st

TTstccxVar

1 11 1

)(11

)(

1

1

1

1

)()(2)0(1 T

Ts

T

s

rT

rTr

T

rT

T

Also

1

111

11

i

Tii

T

t

tiT

t

tit e

ee

Te

TecC

2/2/

2/2/2/11

ii

TiTiTi

ee

eee

T

2/sin

2/sin1 2/1

T

eT

Ti

and

2

1

2

T

t

titecC

2/sin

2/sin1

2/sin

2/sin1 2/12/1

T

eT

Te

TTiTi

T

HT

TT

22/sin

2/sin12

2

2

2

kernelFejer the

2/sin

2/sin22

22

T

THT

where

Thus

dfHT

xVar T

2

2

1

1

1

( )T

s T

T rVar x r

T

Compare with

Basic Property of the Fejer kernel:

If g(•) is a continuous function then :

0

2

0 4lim

gdgHTT

Thus

1

2002limk kT

kkfxVarT

43210-1-2-3-40

10

20

T = 2

The "Fejer " Kernel

43210-1-2-3-40

10

20

T = 5

43210-1-2-3-40

10

20

T =10

The sample autocovariance function

The sample autocovariance function is defined by:

hT

thttx xxxx

hThC

1

1

where

hT

thttx xx

hThC

1

1

T

s

T

ttsst xxa

1 1

otherwise0

,,1 and ,,1 if2

1ThthtshTthts

hTast

or if is known

where

hT

thttx xx

hThC

1

1

T

s

T

ttsst xxa

1 1

otherwise0

,,1 and ,,1 if2

1ThthtshTthts

hTast

or if is known

Theorem Assume is known and the time series is normal, then:

E(Cx(h))= (h),

hCVar x

12

1

11

T h

r T h

rr r h r h

T h T h

ddffhHhT hT

2cos

1 22

and

kChCCov xx ,

2sin

2))((

sin2

))((sin

))((

1

2

kThT

kThT

ddff

kh

2

)(cos

2

)(cos

Proof

Assume is known and the the time series is normal, then:

hxxEhT

hCEhT

thttx

1

1

and

T

s

T

ttsstx xxaVarhCVar

1 1

T

s

T

t

T

s

T

ttsst ttssaa

1 1 1' 1''' ''2

stssst ',',

T

s

T

t

T

s

T

ttsst ttssaa

1 1 1' 1''' ''2

0',', since stssst

hT

t

hT

t

tttthT 1 1'

2)]'()'(2[

)(4

1

hT

t

T

ht

tthththT 1 1'

2)]'()'(2[

)(4

1

T

ht

hT

t

tthththT 1 1'

2)]'()'(2[

)(4

1

T

ht

T

ht

tttthT 1 1'

2)]'()'(2[

)(4

1

)]()(][[)(

1 1

)1(

22

hrhrrrhThT

hT

hTr

1

)1(

2 )()(||

1)(

1 hT

hTr

hrhrrhT

r

hT

and

ddffAhCVar x

2),(2

T

s

T

t

T

s

T

ttsst stssstaa

1 1 1' 1''' ',',

ddffA2

),(2

.0',', since stssst

where

T

s

T

t

stisteaA

1 1

,

T

ht

httihT

t

htti ehT

ehT 11 2

1

2

1

hT

t

tihihi

ehT

ee

12

2sin

2sin

22/1

hT

ehT

ee hTihihi

since

1

1

1

i

Tii

T

t

ti

e

eee

2/2/

2/2/2/1

ii

TTiTi

ee

eee

2/sin

2/sin2/1

T

e Ti

hence

2sin

2sin

2, 2/1

hT

ehT

eeA hTi

hihi

2sin

2sin

22/1

2/2/

hT

ehT

ee Tihhihi

Thus

2sin

2sin

2/cos2/1

hT

hT

he Ti

2sin

2sin

2/cos,

2

2

2

22

hT

hT

hA

2

2

2

2/cos

hTHhT

h

and

hCVar x

hhddffhHhT hT

2/cos

1 22

Finally

hhddffBAkChCCov xx

,,,

Where

2sin

2sin

2/cos2/1

kT

kT

ke Ti

T

s

T

t

stistebB

1 1

,

Thus

,, BA

kThT

kh 2/cos2/cos

2sin

2sin

2sin

2

kThT

Expectations, Variances and Covariances of Linear forms

Summary

Theorem Let {xt:t T} be a weakly stationary time series.Let

Then

and

where

and Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.

