Stat 153 - 7 Oct 2008 D. R. Brillinger

Chapter 6 - Stationary Processes in the Frequency Domain

One model

Another

,...2,1,0 )cos( tZtRX tt

,...2,1,0 )cos( }exp{ tZttRX tt

R: amplitude

α: decay rate

ω: frequency, radians/unit time

φ: phase

2π/ω: period, time units

cos(ω{t+2π/ω}+φ) = cos(ωt++φ)

cos(2π+φ)=cos(φ)

f= ω/2π: frequency in cycles/unit time

6.2 The spectral distribution function

Stochastic models. Have advantages

0mean t.s.stationary : )cos( i). ttt ZZtRX

)()( )cos()( hhtRt ZZ

)U(0,2: )cos( ii). tRX t

paramsin linear ),sin()cos(

)sin()sin()cos()cos()cos(

tt

tRtRtR

2/)cos( 0 2

h hREX t

π=3.14159...

)IU(0,2: )cos( iii).1

jjjj

k

jt tRX

J

jjjh hR

1

2 2/)cos(

Graph like pmf, f, or cdf, F

0

,...2,1,0 ),()cos( hdFhh

tVarXF )(0

6.3 Spectral density function, f.

F, spectral distribution function

sfrequencie ofon distributi continuous )(

)(

ddF

f

0

)()cos( dfhh

0

0 )( df

"f(ω)dω represents the contribution to variance of the components

with frequencies in the range (ω,ω+dω)"

0

)()cos( dfhh

Inversion

10 )cos(2

1)(

kh hf

Properties

f(-ω) = f(ω) symmetric

f(ω+2π) = f(ω) periodic

f(ω) 0 nonnegative

fundamental domain [0,π] (Nyquist frequency)

6.5 Selected spectra

(1). Purely random

0 0

0 2

h

hZh

/)( 2

Zf

white noise

MA(1). Xt = Zt + βZt-1

otherwise

k

kk

X

X

0

1 )1/(

0 )(22

2

222 )1( ZX

)1/()cos(2(11

)( 22

Xf

AR(1). Xt = αXt-1 + Zt |α | < 1

,...1,0 || )( 2 kk k

X

)1/( 222 ZX

))cos(21(/)1( )( 222

X f

Geometric series

1|| ...1)1/(1 2

Appendix B. Dirac delta function

Discrete random variables versus continuous

pmf versus pdf

Sometines it is convenient to act as if discrete is continuous

Random variable X

Prob{X=0} = 1

Prob{X 0} = 0

For function g(x), E{g(X)} = g(0)

Cdf F(x) = 0 x<0

= 1 x 0

pdf δ(x) the Dirac delta function, a generalized function

(x)dx=1, (x)g(x)dx=g(0), (y-x)g(x)dx=g(y)

(0)= (x)=0, x 0 N(0,0)

Sinusoid/cosinusoid.

cos(ω0t+φ) φ: U(0,2π), ω0 fixed

)cos()( 021 kk

This process is not mixing

the values are not asymptotically independent

but it is important

What are f(ω) and F(ω)?

With ω0 known series is perfectly predictable

Review.

γ(h) = Cov(Xt ,Xt+h)

0

)()cos()( dfhh

10 )cos()(2

1)(

khkf

All angles in [0,π]

Case of

Rcos(ω0t+φ) φ: U(0,2π), ω0 fixed

)cos()( 0

221 kRk

df )()cos(k 0

Solve for f(.)

d)()cos(k 00

Consider

= cos(kω0 )

Answer. )()( 0

221 Rf

spectral density - peaks go to infinity

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

frequency (cycles/unit time)

Series1

Infinite spike at ω = ω0

Spectral density

Several frequencies.

Σj Rjcos(ωjt+φj) φj: IU(0,2π), ωj fixed

)()( 221

jjj Rf

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

frequency (cycles/unit time)

Series1

infinite spikes at ωj's

Spectral density

Power spectra are like variances

Suppose {Xt} and {Yt} uncorrelated at all lags, then

fX+Y(ω) = fX(ω) + fY(ω)

Cp. if X and Y uncorrelated then

Var(X+Y) = Var(X) + Var(Y)

Example. Xt = Rcos(ω0t+φ) + Zt

/ )()( 2

0

221

ZRf