chapt7x

# chapt7x - Stat 153 - 13 Oct 2008 D. R. Brillinger Chapter 7...

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

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Fourier frequencies ω p = 2πp/N, p = 0,. ..,N-1 Fourier components Σ t=1 N x t cos 2πpt/N Σ t=1 N x t sin 2πpt/N a p +i b p = Σ t=1 N x t exp{i2πpt/N}, p=0,. ..,N-1 = Σ t=1 N x t exp{iω p t/N}
7.3 Periodogram at frequency ω p I(ω p ) = t=1 N x t exp{iω p t/N}| 2 /πN 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) = N(a p 2 + b p 2 )/4π

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Correcting for the mean X X t - work with
dynamic spectrum

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

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Periodogram properties . Asymptotically unbiased

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## This note was uploaded on 10/19/2009 for the course MATH 611 taught by Professor Jsdkasj during the Spring '09 term at Kansas.

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chapt7x - Stat 153 - 13 Oct 2008 D. R. Brillinger Chapter 7...

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