T

ttt xcL

1

T

ttcLE

1

1

11 1

)()(T

Tr Sssrs

T

s

T

tts

r

ccrstccLVar

2( )C f d

T

t

tit ecC

1

)(

Theorem Let {xt:t T} be a weakly stationary time series.

Let and

T

tttc xcL

1

T

tttb xbL

1

Then

T

s

T

ttsbc stbcLLCov

1 1

)(,

1

1

)(T

Tr Sssrs

r

bcr

dfBC )()(

T

t

tit ecC

1

)(

T

t

tit ebB

1

)( where and

Expectations, Variances and Covariances of Quadratic forms

Theorem Let {xt:t T} be a weakly stationary time series.

Let

Then

T

s

T

ttsst xxaQ

1 1

T

s

T

t

T

Tr Strssst

r

arstaQE1 1

1

1,

dfAdfeaT

s

T

t

stist ),(

1 1

and

T

s

T

t

T

s

T

ttsst ttssaaQVar

1 1 1' 1''' ''2

1

1

1

1,2

T

Tr

T

Tp Ss Stptrsst

r r

aapr

1

1

1

1,, ,,

T

Tr

T

Tp Ss Stptrsrsst pstrstaa

r r

stssst ',',

ddffeaT

s

T

t

stist

2

1 1

2

T

s

T

t

T

s

T

ttsst stssstaa

1 1 1' 1''' ',',

ddffA2

),(2

T

s

T

t

T

s

T

ttsst stssstaa

1 1 1' 1''' ',',

T

s

T

t

stisteaA

1 1

, where

and Sr = {1,2, ..., T-r}, if r ≥ 0,

Sr = {1- r, 2 - r, ..., T} if r ≤ 0,

(h,r,s) = the fourth order cumulant

= E[(xt - )(xt+h - )(xt+r - )(xt+s - )]

- [(h)(r-s)+(r)(h-s)+(s)(h-r)]

Note (h,r,s) = 0 if {xt:t T}is Normal.

Theorem Let {xt:t T} be a weakly stationary time series.

Let

Then

T

s

T

ttsst xxaQ

1 1

T

s

T

ttsst xxbP

1 1

T

s

T

t

T

s

T

ttsst ttssbaPQCov

1 1 1' 1''' ''2,

stssst ',',

ddffBA ),(),(2

T

s

T

t

T

s

T

ttsst stssstba

1 1 1' 1''' ',',

Estimation of the spectral density function

The Discrete Fourier Transform

Let x1,x2,x3, ...xT denote T observations on a univariate one-dimensional time series with zero mean (If the series has non-zero mean one uses in place of xt).

Also assume that T = 2m +1 is odd.

Then

xxt

m

kkkkkt tbta

ax

1

0 )sin()cos(2

m

mkkkkk tbta )sin()cos(

2

1

where

T

tt

T

tktk t

T

kx

Ttx

Ta

11

2cos2

)cos(2

T

tt

T

tktk t

T

kx

Ttx

Tb

11

2sin2

)sin(2

with k = 2k/T and k = 0, 1, 2, ... , m.

The Discrete Fourier transform:

T

tkkttk ibaxcX

1

tic kt expT

2 where

tit kk sincosT

2

k = 0, 1,2, ... ,m.

