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chapt6 - Stat 153 7 Oct 2008 D R Brillinger Chapter 6...

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Unformatted text preview: Stat 153 - 7 Oct 2008 D. R. Brillinger Chapter 6 - Stationary Processes in the Frequency Domain One model Another ,... 2 , 1 , ) cos( ± ± = + + = t Z t R X t t φ ϖ ,... 2 , 1 , ) cos( } exp{ = + +- = t Z t t R X t t φ ϖ α 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 mean t.s. stationary : ) cos( i). t t t Z Z t R X + + = φ ϖ ) ( ) ( ) cos( ) ( h h t R t ZZ γ γ φ ϖ μ = + = ) U(0,2 : ) cos( ii). π φ φ ϖ + = t R X t params in linear ), sin( ) cos( ) sin( ) sin( ) cos( ) cos( ) cos( t t t R t R t R ϖ β ϖ α ϖ φ ϖ φ φ ϖ + =- = + 2 / ) cos( 2 h h R EX t ϖ γ = = π=3.14159... ) IU(0,2 : ) cos( iii). 1 π φ φ ϖ j j j j k j t t R X + = ∑ = ∑ = = J j j j h h R 1 2 2 / ) cos( ϖ γ Graph like pmf, f, or cdf, F ∫ = = π ϖ ϖ γ ,... 2 , 1 , ), ( ) cos( h dF h h t VarX F = = ) ( π γ 6.3 6....
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chapt6 - Stat 153 7 Oct 2008 D R Brillinger Chapter 6...

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