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Stat 153 - 7 Oct 2008 D. R. Brillinger

Stat 153 - 7 Oct 2008 D. R. Brillinger Chapter 6 - Stationary Processes in the Frequency Domain. One model. Another. 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)

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Stat 153 - 7 Oct 2008 D. R. Brillinger

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  1. Stat 153 - 7 Oct 2008 D. R. Brillinger Chapter 6 - Stationary Processes in the Frequency Domain One model Another R: amplitude α: decay rate ω: frequency, radians/unit time φ: phase

  2. 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

  3. 6.2 The spectral distribution function Stochastic models. Have advantages π=3.14159...

  4. Graph like pmf, f, or cdf, F

  5. 6.3 Spectral density function, f. F, spectral distribution function "f(ω)dω represents the contribution to variance of the components with frequencies in the range (ω,ω+dω)"

  6. Inversion Properties f(-ω) = f(ω) symmetric f(ω+2π) = f(ω) periodic f(ω)  0 nonnegative fundamental domain [0,π] (Nyquist frequency)

  7. 6.5 Selected spectra (1). Purely random white noise

  8. MA(1). Xt = Zt + βZt-1

  9. AR(1). Xt = αXt-1 + Zt |α | < 1 Geometric series

  10. Appendix B. Dirac delta function Discrete random variables versus continuous pmf versus pdf Sometines it is convenient to act as if discrete is continuous

  11. Random variable X Prob{X=0} = 1 Prob{X 0} = 0 For function g(x), E{g(X)} = g(0) Cdf F(x) = 0 x<0 = 1 x 0 pdf δ(x) the Dirac delta function, a generalized function  (x)dx=1,  (x)g(x)dx=g(0),  (y-x)g(x)dx=g(y) (0)= (x)=0, x 0 N(0,0)

  12. Sinusoid/cosinusoid. cos(ω0t+φ) φ: U(0,2π), ω0 fixed This process is not mixing the values are not asymptotically independent but it is important With ω0 known series is perfectly predictable What are f(ω) and F(ω)?

  13. Review. γ(h) = Cov(Xt ,Xt+h) All angles in [0,π]

  14. Case of Rcos(ω0t+φ) φ: U(0,2π), ω0 fixed Solve for f(.) Consider = cos(kω0 ) Answer.

  15. Spectral density Infinite spike at ω = ω0

  16. Several frequencies. Σj Rjcos(ωjt+φj) φj: IU(0,2π), ωj fixed

  17. Spectral density infinite spikes at ωj's

  18. Power spectra are like variances Suppose {Xt} and {Yt} uncorrelated at all lags, then fX+Y(ω) = fX(ω) + fY(ω) Cp. if X and Y uncorrelated then Var(X+Y) = Var(X) + Var(Y) Example. Xt = Rcos(ω0t+φ) + Zt

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