Note:

m

kkkkkt tbta

ax

1

0 )sin()cos(2

m

mkkkkk tbtai )cos()sin(

2

1

m

mkkkkk tbta )sin()cos(

2

1

m

mkkkkk tbta )sin()cos(

2

1

Since

m

mkkkkk tbta )cos()sin(

m

mkk

T

tkt ttx

T)sin()'cos(

2

1''

)cos()'sin(2

1'' ttx

T k

T

tkt

m

mk

T

tkt ttx

T 1'' )'sin(

2

0

'2sin

2

1''

T

t

m

mkt T

ttkx

T

Thus

m

kkkkk tbtai

1

)cos()sin(2

1

m

mkkkkkt tbtax )sin()cos(

2

1

m

mkkkkk tbta )sin()cos(

2

1

)sin()cos(2

1titiba kk

m

mkkk

m

mkkkk titX sincos

2

1

m

mkkk tiX exp

2

1

m

mkkk tiX exp

2

1

Summary:The Discrete Fourier transform

T

tkkttk ibaxcX

1

tikt

ketic T

2exp

T

2 where

tit kk sincosT

2 k = 0, 1,2, ... ,m.

m

mk

tik

m

mkkkt

keXtiXx 2

1exp

2

1 and

Theorem

with kk/T)

E[Xk] = 0

dfkXXCov hkThk ,,

2sin

2sin

2sin

2sin

2 and

2

TT

TkT

with kk/T) and hh/T)

dfHT

XVar kTk

22

kernelFejer the

2/sin

2/sin22

22

T

THT

where

Proof Note

titec kkti

tk sincos

T

2

T

2 where

T

tttk xcX

1

Thus

0 since 02

1

μeT

XET

t

tik

k

kXVar

dfCdfecT

t

tit

22

1

)(

T

t

ti keT 1

2

kii

TiTiTi

ee

eee

T

with

22/2/

2/2/2/1

kTi T

eT

with 2/sin

2/sin2 2/1

T

t

titiT

t

tit ee

TecC k

11

2

1

12

k

k

k

i

Tii

e

ee

T

Thus

2

1

2

T

t

titecC

2/sin

2/sin2

2/sin

2/sin2 2/12/1

T

eT

Te

TTiTi

T

HT

TkT

k

k

2

2

2

2

2 2

2/sin

2/sin2

kernelFejer the

2/sin

2/sin22

22

T

THT

where

Thus

dfHT

XVar kTk

22

Also

dfBCXXCov hk ,

with =2(k/T)+

T

t

tiT

t

tTki eT

eT

C11

])/(2[ 22

)2/sin(

]2/)sin[(2 2/1

T

eT

Ti

T

t

tiT

t

tThi eT

eT

B11

])/(2[ 22

2/sin

2/sin2 2/1

T

eT

Ti

with =2(h/T)+

Thus

BC)2/sin(

)2/sin(22

T

T

2/sin

2/sin

T

dfkXXCov hkThk ,,

2sin

2sin

2sin

2sin

2 where

2

TT

TkT

and

Defn: The Periodogram:

2

1

2

1

)cos()sin(2 T

tkt

T

tktkT txtx

TI

222

222 kkkkk XT

XXT

baT

k = 0,1,2, ..., m

with k = 2k/T and k = 0, 1, 2, ... , m.

Periodogram for the sunspot data

0

10000

20000

30000

0 0.5 1 1.5 2 2.5 3

note:

