Stat 153 - 7 Oct 2008 D. R. Brillinger Chapter 6 - Stationary Processes in the Frequency Domain One...
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Transcript of Stat 153 - 7 Oct 2008 D. R. Brillinger Chapter 6 - Stationary Processes in the Frequency Domain One...
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