3. Stat 153 - 13 Oct 2008 D. R. Brillinger
Chapter 7 - Spectral analysis
7.1 Fourier analysis
Xt = μ + α cos ωt + βsin ωt + Zt
Cases
ω known
versus
ω unknown, "hidden frequency"
Study/fit via least squares

4. Fourier frequencies
ωp = 2πp/N, p = 0,...,N-1
Fourier components
Σt=1N xt cos 2πpt/N Σt=1N xt sin 2πpt/N
ap +i bp = Σt=1N xt exp{i2πpt/N}, p=0,...,N-1
= Σt=1N xt exp{iωpt/N}

5. 7.3 Periodogram at frequency ωp
I(ωp ) = |Σt=1N xt exp{iωpt/N}|2 /πN
= N(ap2 + bp2)/4π
basis for spectral density estimates
Can extend definition to I(ω)
Properties
I(ω)  0
I(-ω) = I(ω)
I(ω+2π) = I(ω)
Cp. f(ω)

6. Correcting for the mean
work with

7. dynamic spectrum

8. Relationship - periodogram and acv
f(ω) = [γ0 + 2 Σk=1 γk cos ωk]/π
I(ωp) = [c Σk=1N-1 ck cos ωpk]/π
River height Manaus, Amazonia, Brazil
daily data since 1902

13. Periodogram properties.
Asymptotically unbiased
Approximate distribution
f(ω) χ22 / CLT for a, b
χ22 /2 : exponential variate
Var {f(ω) χ22/2} = f(ω)2
Work with log I(ω)

15. Periodogram poor estimate of spectral density
inconsistent
I(ωj), I(ωk) approximately independent
Flexible estimate,
χ22 + χ χ22 = χν2 , υ = 2 m
E{ χν2} = v Var{χν2} = 2υ

16. Confidence interval. 100(1-α)%
work with logs
Might pick m to get desired stability, e.g. 2m = 10 degrees of freedom
Employ several values, compute bandwidth
m = 0: periodogram

19. Model.
Yt = St + Nt
seasonal St+s = St
noise {Nt}
Buys-Ballot
stack years
column average

22. General class of estimates
Example. boxcar
Its bandwidth is m2π/N

23. Bias-variance compromise
m large, variance small, but
m large, bias large (generally)
Improvements
prewhiten - work with residuals, e.g. of an autoregressive
taper - work with {gt xt}

28. The partial correlation coefficient.
4-variate: (Y1 ,Y2 , Y3 , Y4 )
ε1|23 error of linear predicting Y1 using Y2 and Y3
ε4|23 error of linear predicting Y4 using Y2 and Y3
ρ14|23 = corr{ε1|23 , ε4|23}
pacf(2) = 0 for AR(2) because Xt+1 , Xt+2 separate Xt+1 and Xt+4

30. For AIC see text p. 256

32. Forecasts

33. Elephant seal dives

35. Binary data
X(t) = 0,1
nerve cells firing
biological phenomenon
refractory period

37. Continuous time series. X(t), -< t<
Now Cov{X(t+τ),X(t)} = γ(τ), -< t, τ<
and
Consider the discrete time series {X(kΔt), k =0,±1,±2,...}
It is symmetric and has period 2π/Δt
Nyquist frequency: ωN = π/Δt
One plots fd(ω) for 0  ω  π/Δt, BUT ...