2

1

2

1

)cos()sin(2 T

tkt

T

tktkT txtx

TI

T

tk

T

skst stxx

T 1 1

)sin()sin(2

T

tk

T

skst stxx

1 1

)cos()cos(

T

t

T

skst stxx

T 1 1

)cos(2

1

1 1

)cos(2 T

Th

hT

tkhtt hxx

T

1

1

2 cos( )T

x kh T

C h h

1

1

2 0 2 cos( )T

x x kh

C C h h

1

1

2 k

Th

xh T

e C h

Theorem

dfHIE kTkT )(2

22

)(

dfHIVar kTkT

2

)(,

dfk kkT

2

)(,

dfk kkT

2kTIE

hTkT IICov ,

2

)(,4

dfk hkT

sin sin

2 2 2where ,

sin sin2 2

T

T T

kT

2

)(,4

dfk hkT

In addition: )(4lim kkT

TfIE

)(16lim 22kkT

TfIVar

0,lim hTkT

TIICov

If k ≠ 0

If k ≠ h

Proof Note

stistist

kk eea

T

1

T

1 where

Let

kTI

T

s

T

ttsst xxa

1 1

T

s

T

t

stisteaA

1 1

T

s

T

t

stistiT

s

T

t

stisti eeee kk

1 11 1 T

1

T

1

1

1

Because 2 cos( ) and cos2

ix ixT

T k x kh T

e eI C h h x

T

s

T

t

ststiT

s

T

t

ststi kk ee1 11 1 T

1

T

1

T

t

tiT

s

si kk ee11T

1

T

t

tiT

s

si kk ee11T

1

kkTkkTsTi kke ,,

T

1 2/1

A

kkTkkT kkA ,,T

1,

22

2

1

2

1kTkT HH

dfHH kTkT )(2

1 22

2( )T kH f d

dfAIE kT )(,

Recall

Basic Property of the Fejer kernel:

If g(•) is a continuous function then : 0

2

0 4lim

gdgHTT

2and ( )T k T kE I H f d

2

Thus lim lim ( )T k T kT T

E I H f d

4 ( ) 4 ( )k kf f

The remainder of the proof is similar

Consistent Estimation of the Spectral Density function f()

Smoothed Periodogram Estimators

Defn: The Periodogram:

2

1

2

1

)cos()sin(2 T

tkt

T

tktkT txtx

TI

222

222 kkkkk XT

XXT

baT

k = 0,1,2, ..., m

Properties: )(4lim kkT

TfIE

)(16lim 22kkT

TfIVar

0,lim hTkT

TIICov

If k ≠ 0

If k ≠ h

Spectral density Estimator

2

1

2

1

)cos()sin(2

1

4

1ˆT

tkt

T

tktkT txtx

TIf

Properties:

)(4

1lim kkTT

fIE

)(4

1lim 2

kkTT

fIVar

If k ≠ 0

The second properties states that:

is not a consistent estimator of f():

kTk If

4

1)(ˆ

Periodogram Spectral density Estimator

2

1

2

1

)cos()sin(2

1

4

1ˆT

tkt

T

tktkT txtx

TIf

Properties: )(4

1lim kkTT

fIE

)(4

1lim 2

kkTT

fIVar

If k ≠ 0

The second property states that:

is not a consistent estimator of f():

kTk If

4

1)(ˆ

Asymptotically unbiased

Examples of using packagesSPSS, Statistica

Example 1 – Sunspot data

0

50

100

150

1770 1790 1810 1830 1850 1870

Using SPSSOpen the Data

Select Graphs-> Time Series - > Spectral

The following window appears

Select the variable

Select the Window

Choose the periodogram and/or spectral density

Choose whether to plot by frequency or period

0.0 0.1 0.2 0.3 0.4 0.5

Frequency

-3.679E-1

2.203E4

5.987E4

Per

iod

og

ram

Periodogram of no by Frequency

2.718E0 7.389E0 2.009E1 5.46E1 1.484E2

Period

-3.679E-1

2.203E4

5.987E4

Per

iod

og

ram

Periodogram of no by Period

Periodogram Spectral density Estimator

2

1

2

1

)cos()sin(2

1

4

1ˆT

tkt

T

tktkT txtx

TIf

Properties: )(4

1lim kkTT

fIE

)(4

1lim 2

kkTT

fIVar

If k ≠ 0

The second property states that:

is not a consistent estimator of f():

kTk If

4

1)(ˆ

Asymptotically unbiased

Smoothed Estimators of the spectral density

The Daniell Estimator

d

drrkTk

dT I

df

]12[4

1)(ˆ )(

2

1

2

1

)cos()sin(2 T

tkt

T

tktkT txtx

TI

Properties

d

drrkk

dT f

dfE

]12[

1)(ˆ )(

d

drrkk

dT f

dfVar 2

2)(

]12[

1)(ˆ

hkkd

Tkd

T ffd

khdffCov

2)()(

]12[

12)(ˆ),(ˆ

12 if dkh

1.

