Stat 153 13 Oct 2008 D R Brillinger

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

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

7. 3 Periodogram at frequency ωp I(ωp ) = |Σt=1 N xt exp{iωpt/N}|2 /πN = N(ap 2 + bp 2)/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)

Correcting for the mean work with

dynamic spectrum

Relationship - periodogram and acv f(ω) = [γ 0 + 2 Σk=1 γk cos ωk]/π I(ωp) = [c 0 + 2 Σk=1 N-1 ck cos ωpk]/π River height Manaus, Amazonia, Brazil daily data since 1902

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

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

Confidence interval. 100(1 -α)% work with logs

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

General class of estimates Example. boxcar

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

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. . .