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Time series power spectral density . frequency-side,  , vs. time-side, t

Time series power spectral density . frequency-side,  , vs. time-side, t X(t), t= 0, ±1, ±2,… Suppose stationary c XX (u) = cov{X(t+u),X(t)} u = 0, ±1, ±2, … lag f XX (  )= (1/2 π ) Σ exp {-i  u} c XX (u) -  <  ≤  period 2 π non-negative

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Time series power spectral density . frequency-side,  , vs. time-side, t

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  1. Time series power spectral density. frequency-side, , vs. time-side, t X(t), t= 0, ±1, ±2,… Suppose stationary cXX(u) = cov{X(t+u),X(t)} u = 0, ±1, ±2, … lag f XX()= (1/2π) Σ exp {-iu} cXX(u) - < ≤ period 2π non-negative /2 : frequency (cycles/unit time) cXX(u) = ∫ -pi pi exp{iuλ} f XX() d 

  2. The Empirical FT. Data X(0), …, X(T-1) dXT()= Σ exp {-iu} X(u) dXT(0)= Σ X(u) What is the large sample distribution of the EFT?

  3. The complex normal.

  4. Theorem. Suppose X is stationary mixing, then

  5. Proof. Write Evaluate first and second-order cumulants Bound higher cumulants Normal is determined by its moments

  6. Consider

  7. Comments. Already used to study mean estimate Tapering, h(t)X(t). makes Get asymp independence for different frequencies The frequencies 2r/T are special, e.g. T(2r/T)=0, r 0 Also get asymp independence if consider separate stretches p-vector version involves p by p spectral density matrix fXX( )

  8. Estimation of the (power) spectrum. An estimate whose limit is a random variable

  9. Some moments. The estimate is asymptotically unbiased Final term drops out if  = 2r/T  0 Best to correct for mean, work with

  10. Periodogram values are asymptotically independent since dT values are - independent exponentials Use to form estimates

  11. Approximate marginal confidence intervals

  12. More on choice of L

  13. Approximation to bias

  14. Indirect estimate

  15. Estimation of finite dimensional . approximate likelihood (assuming IT values independent exponentials)

  16. Bivariate case.

  17. Crossperiodogram. Smoothed periodogram.

  18. Complex Wishart

  19. Predicting Y via X

  20. Plug in estimates.

  21. Large sample distributions. var log|AT|  [|R|-2 -1]/L var argAT [|R|-2 -1]/L

  22. Berlin and Vienna monthly temperatures

  23. Recife SOI

  24. Furnace data

  25. Partial coherence/coherency. Mississipi dams RXZ|Y = (R XZ – R XZ R ZY )/[(1- |R XZ|2 )(1- |RZY |2 )]

  26. Advantages of frequency domain approach. techniques for many stationary processes look the same approximate i.i.d sample values assessing models (character of departure) time varying variant...

  27. London water usage Cleveland, RB, Cleveland, WS, McRae, JE & Terpenning, I (1990), ‘STL: a seasonal-trend decomposition procedure based on loess’, Journal of Official Statistics Y(t) = S(t) + T(t) + E(t) Seasonal, trend. error

  28. Dynamic spectrum, spectrogram: IT (t,). London water

  29. Earthquake? Explosion?

  30. Lucilia cuprina

  31. nobs = length(EXP6) # number of observations wsize = 256 # window size overlap = 128 # overlap ovr = wsize-overlap nseg = floor(nobs/ovr)-1; # number of segments krnl = kernel("daniell", c(1,1)) # kernel ex.spec = matrix(0, wsize/2, nseg) for (k in 1:nseg) { a = ovr*(k-1)+1 b = wsize+ovr*(k-1) ex.spec[,k] = mvspec(EXP6[a:b], krnl, taper=.5, plot=FALSE)$spec } x = seq(0, 10, len = nrow(ex.spec)/2) y = seq(0, ovr*nseg, len = ncol(ex.spec)) z = ex.spec[1:(nrow(ex.spec)/2),] # below is text version filled.contour(x,y,log(z),ylab="time",xlab="frequency (Hz)",nlevels=12 ,col=gray(11:0/11),main="Explosion") dev.new() # a nicer version with color filled.contour(x, y, log(z), ylab="time", xlab="frequency(Hz)", main= "Explosion") dev.new() # below not shown in text persp(x,y,z,zlab="Power",xlab="frequency(Hz)",ylab="time",ticktype="detailed",theta=25,d=2,main="Explosion")

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