2.

3.

• Now let T ∞, d ∞ such that d/T 0. Then we obtain asymptotically unbiased and consistent estimators, that is

ffE kd

TdT

)(ˆlim )(

,

.0)(ˆlim )(

,

kd

TdT

fVar

• Choosing the Daniell option in SPSS

0.0 0.1 0.2 0.3 0.4 0.5

Frequency

2.009E1

5.46E1

1.484E2

4.034E2

1.097E3

2.981E3

8.103E3

2.203E4

5.987E4

1.628E5

Den

sity

Window: Unit (5)

Spectral Density of no by Frequencyk = 5

2.718E0 7.389E0 2.009E1 5.46E1 1.484E2

Period

2.009E1

5.46E1

1.484E2

4.034E2

1.097E3

2.981E3

8.103E3

2.203E4

5.987E4

1.628E5

Den

sity

Window: Unit (5)

Spectral Density of no by Periodk = 5

0.0 0.1 0.2 0.3 0.4 0.5

Frequency

1.484E2

4.034E2

1.097E3

2.981E3

8.103E3

2.203E4

5.987E4

1.628E5

Den

sity

Window: Unit (9)

Spectral Density of no by Frequencyk = 9

2.718E0 7.389E0 2.009E1 5.46E1 1.484E2

Period

1.484E2

4.034E2

1.097E3

2.981E3

8.103E3

2.203E4

5.987E4

1.628E5

De

ns

ity

Window: Unit (9)

Spectral Density of no by Period

k = 5

Other smoothed estimators

More generally consider the Smoothed Periodogram

d

drrkTdTk

dT ITrWf 2/)(

~,

)(

and

d

drdT TrW

4

12/,

TrWTrW dTdT 2/2/ ,,

where

Theorem (Asymptotic behaviour of Smoothed periodogram Estimators )

and

Let

0sstst ux

24 tuE

T

Tdlim

0lim T

dT

T

where {ut} are independent random variables with mean 0 and variance 2 with

Let dT be an increasing sequence such that

and

Then

Proof (See Fuller Page 292)

ffE kd

TT

)(~

lim )(

)(~

2/lim )(

1

2, k

dT

d

drdT

TfVarTrW

T

or 02

or 02

2

f

f

Weighted Covariance Estimators

Recall that

where

1

1

)(2T

Th

hkT

kehCI

h

hkehC )(2

1||for 0

1||for 1

1

Th

ThxxThC

hT

ttht

The Weighted Covariance Estimator

where {wm(h): h = 0, ±1,±2, ...} are a sequence of weights such that:

i) 0 ≤ wm(h) ≤ wm(0) = 1

ii) wm(-h) = wm(h)

iii) wm(h) = 0 for |h| > m

h

hmk

wmT

kehChwf

)(

2

1)(ˆ

,

The Spectral Window for this estimator is defined by:

i) Wm() = Wm(-)

ii)

h

himm ehwW

2

1

Properties :

1

dWm

also (Using a Reimann-Sum Approximation)

= the Smoothed Periodogram Estimator

Note:

dIWf TkmkwmT

,

ˆ

2/

2/1,

2ˆT

TrTkmk

wmT IW

Tf

1.

Asymptotic behaviour for large T

dfWT

fE kmkwmT

2ˆ,

2.

dfWT

fVar kmkwmT

22

,

2ˆ

hwmTk

wmT ffCov ,,

ˆ,ˆ3.

