2. 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

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

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

8. 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)"

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

10. 6.5 Selected spectra (1). Purely random white noise

12. MA(1). X t = Z t + βZ t-1

13. AR(1). X t = αX t-1 + Z t |α | < 1 Geometric series

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

17. 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)

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

19. Review. γ(h) = Cov(X t,X t+h ) All angles in [0,π]

20. Case of Rcos(ω 0 t+φ) φ: U(0,2π), ω 0 fixed Solve for f(.) Consider = cos(kω 0 ) Answer.

21. Infinite spike at ω = ω 0 Spectral density

22. Several frequencies. Σ j R j cos(ω j t+φ j ) φ j : IU(0,2π), ω j fixed

23. infinite spikes at ω j 's Spectral density

24. Power spectra are like variances Suppose {X t } and {Y t } uncorrelated at all lags, then f X+Y (ω) = f X (ω) + f Y (ω) Cp. if X and Y uncorrelated then Var(X+Y) = Var(X) + Var(Y) Example. X t = Rcos(ω 0 t+φ) + Z t