Stat 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.

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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

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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)

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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

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

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22 General class of estimates
Example. boxcar

23 Bias-variance compromise
m large, variance small m large, bias large (generally) improvements prewhiten - work with residuals taper - work with {gt xt}

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32 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, BUT ...