2 2

dfWWT hmkm

hk for

1. Bartlett

Examples wm(h) = w(h/m)

Note:

1||0

1||||1

xfor

xifxxw

2/sin

2/sin

2

12

2

m

m

mWm

2. Parzen

w(x) = 1 -2 a + 2a cos(x)

otherwise0

1||2/1 if|]|1[2

2/1|| if||6613

32

xx

xxx

xw

3. Blackman-Tukey

with a = 0.23 (Hamming) , a = 0.25 (Hanning)

0.0 0.1 0.2 0.3 0.4 0.5

Frequency

2.009E1

5.46E1

1.484E2

4.034E2

1.097E3

2.981E3

8.103E3

2.203E4

5.987E4

1.628E5

Den

sity

Window: Unit (5)

Spectral Density of no by Frequency

Daniell

0.0 0.1 0.2 0.3 0.4 0.5

Frequency

5.46E1

1.484E2

4.034E2

1.097E3

2.981E3

8.103E3

2.203E4

5.987E4

1.628E5

Den

sity

Window: Tukey (5)

Spectral Density of no by Frequency

Tukey

0.0 0.1 0.2 0.3 0.4 0.5

Frequency

5.46E1

1.484E2

4.034E2

1.097E3

2.981E3

8.103E3

2.203E4

5.987E4

1.628E5

Den

sity

Window: Parzen (5)

Spectral Density of no by Frequency

Parzen

0.0 0.1 0.2 0.3 0.4 0.5

Frequency

5.46E1

1.484E2

4.034E2

1.097E3

2.981E3

8.103E3

2.203E4

5.987E4

1.628E5

Den

sity

Window: Bartlett (5)

Spectral Density of no by Frequency

Bartlett

1.

Approximate Distribution and Consistency

dfWT

fE kmkwmT

2ˆ,

2.

dfWT

fVar kmkwmT

22

,

2ˆ

hwmTk

wmT ffCov ,,

ˆ,ˆ3.

2 2

dfWWT hmkm

hk for

1.

Note: If Wm() is concentrated in a "peak" about = 0 and f() is nearly constant over its width, then

2.

kk

wmT ffE ,

ˆ

dWT

ffVar mkkwmT

22

,

2ˆ

and

Confidence Limits in Spectral Density Estimation

1.

Satterthwaites Approximation:

2.

where c and r are chosen so that

2,

ˆrk

wmT cf

crcEfE rkwmT

2

,ˆ

rccVarfVar rkwmT

22, 2ˆ

Thus

= The equivalent df (EDF)

k

wmT

kwmT

fE

fVarc

,

,

ˆ2

ˆ

k

wmT

kwmT

fVar

fEr

,

2

,

ˆ

ˆ2

and

kk

wmT ffE ,

ˆ

dWT

ffVar mkkwmT

22

,

2ˆand

Now

Thus

k

wmT

kwmT

fE

fVarc

,

,

ˆ2

ˆ

k

wmT

kwmT

fVar

fEr

,

2

,

ˆ

ˆ2

dW

T

m

2

rfdWT

f kmk /2

Then a [1- 100 % confidence interval for f() is:

Confidence Limits for The Spectral Density function f():

Let and denote the upper and lower critical values for the Chi-square distribution with r d.f. i.e.

22/, r

22/1, r

2/22/1,

222/,

2 rrrr PP

22/1,

,

22/,

, )(ˆ)(

)(ˆ

r

wmT

r

wmT fr

ffr

Estimation of the spectral density function

Summary

cos( ) i hh h f d e f d

0

1 10 cos( )

2 h

f h h

The spectral density function, f() The spectral density function, f(x), is a symmetric function defined on the interval [-,] satisfying

and

1cos( )

