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) f= ω/2π: frequency in cycles/unit time

6. 2 The spectral distribution function Stochastic models. Have advantages π=3. 14159. . .

Graph like pmf, f, or cdf, F

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)"

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

6. 5 Selected spectra (1). Purely random white noise

MA(1). Xt = Zt + βZt-1

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

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

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

Sinusoid/cosinusoid. cos(ω0 t+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)?

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

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

Spectral density Infinite spike at ω = ω0

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

Spectral density infinite spikes at ωj's

Power spectra are like variances Suppose {Xt} and {Yt} 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. Xt = Rcos(ω0 t+φ) + Zt