2

1

2

h

i h

h

h h

h e

Using cos cos

Using cos sin and sin sinie i

Periodogram Spectral density Estimator

2 2

1 1

1 1ˆ sin( ) cos( )4 2

T T

k T k t k t kt t

f I x t x tT

Properties: )(4

1lim kkTT

fIE

)(4

1lim 2

kkTT

fIVar

If k ≠ 0

The second property states that:

is not a consistent estimator of f():

kTk If

4

1)(ˆ

Asymptotically unbiased

1

1

1

1

Note 2 cos( )

2 k

T

T k x kh T

Ti h

xh T

I C h h

C h e

1

1

1

1

1ˆand 4

1 cos( )

2

1

2k

k T k

T

x kh T

Ti h

xh T

f I

C h h

C h e

Smoothed Estimators of the spectral density

Smoothed Periodogram Estimators

d

drrkTdTk

dT ITrWf 2/)(

~,

)(

and

d

drdT TrW

4

12/,

TrWTrW dTdT 2/2/ ,,

where

The Daniell Estimator

( ) 1ˆ ( )4 [2 1]

dd

T k T k rr d

f Id

,

1/ 2

4 [2 1]T dW r Td

The Weighted Covariance Estimator

where {wm(h): h = 0, ±1,±2, ...} are a sequence of weights such that:

i) 0 ≤ wm(h) ≤ wm(0) = 1

ii) wm(-h) = wm(h)

iii) wm(h) = 0 for |h| > m

,

1ˆ ( ) ( )2

kw i hT m k m

h

f w h C h e

1. Bartlett

Choices for wm(h) = w(h/m)

1||0

1||||1

xfor

xifxxw

2. Parzen

w(x) = 1 -2 a + 2a cos(x)

2 3

3

1 6 6 | | if | | 1/ 2

2[1 | |] if 1/ 2 | | 1

0 otherwise

x x x

w x x x

3. Blackman-Tukey

with a = 0.23 (Hamming) , a = 0.25 (Hanning)

The Spectral Window for this estimator is defined by:

i) Wm() = Wm(-)

ii)

h

himm ehwW

2

1

Properties :

1

dWm

also (Using a Reimann-Sum Approximation)

= the Smoothed Periodogram Estimator

Note:

dIWf TkmkwmT

,

ˆ

1 / 2

,1 / 2

2ˆT

wT m k m k T

r T

f W IT

1.

Approximate Distribution and Consistency

dfWT

fE kmkwmT

2ˆ,

2.

dfWT

fVar kmkwmT

22

,

2ˆ

hwmTk

wmT ffCov ,,

ˆ,ˆ3.

2 2

dfWWT hmkm

hk for

1.

Note: If Wm() is concentrated in a "peak" about = 0 and f() is nearly constant over its width, then

2.

kk

wmT ffE ,

ˆ

dWT

ffVar mkkwmT

22

,

2ˆ

and

Then a [1- 100 % confidence interval for f() is:

Confidence Limits for The Spectral Density function f():

Let and denote the upper and lower critical values for the Chi-square distribution with r d.f. i.e.

22/, r

22/1, r

2/22/1,

222/,

2 rrrr PP

22/1,

,

22/,

, )(ˆ)(

)(ˆ

r

wmT

r

wmT fr

ffr

and

kk

wmT ffE ,

ˆ

dWT

ffVar mkkwmT

22

,

2ˆand

Now

Thus

k

wmT

kwmT

fE

fVarc

,

,

ˆ2

ˆ

k

wmT

kwmT

fVar

fEr

,

2

,

ˆ

ˆ2

dW

T

m

2

rfdWT

f kmk /2

Then a [1- 100 % confidence interval for f() is:

Confidence Limits for The Spectral Density function f():

Let and denote the upper and lower critical values for the Chi-square distribution with r d.f. i.e.

22/, r

22/1, r

2/22/1,

222/,

2 rrrr PP

22/1,

,

22/,

, )(ˆ)(

)(ˆ

r

wmT

r

wmT fr

ffr

and

k

wmT

kwmT

fE

fVarc

,

,

ˆ2

ˆ

2m

Tr

W d

rfdWT

f kmk /2